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Probability in the Philosophy of Religion
 0199604762, 9780199604760

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Probability in the Philosophy of Religion

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Probability in the Philosophy of Religion e dite d by

Jake Chandler and Victoria S. Harrison

1

3

Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. If furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © The several contributors 2012 The moral rights of the authors have been asserted First Edition published in 2012 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 978–0–19–960476–0 Printed in Great Britain by MPG Books Group, Bodmin and King’s Lynn Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

Contents Acknowledgements List of Contributors 1. Probability in the Philosophy of Religion Jake Chandler and Victoria S. Harrison

vii viii 1

Part I. Testimony and Miracles 2. Peirce on Miracles: The Failure of Bayesian Analysis Benjamin C. Jantzen

27

3. The Reliability of Witnesses and Testimony to the Miraculous Timothy McGrew and Lydia McGrew

46

4. Does it Matter whether a Miracle-like Event Happens to Oneself rather than to Someone Else? Luc Bovens

64

Part II. Design 5. Can Evidence for Design be Explained Away? David H. Glass 6. Bayes, God, and the Multiverse Richard Swinburne

79 103

Part III. Evil 7. Comparative Confirmation and the Problem of Evil Richard Otte 8. Inductive Logic and the Probability that God Exists: Farewell to Sceptical Theism Michael Tooley

127

144

Part IV. Pascal’s Wager 9. Blaise and Bayes Alan Hájek 10. Many Gods, Many Wagers: Pascal’s Wager Meets the Replicator Dynamics Paul Bartha

167 187

vi conte nt s

Part V. Faith and Disagreement 11. Does Religious Disagreement Actually Aid the Case for Theism? Joshua C. Thurow

209

12. Can it be Rational to Have Faith? Lara Buchak

225

Index of Names Index of Subjects

249 252

Acknowledgements The seed that eventually germinated into this book was sown in a Scottish Youth Hostel during a Reading Party hosted by the Department of Philosophy at the University of Glasgow. We are grateful to the Philosophy Department at Glasgow for providing such a convivial setting for the exploration of ideas. We are also grateful to Professor Igor Douven of the Odysseus Formal Epistemology Project, formerly housed at KU Leuven, for his generous support for a conference ‘Formal Methods in the Epistemology of Religion’, which provided an invaluable opportunity for the development of many of the ideas and arguments that eventually found their way into this book. The book itself, however, would not have come to full maturity without the assistance and conviction of Peter Momtchiloff at Oxford University Press and the two anonymous reviewers provided by the Press. Thanks are also due to Ms Sophie J. Lang of Greenock Academy, who patiently assisted Victoria Harrison with the preparation of the manuscript during her time acquiring work experience in the Philosophy Department at the University of Glasgow. We are also grateful to Dr Ioanna-Maria Patsalidou for her help in preparing the index of subjects and bringing order to the bibliographies. Finally, Jake Chandler would like to thank the Research Foundation—Flanders (FWO) for financial support during the later stages of this project.

List of Contributors Paul Bartha is Professor and Head of the Department of Philosophy at the University of British Columbia. Luc Bovens is Professor and Head of the Department of Philosophy, Logic and Scientific Method at the London School of Economics and Political Science. Lara Buchak is Assistant Professor of Philosophy at the University of California at Berkeley. Jake Chandler is Research Fellow at the Institute of Philosophy, KU Leuven. David H. Glass is Lecturer in Computer Science at the University of Ulster. Alan Hájek is Professor of Philosophy at the Research School of the Social Sciences, Australian National University. Victoria S. Harrison is Reader in Philosophy and Director of the Centre for Philosophy and Religion at the University of Glasgow. Benjamin C. Jantzen is Assistant Professor of Philosophy at Virginia Polytechnic Institute and State University. Lydia McGrew is an independent scholar living in Michigan. Timothy McGrew is Professor and Chair of Philosophy at Western Michigan University. Richard Otte is Professor of Philosophy at the University of California at Santa Cruz. Richard Swinburne is Emeritus Nolloth Professor of the Philosophy of the Christian Religion and Emeritus Fellow of Oriel College at the University of Oxford. Joshua C. Thurow is Assistant Professor of Philosophy at Mount Marty College. Michael Tooley is Professor of Philosophy and Distinguished College Professor of Arts and Sciences at the University of Colorado at Boulder.

1 Probability in the Philosophy of Religion Jake Chandler and Victoria S. Harrison

The past few years have witnessed a remarkable resurgence of interest in the intersection of formal methodology and epistemological theory, manifested in the organization of various conferences and the publication of numerous edited volumes. It is our opinion that this trend is likely to have a profound and positive impact on philosophical research, and will do so beyond the confines of epistemology proper, as recent developments in formal epistemology are brought to bear on relevant debates in further philosophical sub-disciplines. Of course, it must be conceded that the ‘formalization’ of any area of philosophical inquiry does have its potential pitfalls. Most obviously, the technicality of an exposition risks obscuring both shoddy philosophical reflection and dubious empirical assumptions, casting a veneer of false precision and scientific respectability. But the merits are legion, ranging from a welcome disambiguation of everyday discourse, to the enabling of a fast and reliable calculation of the sometimes surprising consequences of a set of apparently anodyne philosophical commitments. Of the numerous formal approaches to the philosophy of rational belief, it is the probabilistic framework that has arguably proven to be the most fruitful to date and remains the dominant approach in contemporary philosophy of science and epistemology.1 In particular, a body of research carried out under the heading of ‘Bayesian confirmation theory’ has, over the past fifty years or so, applied the tools of probability theory to deliver some extremely promising insights into the nature and logic of evidential support.2

1 For overviews of the philosophical literature on the nature of probability, see A. Hájek, B. Fitelson, and N. Hall (2005), A. Hájek (2010), or P. Humphreys (1998) for succinct expositions; for book-length presentations, see D. Gillies (2000) or D. Mellor (2005). A. Eagle’s recent anthology contains many of the essential articles on the subject (Eagle 2010). 2 Probabilistic analyses of various evidential concepts find their roots in R. Carnap (1962). See F. Huber (2007) for a brief recent summary of the achievements and shortcomings of this enterprise. J. Earman (1992) provides a more in-depth and sympathetic coverage of the same territory.

2 probabi lity in the phi lo sophy of re lig ion As probabilistic epistemology continues to blossom into a mature and increasingly sophisticated research programme, it is worth remembering that the philosophy of religion has long proven to be an extremely fertile ground for the application of probabilistic thinking to traditional epistemological debates. Philosophy of religion, as a modern academic discipline, traces its pedigree through ancient and medieval debates about the existence and nature of God. However, the discipline, as we know it today, finds its formative moment in the work of early modern European philosophers such as David Hume (1711–76). Hume’s influence on this area of philosophy has been so pervasive that even today many key debates still take place within the terms he established. And indeed, a number of the essays in this volume can be readily identified as falling within the long tradition of responses to Hume’s formulations of certain key problems faced by theism.3 This is especially so of the essays by Benjamin Jantzen, Timothy and Lydia McGrew, and Luc Bovens in Part I of this volume on ‘Testimony and Miracles’. But Hume also had a profound influence on the approach taken by many modern philosophers of religion to issues concerning cosmology and arguments from design, the subject of the essays by David Glass and Richard Swinburne in Part II. In addition, he set the terms of the modern debate on the problem of evil—the focus of the essays by Michael Tooley and Richard Otte in Part III. In Hume’s day, and indeed until the mid-twentieth century, philosophy of religion was not generally regarded as a distinct area of philosophy clearly distinguishable from other areas. Instead it was practised as part of a unified philosophical reflection that also encompassed areas of general popular interest, such as ethics and politics, as well as more abstract areas like metaphysics (see Harrison 2011). It is frequently overlooked today that all of the seminal figures who shaped the modern Western philosophical tradition were seriously engaged with questions that are now regarded as the unique preserve of philosophy of religion (see Taliaferro 2005). Despite the fact that by the mid-twentieth century, philosophy of religion, along with most of the other sub-branches of philosophy now familiar to us, had become sharply demarcated as a distinct area of philosophical inquiry, the issues addressed within it remained not too far removed from the more general concerns of ‘ordinary’ reflective people. In fact, in an age of increasing popular scepticism about theistic belief, the preoccupation of philosophers of religion with arguments for and against the existence of God echoed the concerns of the non-philosophical but reflective public. Philosophy of religion has never been completely detached from the spirit and concerns of the age in which it is practised. This was as true in Hume’s day as it is in our own. The chapters in this volume are part of a long tradition of philosophical reflection on matters of religious concern. But they are also examples of that form of analytic philosophy of religion which began to emerge from the intellectual culture

3 See David Hume, Dialogues Concerning Natural Religion (1779).

introduction 3 of the late 1960s, after the thawing of the freeze on the discipline effected by the dominance of logical positivism during the 1950s (see Harrison 2007). What emerged in the response to the challenges, such as that presented by the methodology of logical positivism, confronting philosophy of religion in the mid-twentieth century was a newly invigorated discipline which—although it remained by and large focused on traditional themes—was imbued with a new confidence and characterized by a greater rigour than had been evident prior to this time. Part of what contributed to this amplification of concern with analytical rigour was the concentration on linguistic analysis that had crystallized as philosophers of religion sought to recover the integrity of their discipline in the aftermath of positivism. The focus of philosophers of religion on linguistic analysis during this period is well documented and it mirrored the situation in other areas of philosophy.4 However, less well documented and, perhaps, less well appreciated is the key contribution made to the discipline by the intensification of the cross-fertilization of probability theory and philosophy of religion which occurred at this time. Probability theory came to play a central role in the work of many philosophers of religion and, arguably, it was even more influential than linguistic analysis in driving philosophy of religion in the direction of increased analytic rigour. Ironically, one reason that the formative role that the notion of probability has exercised on the philosophy of religion in recent times is often overlooked may be that it has enjoyed such a ubiquitous structuring effect on the content and methodology of the discipline that its very ubiquity has hidden it from notice.5 In the same way that fish rarely notice the water they swim in, philosophers of religion have imbibed notions of probability, and kindred notions of evidence and rational belief, and allowed these to structure their approach to the discipline. The rapprochement between probability theory and philosophy of religion which occurred in the 1970s set the course for the evolution of the latter for the decades to come. The publication of Richard Swinburne’s The Existence of God in 1979 was pivotal to this evolution as it helped to establish probability theory as the dominant mode of discourse within philosophy of religion. The publication of Swinburne’s book heralded a decisive shift in Anglo-American philosophy of religion. This was so both in terms of the analytic rigour henceforth demanded in the field and in the topics that philosophers of religion increasingly came to focus their attention upon. From this point, linguistic analysis dropped out of focus as the dominating concern of the discipline; instead interest converged on precisely those areas of the subject which are most amenable to probabilistic analysis. To give just one example, prior to this 4 See, for example, various essays in vol. 5 of G. Oppy and N. Trakakis’ recent monumental anthology covering the history of philosophy of religion from antiquity to the present day (Oppy and Trakakis 2009). See also B. Mitchell’s well-known collection of essays from the post-positivist period (Mitchell 1971). For a discussion, see V. S. Harrison (2007, ch. 4). 5 Although there have been some attempts to bring the role of probability theory in philosophy of religion and theology more clearly into view. See D. J. Bartholomew (1988) and F. Watts (2008).

4 p robab i l ity i n th e ph i lo s ophy of re l i g i on development the logical problem of evil (as formulated by John Mackie in 1955) was an overriding focus of attention within the discipline; after, however, there was a dramatic loss of interest in this problem as attention shifted to the evidential problem of evil (see Rowe 1979 and 1991). The debate about this latter problem is carried forward in the present volume by the contributions of Michael Tooley and Richard Otte. The change that took place when philosophers of religion shifted attention from the logical to the evidential problem of evil is symptomatic of a more general shift that took place from concern with the notion of proof to a focus on the rationality of belief construed in probabilistic terms. As the notion of probability became more thoroughly assimilated into philosophy of religion, not only was there a shift of attention toward the evidential problem of evil, but there was also renewed interest in other areas of the discipline that invited analysis in inductive terms. One such was the topic of ‘Testimony and Miracles’, already discussed by Hume in the eighteenth century. Richard Swinburne was again ahead of the trend in this respect when his book The Concept of Miracle was published in 1971. The wealth of research that began to appear in the 1980s on religious epistemology also points to the increasing centrality, within philosophy of religion, of the notion of evidence and the related notion of what it is reasonable to believe.6 The movement known as Reformed Epistemology, which has been especially prominent among North American theistic philosophers, forms part of this trend.7 The increasing importance of notions of evidence and explanation can also be identified in the renewed concern with the design argument for the existence of God (see Part II of this volume). However, the fecundity of the recent pairing of philosophy of religion and probability theory has far outstripped the usual debates about evidence and explanation. It has also generated a resurgence of interest in topics such as Pascal’s Wager (see the contributions of Alan Hájek and Paul Bartha to this volume), which is widely regarded as a landmark in the formal understanding of probability as well as decision theory (see Hacking 1975). The influence of this change of focus away from concern with proof and certainty and towards notions of evidence, rational belief, explanation, and, underlying all of these, probability, was so pervasive that in retrospect one might come away with the impression that the trajectory taken by the discipline over the course of the last few decades could have been predicted. Despite this, the recent history of philosophy of religion seen through the lens of the evolution of probability theory is yet to be written, and it is not the purpose of the current volume to fill this lacuna. Nor does the current volume intend to review the trajectory that, under the impetus

6 For a representative collection of essays on religious epistemology from this period, and some critical discussion of them, see R. D. Geivett and B. Sweetman (1992). 7 Notable within this movement are W. P. Alston (see Alston 1991), N. Wolterstorff (see Wolterstorff 1983), and A. Plantinga. The latter has contributed an important trilogy to the literature (see Plantinga 1993a, 1993b, and 2000).

introduction 5 of developments in our thinking about probability, philosophy of religion has taken over the past few decades. Rather it showcases some of the latest work—by both established and new scholars—on key topics in philosophy of religion that reflect the central strategic importance that probability theory has had, and continues to enjoy, on this area of philosophy. The essays contained in this volume demonstrate how far philosophy of religion has come in reaping the benefits of applying probability theory to its traditional domain; they are also suggestive of areas in which there remains much scope for a deeper and yet more rigorous application of the tools of probability and decision theory. In the following sections we briefly review each of the topics with which this volume is concerned. These topics have been chosen because they are all key areas of philosophy of religion to which the contribution of probability theory has been pronounced. We also provide short summaries of each chapter, situating them in the context of the wider debates to which they contribute. We conclude this chapter with some brief comments regarding the future direction in which such studies might lead. We now turn to our first topic.

Part I. Testimony and Miracles Reports of the occurrence of ‘miraculous’ events, such as spontaneous healings, transmutations of substances, and even resurrections, occupy a prominent place in the battery of considerations that are typically adduced in favour of theism. Whilst there is a long tradition of philosophical commentary on such appeals to testimony, as indicated above, it is fair to say that the bulk of the contemporary discussion of the matter has been written in connection with Hume’s eighteenth-century essay ‘Of Miracles’, which appears as Section X of his Enquiries Concerning Human Understanding. Here, Hume offers a number of considerations aiming to demonstrate ‘that no human testimony can have such force as to prove a miracle, and make it a just foundation for any system of religion’ (Hume 1748/77, Section X, Part II, §22). The centrepiece of the discussion is the following famous maxim, drawn from Part I of the essay, in which Hume outlines the way in which he believes the evidential impact of reports of the occurrence of a miraculous event ought to be assessed: no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish: And even in that case, there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior. (Hume 1975: 115–16)

In Part II of the essay, Hume moves on to adduce various factual considerations aiming to undermine the degree to which one might consider the falsity of the testimony to be miraculous. In view of the strength of prior evidence against the

6 probabi lity in the phi lo sophy of re lig ion occurrence of any given miracle, the upshot is then, according to Hume, that testimony to the miraculous ought to be ultimately discounted. Although Hume’s maxim is not couched in probabilistic terms, the quantitative talk of the degree to which an event is miraculous, or again of the degree of strength of an argument, certainly invites a translation into a contemporary probabilistic framework. And indeed, much of the post-Humean literature on the epistemology of testimony to the miraculous has been framed in these terms. In recent years, from the mid-1980s onward, the focus of the discussion has been largely exegetical, with the emergence of a lively debate among philosophers of science regarding the proper probabilistic interpretation and evaluation of Hume’s comments.8 The contributions to the present volume, however, largely steer clear of interpretational issues surrounding ‘Of Miracles’, focusing instead on a number of general questions raised by probabilistic approaches to the problem of miracles.9 In his ‘Peirce on Miracles: The Failure of Bayesian Analysis’, Benjamin Jantzen draws upon three manuscripts of Charles Sanders Peirce to offer a sympathetic overview of the latter’s criticisms of the application of probabilistic reasoning in the assessment of the impact of testimony to the occurrence of miracles. The method under criticism, the Method of Balancing Likelihoods (MBL), which Peirce attributes to Hume, involves a computation of the odds of the occurrence of a miracle, conditional on the claims of n witnesses testifying to its occurrence. In the particular case in which the testimonies satisfy a certain strong independence condition, this can be computed from just the prior probability of the occurrence M of the miracle and the individual probabilities of the various witnesses testifying positively on the respective assumptions of M and not-M. Jantzen finds in Peirce a range of concerns regarding the applicability of MBL. These range from the apparent denial, on the basis of seemingly frequentist considerations, of the possibility of assigning, even in principle, the requisite kinds of conditional probabilities, to the claim that MBL is an empirical failure. The core complaint, however, is that a sampling bias inherent in the collection of witness reports on miracles renders the updating of one’s odds for M by means of MBL a fruitless enterprise. Indeed, for obvious sociological and psychological reasons, one is far more likely to find oneself in the possession of testimonies in favour of M than

8 See P. Millican (MS) for a catalogue of recent proposals. Among the suggestions discussed are those of J. Earman (1993), D. Gillies (1991), R. D. Holder (1998), P. Millican (1993), and J. H. Sobel (1991). J. Earman (2000) provides an important, if somewhat controversial, book-length critique of Hume from a probabilistic perspective. 9 For general overviews of the literature on miracles, see D. Corner (2005), M. Levine (2010), T. McGrew (2010), and J. H. Sobel (2004, ch. 8). These notably address the vexed issue of what exactly does and does not qualify as a miracle. This is an issue that is bracketed by the contributors to the present volume, who rely on an intuitive understanding of the term, as used in the religious context, which uncontroversially applies to the kinds of examples that we gave in the opening paragraph of this section.

introduction 7 one is of testimonies in favour of its negation. And of course, as Jantzen illustrates with a number of examples, this bias in the data can have a dramatic impact on the resulting assessment of odds. As a corrective, Peirce suggests a two-way division of inquiry. First, an abductive stage leads to the formulation of various hypotheses, put forward on the basis of their providing good explanations of the available data. This data is then to be discounted in the second, inductive, phase of the process, during which the collection of a bias-free sample underpins an assessment of the relative merits of the hypotheses under consideration. The point of departure for Timothy and Lydia McGrew’s ‘The Reliability of Witnesses and Testimony to the Miraculous’ is a well-known formula, offered by Condorcet (1783), for updating one’s degree of confidence in the occurrence of a miracle upon learning of the positive testimony of an alleged witness to the event. The focus of their concern is on the evaluation of a crucial term in Condorcet’s formula, namely of the variable that represents the reliability of the witness. According to a school of thought that had considerable influence at the opening of the nineteenth century (see Reid 1822), the testimony of others ought to be considered at least prima facie credible. But the issue then immediately arises of whether or not, in a particular given instance, the default assumption of witness reliability is to be overridden by various countervailing considerations. As the McGrews point out, with respect to the particular context of testimony to the occurrence of the miraculous, one finds a variety of positions on the impact of such further considerations, with some, such as Hume (1748/77), taking these radically to undermine witness credibility and others, such as George Campbell (1839), being considerably more sanguine. The McGrews concur with Venn (1888) in expressing a certain degree of scepticism concerning the possibility of establishing the totality of the facts that would be pertinent to the assessment of witness reliability. They also consider the past track record of a witness’ reliability to be only one part of the story in evaluating witness testimony. Unlike Venn, however, who considers only the Condorcet approach, they do not conclude that these considerations undermine the use of probability in the evaluation of testimony. They argue instead in favour of a Bayesian approach that departs from Condorcet’s method in that it does not require us to calculate witness reliability as a separate factor. Instead, they suggest what one might want to call an ‘holistic’ assessment of the impact of witness testimony based on ‘evidence about the epistemically relevant features of the witness, the situation in which he testifies, and their interaction’. They argue that Bayes factors are the most useful formal tool for modelling this approach and for incorporating the pertinent facts without running the risk of oversimplification. The final contribution on this topic is entitled ‘Does it Matter whether a Miraclelike Event Happens to Oneself rather than to Someone Else?’. In this chapter, Luc Bovens investigates the interesting but largely neglected question of the relative impacts of first-person versus third-person experiences of miraculous events with respect to

8 probabi lity in the phi lo sophy of re lig ion religious belief.10 He sets out to evaluate, from a Bayesian perspective, William Alston’s claim that, to the extent that experiencing first-hand a certain kind of miraculous event would warrant belief in God, so too would obtaining the testimony of a trustworthy third party to the effect that an event of that nature occurred (Alston 1991). Bovens first considers a simple probabilistic model that could at first pass be taken to provide a straightforward vindication of Alston’s claim. He then argues, however, that this model fails adequately to represent the nature of the information received by the subject. In particular, it fails to do justice to the fact that the agent, upon learning that a miracle has occurred, also typically thereby learns that she has learned this fact. In conjunction with details regarding the so-called ‘protocol’, which specifies the range of possible informational updates alongside the probabilities of their occurrence conditional on the various possible states of the environment, this additional information can make a dramatic difference to the conclusions that an agent is entitled to draw. The dangers of failing to take into account this kind of acquisition of higher-order information have long been recognized.11 As Bovens points out, following Pearl (1988), it leads, for instance, to the counterintuitive ‘halfer’ conclusion in Gardner’s Three Prisoners puzzle, a stylistic variant of the well-known Monty Hall puzzle. With these considerations in mind, Bovens offers an alternative model, built on the kinds of protocols that he takes to be typically associated with, on the one hand, first-person experience of a miracle, and on the other, acquisition of third-person testimony to the effect that a miracle has occurred. He notes that the impact of the acquired evidence is greater in the former case than it is in the latter,12 and argues that this undermines Alston’s claim. He concludes, then, that it does matter whether a miracle-like event happens to oneself rather than to someone else.

Part II. Design Whilst theists have often seen indications of divine intervention in reports of extraordinary goings-on, many have claimed to find evidence for the hand of God in more mundane features of our world. In particular, there is a long tradition, stretching back to classical antiquity, of holding that the existence of entities displaying various particular features, such as the capacity for conscious thought, points to the handiwork of a supernatural designer. Paradigmatic writings in this tradition include Cicero’s De Natura Deorum, in the first century bc, Aquinas’ Summa Theologiae, in the thirteenth century, and, in the early nineteenth century, William Paley’s Natural Theology.

10 It is worth noting that, in contrast to the other contributions to this section, Bovens does not discuss the credibility of witness reports of the occurrence of a miracle, but rather the extent to which the veracity of such reports would support the theistic position. Despite this difference in focus, his observations and arguments carry over, mutatis mutandis. 11 See for instance J. H. Sobel (1992) and G. Shafer (1985). 12 Measured in terms of the odds of the existence of God, conditional on the information received.

introduction 9 With the advent of evolutionary thinking in the decades following the publication of Paley’s book, this etiological hypothesis lost a considerable amount of its appeal: after all, the operation of natural selection on random heritable variation seemed to many to promise a perfectly satisfactory, and more parsimonious, alternative explanation of the facts. Yet the debate over the inference to design from features of the biological world appears to have enjoyed a new lease of life in recent years. This has been partly spurred on by the somewhat controversial publications of members of the so-called Intelligent Design (ID) movement, who have, inter alia, questioned the extent to which evolutionary theory can account for the full extent of the data.13 Another factor has been the relatively recent claim that the universe is ‘fine-tuned’ for life, in the sense that, had the values of various physical constants been only slightly different to what they in fact are, life as we know it would not have been possible.14 The contention then is that, although evolution by natural selection may well provide a satisfactory account of the biological data given that these constants have the values that they do, this fine-tuning itself provides evidential support for the intervention of an external agency that the facts of biological evolution do nothing to undermine. In response to this, the so-called ‘multiverse’ objection points to the existence of an alternative, non-theistic and putatively independently plausible hypothesis that could account for the data. On this view, the world is constituted by an ensemble of universes whose vast and varied nature makes the existence of a fine-tuned universe likely. What’s more, the reply goes, it should come as no surprise that we find ourselves contemplating that universe rather than any other universe in the ensemble, since fine-tuning is a precondition for the very existence of observers.15 In his contribution, ‘Can Evidence for Design be Explained Away?’, David Glass discusses an issue that is of obvious relevance to this debate. Suppose that an observation E provides a certain amount of support for each of two logically compatible hypotheses, H1 and H2 , which, if true, would constitute prima facie explanations of E. The subsequent discovery that H1 is true would then undermine the plausibility of H2 . In Glass’ terminology, the prima facie evidence E in favour of H2 has been, to a certain extent, ‘explained away’ by H1 .16

13 See for instance M. Behe (1996); and H. Orr (1997) and S. Sarkar (2007) for replies. Other well-known ID theorists include W. Dembski (1998), whose views on probability and inference have been heavily criticized in B. Fitelson, C. Stephens and E. Sober (1999), and elsewhere. R. T. Pennock’s edited volume contains a sample of views on both sides of the ID debate (Pennock 2001). 14 See J. Leslie (1989, ch. 2) for an accessible presentation of the case for fine-tuning. 15 For recent general overviews of the debate over arguments from design, see, for instance, K. E. Himma (2009) and D. Ratzsch (2010). N. Manson’s (2003) noteworthy anthology also includes a useful introductory chapter. E. Sober (2005) provides a characteristically lucid discussion of some of the issues from a probabilistic point of view. Chapter 2 of N. Bostrom (2002) contains a discussion of some of the probabilistic literature on fine-tuning. 16 The terminology of ‘explaining away’ is due to J. Pearl (1988), who was the first to discuss this kind of phenomenon.

10 p robab i l ity i n th e ph i lo s ophy of re l i g i on Glass offers a probabilistic analysis of the phenomenon, according to which H1 at least partially explains away H2 with respect to E if and only if the probability of H1 conditional on the conjunction of E and H2 is less than the probability of H1 conditional on E alone. The explaining away is then said to be complete if and only if, in addition, the probability of H1 conditional on the conjunction of E and H2 is less than the prior probability of H1 . After outlining the precise probabilistic conditions under which partial and complete explaining away do and do not occur, Glass deploys these results in an attempt to evaluate the extent to which the truth of two non-theistic hypotheses would explain away the existence of complex adaptations and cosmological fine tuning (both of which are typically adduced in favour of theism in the context of the argument from design). The two non-theistic hypotheses he considers are the operation of natural selection and the existence of a multiverse. In the following chapter, entitled ‘Bayes, God, and the Multiverse’, Richard Swinburne, a longtime advocate of the fine-tuning argument (see Swinburne 1990), provides a provocative assessment of the epistemic impact of the multiverse objection. He begins with a concise exposition of his views on the posterior probability of causal explanations of the data, which includes an endorsement of the noteworthy claim that simpler hypotheses ought to be attributed a higher prior probability, ceteris paribus, than their more complex counterparts.17 With this in hand, he turns his attention to assessing the relative prior probabilities and likelihoods of his favoured theistic hypothesis, on the one hand, and its one-universe non-theistic competitor, on the other.18 Arguing on the basis of the joint claims of the goodness of the existence of humankind and the extreme simplicity of theism, he concludes that the resulting balance of posterior probabilities weighs heavily in favour of theism. Finally, in two sections of his chapter devoted to the various incarnations of the multiverse hypothesis, Swinburne argues that although the godless multiverse hypothesis is somewhat simpler than the godless one-universe hypothesis, it remains far more complex—and hence, according to him, far less a priori probable—than the theistic alternative; largely for this reason, he concludes that consideration of the multiverse scenario has little impact on the force of the argument from fine-tuning.

Part III. Evil The five chapters in the two previous parts of this volume have dealt with considerations in favour of theism. We now turn to one of the best-known considerations against it, at least insofar as theism is committed to the existence of a deity of immense power,

17 For a more detailed exposition of his views on this matter, see Swinburne (1997). The locus classicus of this, not uncontroversial, idea is H. Jeffreys (1961); see C. Howson (1988) for a critical discussion. 18 Here we mean ‘likelihood’ in the technical sense of the probability of the data conditional on the truth of the hypothesis.

introduction 11 knowledge, and goodness: namely, the existence of what prima facie appears to be a morally intolerable amount of suffering and hardship in the world. Whilst, as mentioned earlier, some presentations of the argument from evil, as it is known, appeal to a logical incompatibility between these unfortunate circumstances and the theistic hypothesis (for example, Mackie 1955 or McCloskey 1960), it is now widely acknowledged that this is a somewhat implausibly strong premise to proceed from. After all, appearances to the contrary, it may well prove to be the case that, all things considered, allowing the circumstances in question to obtain is morally permissible after all. A more modest and popular endeavour, whose roots—like those of so many of the other arguments considered in this volume—can be traced back at least as far as Hume’s Dialogues, involves arguing from the existence of evil to the mere improbability of theism.19 As mentioned earlier, this so called ‘evidential’ incarnation of the argument from evil is the focus of the two contributions on this topic to the present volume.20 Richard Otte’s ‘Comparative Confirmation and the Problem of Evil’ addresses the question of whether a lack of belief in the existence of a good reason for God to permit the existence of evil (U) does in fact evidentially favour atheism (NG) over various forms of theism. His suggestion is that this may not turn out to be the case and that there may even be grounds to claim that the converse holds. The issue is considered from the vantage point of a number of possible probabilistic analyses of the concept of evidential favouring.21 Amongst these, one finds Ian Hacking’s well-known ‘Law of Likelihood’, which states that an item of evidence E evidentially favours hypothesis H1 over competing hypothesis H2 if and only if the probability of E conditional on H1 is strictly greater than that of E conditional on H2 (see Hacking 1965). But Otte also considers a number of alternatives to the Law of Likelihood that have recently been put up for discussion by Branden Fitelson.22 These alternatives formalize, in various ways, the idea that E evidentially favours H1 over H2 if and only if E raises the probability of H1 more than it does the probability of H2 . Otte argues that on all the analyses of evidential favouring discussed, his guiding question should be answered in the negative: lack of belief in the existence of a good reason for God to permit the existence of evil (U) does not in fact evidentially favour atheism (NG) over various forms of theism. More specifically, he offers considerations that he takes convincingly to undermine the claim that U favours NG over a particular brand of theism, namely, ‘sceptical theism’, that takes the intentions of God to be inscrutable (UG). 19 See Part 11 of Hume’s Dialogues Concerning Natural Religion (Hume 1779). 20 Noteworthy proponents of the evidential argument include W. Rowe (1979, 1991) and

P. Draper (1989). For general overviews of the issues, see M. Tooley (2010) and N. Trakakis (2006). D. Howard-Snyder (1996) provides an important anthology on the topic. 21 Also known as ‘contrastive’, ‘comparative’, or, again, ‘relational’ confirmation. 22 The locus for the various alternatives to the Law of Likelihood is B. Fitelson (2007). J. Chandler (2010) argues that these alternatives flout various intuitive desiderata. See B. Fitelson (forthcoming) for further discussion.

12 p robab i l ity i n th e ph i lo s ophy of re l i g i on He claims that to the extent that one can even assess the relative magnitudes of Pr(U|NG) and Pr(U|UG), there may be a case to hold that the former is no higher, or even strictly lower, than the latter. Merely holding that Pr(U|NG) is no higher than Pr(U|UG) would be sufficient to establish his conclusion on the assumption that the Law of Likelihood provides a suitable account of evidential favouring. However, appealing to a well-known result reported by Fitelson,23 he further notes that even if it were the case that Pr(U|NG) was lower than Pr(U|UG) it would still suffice to establish his conclusion on any of the various alternative analyses of favouring that he considers. However, the standing of sceptical theism is called into question in the following chapter by Michael Tooley, ‘Inductive Logic and the Probability that God Exists: Farewell to Sceptical Theism’. Here Tooley offers a further development of the treatment of the argument from evil advanced in his recent joint volume with Alvin Plantinga (Plantinga and Tooley 2008). Tooley sets out to evaluate a crucial premise in the evidential argument from evil: the claim that our living in a godless world is more probable than not, conditional on the occurrence of what he calls ‘morally problematic’ events. He takes such events to be those whose known properties are, on balance, morally undesirable. He argues that this premise is defensible. More specifically, he shows that, under certain ‘Carnapian’ assumptions regarding prior probabilities (see Carnap 1962), assumptions that he takes to be rationally mandated, one can place a strikingly low upper bound on the prior probability of the following: its being the case that, for every event e in a reasonably small set of n events, the totality of the properties of e is not, on balance, morally undesirable, conditional on its being the case that, for every e, the known subset of the properties of e is on balance morally undesirable. This is a remarkably strong result. Indeed, if one grants Tooley’s Carnapian assumptions about the priors, it holds even under complete ignorance both (a) of the number k of unknown morally relevant properties of the events, and (b) of the proportion r of those properties that are morally desirable rather than not. Tooley obtains what he is after by first establishing an upper bound for the relevant probability given knowledge of both k and r, before generalizing to the cases in which first knowledge of r and then knowledge of both k and r are dropped.

Part IV. Pascal’s Wager The previous sections have all dealt with considerations that are purportedly relevant to the rationality of theistic or atheistic belief by virtue of being indicative of its truth. It has been argued, however, that there are reasons for belief whose rational import lacks 23 Namely the so-called ‘Weak Law of Likelihood’, which is a shared consequence of a number of alternatives to the Law of Likelihood, and states that E favours H1 over H2 if and only if Pr(E|H1 ) > Pr(E|H2 ) and Pr(E| not-H1 ) ≤ Pr(E| not-H2 ). See B. Fitelson (2007).

introduction 13 this evidential character. Such putative reasons are sometimes termed ‘pragmatic’ or ‘prudential’, as they involve facts about the value or perceived value of a belief given different states of nature.24 There are several arguments for theism that appeal to pragmatic considerations.25 Undoubtedly, however, by far the most famous of these are the variants of Blaise Pascal’s so-called ‘Wager’, which appear in the section entitled ‘Infini Rien’ in his Pensées.26 Whilst the general lines of the Wager can be found in the writings of a substantial number of earlier scholars,27 it is widely recognized that Pascal’s presentation is the first one to exhibit a distinctly modern decision-theoretic flavour.28 In an influential article, Ian Hacking (1972) offered what have now become the standard formal reconstructions of three versions of the argument that are allegedly to be found in the Pensées. All three arguments proceed from claims regarding the payoffs, a, b, c, and d, respectively associated with (i) belief in God given God’s existence (handsomely rewarded in the afterlife), (ii) belief in God given God’s non-existence, (iii) non-belief in God given God’s existence (possibly liable to punishment), and (iv) non-belief in God given God’s non-existence. The first variant of the Wager is the ‘Argument from Dominance’, so-called because it proceeds to the conclusion that one ought to believe in God from the simple claim that belief in God ‘weakly dominates’, in decision-theoretic terminology, non-belief: a > c but b ≥ d.29 The Argument from Dominance makes no claim regarding the probability of God’s existence; nonetheless it makes strong claims about payoffs. The second variant of the Wager, however, the ‘Argument from Expectation’, pursues a different strategy. It trades some of the strength of this claim about payoffs for a weak claim about probabilities, namely a claim that the existence and non-existence of God are equiprobable. Given this latter claim, if one merely further assumes that a − c > d − b, it follows that the expected utility of believing in God exceeds that of not doing so. From this comparison of expectations, it is then concluded that one ought to believe in God. The final argument, which is probably the best known version of the Wager, is the ‘Argument from Dominating Expectation’. It drops the premise of equiprobability, 24 It is worth pointing out that this terminology may be somewhat unhelpful insofar as it suggests that the truth of a belief is of no practical significance. ‘Merely pragmatic’ or ‘merely prudential’ may be more apt. 25 For instance, William James advances a pragmatic argument in his essay, first published in 1896, ‘The Will to Believe’ (James 1956); see J. Jordan (2011) for an overview. 26 One commonly-cited edition of the Pensées is L. Lafuma’s (Pascal 1963). Popular translations into English include those of A. J. Krailsheimer (Pascal 1966) and, more recently, H. Levi (Pascal 1995). 27 See J. Ryan (1945) for a discussion of various historical precursors of the Wager. 28 For brief introductions to the basic principles of decision theory, see for instance J. Joyce (2006) or M. Machina (2005), the latter also covering a range of heterodox accounts of rational choice. M. Resnik (1987) provides a gentle, book-length overview aimed at a philosophical audience, as does M. Petersen (2009), who covers essentially the same ground from a similar perspective. A more rigorous, but consequently more demanding, survey is provided in D. Kreps’ excellent textbook (Kreps 1988). 29 More precisely, the conclusion should be that one ought to take steps that would lead one to believe in God. We ignore this distinction in what follows.

14 p robab i l ity i n th e ph i lo s ophy of re l i g i on and secures a comparatively superior expected utility of belief in God by assuming a non-zero probability for the existence of God, a positively infinite value of a, and a finite value of b, c, and d. Indeed, the joint upshot of these assumptions is a positively infinite expectation of wagering for God versus a finite expectation of wagering against. Again, it is then concluded from this that rationality mandates belief in God. Even if one brackets the general debate over the very existence of non-evidential reasons for belief,30 the Wager remains a controversial argument to say the least.31 Most criticisms, however, granting the validity of the various incarnations of the Wager reviewed above, cast doubt instead on the plausibility of the premises involved. Alan Hájek’s contribution to the present volume, ‘Blaise and Bayes’, is not so concessive. Instead he contends that all three of Hacking’s reconstructed variants of the Wager are invalid in their original form and he takes the opportunity, along the way, to set straight a number of mistakes in the secondary literature. His first port of call is the Argument from Dominance. He argues that the argument is invalid as it stands and that, contrary to what has been claimed in the literature, it could only be rescued by adducing one of two further strong premises. One must either assume a positive probability for the existence of God or appeal to what he calls a ‘superduperdominance’, rather than mere weak dominance, of wagering for God, claiming that the best consequence of not wagering for God is strictly inferior to the worst consequence of wagering for God (assuming that a ≥ b and d ≥ c, this amounts to the claim that d < b). He then turns his critical eye onto the Argument from Dominating Expectation. Here he reiterates the worry that he developed in an earlier article, namely that the argument fails due to the possibility of opting for a so-called ‘mixed strategy’, a strategy that merely accords a positive, though non-unit, probability to wagering for God (see Hájek 2003). Given the infinite value of payoff a, any such strategy will, like the ‘pure’ strategy of wagering for God with probability 1, have a positively infinite expected utility. The upshot of this is that wagering for God is not the uniquely rationally permissible choice. In his earlier article, Hájek considered four possible ways of fixing the Wager (see, again, Hájek 2003)—each of which, as he observed, seemed to be incompatible with Pascal’s religious views. In his contribution to this volume, Hájek discusses two further options for rescuing the Wager, both of which involve taking c to be negatively 30 There is a substantial and rapidly growing literature on this issue. Expressions of scepticism can be found in J. Adler (2002), T. Kelly (2002), and N. Shah (2006). For dissent, see A. Reisner (2009). It is worth noting that some of those who deny that there can be non-evidential reasons to believe, simultaneously concede that there can be non-evidential reasons to cause oneself to believe (for example, Shah 2006). 31 For some general discussions of the recent critical literature, see A. Hájek (2008) and P. Saka (2002). Chapter 13 of J. H. Sobel (2004) offers a slightly more technically demanding overview. J. Jordan (1994) provides an important collection of essays on the topic.

introduction 15 infinite. One of these, he argues, appears to have the advantage of being faithful to Pascal’s theology and yields the desired conclusion, given the conservative extension of standard decision theory that he proposes. But the introduction of mixed strategies is not the only expansion of the set of choices with respect to the Wager that is considered in the literature. Another widelydiscussed source of variety is the possibility of wagering for alternative would-be deities. This is the root of Diderot’s so-called ‘Many Gods’ objection (Diderot 1875–7, LIX). By suitably expanding the decision matrix, one can introduce alternative wagers whose desirability is on a par with that of wagering for God. The next contribution to this volume, Paul Bartha’s ‘Many Gods, Many Wagers’, discusses a collection of decision problems which bring into play more than one deity to wager for. The first case discussed is one in which the agent considers a finite set of mutually exclusive theistic possibilities, of the form ‘Only Godn exists’, and a corresponding set of possible wagers, where the payoff for wagering for Godn , given that only Godn exists, is positively infinite and the payoff for wagering for Godm (m  = n) in the same circumstances is finite. Intuitions dictate that one ought to wager for the deity whose existence one judges to be most probable. The principle that delivers this intuition is sometimes known as ‘Schlesinger’s Principle’ (after Schlesinger 1988). It states that, of a set of acts that have positively infinite expected utility, it is only permissible to choose an act that maximizes the probability of achieving an outcome with positively infinite utility. Here Bartha provides a useful summary of his own recent work on relative utility theory—a generalization of classical decision theory devised to cope with decision problems involving infinite utilities (Bartha 2007)—and shows that, contrary to how matters might have originally appeared, Schlesinger’s Principle can be recovered from this framework. However, the bulk of Bartha’s contribution focuses on providing an innovative model of rational deliberational dynamics that handles the somewhat unusual character of the acts involved in Pascalian decision problems. Indeed, in line with Hacking, he understands the act of wagering for—respectively, against—a deity as taking steps that will increase to 1—respectively, decrease to 0—one’s assessment of the probability of that deity’s existence (see Hacking 1972: 188). But since decisions to wager are themselves grounded in such probability assessments, this leaves open the possibility of a rather peculiar kind of decision instability: making a wager that undermines the very grounds that one had for choosing it. To illustrate, consider a Pascalian decision problem involving a partition of two possible states of nature: the state in which a grouchy god exists and the state in which no god at all exists. Grouchy gods reward those who wager against them with a good of positively infinite value. In the remaining cases a merely finite payoff is received. According to standard infinitistic decision theory, one ought to wager against the grouchy god if and only if one grants a strictly positive probability to his existence. But consider an agent whose initial probabilities incline them to wager against. As their

16 p robab i l ity i n th e ph i lo s ophy of re l i g i on confidence in the non-existence of the grouchy deity goes to 1, their initial impetus for the wager is lost: wagering against is rationally self-defeating. There is an interesting parallel with frequency-dependent selection in evolutionary dynamics here. Consider the case of a simple anti-correlation game such as the game of Hawk and Dove (Maynard Smith and Price 1973). An aggressive hawkish strategy yields high dividends when played against a passive, dove-ish strategy, but leads to disastrous consequences when played against itself. Assuming random pairing of players, a low population frequency of hawks yields a low probability of playing against a hawk. This favours the differential replication of hawks at the expense of doves. However, as the frequency of hawks increases, the probability of playing against a hawk increases, and the fitness of playing hawk decreases, accordingly. One could say that, in a certain sense, the (pure) strategy of playing hawk is evolutionarily self-defeating. This parallel is exploited by Bartha in his proposed model of rational deliberational dynamics for Pascalian decision problems.32 He offers a model whose key advantage is that it allows the probabilities of making the different wagers to evolve over time, increasing proportionally to their relative expected utility. He then proposes that a (possibly mixed) wagering strategy is choice-worthy only if its probability vector corresponds to a certain kind of equilibrium in the deliberation dynamics. Bartha wraps up his argument by applying this framework to various ‘many gods’ Pascalian decision problems, observing that a number of theistic wagering strategies (that is, strategies that consist in equal mixtures of wagers for particular deities) turn out to be robustly choice-worthy. This, he concludes, provides a limited defence of Pascal’s Wager.

Part V. Faith and Disagreement The notion of rational belief has, in one way or another, been at work in all of the essays in this volume, although it came to the fore in the previous section. There is a long tradition in philosophy of religion, and in its predecessor philosophical theology, of reflection on the rational standing of religious belief—most especially, belief in the core tenet of theism: that God exists. Within this tradition, opinion has remained polarized over whether faith is or is not rational. As the early Christian apologist, Tertullian (c.160–220), exclaimed: What has Athens to do with Jerusalem?33 Athens, of course, here represents philosophy or reason, while Jerusalem represents faith. While there are scholarly disagreements about exactly how Tertullian’s remark should be interpreted,

32 Despite the novelty of his model, Bartha was not the first to discuss a possible connection between the dynamics of deliberation and the dynamics of evolution by natural selection. B. Skyrms (1994) highlights parallels between both (i) Jeffrey’s evidential decision theory and evolutionary dynamics under correlated interaction, and (ii) Savage’s causal decision theory and evolutionary dynamics under random interaction. 33 See Tertullian, De Praescriptione haereticorum 7.9.

introduction 17 his point is usually taken to be that religious faith does not require the support of reason and, in that sense, it is irrational. Despite its rhetorical appeal, and its ability to explain why there is so much disagreement about matters of religious conviction, this view has never completely won out over the view that faith and rationality can be successfully partnered. Although a Wittgensteinian form of fideism enjoyed undeniable influence within philosophy of religion in the 1970s and 1980s,34 the view that religious belief is irrational (or, in some versions, arational) has never been the dominant position within the discipline. Instead, philosophers of religion have been understandably reluctant to give up the idea that it can be rational to have faith, and many have been more sympathetic to Anselm of Canterbury’s (1033–1109) well-known vision of faith seeking understanding than they have to Tertullian’s position. In recent times the effort to articulate the relationship between faith and reason has increasingly led philosophers of religion to explore different models of rational belief and to test their applicability to the religious domain.35 This project is being pushed forward with the help of increasingly sophisticated ways of thinking about the relationship between rationality and probability. The final two essays in this volume engage with this project and each seeks to expand our understanding of the connection between probability, rationality, and belief in the context of theism. In his ‘Does Religious Disagreement Actually Aid the Case for Theism?’, Joshua Thurow argues against the popular claim that mutually recognized disagreement among rational peers with respect to a given proposition P would invariably lead to suspension of judgement as to whether or not P.36 He contends that the dynamics of belief under revealed disagreement must be sensitive to the justificatory structure of the agents’ beliefs: suspension of judgement is indeed mandated under revealed disagreement, but only with respect to the agents’ basic beliefs, those beliefs from which the remainder of their beliefs are derived. He motivates his claim with an example, the ‘disagreeing detectives case’, whose schematic structure is as follows: Two rational, equally cognitively competent and well-informed agents agree on a basic body of evidence E which, taken alone, would warrant an inference that P. They also agree that adding the further proposition Q to this body of evidence would override the inference, licensing instead a conclusion that not-P. They disagree, however, as to whether or not Q, with one agent holding that Q and the other that not-Q. The upshot of this is a derivative disagreement, in turn, 34 This was inspired by certain remarks in Wittgenstein’s Philosophical Investigations (1959) and his Lectures and Conversations on Aesthetics, Psychology & Religious Belief (1966). D. Z. Phillips and Norman Malcolm were both prominent proponents of this approach to philosophy of religion. See, for example, Phillips (1970, 1971, 1986, and 1988) and Malcolm (1977). 35 The Reformed Epistemology movement, discussed earlier, is part of this trend. 36 In the context of theistic belief, this view has been most notably espoused by R. Feldman (2007). See Thurow’s contribution to this volume for further references.

18 p robab i l ity i n th e ph i lo s ophy of re l i g i on as to whether or not P. After mutual revelation of their respective beliefs, the agents suspend judgement on the basic issue of whether or not Q. And this, Thurow argues, then leads to an agreement that P.37 The structure of this example, he argues, may well be instantiated in the context of religious disagreement. Here he suggests that we could, for instance, find agreement among peers regarding the existence of various witness reports to the occurrence of some miraculous event (E). This, taken alone, may warrant an inference to the existence of God (P). The peers could further agree that were one to grant the claim that various collectively endorsed beliefs are in tension with the claims of theism (Q), the resulting total body of evidence would however, on balance, warrant an inference to not-P. But the peers disagree about whether or not Q, and consequently about whether or not P. After disclosure of their respective beliefs the agents then come to an agreement that P. In addition to these informal considerations, Thurow offers a probabilistic model that he takes to vindicate his claims, drawing on a distinction between foundational and non-foundational aspects of an agent’s probabilistic credence function. On his model, after mutual disclosure of their respective credences, agents average their foundational degrees of belief, then let these opinion shifts at the foundational level percolate through to the remainder of their credences. In the final contribution to this collection, Lara Buchak develops an account of the attitude of having faith in a proposition, and discusses the conditions under which, given this account, one could provide a positive answer to the titular question of her article: ‘Can it be Rational to Have Faith?’ After exploring, and eventually rejecting, a number of alternative proposals, Buchak settles for an account according to which having faith that P entails having a commitment to decline to obtain any further evidence as to whether or not P prior to acting on the supposition that P (in other words, having a preference to remain ignorant of such further evidence). With this account of the attitude of having faith in hand, she turns to the question of the rationality of faith, and in particular, to the rationality of declining to obtain further information prior to acting. As Buchak notes, there are a number of circumstances under which such behaviour is clearly permissible, and indeed mandated by the canons of rationality: there may be various costs associated with the acquisition of further information that outweigh any potential benefits that this acquisition may bring. But, as she also points out, the putative rationality of declining cost-free information sits somewhat uneasily with a well-known result due to I. J. Good (1967). According 37 It is worth noting, as Thurow acknowledges, that this kind of situation would appear to be proscribed

by a condition known in the literature on belief revision as ‘Preservation’. Indeed, according to the Principle of Preservation, it is irrational to suspend judgement on an item of information (such as Q) whose assimilation would lead to the retraction of one’s current beliefs (such as the belief that P). The upshot of this would be that, in view of the fact that adding Q to E would license the inference that not-P, it cannot be the case that Q, taken by itself, licenses the inference that P.

introduction 19 to Good’s Theorem, granting standard expected utility theory and a number of weak auxiliary assumptions, the option of obtaining cost-free information prior to acting is weakly preferable, and in some cases strictly preferable, to the option of declining it. So is it then irrational to have faith, on Buchak’s analysis of the concept? Not necessarily so, she avers. Building on her earlier work, she argues that, on some independently-motivated generalizations of expected utility theory, Good’s result no longer holds: in such frameworks, under certain circumstances, declining cost-free information can be strictly preferable to obtaining it (see Buchak 2010).38

Future Directions Perhaps it is this last topic of ‘Faith and Disagreement’ that points us most squarely towards the future direction of philosophy of religion. Once again, as in the midtwentieth century, philosophy of religion seems to be undergoing a time of rapid evolution in response to a number of pressures. It could be argued that the most important factor impacting the discipline today is the increasingly urgent requirement that it should take into account the diversity of religious conceptions available, while developing to include a broader range of topics than those that fell within its traditional disciplinary purview. That purview was largely limited to issues concerning the existence and nature of God, as God is conceived within Western forms of theism. Such issues are now often regarded as only one part of the domain of inquiry that philosophers of religion might be concerned with (see Schellenberg 2008; Harrison 2010).39 While the chapters in this volume each demonstrate the continued fecundity of ongoing philosophical examination of the central tenets of Western theism, they also invite further studies which will deploy the tools of probabilistic thinking in the context of the expanded domain of the philosophy of religion of the future. It remains to be seen how probability theory can contribute to this exciting and urgent new phase in the development of the discipline. However, judging by the work presented in this volume, we can expect that the interplay of probabilistic reasoning and philosophy of religion—which has been so fruitful over the last several decades—will make a valuable contribution to the emergent new style of philosophy of religion. The chapters contained here have showcased the extent to which philosophy of religion has successfully assimilated probability theory. But they also point the way to future work in philosophy of religion that, building on what has already been achieved, will be positioned to engage even more fully with the very latest refinements that are 38 It is worth noting that T. Seidenfeld (2004) criticizes various generalizations of expected utility theory—in particular the coupling of set-based Bayesianism with the so-called ‘-Maximin’ decision rule—for having precisely this kind of consequence. One person’s modus ponens is another’s modus tollens, as they say. 39 The importance of this trend is evident in the steadily increasing number of books aimed at undergraduates taking seriously ideas and arguments concerning a range of different religious conceptions. Some of the best examples are: C. Taliaferro (1998), K. Yandell (1999), and G. Griffith-Dickson (2000 and 2005).

20 p robab i l ity i n th e ph i lo sophy of re l i g i on even now emerging from the workshops of probability theorists. We conclude this introduction, then, by expressing our confidence that probability theory will continue to enjoy a profound influence on philosophy of religion for some time to come.

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introduction 23 Petersen, M. (2009) An Introduction to Decision Theory. Cambridge: Cambridge University Press. Phillips, D. Z. (1970) Faith and Philosophical Enquiry. London: Routledge & Kegan Paul. (1971) ‘Religious Beliefs and Language-games’. In B. Mitchell (ed.), The Philosophy of Religion. Oxford: Oxford University Press, 121–42. (1986) Belief, Change and Forms of Life. London: Macmillan. (1988) Faith After Foundationalism. London: Routledge. Plantinga, A. (1993a) Warrant: The Current Debate. New York: Oxford University Press. (1993b) Warrant and Proper Function. New York: Oxford University Press. (2000) Warranted Christian Belief. New York: Oxford University Press. and M. Tooley (2008) Knowledge of God. Oxford: Blackwell Publishing. Ratzsch, D. (2010) ‘Teleological Arguments for God’s Existence’. In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2010 Edition): http://plato.stanford.edu/archives/ win2010/entries/teleological-arguments Reid, T. (1822) ‘Inquiry into the Human Mind’. In D. Stewart (ed.), The Works of Thomas Reid. Volume 1. New York: E. Duyckinck, Collins & Hannay, 135–330. Reisner, A. (2009) ‘The Possibility of Pragmatic Reasons for Belief and the Wrong Kind of Reasons Problem’, Philosophical Studies, 145: 257–72. Resnik, M. D. (1987) Choices: An Introduction to Decision Theory. Minneapolis: University of Minnesota Press. Rowe, W. L. (1979) ‘The Problem of Evil and Some Varieties of Atheism’, American Philosophical Quarterly, 16: 335–41. (1991) ‘Ruminations about Evil’, Philosophical Perspectives, 5: 69–88. Ryan, J. (1945) ‘The Wager in Pascal and Others’, New Scholasticism, 19, 3: 233–50. Reprinted in J. Jordan (ed.) (1994) Gambling on God: Essays on Pascal’s Wager. Savage, MD: Rowman & Littlefield, 21–9. Saka, P. (2002) ‘Pascal’s Wager about God’. In J. Fieser and B. Dowden (eds), Internet Encyclopedia of Philosophy: http://www.iep.utm.edu/pasc-wag Sarkar, S. (2007) Doubting Darwin? Oxford: Blackwell. Schellenberg, J. L. (2008) ‘Imagining the Future: How Scepticism Can Renew Philosophy of Religion’. In D. Cheetham and R. King (eds), Contemporary Practice and Method in the Philosophy of Religion: New Essays. London: Continuum, 15–31. Schlesinger, G. (1988) New Perspectives on Old-time Religion. Oxford: Clarendon Press. Seidenfeld, T. (2004) ‘A Contrast between Two Decision Rules for Use with (Convex) Sets of Probabilities: -Maximin versus E-Admissibility’, Synthese, 140 (1/2): 69–88. Shafer, G. (1985) ‘Conditional Probability’, International Statistical Review, 53: 261–77. Shah, N. (2006) ‘A New Argument for Evidentialism’, The Philosophical Quarterly, 56: 481–98. Skyrms, B. (1994) ‘Darwin Meets The Logic of Decision: Correlation in Evolutionary Game Theory’, Philosophy of Science, 61, 4: 503–28. Sobel, J. H. (1991) ‘Hume’s Theorem on Testimony Sufficient to Establish a Miracle’, Philosophical Quarterly, 41: 229–37. (1992) ‘Kings and Prisoners (And Aces)’, Proceedings of the Biennial Meeting of the Philosophy of Science Association. Volume One: Contributed Papers: 203–16. (2004) Logic and Theism: Arguments For and Against Beliefs in God. Cambridge: Cambridge University Press.

24 p robab i l ity i n th e ph i lo sophy of re l i g i on Sober, E. (2005) ‘The Design Argument’. In W. Mann (ed.), The Blackwell Guide to the Philosophy of Religion. Oxford: Blackwell, 117–47. Swinburne, R. (1971) The Concept of Miracle. London: Macmillan. (1979) The Existence of God. Oxford: Oxford University Press. (1990) ‘Argument from the Fine-tuning of the Universe’. In J. Leslie (ed.), Physical Cosmology and Philosophy. London: Prentice Hall, 154–73. (1997) Simplicity As Evidence of Truth. Milwaukee, WI: Marquette University Press. Taliaferro, C. (1998) Contemporary Philosophy of Religion. Oxford: Blackwell. (2005) Evidence and Faith: Philosophy and Religion since the Seventeenth Century. Cambridge: Cambridge University Press. Tertullian (1907) De praescriptione haereticorum. Trans. and intro. by P. de Labriolle. Paris: no publisher. Tooley, M. (2010) ‘The Problem of Evil’. In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2010 Edition): http://plato.stanford.edu/archives/spr2010/entries/evil Trakakis, N. (2006) The God Beyond Belief: In Defence of William Rowe’s Evidential Argument from Evil. Dordrecht: Springer. Venn, J. (1888) The Logic of Chance. New York: Chelsea Publishing Company. Watts, F. (ed.) (2008) Creation: Law and Probability. Aldershot: Ashgate. Wittgenstein, L. (1959) Philosophical Investigations. Trans. G. E. M. Anscombe. Oxford: Blackwell. (1966) ‘Lectures on Religious Belief ’. In L. Wittgenstein, Lectures and Conversations on Aesthetics, Psychology & Religious Belief. Oxford: Blackwell, 53–72. Wolterstorff, N. (1983) ‘Can Belief in God be Rational if it has no Foundations?’ In A. Plantinga and N. Wolterstorff (eds), Faith and Rationality: Reason and Belief in God. Notre Dame: University of Notre Dame Press, 135–86. Yandell, K. (1999) Philosophy of Religion: A Contemporary Introduction. London and New York: Routledge.

PART I

Testimony and Miracles

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2 Peirce on Miracles: The Failure of Bayesian Analysis Benjamin C. Jantzen

I. Introduction The sacred texts of the major monotheist religions contain reference to a great many wonders, prodigies, and mysteries—what modern commentators typically refer to as ‘miracles’. Historically, these miracles were invoked as justification for belief in particular revealed religions; miracles are the marks of true revelation. In The Analogy of Religion, Joseph Butler characterizes the role of miraculous claims in these terms: It is an acknowledged historical Fact, that Christianity offered itself to the World, and demanded to be received, upon the Allegation, i.e. as Unbelievers would speak, upon the Pretence, of Miracles, publickly wrought to attest the Truth of it, in such an Age; and that it was actually received by great Numbers in that very Age, and upon the professed Belief of the Reality of these Miracles. (Butler 1736: 241)

As Butler suggests, many have viewed testimony of the miraculous as rational justification for belief in the genuineness of a prophet, the divinity of an ostensible Messiah, or the supernatural character of a remarkable occurrence. While the relative importance of miracles to the project of natural theology or the rationalization of revealed religion is debatable, testimony of the miraculous offers at least some evidence in favour of religious belief, and as Butler says, ‘it lies upon Unbelievers to shew, why this Evidence is not to be credited’ (Butler 1736: 243). Butler’s challenge can be stated more neutrally as a question about the epistemic value of historical testimony: What credence should we attribute to the occurrence of a miracle when all we have to go on are the historical texts of ‘those who have writ upon the subject’? (Butler 1736: 243). One prominent response is provided by the Bayesian approach to inference. Bayesianism consists of two parts: (i) the identification of probability with rational degrees of belief and (ii) a rule for inductive inference. This rule concerns the way in which probabilities (and by extension, degrees of belief) should be updated upon learning new facts. If P(q|B) is the probability that proposition q is true given background

28 p robab i l ity i n th e ph i lo sophy of re l i g i on knowledge B (called the prior probability of q), then upon learning some new fact e at a later time, one should assign a new probability (the posterior probability) to the proposition q as follows: P(q|B, e) =

P(e|B, q)P(q|B) P(e|B)

(1)

For the Bayesian, the answer to our question about the epistemic value of testimony can be rephrased as a question about the posterior probability of a miracle given testimony. This probability is determined by the prior probability of the miracle, the probability that a witness is accurate given that the miracle really did occur, and the probability of obtaining testimony of the sort proffered by the witness, all conditional on our background knowledge. A detailed account of the Bayesian approach to testimony is provided in Section III. While there are grounds to disagree over the values of the relevant probabilities in the case of miracles, it is generally accepted that the Bayesian approach can be consistently applied to inferences from testimony. That is, the use of the Bayesian rule of induction in the context of miraculous testimony is taken to be no more or less controversial than the Bayesian programme itself. In an overlooked argument spread out over three manuscripts, Charles Sanders Peirce (1839–1914) provides a sweeping critique of the Bayesian approach as it applies to the assessment of historical testimony.1 Insisting that prior probabilities must be determined by facts about the world, such as empirical frequencies, Peirce begins his critique by arguing that there can be no objective probability of a witness’ veracity. If we grant that a suitable probability can be had, says Peirce, then we still have a problem with independence—those conditions which lead one witness into error tend to affect others. These criticisms are developed in Section IV below. Brushing them aside however, still leaves Peirce’s strongest objection: the Bayesian computation is structurally biased, and the very sampling process of history renders this bias ineliminable. This two-part claim—the subject of Sections V and VI—is the most original part of Peirce’s critique. Finally, even if each of the preceding objections can be defeated, Peirce claims that the Bayesian inference scheme is, like any other ‘rule of inference’, subject to empirical verification. He musters some historical evidence, discussed in Section VI, to suggest that the Bayesian approach is an empirical failure. My aim is to present Peirce’s argument in the strongest possible light, and to single out those objections which still have purchase on the Bayesian approach. As it stands, no

1 Throughout this essay I use the notation CP x.xxx to cite material from the Collected Papers of Charles Sanders Peirce (1935). In this convention, (CP 1.234) for instance refers to volume 1, paragraph 234. Using this notation, the three manuscripts are: ‘The Logic of Drawing History from Ancient Documents’ (CP 7.162–7.255); a fragment called ‘Hume on Miracles’ by the editors of Peirce’s collected papers (CP 6.522–6.547); and a piece entitled ‘The Laws of Nature and Hume’s Argument Against Miracles’ reproduced in Wiener et al. (1947). For a more detailed exegesis of the argument contained in these manuscripts, see Jantzen (2009).

pe irc e on m irac le s 29 satisfactory response to the problem of bias in historical sampling can be found in the literature on Bayesian probability.

II. The Context of Peirce’s Critique Peirce had two targets in mind when he penned his critique: German historians—whom we’ll get to in Section VI—and David Hume. Hume thought he had struck upon a general argument for discounting the evidentiary force of miraculous testimony, an argument he eventually published as Chapter X of his An Enquiry Concerning Human Understanding. In outline, the argument runs as follows. The occurrence of a miracle is, in Hume’s terminology, a ‘matter fact’ and such propositions can only be ascribed a probability; they cannot be known with certainty.2 The probability attributed to a proposition is proportioned to the evidence of experience: if a particular effect has always been observed to follow a given cause, then we should have the greatest possible confidence that a future instance of the cause will result in the same effect. Now, a miraculous event is an event that violates some presumptive law of nature—it is an event contrary to all prior experience.3 Thus, the proposition that a miracle occurred should be granted the lowest possible probability. Even the testimony of a perfect witness—one never known to have told a falsehood—can at best be granted a moral certainty that, in Hume’s calculus, precisely cancels our certainty that the miracle did not occur. The result is that no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior. (Hume 1988: 105–6)

Nothing in the above reconstruction (except perhaps the direct quotation) is uncontroversial. There is an enormous literature seeking to establish just what it is Hume meant in the first place. Much of this debate centres on Hume’s definition of a miracle as ‘a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent’ (105). Particularly contentious are the relevant notions of a law of nature and the manner in which we might have epistemic access to volitions of the Deity. However, many authors including Peirce draw an important distinction that allows us to sidestep this debate. The claim that a miracle occurred 2 Hume uses the term ‘probability’ in a way that is difficult to reconcile with the technical sense of the term. For instance, he famously speaks of adding and subtracting probabilities of events to determine a posterior probability. In order to apprehend the gist of Hume’s argument, the reader is invited to understand ‘probability’ in the modern technical sense with the caveat that only non-extremal probabilities are to be admitted for matters of fact. 3 John Earman introduces the helpful term ‘presumptive law’ when reconstructing Hume’s notion of a miracle (see Earman 2000). A presumptive law is simply a law-like regularity for which we possess uniform support. This stands in contrast to an actual law for which exceptions would amount to a logical contradiction.

30 p robab i l ity i n th e ph i lo sophy of re l i g i on is actually the conjunction of two claims: (i) that a particular event occurred, and (ii) that the cause of the event was supernatural. One can focus on the question of whether the purported event took place, where the event is described in neutral naturalistic terms, and ignore the further problem of how that event came to be or how it should be characterized. Like Peirce, I will focus only on the former question. When I speak about the probability that a miracle occurred, I will mean the probability of the event occurring as described, irrespective of the nature of that event (natural or supernatural). The relevant notion of ‘miracle’ is thus an unprecedented or extremely rare event. Peirce interprets Hume as providing a probabilistic inference scheme for which testimony is generally insufficient to establish the occurrence of singular events. That inference scheme is the Bayesian rule for updating probabilities described in the next section.

III. The ‘Method of Balancing Likelihoods’: A Bayesian Approach Suppose we have n witnesses, each of whom testifies to the occurrence of a miraculous event. Bayesianism asserts that a rational agent’s degree of belief in a proposition should conform to the probability of that proposition. By adopting the Bayesian framework, we can use the probability calculus to give a precise answer to our question which now has the following form: Given the testimony of these n witnesses, what is the probability that the event they describe has in fact occurred? Introducing some notation will make the expressions to follow much easier to read. Denote by wi the ith witness, and let m stand for the proposition that a miraculous event (in the sense of an unprecedented event) has occurred. Then the probability that m is the case given all of the witness testimony in favour of m can be denoted by P(m|w1 says “m”, w2 says “m”,. . . wn says “m”). Applying Bayes’ theorem allows us to express this probability as P(m|w1 says “m”, w2 says “m”, . . . , wn says “m”) =

P(w1 says “m”, w2 says “m”, . . . , wn says “m”|m)P(m) P(w1 says “m”, w2 says “m”, . . . , wn says “m”)

(2)

We can make this expression more compact by introducing another notational device. If wi says “m” and m is the case or if wi says “not m” (denoted ∼m) and ∼m is the case, then we write wi +. That is, wi + stands for “witness wi is correct”. Likewise, wi − stands for “witness wi is incorrect”. Now we can write the desired probability in the form P(m|w1 says “m”, w2 says “m”, . . . , wn says “m”) =

P(w1 +, w2 +, . . . , w22 + |m)P(m) P(w1 says “m”, w2 says “m”, . . . , wn says “m”)

(3)

pe irc e on m irac le s 31 Now suppose that each witness is fully independent of all the others conditional on the truth or falsity of m such that P(wi + |m, {w+}) = P(wi + |m) and P(wi − |∼m, {w−}) = P(wi − |∼m), where {w+} is any set of wj + for which i  = j and {w−} is any set of wj − for which i  = j. If these conditions obtain then (3) reduces to P(m|w1 says “m”, w2 says “m”, . . . , wn says “m”)  P(m) ni=1 P(wt + |m) = P(w1 says “m”, w2 says “m”, . . . , wn says “m”)

(4)

The probability appearing in the denominator on the right hand side of Equations (2), (3), and (4)—the probability of the set of witnesses testifying to “m”—is difficult to assess, but can be eliminated by considering odds ratios. Rather than ask for the probability that the singular event occurred we can ask for the odds in favour of the event occurring given the testimony of the n witnesses: O(m|w1 says “m”, w2 says “m”, . . . , wn says “m”) =

P(m|w1 says “m”, w2 says “m”, . . . , wn says “m”) P(∼m|w1 says “m”, w2 says “m”, . . . , wn says “m”)

(5)

If witnesses are independent conditional on the fact of the event, then we can use (4) to rewrite the ratio in (5) strictly in terms of the prior probability of the miracle and the individual probabilities that each witness gives true testimony. The use of the resulting expression to compute the odds in favour of the miracle having occurred and to apportion belief accordingly, is what Peirce calls the ‘Method of Balancing Likelihoods’ (hereafter MBL) (CP 7.176). We can summarize this Bayesian inference scheme as follows: MBL: Given n witnesses w1 , w2 , . . . ,wn providing testimonies that m is the case, where the probability for each witness being correct or incorrect conditional on the occurrence or non-occurrence of the miracle respectively is known and fully independent of every other witness, the odds in favour of m being the case are given by: O(m|w1 says “m”, w2 says “m”, . . . , wn says “m”) =

P(m)P(w1 + |m)P(w2 + |m) · · · P(wn + |m) P(∼m)P(w1 − |∼m)P(w2 − |∼m) · · · P(wn − |∼m)

While Hume’s original argument against miracles is controversial, MBL is a straightforward application of Bayesian probability. Before considering Peirce’s objections to MBL, it is worth considering how he thinks the method captures Hume’s intended argument. To this end, consider the case of a single witness, w1 . According to MBL, the odds in favour of a miracle are given by O(m|w1 says “m”) =

P(m)P(w1 + |m) . P(∼m)P(w1 − |∼m)

By supposition, the prior odds against a miracle (P(∼m)/P(m)) are overwhelming. In order for the posterior odds in favour of the miracle—the odds after we have learned of

32 p robab i l ity i n th e ph i lo sophy of re l i g i on the witness testimony—to be appreciable, the odds against the witness being mistaken (P(w1 + |m)/P(w1 − |∼m)) must be equally overwhelming, and even then the result is roughly even odds. In this sense Hume’s maxim holds.4 Though Peirce doesn’t point it out, MBL actually poses a problem for Hume’s sweeping conclusion when more than one witness is considered. Let’s assume that the odds against a miracle are very great, though finite such that P(m)/P(∼m) = 10−100 . Suppose further that any given witness is only slightly more likely to correctly report a miracle than to give false testimony of one, such that P(wi + |m)/P(wi − |∼m) = 1.001. A single such witness accords with Hume’s claim, since MBL tells us that, even after learning of the witness’ testimony, odds are still overwhelmingly against the miracle having occurred. However, if witnesses are independent of each other conditional on the event, then, as Charles Babbage proved, it is always possible to specify a number of witnesses such that their combined testimony overwhelms the improbability of the miracle.5 In this case, somewhat less than a quarter million witnesses would suffice. Of course, this is an unrealistic number. But so is 10−100 when given as a probability supposedly based upon experience. The point is that no matter how improbable an event (assuming it has some finite prior probability) it is always possible in principle to muster enough independent witnesses to tip belief in favour of the event. For more modest numbers, such as prior odds of 10−12 in favour of the event and odds in favour of an accurate witness report of 3 to 1, then only a manageable 25 witnesses are required. While this line of argument might be tempting for those who would turn MBL against Hume, we’ll see in Section V that it actually opens the door to Peirce’s strongest objection to the entire Bayesian approach to historical testimony. Before we get there, however, there are some preliminary objections to consider.

IV. Against MBL: Probabilities and Independence To begin with, adopting an interpretation of probability akin to Peirce’s precludes attributing probabilities to witness veracity. One can plausibly read Peirce as offering at least two distinct arguments for this claim. In the first, Peirce denies the meaningfulness of the putative probabilities, arguing that there is ‘no such quantity as a real, general, and predictive truthfulness of a witness’ (CP 7.179). This argument hinges critically on 4 If ‘we suppose that the impression made on the mind of the wise man is proportional to the logarithm of the odds as its exciting cause, then the total impression will be’:       P(m) P(w1 + |m) P(m)P(w1 + |m) log = log + log P(∼m)P(w1 − |∼m) P(∼m) P(w1 − |∼m)

(CP 7.165, emphasis in original). That is, if we apportion belief as the logarithm of probability, we can even recover Hume’s talk of adding and subtracting evidence. 5 Babbage proved this result, and used it to argue against Hume in his Ninth Bridgewater Treatise (1838) (see Chapter X and Appendix E). However, he was not careful to distinguish the relevant sense in which witnesses must be independent.

pe irc e on m irac le s 33 Peirce’s account of probability.6 For Peirce, probabilities are objective facts pertaining to arguments—they reflect the limiting frequency with which a particular pattern of inference yields true conclusions when given true premises of a particular type. So for instance, to say that a fair coin has probability 1/2 of coming up heads amounts to the following: if we always infer that the coin will come up heads when told that the coin is subjected to a particular flipping process, we will be right half of the time in the limit. In his mature account, Peirce notes that ‘in the limit’ refers to what would be the case if we flipped the coin an infinite number of times. That is, the frequency to be identified with a probability is not an actual frequency but rather the limiting frequency that would result if the specified inference were applied infinitely many times. In the coin example, probability is to be identified with the limiting frequency of heads in an infinite number of flips. Importantly, Peirce seems to think that the process in question (e.g. flipping a coin in a specified manner) must be repeatable in an actual, temporal series of indefinite length. That is, the limiting frequency concerns what would be the case if we actually repeated the flipping of the coin for a number of times approaching infinity. When it comes to the judgement of witnesses, Peirce asserts there is no such repeatability. Once a witness has passed judgement on, say, the occurrence of a miracle, we cannot ask him to pass judgement on the same sort of miracle again—we cannot repeat the process: ‘His new statements, if he makes any, will necessarily relate to different topics from his old one which he has exhausted; and his personal relation to them will be different. There is, therefore, no argument from what his credibility was in one case, to what it will be in another . . . ’ (CP 7.179). From this fact, Peirce concludes that there is no such thing as the objective probability of witness veracity. Thus, we might reconstruct Peirce’s first objection to witness probability as follows: (i) meaningful probabilities attach only to rules of inference that can in principle be applied infinitely many times (i.e. the premises occasioning the use of the rule must potentially be true on infinitely many occasions); (ii) humans cannot even in principle be asked to repeat relevantly identical judgements when issuing testimony; (iii) therefore, there are no meaningful probabilities corresponding to the accuracy of human judgements. I would argue that this is the best representation of Peirce’s principal objection to witness probabilities, at least as expressed in CP 7.162–7.255. However, this is a weak objection for a number of reasons. Most prominently, one could simply point to a variety of interpretations of probability that avoid this concern by dropping the requirement of the potential repeatability of a stochastic process. For instance, one could continue to insist on objective probabilities and adopt what Donald Gillies calls a ‘single-case propensity theory’, or one could simply embrace the subjective Bayesian approach (see Gillies 2000).

6 Peirce’s seminal writings on probability—both early and late—are collected in J. Buchler (1955: chs 12 and 13).

34 p robab i l ity i n th e ph i lo sophy of re l i g i on There is another way to argue against the plausibility of assigning probabilities in the manner MBL requires. Whereas the above approach focused on the repeatability of witness judgements and argued against the meaningfulness of the assigned probabilities, this alternate approach focuses on the event in question and argues against the epistemic accessibility of the relevant probabilities. Specifically, the relevant reference class for the probability represented by P(w1 + |m) is the set of all events in which w1 offers testimony either for or against the occurrence of m when a miracle of the same sort as m actually occurs. Since miracles are by definition rare, this reference class will be quite small. In fact, Peirce seems to think that it consists of at most one member. This is a plausible reading of what Peirce is getting at when he says: [I]n insurance, though the cause of any one man’s death might be ascertained, yet that would have no relation to the purposes of insurance, and why it is that out of a thousand men insured at the age of thirty, just so many will die each year afterward, is a question not to be answered except that it is due to the cooperation of many causes. It is this which makes the calculation of chances appropriate. . . . But now on the other hand, take a question of history. We do not care to know how many times a witness would report a given fact correctly, because he reports that fact but once. (CP 7.178)

In other words, the event description ‘a man insured at age thirty died within five years’ is sufficiently coarse-grained that the reference class of ‘men insured at age thirty’ contains many instances. However, the reference class ‘a witness who offered testimony on the occasion of the resurrection of Christ’ is so small as to be trivial.7 This does not entail that the associated probabilities are meaningless. But it does mean that there is no way to estimate them with evidence unrelated to the event in question. We cannot even ascertain whether the reference class is empty without knowing whether the miracle in question occurred. If we knew that, we wouldn’t need MBL. As a distinct epistemic objection, Peirce suggests that we have so far failed to condition on some salient facts, namely that the veracity of a witness is dependent upon the nature of the story being reported. As Peirce puts it, ‘[t]he method of balancing likelihoods not only supposes that the testimonies are independent but also that each of them is independent of the antecedent probability of the story’ (CP 7.176). In other words, one’s inclination to report on an event is in part determined by the nature of that event. Generally, the more marvellous or extraordinary an event is, the more likely it is that a witness will report it. If I were out for an evening stroll and witnessed a rain shower, then I would not be very likely to say anything about it. However, if I were to witness a tornado—a much less probable event—then I would be very 7 This is roughly how Cathy Legg reads the quoted passage (see Legg 2001). I should also point out

that, mixed in with the arguments discussed here, Peirce makes an additional objection predicated on the assumption that probability is useless when attempting to settle whether a particular application of an inference yielded a true result (e.g. in determining whether a witness was accurate on a particular occasion). This is consonant with what he says elsewhere about the meaninglessness of the probability of an event. See, for instance, Buchler (1955: 160).

pe irc e on m irac le s 35 likely to issue testimony to this effect. In historical matters, Peirce thinks that ‘this circumstance in itself almost destroys the legitimate weight of any argument from the antecedent improbability [of an event]’ (CP 7.176). That is, while marvellous events are improbable, that very improbability makes it overwhelmingly likely that witnesses will testify to the event. As Peirce notes in passing, however, this concern cuts both ways with respect to witness veracity. While they are more likely to issue testimony, witnesses may also alter their reports to avoid asserting something they deem highly improbable. Thus, the probability of a witness giving accurate testimony conditional on the perceived improbability of a story can be either much greater or much lower than the unconditional probability. This dependence of witness veracity on the perceived improbability of an event can be expected to vary from witness to witness and across events—two people may respond very differently to the same improbable event. It is therefore implausible that we could discover a general rule for estimating the effect of an event’s improbability on the veracity of a witness, and thus implausible that we could estimate the probabilities required by MBL given the available information. Of course, both this and the preceding epistemic argument gain purchase only if we insist on interpreting MBL in terms of objective probabilities, the values of which can be estimated from the available empirical data. Once again, it is possible to evade the concern by endorsing an alternate interpretation of probability, though the options are more limited. For instance, one could still take the Bayesian view, but only at the cost of admitting a broad permissiveness in assigning priors. That is, given the epistemic problems emphasized by Peirce, the Bayesian’s subjective priors cannot be very tightly bound to any empirical constraints. Nonetheless, the position remains viable. All of the above arguments against assigning probabilities to witness veracity obtain only insofar as we insist on a frequentist or propensity account of probability resembling Peirce’s.8 If we shift to an alternate account of probability—such as that of the subjective Bayesian—these worries largely vanish. Both Peirce and I are willing to concede this point and proceed to examine MBL under the assumption that the probabilities considered so far are well-defined and epistemically accessible. Doing so leads immediately to a further difficulty with independence. In order to allow witness probabilities to be assessed one at a time,9 MBL requires that witnesses be independent of one another conditional on the fact of whether m is true or not.10 But this is unlikely 8 Peirce’s position evolved over time from a plainly frequentist account to one which Donald Gillies would call a ‘long-run propensity theory’ (see Gillies 2000). 9 If it is questionable whether or not there exists a non-trivial reference class with respect to which one can define an objective probability P(wi + |m), then it is all the more implausible that one exists for the joint distribution P(w1 +, w2 +, . . . , wn + |m)P(m). Even if one does, it is difficult to see how one could ever have access to that distribution. 10 Commentators on Peirce and Hume alike have not been careful to distinguish which independencies are required. MBL demands that testimonies be independent of one another conditional on the fact of the event, not that they be unconditionally independent. In fact, for MBL to apply it need not be the case that the veracities of the witnesses are unconditionally independent. Cathy Legg says only that the ‘testimonies would have to be independent of each other’, while Kenneth Merrill seems to interpret Peirce as insisting

36 p robab i l ity i n th e ph i lo sophy of re l i g i on to be the case as ‘the same circumstances which lead one witness into error are likely to operate to deceive another’ (CP 7.176). In addition, witnesses are inclined to agree or contradict one another irrespective of the facts for a variety of motivations (e.g. rivalry or common political aims).11 Once again, the subjective Bayesian has an answer: we can at least guess at this mutual influence when assigning our prior conditional probabilities and perform the more complex computation indicated by Equation (3). Whether we accept this solution or simply concede for sake of argument that witnesses are approximately independent in the way MBL requires, the Bayesian approach leaves one more objection unanswered, one which cannot be evaded within the Bayesian framework.

V. Against MBL: A Problem for the Bayesian Peirce’s strongest criticism of MBL amounts to a general indictment of Bayesian inference. To see the criticism, it will help to have in mind the outline of Peirce’s philosophy of science. For Peirce, scientific inference consists of three stages; the first he calls ‘abduction’ (CP 6.525), the second ‘deduction’ (CP 7.203), and the third ‘induction’ (CP 7.206). In the abductive stage, one tentatively adopts hypotheses that stand as viable explanations for whatever surprising facts or phenomena have motivated an inquiry. A hypothesis explains a set of facts just if the truth of the hypothesis would render the facts likely. An abductive inference—an inference from the facts to an explanatory hypothesis—is an inference of the following form: (1) Surprising fact C is observed. (2) If proposition A were true, C would be likely. (3) A is an explanatory hypothesis for C. The abductive stage of scientific inquiry is a filter separating those propositions worthy of testing from those that are not. That a proposition is abductively inferred from a set of facts, however, provides no grounds for attributing any particular degree of confidence to that hypothesis. For Peirce, this is accomplished only by the latter two stages of the process in which the consequences of a given hypothesis are deduced and tested (inductively) against fresh facts. The details of Peirce’s account of these latter two stages need not concern us, save one: the facts used to generate a hypothesis abductively cannot be used to test it inductively. To do so would provide circular justification for the acceptance of the hypothesis. This mistake is what Peirce likely has in mind when he says that Hume ‘has completely mistaken the nature of the true logic of abduction’ (CP 6.537). It isn’t to Hume’s rejection of miracles that Peirce objects, but rather that unconditional independence is necessary if MBL is to work (Legg 2001: 303; Merrill 1991: 99). Peirce, however, correctly emphasizes that unconditional independence is unnecessary (CP 7.168). 11 William Kruskal emphasizes the implausibility of assuming the independence of testimonies in the context of Hume’s argument (see Kruskal 1988).

pe irc e on m irac le s 37 the inference scheme of MBL. In the application of this scheme to the question of a particular miracle, the hypothesis that a miracle occurred is abductively inferred from given witness testimony. To use that same testimony to assess the probability of the hypothesis that a miracle occurred is to commit oneself to a viciously circular justification. We can state the problem without worrying specifically about history, miracles, or witnesses. As Peirce puts it, were it the case that MBL can be used to apportion belief in an event given only a set of testimonies concerning that event, then a man who merely knew of a certain urn of balls that a hundred white drawings had been made from it, would, in the absence of all information in regard to the black drawings, be entitled to a definite intensity of ‘belief ’ in regard to the next drawing, and not only so, but the degree of this ‘belief ’ would remain quite unaffected by the further information that the number of black drawings that had ever been made from the urn was zero! (Wiener et al. 1947: 223)

When seen in terms of abduction, the complaint is as follows. Knowing that a certain number of white drawings had been made from the urn supports the abductive inference that the urn contains some number of white balls. However, this same knowledge by itself cannot be used to estimate the relative proportion of white balls in the urn or, derivatively, the probability that the next one drawn will be white. This is roughly what MBL suggests can be done. But this is to ignore the fact that the white drawings under consideration were all, in a sense, selected for consideration because they were white drawings. Whether or not I am correctly reading Peirce’s diagnosis of Hume’s mistake, I claim that the above interpretation presents a general problem for Bayesian inference. At the core of Bayesian epistemology is a rule for updating degrees of belief as represented by a probability distribution over a set of propositions. The rule states that upon learning evidence E, we ought to update all of our probability assignments such that P(E) = 1, and the new, posterior probability of a hypothesis H is assigned the value of the old prior conditional probability P(H|E) according to Equation (1) above. Given that the prior probabilities were coherent, the resulting posterior probability is supposed to reflect a rational degree of belief. The naïve Bayesian assumes that at any one time, our probability distribution is complete. That is, we are supposed to have assigned a definite probability to every hypothesis conditional on any other set of propositions in our language. This is obviously unrealistic. The set of hypotheses which any one person has explicitly considered and for which he may be said to possess a definite degree of belief is quite limited—there are many hypotheses in his language which he has not included in the domain of his subjective probability function. The recognition that we occasionally add hypotheses to the domain of our subjective probability functions is the basis of an objection to Bayesian confirmation theory called the ‘problem of new hypotheses’. As originally stated, the problem is that the introduction of new hypotheses instigates changes in an agent’s probability distribution that are not part of the basic Bayesian scheme—in suddenly considering

38 p robab i l ity i n th e ph i lo sophy of re l i g i on a new hypothesis, the agent does not learn anything new about the world and so no adjustment of his distribution ought to take place (see Chihara 1987). Framed in this way, the objection from new hypotheses amounts to the claim that the Bayesian account of confirmation omits an important sort of learning. One response to the problem of new hypotheses is simply to embrace it. Richard Otte, for instance, suggests that Bayesian confirmation theory merely requires that changes in our subjective probability distributions be precipitated by experience, and the act of thinking of a new hypothesis can be viewed as an experience (see Otte 1994). Thus, the Bayesian can consistently embrace a more descriptively realistic model of confirmation in which agents consider only a limited set of hypotheses at a given time and update their degrees of belief when new hypotheses are considered. In making this suggestion, however, Otte notes that the Bayesian stance provides no rule to follow when expanding the domain of the probability function (other than probabilistic coherence). Herein lies a problem, the very problem to which Peirce points. The only new hypotheses we tend to consider are those which we think are probable, conditional on some evidence we already possess. That is, we only choose to expand the domain of our probability functions when doing so will give a high posterior in favour of the added hypothesis. Peirce’s objection to MBL can be recast in general terms apropos of this process: the way in which we arrive at new hypotheses on the basis of evidence biases our assessment of the probability of that hypothesis using the same evidence. The general lesson to be drawn is that when it comes to newly introduced hypotheses, we would be positively irrational to apportion belief via Bayesian conditionalization on the very evidence that motivated the introduction.12 There are some obvious ways in which the Bayesian can adjust for the bias introduced with new hypotheses. First, he could simply insist that one suspend judgement until new evidence pertinent to the introduced hypothesis becomes available. That is, one could insist that expanding the domain of the probability function involves two stages, not a single adjustment. In the first stage, a new hypothesis is tentatively identified and attributed a coherent prior probability (as are all its logical consequences if one’s Bayesian model assumes logical omniscience). In the second stage, additional evidence collected or recognized only after the hypothesis has been added to the domain is used to update the prior. Only after updating on this new evidence should the agent begin to take the assigned probability of the new hypothesis seriously. This would be the Bayesian analogue of Peirce’s division of inquiry into abductive and

12 Note that this is distinct from the problem of old evidence. There the concern is that the Bayesian cannot learn in instances where it looks like she should be able to. Conditioning on old evidence (with a probability of unity) cannot affect the probability assigned to a proposition, yet we can think of instances where it seems like it should. The problem I’m pointing to here is that the Bayesian claims to learn too much when nothing has, in fact, been learned. In expanding the domain of the probability function, conditional probabilities are in effect rigged to favour the added hypothesis. Until additional evidence is considered, it would be a mistake to think we have learned anything about the probability of the newly introduced hypothesis.

pe irc e on m irac le s 39 inductive steps. Similarly, one could attempt a sort of re-sampling procedure akin to a statistical ‘jackknife’ procedure.13 The idea would be to set aside some portion of the available evidence. After constructing a new hypothesis on the basis of the remaining portion, a posterior could be computed using the evidence previously set aside.14 The latter would in general be unbiased with respect to the introduced hypothesis, and one could take seriously the degree to which it confirms the new hypothesis without worrying about Peirce’s objection. However, if we are forced to draw conclusions solely from historical testimony about singular events, then neither of these strategies can help. This is because the pool of available evidence—the historical testimony—is doubly biased. The historical record is not an even-handed report of all events no matter how mundane. What get preserved are typically just the facts that support interesting hypotheses. In order to adopt the first strategy and suspend judgement, we would have to appeal to some other class of evidence since no more testimony is forthcoming. There is nothing wrong with doing so, but one would then concede the point that no rational degree of belief can be assigned on the basis of the testimony alone. If we try to stick to the testimony we’re given and attempt a re-sampling technique, we will be stymied by the fact that the evidence—the available testimony—was not selected for historical preservation independent of the hypothesis under consideration.

VI. Against MBL: The Sampling Bias of History To see why historical testimony poses a special problem with respect to Bayesian confirmation, we need to expand on the computation underlying MBL. Notice that, in stating MBL, we only explicitly acknowledged those witnesses testifying to the occurrence of a given miracle. There may be other actual or potential witnesses that would deny the miracle. In order to account for all possible testimony, we need to amend MBL as it was stated above. There is no new math here, only more witnesses to consider: Full MBL: Given n witnesses w1 , w2 , . . . , wn of whom k provide testimonies that m is the case and n − k provide testimony that m is not the case, and where the probability for each witness being correct conditional on the occurrence of the miracle is known and independent of every other witness15 such that P(wi ± |m, {w}) = P(wi ± |m) and 13 See, for example, J. Shao and D. Tu (1995). 14 Note that the procedure as I’ve described it is only very superficially related to the jackknife. In the

latter, one repeatedly computes an estimator, leaving out one or more samples each time. In doing so, one can estimate the bias and variance of the estimator and (partially) compensate. There cannot be an exact analogue in the case of new hypotheses, since the hypothesis introduced is a function of the available evidence. Thus, it cannot be held constant through multiple iterations using different subsets of the evidence. 15 The independence conditions stated in Full MBL are a little stronger than actually required for the given equality to hold. However, the weaker conditions are unwieldy and no harm is done to the argument that follows in assuming the stronger independence.

40 p robab i l ity i n th e ph i lo sophy of re l i g i on P(wi ± |∼m, {w}) = P(wi ± |∼m), where {w} is any set of wj + and/or wj − for which i  = j, the odds in favour of m being the case are given by:

O(m|w1 says “m”, w2 says “m”, . . . , wn says “m”)   P(m) ki=1 P(wi + |m) nj=k+1 P(wj − |m) =   P(∼m) ki=1 P(wi − |∼m) nj=k+1 P(wj + |∼m) When I say there may be other witnesses, I mean to suggest that there might have been other testimonies in the record had they been collected. History tends to preserve only the testimonies that support a prized proposition; people tend to inquire only of those witnesses who report the singular, the unusual, or the surprising. Of all possible testimonies concerning a purportedly miraculous event, we only have available some of those that support an abductive inference to the occurrence of a miracle because they support that inference, and we have none that deny it. If we use only these testimonies then we bias the calculation of MBL. A couple of examples might help to make the point. Throughout the 1980s, the Hudson Valley of New York was under invasion from space. At least, that’s what one might have thought judging from the reports of extraordinary UFO sightings in local newspapers, radio broadcasts, and television shows. The ‘Westchester Boomerang’, as it was called, ‘moved slowly and silently and was easily as big as a football field—some witnesses said as big as three football fields. That would make it anywhere from 300 to 900 feet long, far longer than any aircraft manufactured in the United States or any other country’ (Hynek et al. 1987: vii). Assessing whether these events took place as described is a problem exactly analogous to that of deciding whether a miracle occurred on the basis of testimony. The question at issue in this case is whether or not such an extraordinary object actually moved through the skies of New York. So let m now denote the proposition that ‘a solid object greater than 100 meters in width moved slowly and silently through the sky on the night of October 28, 1983’. We have available the testimony of, say, k = 20 witnesses (the usual complement for one of the books published at the time). Let us grant that each of these witnesses is highly reliable in the sense that P(wi ≤ k + |m) = 0.9 and P(wi ≤ k − |∼m) = 0.01. That is, if there really was an enormous craft in the sky, each witness was very likely to have reported it. If there wasn’t, then there was only a 1 in 100 chance that each of these witnesses would have provided false testimony to the contrary. I don’t care to speculate what the prior odds, P(m)/P(∼m), of a giant airborne craft drifting through Westchester County might be, so let’s just call that odds ratio x. In this case, Full MBL tells us that upon learning of the testimony of the 20 witnesses, the posterior odds in favour of m are 10 39 x. The testimony is overwhelmingly confirmatory. If x is a modest odds ratio like 1 in a billion, then we should be morally certain (according to MBL) that the Westchester Boomerang is real. However, Peirce would argue that there is something wrong with the above calculation. The only reason that we are entertaining the possibility that m is true is because we

pe irc e on m irac le s 41 have access to those 20 testimonies. The reason we have those testimonies is because the hypothesis they support was interesting enough to motivate someone to collect them into a book and pass them on to posterity. But we have neglected the thousands of other motorists and residents who could have or should have witnessed the giant object if events transpired as m asserts. Suppose there are only 100 remaining witnesses who also looked at the relevant part of the sky on the night in question, bringing our total number of witnesses to n = 120. Suppose further that these remaining witnesses are not very reliable such that P(wk < i ≤ n − |m) = 0.75 and P(wk < i ≤ n + |∼m) = 0.9. These witnesses were inclined to deny m no matter what. Even if the witnesses left out of our book are as poor as we have assumed, then it is still the case that Full MBL entails a posterior odds in favour of m equal to 10−41 x. That number has a very large, negative exponent—these overlooked witnesses, when combined with our enthusiastic UFO spotters, overwhelmingly disconfirm the truth of m. Of course, there are other sorts of evidence we could and should take into consideration—there is no need to focus solely on testimony. The point is that by using only the given testimony, we will necessarily overestimate the probability of the hypothesis. A more prosaic but far more relevant example concerns medical diagnosis. Imagine that Smith goes to the doctor for a routine check-up. Out of a battery of standard tests, two suggest that Smith is suffering from coronary heart disease (CHD). That is, Smith’s doctor comes to entertain the new hypothesis that Smith has CHD on the basis of two of the tests in the standard battery. Suppose that for each of these tests, the probability that the outcome is accurate given that Smith really has heart disease, P(w1,2 + |m), is 0.6. Similarly, P(w1,2 − | ∼ m) = 0.5. Then according to MBL, the results of these tests should increase the odds in favour of Smith having CHD by a factor of 1.44. If the doctor left it at that, depending upon what the prior odds for CHD were, then she might believe Smith to have CHD and take drastic action. However, the two tests being considered (and the associated probabilities used in the MBL calculation) necessarily confirm the doctor’s CHD hypothesis since they were used to generate it. If the doctor was to include even a single more specific test with a negative result, her conclusion would be radically different according to MBL. Consider such a test w3 for which P(w3 − |m) = 0.05 and P(w3 + |∼m) = 0.9. Combining all of the evidence, she would find that the odds in favour of heart disease are reduced; they are a mere 0.08 times the prior odds. In a sense, the objection being made with respect to testimonies is an application of Babbage’s argument. As I mentioned above, Babbage urged against Hume the fact that, for any non-zero prior probability for an event to occur—no matter how tiny—it is always possible to produce a finite number of witnesses (with a finite probability to give accurate testimony) to overcome that small prior and favour the occurrence of the event. The rather obvious upshot is that the answer MBL yields depends on how many witnesses we consider, and whether they testify in favour or against an event. Peirce’s claim is that the testimonies preserved in the historical record are doubly biased. First, they necessarily confirm the hypotheses abductively inferred from them. Second, they

42 p robab i l ity i n th e ph i lo sophy of re l i g i on are not random samples of all possible witnesses. They are biased in favour of the event they describe, the extraordinary nature of which accounts for the preservation of the testimony. When applied to a set of testimonies without regard for how representative such a sample is, the probabilistic formula of MBL yields absurd conclusions. It should be noted that Peirce’s objection does not specifically support or condemn the occurrence of miraculous events since the calculations of MBL might be biased in either direction. In the UFO example, the initial calculation was biased in favour of the ‘miracle’ (the encounter with an enormous flying craft) because only the interesting eyewitness accounts were preserved in books. But Peirce also considers instances in which MBL invites an irrational bias against a conclusion. In particular, this is the problem he sees with regard to the study of ancient history in the late nineteenth century, particularly in the German schools. As he says, ‘[i]t is one of the consequences of German preeminence in science and philosophy . . . that subjective ways of deciding questions are, at this time, far too highly esteemed’ (CP 7.175). The principal ‘subjective way’ of deciding questions that Peirce wants to pin on the German historians and philologists is MBL. In their case, the problem is not the method as applied to a set of testimonies in the written record, but rather to the various ‘arguments’ brought to bear on the truth of a historical claim by various historians. For Peirce, the judgement of a witness as reflected in his testimony is the application of a rule of inference. If we can attach an objective probability to the truth of a witness’ testimony, then it is only because there is a determinate limiting frequency with which that particular rule of inference yields a true conclusion given true premises. The same goes for the various arguments a historian musters for or against a given conclusion. Each of these arguments is an application of a rule of inference and has some probability associated with yielding true conclusions. For instance, Peirce singles out the arguments of William Petrie against the ancient claims of, amongst others, Manetho, an Egyptian historian of the third century bc. In Peirce’s reading, Petrie treats the first three dynasties of Egypt as entirely mythical, dismissing the chronology provided by Manetho.16 To arrive at his conclusion, Petrie weighs a number of arguments against the claim that the dynasties unfolded as reported in ancient texts—just those arguments which apparently convinced him of the mythical status of these dynasties in the first place (Petrie 1897: ch. 2). Peirce claims that an application of MBL to these arguments (supposing there is some way to attach definite probabilities to each of them) amounts to a biased confirmation of Petrie’s thesis. To put it in the general Bayesian terms I suggested above, Petrie expands his probability space to consider the proposition that the early dynasties were mythical. He does this because that hypothesis is made likely by evidence already at his disposal. He then

16 See the editorial footnote on p. 107 of CP 7. A more charitable reading of his book would attribute to Petrie a laudable scepticism, not a positive denial of the existence of the first dynastic kings.

pe irc e on m irac le s 43 makes the mistake of using the same evidence to determine a degree of confidence in the new hypothesis.

VII. Against MBL: Empirical Disconfirmation A few years after he published the text in which he asserted the mythical status of the first three Egyptian dynasties, Petrie himself discovered the tomb of Menes, a First Dynasty king (see Petrie 1901). In Peirce’s view, this was but one instance of the empirical disconfirmation of MBL. Because MBL is a rule of inference—in this case, a rule by which we infer the posterior probability of a proposition—it is subject to empirical test. If a rule of inference is sound, then most or all of the time it leads from true premises to a true conclusion. This means that, like a hypothetical law of physics, MBL is testable. As his final criticism of MBL, Peirce urges that the method as it has been used with regard to ancient history has been disconfirmed by archaeological evidence. Those scholars who employed MBL to judge historical texts ‘were found to be more or less fundamentally wrong in nearly every case, and in particular that their fashion of throwing all the positive evidence overboard in favour of their notions of what was likely, stands condemned by those tests’ (CP 7.182). Peirce doesn’t discuss the issue, but falsifying a proposition favoured by MBL does not in itself demonstrate that a false conclusion was drawn. MBL is an inference rule that licenses the assignment of a probability on the basis of testimony—it allows one to infer a rational degree of confidence in the hypothesis supported by the testimony. Showing this hypothesis to be false—as was the case with Petrie—does not contradict the claim that it was probably true. To make a convincing empirical case against MBL, one needs to demonstrate that hypotheses favoured by the method turn out to be false at a greater rate than they should, given the assigned probabilities. With regard to historical testimony, Peirce provides only a small sample: ‘On five occasions in my life, and on five occasions only, I have had an opportunity of testing my Abductions about historical facts, by the fulfillment of my predictions in subsequent archeological or other discoveries . . . ’17 This is hardly enough to draw the sort of firm conclusion Peirce wants. Furthermore, it’s hard to see how we could extract enough examples from the historical record to determine a meaningful rate of failure for MBL as a function of the probability it assigns to favoured hypotheses. Of course, there are other sorts of propositions to which one could apply the Bayesian MBL, other than the assessment of historical testimony. For instance, one may use it to assess the probability of the guilt of the accused at trial, or to assess the probability of life on other worlds. It is thus possible that a rich enough source of data on which to make the empirical argument exists—there may be a field in which frequent application of the inference scheme can 17 The quote is from one of Peirce’s Lowell lectures. It is reproduced in an editorial footnote on p. 107 of CP 7.

44 p robab i l ity i n th e ph i lo sophy of re l i g i on be paired with empirical confirmations or refutations. But Peirce doesn’t give us one. Thus, we must take this last criticism of MBL to be merely a potential criticism—the data aren’t in. Of course, the theoretical difficulties MBL faces are immense, but this only emphasizes the contribution to be made by a large-scale examination of its empirical success—perhaps the best way to counter the criticism of the preceding sections is to show empirically that MBL is successful.

VIII. Conclusion If the objection developed in sections V and VI is right, the Bayesian approach to attributing a degree of belief to propositions supported by a fixed set of testimonies is fundamentally bankrupt—there is an ineliminable bias of testimonies in favour of the hypothesis whose introduction they collectively support. Given that this is the case, how should we go about assessing the truth of claims concerning miracles? Peirce provides a well-developed alternative, what he thinks is the proper scientific method.18 According to this approach, one first constructs an abductively valid hypothesis, one that makes the testimonies and all other facts at hand unsurprising. Rather than attempt to simply compute a probability of the truth of this hypothesis on the basis of our prior probabilities as MBL would have it, Peirce claims that we should gather independent data for an inductive phase of inquiry. The sort of induction in this case is what Peirce considers a second-order induction over consequences of the hypothesis (the weaker sort of induction).19 The idea is that we deduce (perhaps only probabilistically) as many consequences from our hypothesis as we can and try to verify or refute each one. The more consequences that are verified, the greater is our justification in believing a miracle occurred. It is worth noting that this option is open to the Bayesian as long as he is willing to drop the assumption that his updated probability distribution reflects a rational degree of belief when the updating is prompted by the introduction of new hypotheses. As I suggested above, there may be other viable alternatives for the Bayesian. For instance, it is conceivable that the Bayesian can invoke a procedure analogous to the statistical jackknife in order to avoid the bias inherent in a single historical sample of testimonies. What such a method might look like in detail is difficult to say, but the possibility is an interesting one. Rather than attempt to resolve the systematic problem with the Bayesian approach, one might instead attempt to argue against bias on a case by case basis—perhaps there is reason to think that one set of testimonies or another is effectively a random sample of all possible testimonies concerning an event. Whatever fix one might be inclined to pursue for the Bayesian programme in general, the lesson we can take from Peirce is that the specific approach of MBL gets us nowhere in assessing the historical testimony for miracles. 18 Peirce’s positive account is well summarized in Legg (2001). 19 Peirce calls this ‘abductory induction’. See, for example, Buchler (1955).

pe irc e on m irac le s 45

References Babbage, C. (1838) The Ninth Bridgewater Treatise. London: John Murray. Buchler, J. (ed.) (1955) Philosophical Writings of Peirce. Mineola, NY: Dover Publications, Inc. Butler, J. (1736) The Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature. London: James, John, and Paul Knapton. Chihara, C. S. (1987) ‘Some Problems for Bayesian Confirmation Theory’, The British Journal for the Philosophy of Science, 38: 551–60. Earman, J. (2000) Hume’s Abject Failure: The Argument Against Miracles. New York: Oxford University Press. Gillies, D. (2000) ‘Varieties of Propensity’, British Journal for the Philosophy of Science, 51: 807–35. Hume, D. (1988) An Enquiry Concerning Human Understanding. Amherst, NY: Prometheus Books. Hynek, J., P. Imbrogno, and B. Pratt (1987) Night Siege: The Hudson Valley UFO Sightings. New York: Ballantine Books. Jantzen, B. (2009) ‘Peirce on the Method of Balancing “Likelihoods” ’, Transactions of the Charles S. Peirce Society, 45: 668–88. Kruskal, W. (1988) ‘Miracles and Statistics: The Casual Assumption of Independence’, Journal of the American Statistical Association, 83: 929–40. Legg, C. (2001) ‘Naturalism and Wonder: Peirce on the Logic of Hume’s Argument against Miracles’, Philosophia, 28: 297–318. Merrill, K. R. (1991) ‘Hume’s “Of Miracles”, Peirce, and the Balancing of Likelihoods’, Journal of the History of Philosophy, 29(1): 85–113. Otte, R. (1994) ‘A Solution to a Problem for Bayesian Confirmation Theory’, British Journal for the Philosophy of Science, 45: 764–9. Peirce, C. S. (1935) Collected Papers of Charles Sanders Peirce. Cambridge, MA: Harvard University Press. Petrie, W. (1897) A History of Egypt from the Earliest Times to the XVIth Dynasty. New York: Charles Scribner’s Sons. (1901) The Royal Tombs of the Earliest Dynasties: 1901. Part II. London: The Egypt Exploration Fund. Shao, J. and D. Tu (1995) The Jackknife and Bootstrap. New York: Springer-Verlag. Wiener, P. P., C. S. Peirce, and S. P. Langley (1947) ‘The Peirce–Langley Correspondence and Peirce’s Manuscript on Hume and the Laws of Nature (At the Smithsonian Institution)’, Proceedings of the American Philosophical Society, 91: 201–28.

3 The Reliability of Witnesses and Testimony to the Miraculous Timothy McGrew and Lydia McGrew

I. The Condorcet Formula and Bayes’ Theorem There is a sense, as obvious as it is difficult to pin down, in which some people are more reliable witnesses than others. All else being equal, the testimony of a diligent, skilled, alert witness is obviously to be preferred over that of someone lazy, inept, or inattentive. The temptation to build reliability into a formal analysis of testimony is, therefore, very strong. In the fifth part of his memoir ‘On the probability of extraordinary facts’ (Condorcet 1783),1 the Marquis de Condorcet reduces the credibility of an event, given testimony in its favour, to a formula in which the two variables are the reliability or truthfulness of the witness (t) and the antecedent probability of the event (p): pt pt + (1 − p)(1 − t) This formula deserves close examination. At first glance, it resembles a common form of Bayes’ Theorem, namely, P(H)P(tH|H) P(H)P(tH|H) + P(∼H)P(tH|∼H) where H is the event in question and tH is testimony that H. On closer inspection, we see that translating Condorcet’s formula into a special case of Bayes’ Theorem is not merely a notational matter. Because in Condorcet’s formula everything turns on t, the specific number representing the reliability of the witness, the formula can be treated as a special case of Bayes’ Theorem only if we make three substantive, and limiting, assumptions. First, and critically, Condorcet is assuming that P(tH|H) and P(t∼H|∼H) are identical—that the witness is equally likely to tell the truth regarding H regardless 1 Condorcet’s article was actually published in 1786.

te sti mony to th e m i rac ulou s 47 of whether it occurs. To dispense with this assumption, we may write more generally: pt pt + (1 − p)(1 − t∗ ) where t∗ is P(t∼H|∼H). Now Condorcet’s unmodified formula appears as a special case of this modified formula when t = t∗ . Second, even the modified formula presupposes that the witness is forthcoming: given that t∗ = P(t∼H|∼H), we can use (1 − t∗ ) as the value for P(tH|∼H) only if P(tH|∼H) + P(t ∼ H|∼H) = 1. In effect, we are assuming that the witness would not be silent on the subject of H had it not occurred. Third, for the formula to be equivalent to Bayes’ Theorem we must assume that the subject in question and the testimony regarding it are to be treated as binary alternatives—as if we posed the question, ‘Did H occur?’ and the witness was restricted to answering ‘Yes’ or ‘No’. In some cases, this is correct or at least a useful approximation, as when we consider reports regarding the outcome of a coin flip. (Here we approximate by disregarding the chance of the coin’s standing on edge or otherwise failing to land heads or tails.) But as we shall discuss later, in many other cases of great interest, the testimony about the event is rich and detailed, and Condorcet’s formula is unable to model these epistemically important aspects of the situation.

II. The Condorcet Approach to Testimony to Miracles Despite the rather rigid assumptions built into the Condorcet formula, the apologetic use of probability in the eighteenth and nineteenth centuries bears a strong resemblance to Condorcet’s modelling of testimony. Apologetic applications consisted chiefly of the computation of high values for what Condorcet would call t, based on the combination of the testimony of multiple independent witnesses, where each individual witness has some positive degree of reliability. Hence, as the Cabinet Cyclopaedia article on ‘Credibility’ (1819) says, ‘[T]he evidence of testimony can overcome any degree of improbability, however great, which can be derived from the nature of the fact’. And the author adds: ‘We wish the candid reader to apply this mode of argumentation to the Gospel testimony’. The application of Condorcet’s modelling of testimony to apologetics reaches its most rigorous formal expression in the work of Charles Babbage in the Ninth Bridgewater Treatise, published in 1838. Babbage, arguing against Hume, demonstrates formally that combined independent testimony can indeed overcome any finite low prior for a miracle. Babbage’s formula representing the impact of testimony is a version of Condorcet’s, and his concept of the positive evidential force of individual testimony is tightly tied to reliability. All of Babbage’s calculations involve a supposition regarding how often the witnesses speak truth and falsehood—e.g., ‘Let us now suppose each

48 p robab i l ity i n th e ph i lo sophy of re l i g i on witness to state one falsehood for every ten truths . . . ’—and (1 − 1/p) in Babbage’s calculations is identical to Condorcet’s t. Babbage apparently cannot conceive of a witness’ testimony as having positive evidential force unless the witness is known to tell the truth more often than not. He reasons to this point from the definition of the probability of an event: ‘Now, an event whose probability is, in mathematical language 1/p, will be called probable or improbable, in ordinary language, according as p is less or greater than 2’ (Babbage in Earman 2000: 208). Babbage treats 1/p as the probability that an individual witness speaks falsely, and when he calculates the number of witnesses necessary to overcome some given low prior probability of miracles, he observes, if ‘p is any number greater than two, this equation [for the number of witnesses required] can always be satisfied’. He concludes: It follows, therefore, that . . . however great the quantity of experience against the occurrence of a miracle, (provided only that there are persons whose statements are more frequently correct than incorrect, and who give their testimony without collusion,) a certain number n can ALWAYS be found; so that it shall be a greater improbability that their unanimous statement shall be a falsehood, than that the miracle shall have occurred. (Babbage in Earman 2000: 209, emphasis in original)

III. The Reliability of a Witness: The Access Problem and the Common Sense School’s Solution From the standpoint of the debate over Christianity, or indeed from the standpoint of historical inquiry generally, the focus on reliability at once raises a difficulty. For, as is true in many historical cases, the original witnesses testifying to the Christian miracles are long dead, and a dead witness cannot be deposed or questioned. Nor do we have, in most historical cases, direct information regarding a witness’ character and honesty nor data concerning his track record of truth-telling in other, non-controversial situations. Standard legal treatises of the eighteenth and nineteenth centuries note that the demeanour of a witness on the stand and under cross-examination is pertinent to the judgement of his reliability. Lacking such clues, how should one determine a value for t in Condorcet’s equation? One approach, represented by the Scottish Common Sense school of Thomas Reid, was, in effect, to set a high default value for t based on our ‘propensity to speak truth, and to use the signs of language, so as to convey our real sentiments’ (Reid 1822: 306). We are, says Reid, conscious of this propensity at work in ourselves; and it is a natural and reasonable extension to attribute it by analogy to others. Reid terms this the principle of veracity, and its counterpart on the side of the hearers of testimony, the tendency to accept the reports of others at face value, he calls the principle of credulity. Put thus baldly, Reid’s assessment sounds more than merely optimistic, though elsewhere Reid makes it plain that he is not endorsing rank credulity (see Reid 1855: 439). He allows that a man may falsify his testimony when he has motives to do so.

te sti mony to th e m i rac ulou s 49 This consideration, however, cuts both ways and can make testimony against one’s interests especially valuable: ‘If there be no appearance of any such motive, much more if there be motives on the other side, his testimony has weight independent of his moral character’ (Reid 1855: 439). For Reid, then, determining the reliability of testimony is principally a matter of determining whether the witness is motivated to promulgate falsehood. In the absence of information bearing on that point, however, the witness may be presumed trustworthy, and the principle of credulity applies in a straightforward fashion. This conception of the prima facie credibility of testimony is not restricted to the philosophers and theologians: it finds expression in the legal literature as well. For example, the eminent legal authority Thomas Starkie writes: In ordinary cases where a witness stands wholly unimpeached by any extrinsic circumstances, credit ought to be given to his testimony, unless it be so grossly improbable as to satisfy the jury that he is not to be trusted. Thus, notwithstanding the general presumption of law in favor of innocence, a defendant may be convicted of a heinous and even improbable crime upon the testimony of a single witness. (Starkie 1876: 834–6)

On the strength of Starkie’s discussion, A. H. Strong sets it down as a principle that a witness, like a defendant, is innocent until proven guilty. Strong’s own statement shows how neatly the Anglo-American law of evidence in the nineteenth century dovetails with a Reidian analysis and how such an analysis was applied directly to the question of testimony to the miraculous. In a subsection titled ‘Principles of Historical Evidence Applicable to the Proof of a Divine Revelation’, Strong argues: In the absence of circumstances which generate suspicion, every witness is to be presumed credible, until the contrary is shown; the burden of impeaching his testimony lying upon the objector. The principle which leads men to give true witness to facts is stronger than that which leads them to give false witness. It is therefore unjust to compel the Christian to establish the credibility of his witnesses before proceeding to adduce their testimony . . . (Strong 1907: 143)

Defending the evidence for the Christian miracles in response to David Hume, George Campbell boldly claims prima facie value for testimonial evidence on the basis of a principle very much like the one Reid would articulate later: Where we have testimony to an event, says Campbell, ‘there is the strongest presumption in favor of the testimony, till properly refuted by experience’ (Campbell 1839: 18). Campbell expands upon the point in the well-known example of the ferry: I have lived for some years near a ferry. It consists with my knowledge, that the passage-boat has a thousand times crossed the river, and as many times returned safe. An unknown man, whom I have just now met, tells me, in a serious manner, that it is lost; and affirms, that he himself, standing on the bank, was a spectator of the scene; that he saw the passengers carried down the stream, and the boat overwhelmed. No person, who is influenced in his judgment of things, not by philosophical subtilties, but by common sense, a much surer guide, will hesitate to

50 p robab i l ity i n th e ph i lo sophy of re l i g i on declare, that in such a testimony I have probable evidence of the fact asserted. (Campbell 1839: 20–1)

Here we see an initial response to the problem of evaluating witness reliability, even in the case of improbable stories: Witnesses are to be taken to be reliable unless there is reason to think otherwise; hence, even improbable events may be justified by sufficient and otherwise unimpeached testimony.

IV. The Reliability of a Witness: The Subject Matter Problem and the Apologetic Response Not all authors were willing to concede to the Scottish Common Sense school the prima facie credibility of a witness to an unusual event. In a notable but largely neglected work on judicial evidence, Jeremy Bentham comes down heavily on the side of Hume in dismissing testimony, even multiple testimony, that runs contrary to what we know of physical laws: In the case of an apparently anti-physical fact, reported by a writer or a number of writers in a distant period; to render it more credible that he should either have been a deceiver or deceived, than that the fact was true, it is not necessary that it should appear that he was acted upon by this or that particular cause of delusion, or that he had this or that point to gain, this or that specific advantage to reap, from the lie. All men are, occasionally, exposed to seduction in this way, to the temptation of swerving from the truth, by all sorts of motives. (Bentham 1827: 311, see also 318)

The application to the argument from miracles is perfectly clear: there is no need to specify the motives by which the first Christian witnesses were (on Bentham’s view) moved to deception or the causes by which they were duped; the scientific incredibility of their tale is, of itself, sufficient to assure us that one alternative or the other is true. And though his book is ostensibly a contribution to the theory of law—he nowhere mentions Jesus, or the resurrection, or the apostles—it is an application Bentham most certainly intended, as some references to Hume in the context make plain.2 But he does not move beyond the level of insinuation as to the nature of the motives or causes in question.3 Hume, at least, is more forthright. The trouble with the actual testimony we have to the miraculous is, at least in part, that the subject matter is religious—and that fact poisons everything. [I]f the spirit of religion join itself to the love of wonder, there is an end of common sense; and human testimony, in these circumstances, loses all pretensions to authority. A religionist may be 2 See, for example, the reference to Alexander of Paphlagonia on p. 309. 3 In this respect, the similarity between Bentham and Berkeley’s Alciphron is striking. See the sixth

dialogue in Fraser (1901: 285).

te sti mony to th e m i rac ulou s 51 an enthusiast, and imagine he sees what has no reality: He may know his narrative to be false, and yet persevere in it, with the best intentions in the world, for the sake of promoting so holy a cause: Or even where this delusion has not place, vanity, excited by so strong a temptation, operates on him more powerfully than on the rest of mankind in any other circumstances; and self-interest with equal force. (Hume 2000: 89)

Thus Hume anticipates, and attempts to confute, the principle of veracity. For our purposes, the critical move here is the invocation of the topic of testimony as a ground for devaluing or dismissing it. Is the consequence just, because some human testimony has the utmost force and authority in some cases, when it relates the battles of Philippi or Pharsalia for instance, that therefore all kinds of testimony must, in all cases, have equal force and authority? (Hume 2000: 95)

He goes further, expressing some of the motives that might suffice to induce a man to bear false witness: The wise lend a very academic faith to every report which favours the passion of the reporter, whether it magnifies his country, his family, or himself, or in any other way strikes in with his natural inclinations and propensities. But what greater temptation than to appear a missionary, a prophet, an ambassador from heaven? Who would not encounter many dangers and difficulties in order to attain so sublime a character? (Hume 2000: 95–6)

In attempting to give a specific rationale for the weakness of testimony in a religious context, however, Hume put a weapon into the hands of his critics. George Campbell swiftly turned the argument around: [T]here is one circumstance which he has overlooked, and which is nevertheless of the greatest consequence in the debate. It is this, that the prejudice resulting from the religious affection, may just as readily obstruct as promote our faith in a religious miracle. What things in nature are more contrary, than one religion is to another religion? They are just as contrary as light and darkness, truth and error. The affections with which they are contemplated by the same person, are just as opposite as desire and aversion, love and hatred. The same religious zeal which gives the mind of a Christian a propensity to the belief of a miracle in support of Christianity, will inspire him with an aversion from the belief of a miracle in support of Mahometanism. The same principle which will make him acquiesce in evidence less than sufficient in one case, will make him require evidence more than sufficient in the other. . . . [I]f the faith of the witnesses stood originally in opposition to the doctrine attested by the miracles; if the only account that can be given of their conversion, is the conviction which the miracles produced in them; it must be a preposterous way of arguing, to derive their conviction from a religious zeal, which would at first obstinately withstand, and for some time hinder such conviction. On the contrary, that the evidence arising from miracles performed in proof of a doctrine disbelieved, and consequently hated before, did in fact surmount that obstacle, and conquer all the opposition arising thence, is a very strong presumption in favour of that evidence; just as strong a presumption in its favour, as it would have been against it, had all their former zeal, and principles, and prejudices, co-operated with the evidence, whatever it was, in gaining an entire assent. (Campbell 1839: 48)

52 p robab i l ity i n th e ph i lo sophy of re l i g i on Campbell’s response to Hume is both pertinent and epistemically important: the motives in the case of the Christian testimony are on the other side, and this gives the testimony of the apostles a weight independent of their character. But it is important to realize that at this point we have moved well beyond Campbell’s initial claim that testimony is to be credited unless there is direct evidence against it. We are no longer treating a witness as prima facie reliable and assigning a high value to t on the basis of some general principle of credulity. Rather, we are looking at the specifics of the historical case under consideration and deciding not so much that the witnesses were reliable as individuals as that the circumstances surrounding their testimony require us to give it strong evidential weight. The principle of credulity appears to give an opening to Hume by failing to take account of the specifics of the case. Hume argues that the nature of the event testified to can render witnesses unreliable—not to be trusted. And it is only by moving yet more deeply into the specifics of the situation that the defender of miracles is able to answer him. But as we shall see, for this we need something better than Condorcet’s formal approach.

V. The Reliability of a Witness: Venn’s Rejection of the Probabilistic Approach to Testimony In The Logic of Chance (1888), John Venn uses a rather Humean approach to argue that the theory of probability cannot be applied to cases of testimony at all. Venn particularly objects to the application of probability to testimony most typical of its use in apologetics: ‘Here is a statement made by a witness who lies once in ten times, what am I to conclude about its truth?’ (1888: 398). He dismisses such calculations not merely because he is sceptical of what he drily calls ‘the possibility of thus assigning a man his place upon a graduated scale of mendacity’, but because ‘the circumstances under which the statement is made. . . are of overwhelming importance’ and ‘would make any sensible man utterly discard the assigned average from his consideration’ (1888: 399). As a committed frequentist regarding probability, Venn sees the problem as one of specifying an appropriately tight reference class. But there are so many variables attending any particular instance of testimony that there is no such thing as ‘the right reference class’—and, hence, no such thing as the right values of t for any specific application of Condorcet’s formula. In particular, Venn believes that there is a great difficulty in applying a probabilistic method to what he calls ‘extraordinary stories’. (There can be no question that he has in mind, inter alia, the argument for miracles from testimony.) Simply knowing that a witness is generally reliable, he argues, is not enough to lead one to give much rational weight to his testimony when he testifies to an extraordinary story. He illustrates the problem with an example regarding his gardener, whom he trusts as to all ordinary matters of fact.

te sti mony to th e m i rac ulou s 53 If he were to tell me some morning that my dog had run away I should fully believe him. He tells me however that the dog has gone mad. Surely I should accept the statement with much hesitation, and on the grounds indicated above. It is not that he is more likely to be wrong when the dog is mad; but that experience shows that there are other complaints (e.g. fits) which are far more common than madness, and that most of the assertions of madness are erroneous assertions referring to these. . . . Practically I do not think that any one would feel a difficulty in thus exorbitantly discounting some particular assertion of a witness whom in most other respects he fully trusted. (Venn 1888: 415–16)

Venn expressly discusses the modified Condorcet formula in this connection, and in the case of the gardener he focuses his complaint on t∗ (1888: 420–1, n. 1). By making t∗ arbitrarily small, one can reduce the overall probability of the event attested to, given the testimony, to a minimum. In the limiting case where t = 1 and t∗ = 0, the entire formula collapses to p—the testimony makes no difference to the probability of the event. A witness who is certain to testify to an event regardless of whether it transpired is worthless. In the case of the gardener, Venn argues that t∗ should be small—that the gardener is unreliable when the event has not occurred—because experience shows that fits can generate a false positive report of madness, thus accounting for the evidence (a report of madness) without supposing the testimony to be true. He acknowledges that one could finesse the probabilistic formalism by introducing the modified Condorcet formula and that this would permit one to model the type of situation he has in mind, but he says, ‘[t]he determination however of t∗ would demand, as I have remarked, continually renewed appeal to experience. In any case the practical methods which would be adopted . . . seem to me to differ very much from that adopted by the mathematicians, in their spirit and plan’ (Venn 1888: 421, n. 1).4 Moreover, Venn actually claims that for some witnesses—scientists, in particular—we should be more willing to accept their testimony when it adduces an extraordinary event than when they testify to an ordinary one (Venn 1888: 421–2). His greatest concern, then, is to argue that a numerical value for t∗ is simply impossible to determine and, more generally, that probabilistic methods do not apply to testimony. ‘We must’, Venn concludes, reject all attempts to estimate the credibility of any particular witness, or to refer him to any assigned class in respect of his trustworthiness, and consequently abandon as unsuitable any of the numerous problems which start from such data. (Venn 1888: 403)5

The publication of Venn’s critique marks the end of the golden age of the probabilistic analysis of testimony. For nearly a century thereafter, the subject received relatively scant notice, attracting the attention of Keynes in 1921 only long enough for him to point out the fallacies of earlier writers on the subject (see Keynes 1921: 180–4). 4 We have silently changed Venn’s notation to conform to ours here. 5 Interestingly, the 1876 edition has ‘abandon as insoluble’.

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VI. Moving Beyond Reliability: Contemporary Formal Representations of Testimony In recent times, the application of probability theory to testimony to the miraculous has undergone something of a revival. Rodney Holder, departing from the Condorcet approach, uses instead Bayes’ Theorem itself. Like Charles Babbage, Holder has his sights on David Hume; and his formal point, like Babbage’s, is that contra Hume it is indeed possible in principle for a miracle to be established by testimony, since it is possible to calculate the number of witnesses necessary to overcome some given low prior probability of a miracle (see Holder: 1998). Holder, like Babbage, makes the simplifying assumption that a witness’ reliability—construed as a feature of the witness—is formally crucial. He suggests that we suppose for the sake of argument that ‘W tells the truth about any matter with probability t , and he treats P(T|M) as equal to t (Holder 1998: 52). But at this point Holder takes pains to emphasize something which is neither comfortably accommodated by Condorcet’s formula nor mentioned by Babbage: Contrary to one of the assumptions needed to make Condorcet’s formula a special case of Bayes’ Theorem, a witness’ testimony regarding an event may not be binary. That is to say, the witness may not say simply that M occurred or that M did not occur. Specifically, Holder discusses a situation in which, if W speaks falsely, there are n possible false stories he may tell. In that case, if W’s false testimony is taken to range randomly over the possible falsehoods, P(T|∼M) is not simply equal to 1 − t but is instead equal to (1 − t) × (1/n). This situation, of course, can give us a much lower P(T|∼M) than we would obtain even from a highly reliable witness if P(T|∼M) were simply equal to 1 − t. Consider the evidential force of T as modelled by the odds form of Bayes’ Theorem: P(T|M) P(M) P(M|T) = × P(∼M|T) P(T|∼M) P(∼M) Here we see that the ratio of the posterior probabilities is the product of the ratio of the prior probabilities and the likelihoods. The ratio of the likelihoods, P(T|M)/P(T|∼M), is the Bayes factor, which shows the evidential power of T vis-à-vis M and ∼M. Clearly, a low P(T|∼M) is helpful for obtaining a Bayes factor that favours M, so Holder’s idea of multiple possible false stories is highly pertinent to the evidential force of T. The modified Condorcet formula can allow for the possibility of multiple possible false stories by representing P(T|∼M) as 1 − t∗ . But that representation implies that t∗ is some other specific number—P(t ∼ M|∼M)—which, subtracted from 1, gives us the probability that W will say that M occurred if he speaks falsely. In fact, however, where P(T|∼M) is actually obtained as Holder suggests (by multiplying the probability that W speaks falsely by the number of possible false stories he could tell), 1 − t∗ is a highly artificial way of representing P(T|∼M). Holder’s use of non-binary testimony therefore represents a calculation of the probability of T on ∼M that is uniquely

te sti mony to th e m i rac ulou s 55 Bayesian and can be accommodated only awkwardly within anything like Condorcet’s formula. Moreover, even witnesses that are individually unreliable may not all be expected to tell the same false tale; therefore, the possibility of multiple possible falsehoods moves us beyond the strict requirement for positively reliable witnesses. Holder’s analysis still suffers some of the constraints and difficulties imposed by the crucial use of a number indicating reliability. For example, he still treats P(T|M) as equal to some t representing the witness’ reliability in general, and we have already noted some of the difficulties of obtaining such a number. Moreover, the notion of multiple false stories a witness could tell, while intriguing and formally important, is sometimes artificial when applied to real-life situations, especially if the multiple false stories must all be treated as equiprobable. In particular, it seems often not to be clearly relevant to the question of miracles, where it is rather difficult to think of a witness’ false testimony to a miracle as ranging over a large number of possible false miracle stories he could tell. Holder’s simplifying use of traditional considerations of reliability therefore restricts his resources for estimating a strongly positive Bayes factor for individual witness testimony. The only two possible sources in his analysis for a positive Bayes factor are high witness reliability and a large number of possible false stories over which false witness testimony might range. A more decisive move away from dependence on t occurs in the work of John Earman, whose approach is purely Bayesian. Most importantly, what Earman calls his ‘minimal reliability condition’ simply is a positive Bayes factor for the witness’ testimony to the miracle (Earman 2000: 55). Earman particularly stresses that, for a witness’ testimony to be positive evidence for a miracle, it is not necessary that the witness be reliable in an absolute sense—for example, that P(T|∼M) < .5. It is also not necessary that P(T|M) > .5. Earman’s formal point—that an unreliable individual witness may nonetheless give testimony that is positive evidence for his story—can be exemplified in a case of multiple possible false stories already discussed. But Earman mentions neither a number t which is the witness’ general reliability (the number of times he tells the truth, or the probability that he tells the truth about any matter on which he happens to speak) nor a low likelihood for ∼M arrived at in the way Holder describes (by multiplying 1 − t by the number of possible falsehoods). A comparison of Earman’s notion of reliability with Condorcet’s or even Babbage’s reveals that Earman is making a rather radical departure from both. Babbage’s formalism expressly requires individual witnesses with a reliability greater than 1/2. Earman does quote Babbage as asking us to imagine witnesses who ‘speak the truth more frequently than falsehood’ without expressly noting the difference between himself and Babbage at just this point (2000: 54). It is likely that the reason is that Earman, like Holder and Babbage, is more concerned with the falsehood of Hume’s insistence that testimony cannot establish a miracle than with the question of reliability as Babbage conceives it. But Earman makes it clear elsewhere that he is not making this assumption of reliability.

56 p robab i l ity i n th e ph i lo sophy of re l i g i on Earman’s phrase ‘minimal reliability condition’, then, is little more than a terminological convention, since he never asks us to imagine that we know how often a witness speaks the truth or to assign a number t which represents the witness’ reliability, nor does the witness have to be positively, non-comparatively reliable to any degree, however minimal, to satisfy Earman’s condition. Only a rhetorical trace remains in Earman’s discussion of the notion that reliability is to be construed primarily as a feature of the witness. When Earman describes his one area of agreement with Hume and his own scepticism about miracles, he applies his minimal reliability condition like this: Now again I do not believe that there is any, in principle, unbreachable obstacle to satisfying the minimal reliability condition for witnesses to religious miracles or UFO abductions. But I do believe, in a way that I cannot articulate in detail, that these cases are in fact relevantly similar to the case of faith healing where there is a palpable atmosphere of collective hysteria that renders the participants unable to achieve the minimal reliability condition—indeed, one might even say that a necessary condition for being a sincere participant in a faith healing meeting is the suspension of critical faculties essential to accurate reporting. . . . I acknowledge that the opinion is of the kind whose substantiation requires not philosophical argumentation and pompous solemnities about extraordinary claims requiring extraordinary proofs but rather difficult and delicate empirical investigations both into the general workings of collective hysteria and into the details of particular cases. (Earman 2000: 61)

It is interesting to note that Earman talks about people as ‘achiev[ing]’ the minimal reliability condition and that, when he expresses his scepticism, he does so not by imagining people who are level-headed but are deceived, but rather by imagining people who have suspended their critical faculties. Thus, reliability (or here, more properly, unreliability) is still to some slight degree treated in this passage as a feature of the witness rather than simply as a Bayes factor that includes features both of the witness and of the epistemic context.

VII. Moving Beyond Reliability: Bayes Factors Alone We propose a final break with the epistemic model suggested by the Condorcet formula, a break that is very much in the spirit of Earman’s analysis but makes the de-emphasis of individual witness reliability—in the sense of the proportion of times the witness tells the truth—more explicit. A positive Bayes factor for a witness’ testimony is not reducible to an estimate of the witness’ reliability, nor need an estimate of witness reliability play a crucial role in estimating a Bayes factor for the impact of the testimony on the probability of the event. The point is not that information about a witness’ reliability is irrelevant to the evaluation of his testimony. Obviously, if information is available about the witness’ truthfulness in the past, it should be taken into account in evaluating his new testimony. But the spirit of Venn at this point rises up to remind us that even when we have such

te sti mony to th e m i rac ulou s 57 data about a witness, its relevance may be mitigated by the special nature of the new event to which he testifies. And though Venn was much too inclined to inductive scepticism, we can acknowledge what is true in his discussion simply by pointing out that when there is special reason to suspect that the new situation is relevantly different from others, the Bayes factor should take account of that fact. If, for example, we have special evidence that the witness will be more inclined to lie about this case than about others in his track record or that he is more likely to be mistaken about this case than about others, we should not grant his testimony as much positive weight as we would if the circumstances were otherwise. But that only brings us back to the importance of taking multiple types of evidence into account in the Bayes factor and not calculating it mechanically from a previous track-record of truth-telling on the part of the witness. The theoretical model we advocate allows us, pace Venn, to apply probability theory to testimony while at the same time acknowledging the importance of the sorts of particular circumstances Venn emphasizes and, indeed, taking account of them formally. The Bayes factor for a given witness’ testimony should represent not a feature of the witness alone (reliability) but rather a function of our evidence about the epistemically relevant features of the witness, the situation in which he testifies, and their interaction. This approach has many advantages. First, it allows us to move beyond binary testimony without forcing the analysis into the ‘many possible equiprobable false stories’ model when that model does not naturally apply. This is particularly important, because non-binary testimony can produce a Bayes factor favourable to the event—by way of a low probability for the specific testimony if the event had not occurred—even when its favourable effects would not normally be explained by saying that if the witness spoke falsely he might tell many different equally probable false stories. For example, consider the issue of detail in testimonial evidence. Where testimony is detailed, many possible sub-hypotheses under ∼M give near-zero probability to the testimony. If a particular person is said to have been raised from the dead, and the testimony alleges identifying scars on the putatively resurrected person’s body, any deception of the witness would, to account for the evidence, need to involve faking the scars as well. This lowers the likelihood for the sub-hypothesis in which the witness was deceived by someone else. The non-binary nature of the testimony is epistemically relevant, because the witness did not simply say ‘Yes’ to a general question such as, ‘Did you see so-and-so risen from the dead?’ but rather described the alleged meeting in more detail. Hume illustrates the attractiveness of detailed testimony when, in the first edition of his essay ‘Of Miracles’, he attempts to describe an alleged miracle at Saragossa relayed by the Cardinal de Retz (Hume 2000). Hume implies that the testimony was detailed, saying, ‘When the Cardinal examin’d it, he found it to be a true, natural Leg like the other’ (Hume 2000: 261). But the Cardinal actually says merely, ‘[J]e l’y vis avec deux,’—‘I saw him with two’—which leaves open multiple possibilities for fakery on the part of those presenting the ‘miracle’ to the Cardinal (De Retz 1718: 295).

58 p robab i l ity i n th e ph i lo sophy of re l i g i on The consideration of more or less detail in testimony is relevant by way of background considerations, a point we can see easily in the evaluation of purported healing miracles. When it comes to testing the hypothesis that a patient was not really ill before the supposed miracle or was not really healed afterwards, detailed medical tests regarding either of these questions (when such tests are available) are much more epistemically helpful than a simple binary answer from a medical person to the question, ‘Was this person healed of cancer?’. In the absence of reliable and detailed information showing that the individual was as ill as alleged, later testimony that the patient was well is not convincing as to the occurrence of a miracle: P(T|∼M) remains fairly high. Hume also falls into an error here when he cites as purportedly well-attested miracles the cures alleged at the tomb of the Abbé Paris (Hume 2000: 93–4). In several notable cases (Anne le Franc, the Sieur le Doulx, the widow de Lorme, Don Alphonso), the prior condition had been invented, exaggerated, faked, or was already under independent medical treatment (see Adams 1767: 81–97). Initially less detailed allegations of miracle leave open such possible explanations, which may then come to light when a more detailed investigation is made. Non-binary evidence involving detail is relevant to ruling out alternative theories such as honest mistake on the part of the witness. A witness who gives many details of his supposed meeting with a person previously dead has clearly not simply made a mistake from afar on the basis of a general physical resemblance. The mistake theory has extremely poor likelihood for the actual testimonial evidence. Similarly, where detailed testimony is present, we cannot say that the witness did not intend to claim a miracle or that his experience may be readily explained in some ordinary fashion. If, for example, a witness testifies to a mere experience of feeling that all is well, feeling confident that the dead person lives on, or feeling the dead person’s presence, this may easily be explained non-miraculously. Even a vaguely described experience of ‘seeing’ the dead person may in some circumstances be taken to be a metaphoric claim or be attributed to a dream. A brief experience of wakeful ‘seeing’ requires a less severe hallucination to account for non-miraculously. But when the actual testimony alleges a prolonged meeting with multiple witnesses at the same time, in normal perceptual conditions, with interactions among all of those present, such non-miraculous hypotheses have low likelihood for the testimony. It can be accounted for only by some much stronger hypothesis such as mass hallucination or intent to deceive on the part of all the witnesses, which will usually have a concomitantly lower probability (and sometimes extremely low probability) conditional on ∼M.6

6 A common sceptical strategy in the controversy over miracles is to focus attention on auxiliary hypotheses which, conjoined with ∼M, give a likelihood for the evidence approximately equal to that of M. However, for any such auxiliary H1 , the relevant contrast is not between P(T|M) and P(T|∼M & H1 ) but between P(T|M) and P(T|∼M). If P(H1 |∼M) > P(T|∼M).

te sti mony to th e m i rac ulou s 59 If our formalism does not build in an assumption of binary testimony, it is able to model the effect of minor discrepancies among witnesses—for example, as evidence against collusion in a falsehood. It is common to find in multiple independent (or even partially independent) testimonies to the same real event that there are discrepancies of detail and even outright contradictions, but so far from implying by themselves that the testimony is false, such discrepancies can count in favour of the truth of the testimonies by way of our experiential knowledge regarding the normal range and type of discrepancies found among truthful witnesses. As Starkie points out, ‘[I]t has been well remarked . . . that “the usual character of human testimony is substantial truth under circumstantial variety”’ (Starkie 1876: 831). A Bayes factor approach that does not depend on reliability calculations allows us to abandon the assumption of forthcomingness. This assumption might seem uncontroversial, for if a person actually did witness something that seemed to him to be a miracle, he would probably not keep it to himself. But when forthcomingness is built into the formalism, it applies across the board, to situations where nothing whatsoever occurs as much as to situations where a witness does have some extraordinary experience. And if nothing happens, the assumption that the witnesses will speak is implausible in many circumstances. Most of us do not go around from day-to-day actively testifying that no miracle has happened to us. Suppose that we use either Condorcet’s formula or a Bayesian formula in which t plays a crucial role—as, for example, in Holder’s model where P(T|M) is t and where P(T|∼M) is either 1 − t or else is (1 − t) multiplied by 1/n, where n is some number of possible false stories the witness could tell. There simply is no place in the probability space thus interpreted for witnesses who make no report. This limitation causes serious problems for an accurate estimate of P(T|∼M). As we have argued elsewhere, in the case of the resurrection of Jesus, the majority of the probability space under ∼R goes to situations in which all goes on as normal (McGrew and McGrew 2009: 619). The women do not testify that they saw Jesus; the disciples do not testify to their meetings; there is no empty tomb. Jesus’ followers would, one assumes, simply have gone on with their lives and eventually got over their grief at his death if he had not risen and if they had had no special experiences such as those recorded in the gospels. The assumption that contemporaries will make some remark on the occurrence or non-occurrence of a miraculous event is safest where a miracle has been publically predicted. An example would be the (non-)resurrection of Dr Emms on May 25, 1708 at Windmill Hill, as predicted by John Lacy, Esq., in a pamphlet published earlier in the year (Parkinson 1856: 229, n. 1). Unsurprisingly, there is ample testimony to the failure of Dr Emms to rise from the dead, including an attempted explanation by Lacy himself. A more common situation that would come close to fulfilling the requirement of public prediction would be a previously announced healing service, where we might expect witnesses to testify to their disappointment if they saw nothing out of the ordinary.

60 p robab i l ity i n th e ph i lo sophy of re l i g i on But there is no reason to think that all miracles will be announced in this way, and indeed such a set-up is particularly open to the charge that witnesses were psychologically ‘worked up’ and hence were inclined to report what they expected to see. Particularly in situations where no miracle was especially expected by adherents of a religion and where the witnesses have nothing to gain by lying, the most likely behaviour in the absence of a miracle is simple silence, and the probability of testimony either that a miracle has happened or that a miracle has not happened is quite low given ∼M. The pure Bayes factor approach permits us to take into account numerous situational considerations that require an abandonment of the Condorcet assumption of equireliability. Nor are we required, once we are working with Bayes factors alone, to fit such considerations into the artificial construct of some number (1 − t∗ ) to represent the witness’ reliability in the new situation. Instead, we can adjust the Bayes factor directly by estimating the relevance and effect of the details of the surrounding situation. Suppose, for example, that a witness is under duress either to testify in a particular way or not to do so. Or suppose that he knows that he is unlikely to be believed and therefore realizes that telling the story would be a waste of time. Many factors can affect witness interests, and considerations of witness interests are epistemically relevant though they do not refer to a feature of the witness alone but rather to the witness’ relationship to his circumstances. And as Campbell points out in his rejoinder to Hume discussed above, such factors may cut both ways, making P(T|∼M) either higher or lower than (1 − t). The exception built into Voltaire’s maxim that ‘All that is not demonstrated to the eyes, or recognized as true by those clearly interested to deny, is at most only probable’, suggests the very point that Campbell is exploiting: testimony against the witness’ own interest is particularly valuable, as then P(T|∼M) is apt to be much lower (and, consequently, the Bayes factor in favour of M much higher) than in a case where the testimony is not against the witness’ interest (Voltaire 1869: 609, emphasis added). The nature of the event testified to and the circumstances surrounding it are relevant to the Bayes factor in ways that may well violate equireliability, though they do not require us to treat the witness as unreliable per se. Venn’s example of his gardener is pertinent here. Venn’s point is not that his gardener is an unreliable person; he is testifying as truly as he can to what he sees. But Venn’s special knowledge of the symptoms of fits and of rabies allows him reasonably to give the gardener’s testimony less weight when he states that the dog has rabies than when he states that the dog has run away.7 7 Venn is incorrect, however, in believing that the issue is the extraordinary nature of the gardener’s story

when he says that the dog is mad. The discounting of the gardener’s testimony has to do not with the lower prior probability of the dog’s being mad than of the dog’s having run away but with the comparative likelihoods. According to Venn’s account of fits and madness, the gardener’s diagnosis is particularly liable to error, so that the comparative likelihoods do not strongly favour the accuracy of the gardener’s diagnosis when he says that the dog is mad.

te sti mony to th e m i rac ulou s 61 Similarly, some types of events are not open to direct inspection; witnesses cannot attest to them directly and therefore may be reliable in general but mistaken in their interpretation of what they have seen in this type of case. A witness to a purported miraculous healing, for example, may not be in a position to say whether the person actually was ill prior to the event, or he may sincerely believe that the healing could not have taken place naturally, when in fact spontaneous healing was not particularly improbable. In some situations there are ample resources for fraud perpetrated upon the witnesses themselves, which can cause even generally reliable witnesses to commit honest error. Conversely, the realization that these circumstances do not obtain can strengthen the Bayes factor in favour of M. As in the case of detailed testimony, situational considerations can rule out alternative explanations of the evidence given ∼M. Suppose, for example, that a witness testifies to something definite and specific that either did or did not happen within his own sight. If opportunities for misinterpretation and resources for deceiving the witnesses are few, if the witness was in a good position to have strong empirical evidence about what occurred, these considerations will give low probability to the evidence under some otherwise plausible sub-hypotheses of ∼M.8 Under many such circumstances the only remotely plausible resources for explaining the evidence given ∼M will be found either in the theory that the witnesses themselves are perpetrating a deliberate fraud or in the supposition that they have suffered severe hallucinations. A consideration of Bayes factors rather than witness reliability per se thus enables us to make a properly sharp distinction between a ‘witness’ (in one sense) who suffers hardship or death for a political, religious, or ideological cause and one who suffers for his stubborn testimony to some actual, concrete event that he claims to have seen and would be in a position to know about. If we think of any adherent to a religion or ideology as a ‘witness’ and consider only his reliability, we may find ourselves thinking of a great many sincere and (in one sense) reliable people who have died for beliefs that are in fact false. But this consideration should have little impact on our consideration of witness testimony in the relevant sense, which is more like the notion of witness testimony in a legal context—testimony to specific, personally witnessed events (see Jenkin 1734: 529–31).

VIII. Beyond Credulity and Scepticism By abandoning the search for the abstract reliability number t, we avoid a false dilemma—a push to adopt either a naïve or a sceptical stance that often implicitly lies behind challenges to evidence for miracles. On the one hand, if we argue with 8 It is interesting to reflect that some reported miracles would be easier to fake in our own time than they would have been in a time or place with far fewer technological resources.

62 p robab i l ity i n th e ph i lo sophy of re l i g i on Reid that men in general are likely to tell the truth, and if we attempt simply to transfer this notion of the general truthfulness of mankind to an expectation of truth-telling about a miracle, we may seem to be endorsing credulity in the worst sense of the word. G. K. Chesterton appears to fall into just this error when he says that if he believes an apple woman about apples, he should also believe her when she says that she witnessed a miracle (Chesterton 1945: 279). On the other hand, if the nature of the event testified to influences the weight we give to the testimony itself, and if we cannot grant serious evidential force to testimony to a miraculous event, we are forced into Humean scepticism about miracles and testimony. But if we consider the specific circumstances of individual cases and assign a Bayes factor to the testimony accordingly, we are not obligated to adopt either a credulous or a sceptical stance a priori towards testimony to the miraculous. We may be able to see that a given witness in some historical case had a strong interest in making his testimony even if it was false. We may be able to see that the event in question was of such a sort that the witness could well have been mistaken about what he thought he saw. But on the other hand, we might be able to tell from an examination of the circumstances that the event in question was not of a sort that allowed for easy mistake and that the witness in question did not stand to gain but rather was likely—and must have known that he was likely—to suffer for making his testimony. It is true, as the sceptic would point out, that for many extraordinary stories the Bayes factor for testimony—even that of a previously truthful witness—will be less favourable to the event than it would be if he attested to some ordinary event. We may justly consider that Chesterton’s apple woman has more motivation to tell him a falsehood (perhaps as a joke, perhaps as a pious fraud) about a miracle than about the statement that one type of apple is better for pies than another type. If this evaluation is reasonable, a Bayes factor representing her testimony about the apples is more favourable to the truth of her statement than one representing her testimony about the miracle. On the other hand, if the apple woman is about to die a grisly death for her statement about a miracle, if she had direct access to the event and was unlikely to be mistaken about the sort of event she is attesting to, this will be a far different matter. These sorts of considerations are far more pertinent than the notion that men generally tell the truth. They apply even when we have no track record of a particular witness’ reliability, and they avoid the problems that can be raised about the relevance of such a track record even when we have one. George Campbell showed sound epistemic instinct when he argued that the apostles’ motives were not, as Hume would have it, such as to make them more likely to lie about the resurrection than about ordinary matters of fact. But to understand Campbell’s pertinent response probabilistically, and to avoid the false dilemma of credulity and scepticism, we must move beyond a narrow use of reliability and instead, in a process for which there is no simple algorithm, weigh up a wealth of detail relevant to the force of testimonial evidence.

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References Adams, W. (1767) An Essay in Answer to Mr. Hume’s Essay. 3rd edition. London: B. White. Babbage, C. (1838) The Ninth Bridgewater Treatise. London: John Murray. Bentham, J. (1827) Rationale of Judicial Evidence. Volume 3. London: Hunt & Clarke. Campbell, G. (1839) A Dissertation on Miracles. London: Thomas Tegg. Chesterton, G. K. (1945) Orthodoxy. New York: Dodd, Mead & Company. De Condorcet, N. (1783) ‘Sur le probabilité des faits extraordinaires’. In Histoire de l’Académie Royale des Sciences. Paris: DuPont. De Retz, J. (1718) Memoires du Cardinal De Retz. Volume 3. Amsterdam: n.p. Earman, J. (2000) Hume’s Abject Failure. Oxford: Oxford University Press. Fraser, A. C. (ed.) (1901) The Works of George Berkeley D.D. Volume 2. Oxford: Clarendon Press. Good, J. M., O. Gregory, and N. Bosworth (eds) (1819) Pantalogia: A New Cabinet Cyclopaedia. Volume 3. London: J. Walker. Holder, R. (1998) ‘Hume on Miracles: Bayesian Interpretation, Multiple Testimony, and the Existence of God’, British Journal for the Philosophy of Science, 49: 49–65. Hume, D. (2000) An Enquiry Concerning Human Understanding. Ed. T. L. Beauchamp. New York: Oxford University Press. Jenkin, R. (1734) The Reasonableness and Certainty of the Christian Religion. 6th edition. Volume 2. London: J. J. & P. Knapton et al. Keynes, J. M. (1921) A Treatise on Probability. London: Macmillan and Company. McGrew, T. and L. McGrew (2009) ’The Argument from Miracles: A Cumulative Case for the Resurrection of Jesus of Nazareth’. In J. P. Moreland and W. Lane Craig (eds), The Blackwell Companion to Natural Theology. Oxford: Blackwell. Parkinson, R. (ed.) (1956) The Private Journal and Literary Remains of John Byrom. Volume 2, Part I. Manchester: Charles Simms. Reid, T. (1822) Inquiry into the Human Mind. In D. Stewart (ed.), The Works of Thomas Reid. Volume 1. New York: E. Duyckinck, Collins & Hannay, 135–330. (1855) Essay on the Intellectual Powers of Man. 6th edition. Boston: Phillips, Sampson & Co. Starkie, T. (1876) A Practical Treatise of the Law of Evidence. 10th edition. Philadelphia: T. and J. W. Johnson & Co. Strong, A. H. (1907) Systematic Theology. Volume 1. Philadelphia: American Baptist Publication Society. Venn, J. (1888) The Logic of Chance. New York: Chelsea Publishing Co. Voltaire (1869) Oeuvres complètes de Voltaire. Volume 5. Paris: Chez Firmin Didot Frères.

4 Does it Matter whether a Miracle-like Event Happens to Oneself rather than to Someone Else? Luc Bovens

I. An Ancient Controversy When Jesus’ disciples tell Thomas that they have seen the risen Lord, he notoriously answers: ‘Unless I see the nail marks in his hands and put my finger where the nails were, and put my hand into his side, I will not believe it’ (John 20:25).1 When Jesus then appears to him in person, Jesus admonishes him: ‘Because you have seen me, you have believed; blessed are those who have not seen and yet have believed’ (John 20:29). This biblical passage raises an interesting puzzle. Should one attach the same evidential weight to miracle-like events when they happen to other people as when they happen to oneself? Does testimony have the same evidential weight as personal experience of miracle-like events? It is not untypical for people to have a religious conversion when they themselves are touched by the hand of God in some way or other, while they attach little weight to testimony from others of events that seem at least equally miraculous. Is this a reasonable attitude? There is one track of response that goes back to Hume’s famous essay ‘Of Miracles’ (1748) and was recently discussed by John Earman (2000). The core idea is that third-person reports, and in particular a chain of third-person reports, tend to be less reliable than first-person experiences. If this is the case, then it is clearly reasonable to attach more weight to personal experience than to testimony. Thomas’ response fits in this category—he does not consider the reports of the other disciples to be reliable and Jesus admonishes him for his lack of trust. But now suppose that, in some particular case, a third-person report of a miracle-like event is equally reliable as one’s first-person experience. After all, there are sources that 1 New International Version (1984 edition).

m irac le - l ike eve nt s 65 we can trust and, furthermore, our own senses can be subject to delusions. What are we to say then? Might it still be reasonable to attach more weight to our personal experience than to testimony? William James and William Alston are at loggerheads with respect to this question. They discuss the evidential value of mystical experiences, but at least judging from Alston’s examples, these mystical experiences are very much like our miracle-like events. For James, testimony has less evidential value than personal experience. He writes that ‘(1) mystical states, when well developed, usually are, and have the right to be, absolutely authoritative over the individuals to whom they come’ and ‘(2) [n]o authority emanates from them which should make it a duty for those who stand outside of them to accept revelations uncritically’ (James 1902: 14). Alston spells out the following interpretation of James’ position: a religious belief P is justified when formed directly on the basis of personal mystical experience, but it is not justified when formed on the basis of testimony by a person who came to be justified in believing P on the basis of personal mystical experience (Alston 1991: 280). Alston objects to this position. He submits that if X is justified in believing some proposition P on grounds of a personal experience, then Y is no less justified in believing P on grounds of X’s testimony, provided that Y is justified in believing that X is competent, reliable, and authoritative. This principle, according to Alston, is acceptable for empirical beliefs and to reject it for religious belief is to hold up a double standard. Mystical experiences are notoriously less reliable than ordinary sensory experiences, but this problem is equally prevalent for personal experience as it is for experiences that we learn about through witness reports. He writes: ‘[I]f those not blessed with first hand experience of God cannot become justified in their belief about God from testimony of those who are so blessed, then we are of all men the most miserable’ (1991: 282–93).

II. The Naïve Argument Let us first see what the argument might be on Alston’s side. We could proffer the following simple Bayesian argument. Let a miracle-like event be an event that is seemingly indicative of the existence of an all-good, all-knowing, and all-powerful being. Such an event might occur in a naturalistic world but this would be very improbable. Let E(i) be the proposition that such an event happened to a particular person i. For simplicity, assume that the experience as well as the report of such a miracle-like event are both veridical—i.e. E(i) is true. Let G be the proposition that there is a God. Then for any particular person i the following holds: 1 > P(E(i)|G) = r >> P(E(i)|¬G) = x ≈ 0 but x  = 0.

(1)

Let me explain: if there is a God, then there is chance r (though not a certainty) that he will reveal himself to a particular person i during her life-time through some

66 p robab i l ity i n th e ph i lo sophy of re l i g i on miracle-like event. If there is no God, then the occurrence of such an event in this person’s life-time is close to 0, yet it is not impossible in a naturalistic world. Let my prior odds for the existence of God be O(G) = P(G)/P(¬G). For simplicity, let us set these odds at 1—I find it equally likely that there is or that there is not a God. Then my posterior odds that there is a God, after learning that a miracle-like event happened to me or that a miracle-like event happened to, say, John Smith is simply: O(G|E(i)) =

P(E(i)|G) O(G) = r/x P(E(i)|¬G)

(2)

So my posterior odds that there is a God are magnified by factor r/x, whether the miracle-like event happened to i = me or to i = John Smith.

III. Protocols The simple Bayesian argument in the previous section commits a standard mistake in Bayesian updating, which has been discussed by Glenn Shafer (see Shafer 1985).2 When we are informed of some proposition by a reliable informant, we do not only learn the proposition in question, but we also learn that this item of information was provided to us as one of the items of information that might have been provided to us in the context in question. The protocol is the information-generating process. It determines the various items of information that may be provided to us and our credence of receiving each such item of information given various states of the world. We should then update not on the proposition in question, but rather on the fact that we learned that it was this item of information that was provided to us rather than any other that could have been provided to us in the context in question. This point is particularly salient in the Three Prisoners Problem. Suppose that you, prisoner a, are one of three prisoners a, b, and c. You know that exactly one of the three prisoners will be executed and you assign an equal ex ante credence to each prisoner being executed. Let A be the proposition that a will be executed, B the proposition that b will be executed, and C the proposition that c will be executed. Then P(A) = 1/3. There is a guard whom you know will answer your question in a truthful manner. You ask her to name exactly one of the other two prisoners who will not be executed. Clearly, she can always name one such prisoner. If it is b who will be executed, she will name c, if it is c, she will name b and if it is a (i.e. you) who will be executed, then she will flip a coin to determine whether to name either b or c. You know all this. The guard answers ‘b will not be executed’. It would be erroneous to reason as follows. You learned the proposition ¬B, i.e. that B will not be executed. So you update on the proposition that you learned, viz. ¬B:

2 My argument in this section builds on Bovens and Leeds (2002) and Bovens and Ferreira (2010).

mirac le - l ike eve nt s 67 P(A|¬B) =

P(¬B|A)P(A) 1 × 1/3 = = 1/2 P(¬B) 2/3

(3)

This answer is incorrect—clearly you should not change your credence that you will be executed from 1/3 to 1/2. The information that the guard provides you should not affect your ex ante credence that you will be executed. What is the correct way of reasoning about this problem then? What you learn from the guard is not just the proposition that b will not be executed. You learn more. You learn that the item of information ‘b will not be executed’ has been provided to you (by a truthful guard) as one of the two items of information that might have been provided to you, viz. ‘b will not be executed’ and ‘c will not be executed’ in the context in question. We construct the conditional probability table in which INF is the variable that will take on ‘¬B’ if the item of information that is provided to you is ‘b will not be executed’ (etc.) and @ is the variable that will take on the value A if A is true in the actual world (etc.) in Table 4.1. We now update on ‘¬B’, i.e. that the guard has provided you with the information ‘b will not be executed’: P(@ = A|INF = ‘¬B‘) =

P(INF = ‘¬B’|@ = A)P(@ = A) P(INF = ‘¬B’)

=

P(INF = ‘¬B’|@ = A)P(@ = A)  P(INF = ‘¬B’|@)P(@) @=A,B,C

=

1/2 × 1/3 = 1/3 1/3(1/2 + 0 + 1)

(4)

This is obviously the correct result. The answer of the guard should not affect your credence that you will be executed (see Pearl 1988: 58–9 with reference to Gardner: 1961). So how does this insight apply to our problem? One could ask: What is the typical protocol that leads someone to learn that a miracle-like event happened to herself? What is the typical protocol that leads someone to learn that a miracle-like event happened to John Smith (that is, a particular person other than herself)? Let me introduce some terminology. Let a holy person be a person to whom some miracle-like event has happened. Let a prophet be a person who knows who is and who is not a holy person within a particular population. Table 4.1. The protocol in the Three Prisoners Problem @=

Three Prisoners P(INF | @) INF =

‘¬B’ ‘¬C’

A

B

C

1/ 2 1/ 2

0 1

1 0

68 p robab i l ity i n th e ph i lo sophy of re l i g i on Now people who learn about John Smith being a holy person typically do so in a particular way. They tend to consult prophets and ask them about whether there is anyone who has been the subject of miracle-like events. Furthermore, let us suppose that, within the context in question, a person in search of religious truth consults a single prophet who has intimate knowledge of the lives of n people so that she knows whether miracle-like events happened or did not happen in each person’s life. Either the prophet will say ‘(A miracle-like event happened to) None’ or she will provide a single name, e.g. ‘John Smith’. If she provides a single name, then this does not mean that a miracle-like event only happened to exactly one person. It could have happened to multiple people, but, for the sake of simplicity, let us suppose that the prophet will just provide exactly one name by randomizing under a uniform distribution over the persons to whom a miracle-like event happened. So suppose that there are two people of whose lives the prophet has intimate knowledge, viz. John Smith and Mary Smith. Then the prophet will answer one of the set {‘None’, ‘JS’, ‘MS’}. Let us call this protocol ‘Beatist’ since it describes a person who, in her search for religious truth, takes an interest in whether there do or do not exist holy persons— i.e. beati—in this world. What about conversions on grounds of miracle-like events happening to oneself? What is the typical way in which one comes to learn about a miracle-like event happening to oneself? What comes to mind is the story of St Paul. St Paul persecuted early Christians and he only converts after being blinded and receiving a vision of the resurrected Christ. Now St Paul did not knock on a prophet’s door to learn about holy people—he could not care less about the existence of holy persons before his conversion. Rather, through life’s lessons, St Paul would either be confronted by a miracle-like event or not. His own life would provide one answer out of the set {‘Miracle-like event happening to St Paul’, ‘No miracle-like event happening to St Paul’} or, for short, {‘SP’, ‘¬SP’}. Let us call this protocol ‘St Paul’. In Tables 4.2 and 4.3, I construct conditional probability tables for these protocols in which INF is the random variable that can take on the various items of information that may be in my data base, given my epistemic interests, and @ is the random variable that takes on the value G, if God exists or ¬G if God does not exist. Table 4.2 may require some clarification. The chance of no miracle-like event happening to either JS or MS conditional on God’s existence is (1 − r)2 (on the assumption that miracle-like events happening to JS and MS are independent, Table 4.2. The Beatist protocol @=

Beatist P(INF|@) INF =

G ‘None’ ‘JS’ ‘MS’

(1 − r)2

1/ 2(1 − (1 − r)2 ) 1/ 2(1 − (1 − r)2 )

¬G (1 − x)2 1/ 2(1 − (1 − x)2 ) 1/ 2(1 − (1 − x)2 )

mirac le - l ike eve nt s 69 Table 4.3. The St Paul protocol @=

St Paul P(INF|@) INF =

‘SP’ ‘¬SP’

G

¬G

r 1−r

x 1−x

conditional on God’s existence or non-existence). Similarly, the chance of no miracle-like event happening to either JS or MS conditional on God’s non-existence is (1 − x)2 . Furthermore, the chance of a miracle-like event happening to at least one of JS or MS conditional on God’s existence is (1 − (1 − r)2 ). Now if it happened to at least one person then there is an equal chance that the prophet will say ‘JS’ or that she will say ‘MS’ and so the chance that the prophet will say ‘JS’ conditional on God’s existence is ½(1 − (1 − r)2 ). Or here is another way to see this. The chance that the prophet will say ‘JS’ equals the chance that it happened to JS and not to MS (i.e. r(1 − r)) plus the chance that it happened to both JS and MS and that JS’s was drawn by randomization (i.e. ½r 2 ). Now r(1 − r) + ½r 2 = ½(1 − (1 − r)2 ) = r − ½r 2 . And similarly for the other entries in Table 4.2. We can now calculate: (Beatist)

OBeatist (@ = G|INF = ‘JS’) = P(INF = ‘JS’|@ = G) 1/2(1 − (1 − r)2 ) O(@ = G) = P(INF = ‘JS’|@ = ¬G) 1/2(1 − (1 − x)2 )

(St Paul)

OSt Paul (@ = G|INF =‘SP’) = P(INF = ‘SP’|@ = G) O(@ = G) = r/x P(INF = ‘SP’|@ = ¬G)

We can generalize Beatist for a prophet who has intimate knowledge of the lives of n people: 1/n(1 − (1 − r)n ) 1/n(1 − (1 − x)n ) To see that under the constraints in (1) and for n > 1, OSt Paul (@ = G|INF = ‘SP’) > OBeatist (@ = G|INF = ‘JS’), a proof in the appendix of the following fact is included:

(Beatist)

OBeatist (@ = G|INF = ‘JS’) =

OStPaul (@ = G|INF = ‘SP’) r (1 − (1 − x)n ) = >1 OBeatist (@ = G|INF = ‘JS’) x (1 − (1 − r)n )

(5)

To get a feel for the numbers, let us set r = .50, x = .01. Then the odds for St Paul are 50 and for Beatist are a function of n, as we can see in Figure 4.1. For n = 5, the odds are 19.77. As n approaches infinity, the limit for the odds is 1 (i.e. the prior odds) for Beatist. This is easy enough to explain. If there is a large number of people to whom

70 p robab i l ity i n th e ph i lo sophy of re l i g i on 50

40

30

20

10

0 0

5

10

15

20 n

Figure 4.1 The posterior odds of the God-hypothesis in Beatist as a function of n with n being the number of people of whom the prophet has intimate knowledge for r = .5 and x = .01

miracle-like events may happen, then the evidentiary value decreases as there is bound to be someone to whom something improbable happened, even in the absence of there being a self-revealing God. It is also clear from the equations (Beatist) and (St Paul) that the odds are increasing in r—the more God is a revealing God, the more a miracle-like event is evidence for his existence, ceteris paribus—and decreasing in x: the more improbable the miracle-like event is in a naturalistic world, the more its occurrence is evidence for God’s existence, ceteris paribus. Furthermore, as x approaches 0, the posterior odds go to infinity.

IV. Discussion Let us assume that we do not question the actual occurrence of the miracle-like event and that what is at stake is the rationality of religious belief on grounds of a veridical experience of, or a veridical report about, miracle-like events. Now of course miracle-like events come in various shapes—they range all the way from Jesus rising from the dead to a synagogue remaining standing in an earthquake. Clearly x is much closer to 0 for the former than for the latter. We have seen that, as x approaches 0, the posterior odds go to infinity in the limit for both the St Paul and the Beatist protocols. So the interesting cases are the cases in which x is low but we are in the realm of uncanny events that seem to have the mark of an all-good, all-knowing, and all-powerful being. I also bracket the case in which a person claims that she feels that God ‘spoke to her’ through some miracle-like event happening to herself. Whether one feels called upon

mirac le - l ike eve nt s 71 by God in experiencing a miracle-like event, is not a function of its evidentiary value. To speak with Pascal, such matters may rest on reasons of the heart that the intellect does not understand. My argument comes in when someone wishes to make her faith contingent on the evidentiary value of a miracle-like event that is an uncanny event and that seems to have God’s name written on it. In this case, the protocol of how we come to learn about such miracle-like events does matter a great deal. I have spelled out typical protocols of how one comes to learn a proposition that a miracle-like event happened to another person and of how one comes to learn a proposition that a miracle-like event happened to oneself. And this difference in protocol matters a great deal to the evidentiary value of the proposition. But how typical is typical? I claimed that St Paul was a typical protocol for someone who lets her religious beliefs be affected by miracle-like events happening to herself. But does it need to be the case that, if I change my religious beliefs on grounds of a miracle-like event happening to myself, then I was only open to miracle-like events happening to myself? Not really. We could imagine a St Paul who is just as open to a prophet reporting on the existence of a holy person amongst the people of whose lives he has intimate knowledge. However, the prophet remained silent before the miracle-like event happened to St Paul and the prophet’s confirmation of the miracle-like event that happened to St Paul is the only evidence St Paul has to go on. Then we would need to analyse the person who experienced a miracle-like event herself based on the Beatist protocol and not on the St Paul protocol. But when someone converts on the basis of miracle-like events happening to herself, she is typically portrayed as a person who is not consulting prophets, who is not open to such events happening to others. She is a St Paul, i.e. the kind of person whose religious beliefs can only be affected by miracle-like events happening to herself. I claimed that Beatist was a typical protocol for someone who lets her religious beliefs be affected by miracle-like events happening to other people. But does it need to be the case that if I change my religious beliefs on grounds of a miracle-like event happening to someone else, then I was open to the reports of a prophet reporting on a holy person? Again, not really. It might well be the case that I am completely tuned into exactly one person, other than myself, in this world and only miracle-like events happening to that person could affect my religious faith. Then we would need to analyse the person who changes his religious beliefs on grounds of a miracle-like event happening to that person as a St Paul and not as a Beatist. But when someone converts on the basis of miracle-like events happening to other people, she typically is open to prophets reporting on the existence of some holy person in a population of people. Why did we constrain the prophet in the Beatist protocol to naming the name of one holy person? Might a real-life Beatist not wish to hear about other people to whom miracle-like events have happened? Sure—and we could spell out a protocol for the possible answers of the prophet which includes multiple holy persons. But note that,

72 p robab i l ity i n th e ph i lo sophy of re l i g i on in the case at hand, we are updating on the information ‘John Smith’ and not on the information ‘John Smith and nobody else’. So we would need to stipulate some decision procedure for the prophet which is such that the prophet saying ‘John Smith’ is not tantamount to a miracle-like event happening to John Smith and to nobody else. And this would require additional stipulations. We chose the simplest model, that is, to just let the prophet name exactly one holy person. Not only is this the simplest model, there is also some basis in Beatist psychology for the assumption of naming a single person. Many Beatists are satisfied when they hear of the existence of one holy person. Or at least they do not try to determine whether there are holy persons outside their faith. Once they have determined that Jesus is a holy person, they do not care about finding out whether other faiths rest on reliable holy-person accounts as well. I have constructed a dichotomy of protocols. But of course in the real world these protocols are only the extreme points on a continuum. Furthermore the prophet is just a personification of the societal lore of reliable reports on miracle-like events. And this societal lore may afford a more or less direct connection between the people who experienced a miracle-like event and the prospective believers. Some people are more open to this lore than others. For example, some may be open to miracle-like events happening not only to themselves but also to close acquaintances. That places them somewhere in between a St Paul and a Beatist. I have provided an analysis of the extreme points on this continuum and we simply need to interpolate for the intermediate cases. As an aside, let me remark that my argument is also relevant to miracle-like events happening to followers of cult-figures whose beatification and sainthood is under consideration. José Manuel Almuzara is leading a campaign to beatify Antoni Gaudí. The occurrence of an uncontested miracle-like event would help the cause. Almazura hopes that with Pope Benedict XVI’s visit to Barcelona the number of people praying to Gaudí will go up and along with it the chances of a miracle-like event: ‘It is not the same if only 50 people are praying to him for help as when five million are praying’, he is quoted as saying (Tremlett 2010). In this case the difference is not whether we are open to reports from 50 or 5 million people, but rather the difference is between whether the desired miracle-like events might happen to 50 or to 5 million people. Now it may well be that the chances of a miracle-like event happening to at least one of Gaudí’s followers may go up as the numbers increase. But my argument applies to this case as well: One should not forget that, with increasing numbers of followers, the evidential value of such a miracle-like event would go down. Let us return to the central lesson, viz. if a person wants to change her religious beliefs on the basis of the evidentiary value of miracle-like events (where miracle-like events are to be understood as low-probability events that seem to bear the hand of an all-good, all-knowing, and all-powerful being), then it does make a difference whether she goes through the world with an openness to such events happening to herself or whether she goes through the world with an openness to reports from reliable sources who provide access to a large database of potential subjects of miracle-like events.

mirac le - l ike eve nt s 73 There are miracle-like events which are such that, if they happen to someone with a St Paul-like attitude, then this should have great evidential value for her, but if someone with a Beatist-like attitude were told by a fully reliable judge that they happened to, say, a thirteenth-century monk, then this should have little evidential value for her. There remains the question of what the lesson is for my own belief–updating? What am I entitled to believe on grounds of miracle-like events happening to others as opposed to myself? After all, I am just one person with one attitude. Certainly—but I am also not completely transparent to myself and I can learn about what kind of person I am from the fact that I find myself pondering one thing or another. If I am pondering what I should believe on grounds of a miracle-like event that happened to a thirteenth-century monk and that has been recounted to me by a reliable source, then I am probably more of a Beatist than a St Paul. And if I am pondering what I should believe on grounds of a miracle-like event that has happened to me, then I am probably more of a St Paul than a Beatist. And, as we have seen, miracle-like events have more evidential value to a St Paul than to a Beatist. So it is perfectly reasonable to let miracle-like events that happen to myself have more evidential value than miracle-like events that happen to someone else. And this vindicates James against Alston.3

Appendix (by Claus Beisbart) Theorem. If r and x are real numbers with 1 > r > x > 0 and n is an integer greater than 1, r (1 − (1 − x)n ) > 1. then x (1 − (1 − r)n ) Proof. Since 1 > r > 0 and thus 1 − (1 − r)n > 0, r (1 − (1 − x)n ) >1 x (1 − (1 − r)n )

(1)

1 − (1 − x)n 1 − (1 − r)n > x r

(2)

holds if and only if

n

Define f (z) = 1−(1−z) . So we need to show that f (r) > f (x) for 1 > r > x > 0. This is so z if f (z) is a strictly monotonically decreasing function in the range (0,1), or, in other words, if f  (z) < 0 for z ∈ (0, 1). We calculate: f  (z) =

nz(1 − z)n−1 − (1 − (1 − z)n ) z2

(3)

3 I am grateful to Claus Beisbart, Christopher Clarke, Stephen Leeds, Hannah Woolf, and in particular to Jake Chandler and an anonymous referee, for comments on earlier drafts. Many thanks also to Claus Beisbart for providing the proof in the Appendix. My research was partially supported by the Swedish Collegium for Advanced Study (SCAS) and by the Dutch Organisation for Scientific Research (NWO), project nr. 236-20-005.

74 p robab i l ity i n th e ph i lo sophy of re l i g i on The denominator in (3) is clearly positive, so we need to show that the numerator is negative for z ∈ (0, 1). Note the following equality: nz(1 − z)n−1 − (1 − (1 − z)n ) = (nz + (1 − z)) (1 − z)n−1 − 1

(4)

= ((n − 1)z + 1) (1 − z)n−1 − 1 Hence we need to show that ((n − 1)z + 1) (1 − z)n−1 < 1

(5)

for n is an integer greater than 1. We show this by means of mathematical induction. Clearly for n = 2, (5) holds since (z + 1)(1 − z) = 1 − z2 < 1, since z2 ∈ (0, 1) for z ∈ (0, 1). Now suppose that (5) holds for some integer n > 1. Then we will show that it also holds for n + 1. So we need to show that ((n + 1 − 1)z + 1) (1 − z)n+1−1 < 1

(6)

Note the following equality: ((n + 1 − 1)z + 1) (1 − z)n+1−1

(7.1)

= ((n − 1)z + z + 1) (1 − z)n−1 (1 − z)

(7.2)

= ((n − 1)z + 1) (1 − z)n−1 (1 − z) + z(1 − z)n−1 (1 − z)

(7.3)

= ((n − 1)z + 1) (1 − z)n−1 (1 − z) + z(1 − z)n

(7.4)

From (5), we know that the first addend in (7.4) is smaller than (1–z). Hence, the expression in (7) is smaller than (1 − z) + z(1 − z)n . We also know that (1 − z)n < 1 for z ∈ (0, 1). Hence, z(1 − z)n < z, and so (1 − z) + z(1 − z)n < (1 − z) + z = 1. It follows that the expression in (7) is indeed smaller than 1 and so that the induction step in (6) holds. This concludes the proof.

References Alston, W. P. (1991) Perceiving God: The Epistemology of Religious Experience. Ithaca, NY: Cornell University Press. Bovens, L. and J. L. Ferreira (2010) ‘Monty Hall Drives a Wedge between Judy Benjamin and the Sleeping Beauty: A Reply to Bovens’, Analysis, 70(3): 473–81. and S. Leeds (2002) ‘The Epistemology of Social Facts: The Evidential Value of Personal Experience versus Testimony’. In G. Meggle (ed.), Social Facts and Collective Intentionality. Frankfurt A-M: Haensel-Hohenhausen, 35–43. Earman, J. (2000) Hume’s Abject Failure: The Argument Against Miracles. New York: Oxford University Press. Gardner, M. (1961) Second Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster. Hume, D. (1910) ‘Of Miracles’. In David Hume, An Enquiry Concerning Human Understanding. Harvard Classics 37, Section X. London: Collier & Son. Originally published in 1748. James, W. (1902) The Varieties of Religious Experience. New York: The Modern Library.

mirac le - l ike eve nt s 75 Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems—Networks of Plausible Inference. San Francisco: Morgan Kaufmann. Shafer, G. (1985) ‘Conditional Probability’, International Statistical Review, 53: 261–77. Tremlett, G. (2010) ‘New Hope in Campaign to Make Antoni Gaudí a Saint’, Guardian, 7 Nov. Available at: http://www.guardian.co.uk/world/2010/nov/07/antoni-gaudi-sagradafamilia-miracle

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PART II

Design

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5 Can Evidence for Design be Explained Away? David H. Glass

I. Introduction If it is granted that there is some initial plausibility to a design argument in a given domain, then an obvious way to counter it is to provide an alternative explanation for the phenomenon in question. In so far as there is reason to believe the alternative explanation, this is usually considered to undermine the design argument. In the case of biology, for example, evolution is generally taken to render obsolete design arguments such as that of William Paley. However, in most cases design is compatible with the alternative explanation, but if this is so, why not accept both design and the alternative explanation? The obvious answer is that there is no need to infer two explanations when one will do. When I learn that my children were playing in the study, the hypothesis that there has been a burglary becomes redundant as an explanation for the untidiness. In general, however, there is a question as to when one explanation is good enough to render the other redundant. Are there cases where it is appropriate to infer both design and an alternative explanation? If so, what are the conditions under which this occurs? A Bayesian framework is adopted to address these issues, although most of the results depend on the comparison of relevant likelihoods and so there is some similarity with the approaches of E. Sober (2005) and T. McGrew (2004). It may not be immediately obvious how explanation, which plays a central role in this paper, relates to Bayesian reasoning or even whether it relates to it at all. An obvious way to try to relate them is to link explanation with the likelihood in Bayes’ theorem, i.e. the probability of the evidence given the hypothesis, although to simply equate high likelihood with good explanation is too simplistic. Explanatory considerations can also come into play in other ways within a Bayesian context such as assessment of priors, determining what evidence is relevant and guiding hypothesis construction (see Lipton 2004 ch. 7 for discussion of these points). The main focus in this chapter, however, is on what is known in the artificial intelligence literature as explaining away (Pearl 1988). The basic idea is that even if two explanations are marginally independent they can become

80 p robab i l ity i n th e ph i lo sophy of re l i g i on negatively dependent when one conditions on the evidence they explain.1 Thus, if one of the explanations is found to be true, this lowers the probability of the other explanation. Suppose that a particular design hypothesis D has a certain prior probability P(D) and that it receives confirmation from some evidence E so that P(D|E) > P(D). Further suppose that an alternative explanation A is found to be true. There are two very distinct ways in which A can be said to undermine D. In the first case, the probability of D conditional on A and E is no higher than its prior probability, P(D|E, A) ≤ P(D), in which case the initial confirmation of D has been completely negated. In the second case, the probability of D conditional on A and E is higher than its prior probability but lower than its probability conditional on E, P(D) < P(D|E, A) < P(D|E), in which case A undermines D to some extent but the design argument still has some force. Perhaps there are even cases where A confirms D and so enhances the design argument. Of course, providing a satisfactory alternative to design is not the only way to defeat a design argument. In the biological case, a distinction is often drawn between Humean and Darwinian responses to design. Humean objections are not based on the ability of science to provide alternative explanations but on identifying defects in the design argument itself. Thus, if Humean objections are valid, it is not necessary to have a scientific explanation of biological complexity or fine-tuning in order to reject design, and so there was no need to infer design even before Darwin. By contrast, the Darwinian response to design is not satisfied with the philosophical arguments of Hume, but is only confident in rejecting design when a scientific explanation of the evidence in question is in place. Another objection to some versions of the design argument arises from observational selection effects (OSE). Although typically discussed in the context of fine-tuning, OSEs are also relevant to some arguments based on biological complexity, such as the origin of life. Sober (2005) appeals to an OSE in his rejection of the fine-tuning argument, but there seems to be some consensus that an OSE is insufficient on its own to counter the fine-tuning argument (see for example Weisberg 2005) and that it is only effective when combined with a multiverse hypothesis. Neither Humean objections nor OSEs will be considered in this paper. The structure of the chapter is as follows. In Section II, the focus is not on the design argument but on explaining away in the general context of two hypotheses which provide potential explanations of a piece of evidence. In particular, Section II defines different types of explaining away and presents the conditions under which they occur when a) the hypotheses are marginally independent, and b) they are marginally dependent. The results in Section II are then applied to design arguments based on biological complexity (Section III) and fine-tuning in cosmology (Section IV) where the alternative explanations are evolutionary processes and a multiverse scenario

1 A and B are said to be marginally independent if P(A|B) = P(A), i.e. they are unconditionally independent. A and B are said to be conditionally independent given C if P(A|B, C) = P(A|C).

can evi de nce for de si gn b e e x p lai ne d away ? 81 respectively. It is argued that while opponents of design can appeal to alternative explanations to weaken the strength of the design argument, it is much more difficult to show that such a strategy renders design obsolete. The reasons for drawing this conclusion and the impact of the alternative explanations on design differ in the two contexts considered.

II. Explaining Away Our goal is to consider how evidence E, which counts in favour of a design hypothesis D, can be explained away by an alternative hypothesis A. Although we refer to A as an alternative hypothesis, we will not assume that it is incompatible with D, where incompatibility is understood in the weak sense of the joint probability being zero, i.e. we will not assume that P(A, D) = 0. Nevertheless, learning that A is true can explain away E as evidence for D and conversely learning that D is true can explain away E as evidence for A. Before considering specific design contexts, we shall consider the phenomenon of explaining away in a more generic manner. Simple Bayesian networks can be used to represent the relationships between E, D and A. In order to define a Bayesian network, we first need to define some terms from graph theory. First of all, a directed graph consists of a set of nodes and a set of ordered pairs of nodes known as directed edges. If there is a directed edge from node A to node B, A is referred to as a parent of B and B as a child of A. This is represented by an arrow from node A to node B. A directed path is a sequence of directed edges such as (A, B), (B, C), (C, D) in which each directed edge starts with the node that ends the previous edge. A directed cycle is a directed path from a node to itself. Finally, a directed acyclic graph (DAG) is a directed graph which has no directed cycles. This enables us to define a Bayesian network which consists of a DAG, where the nodes represent random variables, and a joint probability distribution which satisfies the Markov condition with respect to the DAG so that each variable is conditionally independent of its non-descendents given its parents in the DAG. This joint probability distribution can be expressed as the product of the conditional probability distribution of each node given its parents in the DAG and this is how the probabilities are specified in the Bayesian network. It follows from the Markov condition that if there is no directed edge between two nodes they are conditionally independent given some (possibly empty) set of other nodes (for further details see, for example, Pearl 1988; Cowell et al. 1999). Figure 5.1 shows the graphs of two Bayesian networks to represent two scenarios where E represents a piece of evidence and D and A are hypotheses to account for it. In Figure 5.1(a) the hypotheses D and A are marginally independent of each other while in Figure 5.1(b) this is not the case. Here we will consider binary nodes which can have values D and ¬D for example. In Bayesian confirmation theory, E is said to support or confirm D if E increases the probability of D so that P(D|E) > P(D). It is easy to show that

82 p robab i l ity i n th e ph i lo sophy of re l i g i on D D

A

A

E E

(a)

(b)

Figure 5.1 Two Bayesian network structures to represent the relationship between two hypotheses, D and A, and evidence E.

P(D|E) > P(D)

iff

P(E|D) > P(E|¬D),

(1)

provided P(E) > 0 and P(D) > 0, where P is a probability distribution. These conditional probabilities would not be specified in the conditional probability distributions for the Bayesian networks in either Figure 5.1(a) or Figure 5.1(b) since E has two parents (D and A). Nevertheless, they can be easily obtained from probabilities which are part of the specification. When D and A are marginally independent, as in Figure 5.1(a), they can be expressed as follows, P(E|D) = P(E|D, A)P(A) + P(E|D, ¬A)P(¬A) P(E|¬D) = P(E|¬D, A)P(A) + P(E|¬D, ¬A)P(¬A).

(2a) (2b)

In typical cases P(E|D, A) > P(E|¬D, A) since the evidence is more likely to occur when both hypotheses are true and P(E|D, ¬A) > P(E|¬D, ¬A) since D and A are considered to be the only plausible hypotheses. If these inequalities do hold, then from Equations (2a) and (2b) it follows that P(E|D) > P(E|¬D) and so the evidence E confirms D. In fact, it is often the case that P(E|D, A) > max{P(E|¬D, A), P(E|D, ¬A)}

(3)

P(E|¬D, ¬A) < min{P(E|¬D, A), P(E|D, ¬A)}.

(4)

and

Of course, there is no necessary requirement that these conditions hold, but if they do D and A are both confirmed by evidence E.

can evi de nce for de si gn b e e x p lai ne d away ? 83 When D and A are not marginally independent, as in Figure 5.1(b), the picture is not quite so straightforward. P(E|D) and P(E|¬D) are now given by, P(E|D) = P(E|D, A)P(A|D) + P(E|D, ¬A)P(¬A|D) P(E|¬D) = P(E|¬D, A)P(A|¬D) + P(E|¬D, ¬A)P(¬A|¬D).

(5a) (5b)

If A is positively dependent on D so that P(A|D) > P(A|¬D), this increases (decreases) the first (second) term in Equation (5a) relative to the corresponding term in Equation (2a) since P(A|D) > P(A). Similarly, it decreases (increases) the first (second) term in Equation (5b) relative to the corresponding term in Equation (2b) since P(A|¬D) < P(A). Assuming inequalities (3) and (4) hold, this enhances the extent to which E confirms D relative to the independent case. If, on the other hand, A is negatively dependent on D, this would decrease the extent to which E confirms D and could even result in D being disconfirmed by E. Let us suppose that E does in fact confirm D. This means that discovering that E is true, increases the probability of D. But suppose now that we also discover that A is true. What impact would this have on the probability for D? To address this question, we consider two types of explaining away. Definition 1. Let each of D and A be a hypothesis such that if it were true it would provide an explanation for evidence E. Hypothesis A is said to partially explain away evidence E for hypothesis D if and only if A is true and P(D|E, A) < P(D|E), where P is a probability distribution. Definition 2. Let each of D and A be a hypothesis such that if it were true it would provide an explanation for evidence E. Hypothesis A is said to completely explain away evidence E for hypothesis D if and only if A is true and P(D|E, A) ≤ P(D), where P is a probability distribution. Note that in both cases it is another hypothesis that does the explaining away. It is quite possible that additional evidence might play a similar role in lowering the probability of D without providing an explanation of the evidence E. Suppose, for example, that evidence F comes to light that counts against D and suppose also that F is conditionally independent of A and E given D so that P(F|D, E, A) = P(F|D). In this case, P(D|E, F) < P(D|E) and possibly P(D|E, F) < P(D) even though it is not an instance of explaining away because F is not a hypothesis that explains E.2 The following results establish the conditions under which these two types of explaining away occur.3 2 In this case, since F lowers the probability of D it may well increase the probability of A via explaining away, but that is another matter. Another way in which additional evidence might impact one hypothesis directly and the other indirectly will be considered later. 3 For all the results which follow, P will be assumed to represent a probability distribution such that P(D)  = 0, 1, P(A)  = 0, 1, P(E|D)  = 0 and P(E|A)  = 0. Proofs of theorems are provided in the appendix.

84 p robab i l ity i n th e ph i lo sophy of re l i g i on Theorem 1. Evidence E for a hypothesis D is partially explained away by another hypothesis A if and only if A is true and P(E|D, A) · P(E|¬D, ¬A) · P(A|D) · P(¬A|¬D) < P(E|¬D, A) · P(E|D, ¬A) · P(A|¬D) · P(¬A|D).

(6)

Corollary 1. If two hypotheses D and A are marginally independent, then evidence E for hypothesis D is partially explained away by hypothesis A if and only if A is true and P(E|D, A) · P(E|¬D, ¬A) < P(E|¬D, A) · P(E|D, ¬A).

(7)

Theorem 2. Evidence E for a hypothesis D is completely explained away by another hypothesis A if and only if A is true and P(E|D, A) · P(A|D) ≤ P(E|¬D, A) · P(A|¬D).

(8)

Corollary 2. If two hypotheses D and A are marginally independent, then evidence E for hypothesis D is completely explained away by hypothesis A if and only if A is true and P(E|D, A) ≤ P(E|¬D, A).

(9)

Clearly, in the typical case described in expressions (3) and (4) the evidence for D will not be completely explained away by A when D and A are marginally independent. Or, to put it another way, even if A is true D will still be confirmed by E since P(D|E, A) > P(D). Let us call this the residual confirmation of D by E after A is discovered to be true. Even if there is some residual confirmation, it could be that the degree of confirmation is very small in which case A could be said to almost completely explain away the evidence E for D. To quantify this, we can define the degree of residual confirmation. Definition 3. Let each of D and A be a hypothesis such that if it were true it would provide an explanation for evidence E. Suppose that E confirms D. The degree of residual confirmation of hypothesis D by evidence E once another hypothesis A is discovered to be true is given by log [P(D|E, A)/P(D)], where P is a probability distribution. The degree of residual confirmation is a particular application of the log-ratio measure of confirmation, according to which the degree of confirmation of a hypothesis H by evidence E is log [P(H|E)/P(H)]. Generally, measures of confirmation are used to quantify the degree of confirmation of a hypothesis by evidence, but here it is the degree of confirmation of hypothesis D by the conjunction of evidence E and the alternative hypothesis A. There are many other measures of confirmation which could be used (see Fitelson 1999), but the log-ratio measure has been selected as it captures the relationship between the prior probability of the hypothesis, P(D), and the probability conditional on the evidence and alternative hypothesis, P(D|E, A), in an intuitive way. It is easy to show that the degree of residual confirmation has a positive value if P(D|E, A) > P(D), a value of zero if P(D|E, A) = P(D) and a negative value if P(D|E, A) < P(D). The degree of residual confirmation can be compared with the

can evi de nce for de si gn b e e x p lai ne d away ? 85 degree of confirmation of D by E which is given by log [P(D|E)/P(D)] when the log-ratio measure is used. In the case where D and A are marginally independent, the degree of residual confirmation is given by,     P(D|E, A) P(E|D, A) log = log P(D) P(E|A)   P(E|D, A) = log P(E|D, A)P(D) + P(E|¬D, A)P(¬D) −1  P(E|¬D, A) , (10) P(¬D) = log P(D) + P(E|D, A) assuming P(E|D, A)  = 0. In this case the degree of residual confirmation depends on the prior probability of D and increases monotonically with the likelihood ratio, P(E|D, A) , P(E|¬D, A)

(11)

assuming P(E|¬D, A)  = 0. If we drop the assumption of marginal independence between D and A, it is easy to show that log [P(D|E, A)/P(D)] can be expressed as,    −1 P(D|E, A) P(E|¬D, A) P(A|¬D) log , (12) = log P(D) + P(¬D) P(D) P(E|D, A) P(A|D) assuming P(E|D, A)  = 0 and P(A|D)  = 0. In this case, the degree of residual confirmation increases not only with the likelihood ratio expressed in (11) but also with P(A|D)/P(A|¬D) assuming P(A|D)  = 0.4 Clearly, these two factors could pull in opposite directions so that, for example, the likelihood ratio in expression (11) is greater than one and so contributes positively to the residual confirmation, while a negative dependence between A and D contributes negatively. Example 1 On arriving home I discover that the study is very untidy: books are lying on the floor instead of on the shelves, papers are spread all over the room, drawers have been pulled out of the desk, the lamp has been knocked over, etc. Let us call this the evidence E. Two hypotheses spring to mind: B, a burglary, to which we will assign a prior probability of P(B) = 0.01, and C, my children have been playing in the study, to which we will assign a prior probability of P(C) = 0.1. It is possible that my children were playing in the study after (or before) a burglary had taken place and so we will assume marginal independence between B and C. The evidence is more or less as I would expect had there been a burglary, slightly less 4 The log-likelihood ratio measure of confirmation could have been used instead of the log-ratio measure,  P(E,A|D) . This would give the log in which case the degree of residual confirmation would be log P(E,A|¬D)  P(E|D,A)P(A|D) of expression (11) in the independent case and log P(E|¬D,A)P(A|¬D) in the general case. In fact, the

log-likelihood measure has certain advantages over the log-ratio measure (Eells and Fitelson 2002), but the log-ratio measure has been selected for the reasons specified in the text.

86 p robab i l ity i n th e ph i lo sophy of re l i g i on likely if my children had been playing in the study and extremely unlikely if neither of these hypotheses is true. Let us assign probabilities as follows: P(E|B, C) = 0.9, P(E|B, ¬C) = 0.8, P(E|¬B, C) = 0.6, P(E|¬B, ¬C) = 0.001. Conditioning on the evidence E, we find that P(B|E) ≈ 0.118 and since this is greater than P(B) the evidence confirms B. The degree of confirmation of B by E in this case is log(11.8). Suppose that I then discover that my children had in fact been playing in the study. Conditioning on both E and C, we obtain P(B|E, C) ≈ 0.0149 which is not much higher than the prior probability of B. In this case, C has partially explained away the evidence for B and it has almost completely explained it away as indicated by the degree of residual confirmation which is just log(1.49).5 Example 2 Consider example 1 again, but now with a modification. Let’s suppose that the evidence is not at all what I would expect if my children had been playing in the study because, although they could have pulled books off the shelves and spread papers over the floor, it is unlikely (although not impossible) that they could have pulled the drawers out of the desk. So let us assign P(E|B, C) = 0.8 and P(E|¬B, C) = 0.01 and leave all the other probabilities unchanged. Conditioning on the evidence E, we find that P(B|E) ≈ 0.81 and since this is greater than P(B) the evidence confirms B. The degree of confirmation of B by E in this case is log(81). Suppose again that I then discover that my children had in fact been playing in the study. Conditioning on both E and C, we obtain P(B|E, C) ≈ 0.447 which is still much greater than the prior probability of B. In this case, C has partially explained away the evidence for B, but it has certainly not completely explained it away as there is still considerable confirmation of B as indicated by the degree of residual confirmation which is log(44.7). Example 3 Let’s add one further twist. Suppose now that there’s a lock on the door of the study to prevent my children from playing in it, although it is very basic and certainly wouldn’t prevent a burglar from gaining access. The only problem is that I don’t always remember to lock the door so there is still a chance my children could get in. However, if a burglar had gained access, he presumably wouldn’t have locked the door after him so this would increase the chances of my children getting in. Let us now assign probabilities P(C|B) = 0.5 > 0.01 = P(C|¬B), which means that P(C) = 0.0149, but otherwise leave the probabilities as in Example 2. Conditioning on the evidence E, we find that P(B|E) ≈ 0.881 and since this is greater than P(B) the evidence confirms B. The degree of confirmation of B by E in this case is log(88.1). Suppose again that I then discover that my children had in fact been playing in the study. Conditioning on both E and C, we obtain P(B|E, C) ≈ 0.976 which is even greater than P(B|E). In this case, C does not even partially explain away the evidence for B, but actually enhances the confirmation of B as indicated by the degree of residual confirmation which is log(97.6). 5 As noted earlier, the degree of residual confirmation depends on the prior probability. In this case, the probability of B given E and C is just 0.0149, which represents a very small increase over the prior probability of B which was 0.01. By contrast, a degree of residual confirmation of log(1.49) would be much more significant if the prior probability was 0.5.

can evi de nce for de si gn b e e x p lai ne d away ? 87 In Example 3, there are two ways in which learning that hypothesis C is true affects the probability of B. First, there is a negative influence due to partial explaining away that tends to lower the probability of B as in Example 2. However, there is also a positive influence due to the positive dependence between B and C that tends to raise the probability of B. It turns out that this is more than sufficient to compensate for the negative influence and overall results in an enhanced degree of confirmation of B. So far in our discussion of explaining away we have assumed that A is discovered to be true, but explaining away can occur even if A has not been established with certainty. Suppose there is evidence F which confirms A, i.e. P(F|A) > P(F|¬A), but is not directly relevant to D. Since A can explain away the evidence E for D, F can undermine D. We can define this more formally as follows. Definition 4. Let each of D and A be a hypothesis such that if it were true it would provide an explanation for evidence E. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A. We say that hypothesis D is partially undermined via explaining away by F if and only if P(D|E, F) < P(D|E), where P is a probability distribution. Definition 5. Let each of D and A be a hypothesis such that if it were true it would provide an explanation for evidence E. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A. We say that hypothesis D is completely undermined via explaining away by F if and only if P(D|E, F) ≤ P(D), where P is a probability distribution. Having defined these concepts, we can now proceed as before to consider the conditions under which partial and complete undermining occur. Once again we consider two cases: one where D and A are marginally independent and one where they are not. The structures of the corresponding Bayesian networks are shown in Figure 5.2. Theorem 3. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A and that P(F|A) > P(F|¬A). Hypothesis D is partially undermined via explaining away by F if and only if P(E|D, A) · P(E|¬D, ¬A) · P(A|D) · P(¬A|¬D) < P(E|¬D, A) · P(E|D, ¬A) · P(A|¬D) · P(¬A|D).

(13)

Note that this is exactly the same condition under which A partially explains away evidence E for hypothesis D as in theorem 1. Corollary 3. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A, that P(F|A) > P(F|¬A) and that A and D are marginally independent. Hypothesis D is partially undermined via explaining away by F

88 p robab i l ity i n th e ph i lo sophy of re l i g i on D D

A

A

F

E

E

(a)

F

(b)

Figure 5.2 Two Bayesian network structures to represent the relationship between two hypotheses, D and A, and evidence E and F.

if and only if P(E|D, A) · P(E|¬D, ¬A) < P(E|¬D, A) · P(E|D, ¬A).

(14)

Note that this is exactly the same condition under which A partially explains away evidence E for hypothesis D when A and D are marginally independent as in Corollary 1. Theorem 4. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A and that P(F|A) > P(F|¬A). Hypothesis D is completely undermined via explaining away by F if and only if  P(F|A) · P(E|D, A) · P(A|D) − P(E|¬D, A) · P(A|¬D) ≤ P(F|¬A) ·  (15) P(E|¬D, ¬A) · P(¬A|¬D) − P(E|D, ¬A) · P(¬A|D) . Note that if P(F|¬A) = 0, then P(A|F) = 1 and the condition becomes P(E|D, A) · P(A|D) ≤ P(E|¬D, A) · P(A|¬D). This condition is identical to expression (8) in theorem 2 which is correct since conditioning on F in this case is equivalent to conditioning on A. Corollary 4. Suppose that evidence F is conditionally independent of evidence E and hypothesis D given hypothesis A, that P(F|A) > P(F|¬A) and that A and D are marginally independent. Hypothesis D is completely undermined via explaining away by F if and only if  P(F|A) · P(A) P(E|D, A) − P(E|¬D, A) ≤ P(F|¬A) ·  P(¬A) P(E|¬D, ¬A) − P(E|D, ¬A) . (16)

can evi de nce for de si gn b e e x p lai ne d away ? 89 Note that if P(F|¬A) = 0, then the condition becomes P(E|D, A) ≤ P(E|¬D, A). This condition is identical to expression (9) in Corollary 2 which is correct since conditioning on F in this case is equivalent to conditioning on A. As before, we can consider the degree of residual confirmation of D by E taking into account the additional evidence F. When no assumption of marginal independence between D and A is made, this can be expressed as,     1 P(D|E, F) = log , (17) log P(D) P(D) + L × P(¬D) where L=

P(E|¬D, A)P(F|A)P(A|¬D) + P(E|¬D, ¬A)P(F|¬A)P(¬A|¬D) . P(E|D, A)P(F|A)P(A|D) + P(E|D, ¬A)P(F|¬A)P(¬A|D)

(18)

Note that if P(F|¬A) = 0, then the second term in both numerator and denominator for L disappears and Equation (17) becomes identical to Equation (12). Equations (17) and (18) also apply in the case where F does not confirm A, i.e. P(F|A) ≤ P(F|¬A). In the extreme case where P(F|A) = 0, the first term in both numerator and denominator for L disappears.

III. Design in Biology Let us now consider the application of the general results on explaining away to design arguments based on biological complexity. Here I will focus on the existence of complex living organisms and explore whether this provides evidence for life having been designed by God as typically understood by theists, i.e. omniscient, omnipotent and perfectly good. The alternative explanation will be that life is the result of natural processes including, in particular, processes described by the theory of evolution. It seems clear that there is no logical incompatibility between these two explanations, but it is often assumed that evolution explains away evidence that might otherwise provide support for the design hypothesis. In terms of notation, I will represent the hypotheses and evidence as follows: D : Living organisms are the result of Design (by God). N : Living organisms are the result of Natural processes (including evolutionary processes). E : Complex living organisms exist. For present purposes, it is unnecessary to define exactly what is meant by ‘complex living organisms’. Some possibilities include eukaryotic life, organisms that possess a central nervous system or organisms that possess consciousness. A word of clarification about hypothesis N is necessary. The idea in N is that evolutionary processes account for living organisms, but not for the origin of life in the first place, which is attributed to some as yet unknown natural processes. Background knowledge will be assumed

90 p robab i l ity i n th e ph i lo sophy of re l i g i on to contain the laws of physics (both their form and the values of physical constants contained in them), current knowledge of the conditions of the early universe and whatever additional knowledge about the hypotheses might be required to understand them and to make judgments about the probabilities they confer on the evidence, but for simplicity this will be suppressed in the notation. In this way any confirmation for the design hypothesis will be confirmation over and above any that is received from fine-tuning arguments or the order of the universe. Note that background knowledge does not include the existence of any living organisms and so the model allows for the possibility that there might have been no living organisms, in which case D, N, and E would be false. Before investigating the relevance of explaining away in this context, it is worth considering some limitations to the models proposed in this section since they are based on D, N, and E and so other evidence is excluded. There is, of course, plenty of other biological evidence that could be cited as relevant to N, but since N will be assumed to be true for the purpose of investigating whether it explains away E as evidence for D, this other evidence need not be considered here. In effect, this means that all biological evidence apart from E is considered to be conditionally independent of D given N. It might be objected that this assumption is not warranted since there are certain features of biology, such as suboptimal design and the existence of animal suffering, which might be thought to count against D. A detailed response to this objection would take us well beyond the scope of this paper, but here two issues are particularly important.6 First, if these additional features of biology count against design, they would do so even if there were no alternative explanation N to account either for them or for E. However, as discussed in the introduction, the starting point for this paper is the assumption that the design argument has some force in the absence of a suitable alternative explanation. For someone who rejects this assumption, an alternative explanation is unnecessary and so explaining away is not relevant. In fact, for such a person, an alternative explanation might make matters worse since this explanation might itself confirm design, which brings us to the second issue. One way of introducing these additional features of biology into the picture would be to argue that God would be unlikely to use evolutionary processes in the first place and so P(N|D) would be low. However, there are still reasons for thinking that P(N|D) is greater than P(N|¬D) as we shall see and this means that N actually confirms D. It is also worth noting that even if these issues warrant a more detailed

6 Here are several points that might be made in response. First, assessments of suboptimality are not straightforward since they tend to assume a direct correspondence between the designer’s goals and those of a human engineer. Second, animal suffering is a more serious matter, but theists can draw on whatever resources are available to them in addressing the problem of evil and so it is arguably better thought of in that context. Third, one way to take these issues into account would be by construing the design hypothesis more broadly so as to include other possible designers. It is also worth remembering that insofar as these issues provide evidence in favour of evolution, this is already taken into account by the fact that we will simply assume N to be true for the purpose of investigating whether it explains away E as evidence for D.

can evi de nce for de si gn b e e x p lai ne d away ? 91 model where they might reduce the probability of design somewhat, this is quite compatible with design being strongly confirmed overall. For these reasons and to obtain simpler probabilistic models, these factors will not be included in the models presented here. The first question to address is whether E confirms D, i.e. is P(E|D) > P(E|¬D)? An objection here, due to Elliott Sober (2005), is that nothing can be said about the likelihood of design without independent evidence as to the putative designer’s goals and abilities. There are at least two lines of response to this. The first is that since in the present case the designer is God as typically understood by theists, it is possible to argue that God would have reason to create embodied rational agents (Swinburne 2004, ch. 9), that complexity is required for such agents, and hence that God would have reason to bring about complex living organisms. Note that although this requires some knowledge of the designer’s goals it does not require specifying precise goals concerning the kinds of complex organisms or the processes by which they would be created.7 A second line of response is due to Timothy McGrew (2004), who argues that knowledge of the kinds of complexity that human designers bring about can be used to get a handle on the likelihood of design in cases where there are similar kinds of complexity that could not have been designed by humans. I will not pursue these arguments here, but will assume that arguments along these lines are sufficient to show that Sober’s viewpoint is too sceptical. Given this assumption, does E confirm D? The above considerations of Swinburne and McGrew give some reason to think that it does, but how does the alternative explanation N affect the picture? Let us assume for the moment that D and N are marginally independent so that the structure of the relevant Bayesian network is given by Figure 5.1(a) and the expressions for P(E|D) and P(E|¬D) are given in Equations (2a) and (2b) with A replaced by N. If we assume that D and N are the only plausible explanations of E, then we can assume that P(E|¬D, ¬N) is extremely low and so the arguments of Swinburne and McGrew give us good reason to think that P(E|D, ¬N) > P(E|¬D, ¬N).8 This corresponds to the second term on the 7 Hence it does not require the much closer correspondence between the designer’s goals and those of human engineers which would be required to show that suboptimality counts against design. 8 The case where D and N are both false could be considered as another hypothesis, the catch-all hypothesis. This hypothesis may well have a high prior probability, but assuming P(E|¬D, ¬N) is extremely low, then provided the prior probability of D and N are not too low, P(¬D, ¬N|E) will be very low. This is similar to the way in which the catch-all hypothesis that there was no burglary and my children were not playing in the study has a high prior probability, but is extremely improbable given the evidence of the untidy study. This point is quite compatible with there being some other hypothesis within the catch-all that has a high likelihood provided it has a very low prior probability. It could, however, be argued that God is not the only plausible designer and so it cannot be assumed that D and N are the only plausible hypotheses. A simple solution would be to construe the design hypothesis more broadly so as to include other plausible designers, i.e. other possible intelligent beings with sufficient knowledge and power to bring about living organisms and whose prior probabilities of doing so are not too low. This modification would have no impact on McGrew’s arguments and it seems plausible that a weakened version of Swinburne’s argument would still apply provided God still had a reasonable probability compared with other designers.

92 p robab i l ity i n th e ph i lo sophy of re l i g i on right-hand side of Equations (2a) and (2b). Furthermore, their arguments also give us good reason to think that P(E|D, N) > P(E|¬D, N) since multiple explanations typically increase the probability of the evidence. At the very least, a good reason would need to be given for rejecting this inequality. Since this corresponds to the first term on the right hand side of Equations (2a) and (2b), it follows that P(E|D) > P(E|¬D). Let us now suppose that we know N to be true, i.e. we know that living organisms are the product of natural processes including those described by the theory of evolution. First we must ask whether this would partially explain away the evidence for design. Here Corollary 1 is relevant and since we are assuming that P(E|¬D, ¬N) is extremely low, it seems reasonably clear that N does partially explain away evidence for design. But does N completely explain away the evidence for design? Here Corollary 2 is relevant and, given the discussion above, there is no good reason to think that P(E|D, N) ≤ P(E|¬D, N), so it seems unlikely that the evidence for design has been completely explained away. Hence, it seems there is still residual confirmation of design even if N is known to be true, but perhaps this residual confirmation is very small. The degree of the

residual confirmation is given by log P(D|E, N)/P(D) and so can be obtained by replacing A with N in Equation (10). From this equation we can see that the degree of residual confirmation can be expressed in terms of P(D) and the likelihood ratio P(E|D, N)/P(E|¬D, N). Hence, the degree of residual confirmation of design depends to a large extent on the value of this likelihood ratio. We have already seen that there is reason for thinking that this ratio is greater than one, but it would be difficult to be any more specific about the value of P(E|D, N), the probability of complex living organisms given design and evolutionary processes. However, there are some reasons for thinking that P(E|¬D, N) may well be very low. The most obvious reason is that appears to be the way in which evolution is extremely sensitive to contingent events, particularly in terms of extinctions. Stephen Jay Gould, in particular, has emphasized this ‘radical contingency’ in books such as Wonderful Life (Gould 1989). Gould claims that if the tape of life were to be re-run, it would be very unlikely to give rise to creatures such as human beings. This view has been challenged by others including Simon Conway Morris (2003), who emphasizes the role of convergence in evolution and claims that the evolution of intelligence was almost inevitable. Nevertheless, despite the significance of convergence, it certainly seems plausible that there have been important transitions in the history of life that would be very improbable given our knowledge of the mechanisms of evolution and the complexity involved in the transitions. Indeed, this does not seem to be controversial. Even the arch critic of design, Richard Dawkins, acknowledges that certain steps in the development of complex life are improbable. Although not directly relevant at this stage in our discussion, Dawkins refers to the ‘initial stroke of luck’ in the origin of life and goes on to say that ‘it may be that the origin of life is not the only major gap in the evolutionary story that is bridged by sheer luck, anthropically justified’ (Dawkins 2006: 140). He suggests two further hurdles as well:

can evi de nce for de si gn b e e x p lai ne d away ? 93 Mark Ridley . . . has suggested that the origin of the eucaryotic cell . . . was an even more momentous, difficult and statistically improbable step than the origin of life. The origin of consciousness might be another major gap whose bridging was the same order of improbability. (Dawkins 2006: 140)

Clearly, the improbability of the origin of the eukaryotic cell is relevant to P(E|¬D, N) and if we were to consider the evidence E to be not merely the existence of complex living organisms but organisms possessing consciousness, then Dawkins’ comments about consciousness would also be relevant. These considerations give us some reason to think that P(E|¬D, N) is indeed very low.9 This, however, is not necessarily a problem for evolution. As Sober (2002) correctly points out, there is no probabilistic version of modus tollens and so a very low value of P(E|¬D, N) does not necessarily mean that the evolutionary hypothesis without design is improbable. But given that P(E|D, N) is not too low, this does give us reason to think that the likelihood ratio, P(E|D, N)/P(E|¬D, N), is significantly greater than one and hence that the degree of residual confirmation of design, log [P(D|E, N)/P(D)], is also significant. So far we have made the assumption that D and N are marginally independent, but is this a reasonable assumption? Recall that N is the hypothesis that living organisms are the result of natural processes (including evolutionary processes). We can then think of P(N|D) as the probability that such processes produce living organisms given that living organisms have been designed and P(N|¬D) as the probability that such processes produce living organisms given that they have not been designed. This raises the question of the origin of life. Given the complexity of even the simplest lifeforms, there does seem to be a case for saying that P(N|¬D) is very low, as Dawkins’ remarks suggest.10 This means that there is good reason to think that P(N|D) > P(N|¬D). With this dependence between D and N, the structure of the Bayesian network in Figure 5.1(b) is now relevant and Theorem 1 becomes relevant when assessing whether N partially explains away the evidence for design. Replacing A with N in Equation (6), it now becomes less clear that N even partially explains away the evidence for design. 9 It could be argued that P(E|¬D, N) would not be low when the the size of the universe and the possibly large number of planets capable of supporting life are taken into account, which is presumably what Dawkins has in mind when he refers to ‘sheer luck, anthropically justified’. Without more detailed knowledge of the probabilities involved in evolutionary processes and the number of suitable planets, it is difficult to evaluate this claim, but let us suppose that the origin of life somewhere in the universe is quite probable once these factors are taken into account. Whether P(E|¬D, N) is low now depends on what kind of complex life E refers to. If E just referred to any kind of living organism, then clearly P(E|¬D, N) would be quite high. If E refers to eukaryotic life, it might not be so high. Given the improbability of the transition to eukaryotic life, this transition may well only occur on a small proportion of planets where life exists and so unless the number of such planets is large, the transition might not occur at all. Similarly, P(E|¬D, N) becomes lower as the complexity involved in E increases. 10 By ‘complexity’ here I do not mean the same kind of complexity as in the evidence E. There are various ways to make E precise, eukaryotic life or intelligent life for example, but the idea is that the complexity involved in E goes beyond that of the simplest living organisms. By using the term ‘complexity’ here I mean something like Dawkins’ term ‘organized complexity’ which he also applies to the simplest living organisms.

94 p robab i l ity i n th e ph i lo sophy of re l i g i on Nevertheless, if the probability of complex living organisms given neither evolutionary processes nor design, i.e. P(E|¬D, ¬N), is sufficiently low, then partial explaining away will still take place. In this case, Theorem 2 is relevant in determining whether complete explaining away occurs, and it is clear that it does not. The extent of residual confirmation of design in this case is given by replacing A with N in Equation (12). This residual confirmation is boosted relative to the independent case since P(N|D) > P(N|¬D). Thus, as in Example 3, there are two ways in which N influences the probability of D. First, there is a negative influence due to partial explaining away that tends to lower the probability of D as in the independent case. Second, there is also a positive influence due to the positive dependence between D and N that tends to raise the probability of D. While this second influence may not be sufficient to compensate for the negative influence, and so it does not prevent partial explaining away, there is still a reasonable degree of residual confirmation of D. Overall, it seems that even though an explanation in terms of evolutionary processes may well partially explain away the evidence for design in the case of biological complexity, there are reasons for believing that the argument still retains force.

IV. Fine-tuning in Cosmology We turn now from biology to consider the argument for design based on fine-tuning. The story is familiar: if the value of any one of a number of physical constants had been slightly different, carbon-based life would be impossible (see for example Carr 2007). This raises the question as to how this remarkable precision is to be accounted for and since blind chance is not up to the task, other hypotheses must be pursued. The two main candidates are the design hypothesis and the multiverse hypothesis. Design is straightforward enough: an intelligent designer deliberately sets the values of the physical constants. The idea of a multiverse is that our universe is only one of many, perhaps infinitely many, universes with each having its own values for the constants. Assuming there are sufficiently many universes within the multiverse and assuming a suitable distribution of the values of the constants, then it would be expected that some universes would be suitable for life. And, not surprisingly, we find ourselves in one such universe since we could not exist in any of the others.11 The hypotheses and evidence can be represented as follows: D : A Designer (i.e. God) exists. M : A multiverse exists. E : There exists a universe with physical constants suitable for life. 11 In this section, I am assuming that the relevant probabilities can be defined. This has been called into question by McGrew et al. (2001). The basic problem is that fine-tuning arguments seem to assume a uniform distribution over an infinite space of possible values of the constants in which case the uniform distribution is not normalizable. For the purposes of this article, I will assume that there is a resolution to this problem, such as the measure-theoretic approach of Koperski (2005).

can evi de nce for de si gn b e e x p lai ne d away ? 95 By saying that the constants are suitable for life, this means that they are such that had they been slightly different, life would be impossible. Background knowledge will be assumed to contain the existence of a universe including laws of physics, but not the values of the constants in those laws, and whatever additional knowledge about the hypotheses might be required to understand them and to make judgments about the probabilities they confer on the evidence. In this way any confirmation for the design hypothesis will be confirmation over and above any that is received from the order of the universe understood in terms of regularity arising from the existence of laws of physics. As before, background knowledge will be suppressed in the notation for simplicity. We will also assume that the multiverse scenario is such that the existence of a universe with constants suitable for life is almost guaranteed. The first question to address is whether E confirms D, i.e. is P(E|D) > P(E|¬D)? To start with, we shall make the simplifying assumption that D and M are marginally independent so that the structure of the relevant Bayesian network is given by Figure 5.1(a) and the expressions for P(E|D) and P(E|¬D) are given in Equations (2a) and (2b) with A replaced by M. The starting point for fine-tuning is that it is extremely unlikely that the values of the constants would be suitable for life if neither the design nor multiverse hypotheses were true and so we can assume that P(E|¬D, ¬M) is close to zero, and hence that P(E|D, ¬M) > P(E|¬D, ¬M).12 Given the nature of the multiverse hypothesis, we can assume that P(E|D, M) ≈ P(E|¬D, M) ≈ 1. Assuming that P(M) is not too high, it follows from Equations (2a) and (2b) that P(E|D) > P(E|¬D) and so design is confirmed by the evidence. Let us now suppose that we know M to be true. First we must ask whether this would partially explain away the evidence for design. Here Corollary 1 is relevant and since P(E|¬D, ¬M) is close to zero, it seems clear that partial explaining away would occur. In contrast to the biological case, however, complete explaining away would also occur since in this case Corollary 2 would apply and, as we have seen P(E|D, M) ≈ P(E|¬D, M) ≈ 1. Even if P(E|D, M) is very slightly greater than P(E|¬D, M), it is clear from Equation (10) that there would be almost no residual confirmation. Two assumptions need to be relaxed, however. First, it is not obvious that D and M are marginally independent; second, instead of assuming that M is known to be true, it would be worth exploring the impact of some independent evidence F that supported M.13 Assuming that F would be conditionally independent of D and E given M, Figure 5.2(b) would then provide the relevant structure for the Bayesian network.

12 As in the biological case, the possibility of designers other than God could also be considered. See note 8. 13 It would be difficult to say what sort of evidence would provide support for M, where M is understood to be a multiverse with sufficiently many universes with different physical constants so that it is almost guaranteed that at least one universe would be suitable for life. It seems clear that there could be no direct access to such universes, but perhaps indirect evidence could provide some support.

96 p robab i l ity i n th e ph i lo sophy of re l i g i on If the design and multiverse hypotheses are not marginally independent, what is the relationship between them? Recall that the multiverse hypothesis requires sufficiently many universes with different physical constants so as to almost guarantee that at least one universe would be suitable for life. There are a number of reasons for thinking that this hypothesis would be more probable given design than non-design. One reason for this is that inflation theory, which at present is a key component of favoured mechanisms for generating multiple universes, seems to require fine-tuning. Robin Collins argues that the inflationary multiverse scenario requires a number of components to be in place, without any one of which it would still ‘almost certainly fail to produce a single life-sustaining universe’ (Collins 2007: 465). These components are a mechanism to supply the energy needed for the bubble universes (the inflaton field), a mechanism to form the bubble universes, a mechanism to convert the energy of the inflaton field to normal mass/energy, and a mechanism that allows enough variation among the universes. In addition, he argues that appropriate background laws, such as gravity, and physical principles, such as the Pauli exclusion principle, would need to be in place to support life. The basic idea seems to be that factors similar to those which motivate the fine-tuning argument in the first place would come into play in the case of a multiverse. If we knew that our universe was only one of a very large number of universes, very few of which were capable of supporting life, and if we knew that the existence of such universes depended on the existence of various mechanisms which were finely-tuned to generate them, this might provide the basis for a design argument. If Collins is right about this, it seems likely that similar considerations would apply to any multiverse hypothesis and so there would be good reason to think that P(M|D) > P(M|¬D). How do these changes affect the conclusions that were drawn based on the simpler model where D and M were considered to be marginally independent and M was assumed to be true? Theorem 3 provides the conditions under which D is partially undermined via explaining away by F, with M replacing A. Given that the term P(E|¬D, ¬M) on the left-hand side of Equation (13) is extremely low, it is likely that partial undermining does occur. Collins’ argument provides some reason for thinking that P(M|¬D) might also be very low, but it is not clear that it would be sufficiently low to prevent partial undermining. It does, however, seem that complete undermining via explaining away by F would not occur. Here, Theorem 4 is relevant. Given an extremely low value for P(E|¬D, ¬M), this strongly suggests that the right hand side of expression (15) is negative. By contrast, given that P(E|D, M) ≈ P(E|¬D, M) ≈ 1, the left-hand side would be positive if Collins’ argument is sound, but even without Collins’ argument there does not seem to be any reason for thinking that the left-hand side is negative. Hence, it seems that inequality (15) does not hold and so complete undermining via explaining away does not occur. The degree of residual confirmation in this case would be given by Equations (17) and (18) with M replacing A in (18). To simplify the expression

can evi de nce for de si gn b e e x p lai ne d away ? 97 for L in (18) we can set P(E|¬D, ¬M) = 0 and P(E|D, M) = P(E|¬D, M) = 1. This gives, L=

P(F|M)P(M|¬D) . P(F|M)P(M|D) + P(E|D, ¬M)P(F|¬M)P(¬M|D)

(19)

Provided L < 1 there is some residual confirmation and the smaller the value of L the greater the degree of residual confirmation. Suppose for the sake of argument that the probability of the evidence F in the absence of the multiverse were 0, i.e. P(F|¬M) = 0. In that case we would have L = P(M|¬D)/P(M|D) assuming P(F|M)  = 0 and so whether there was any residual confirmation would depend on the strength of Collins’ argument that P(M|D) > P(M|¬D). This is a very extreme case, however, since it would essentially amount to empirical proof of a multiverse scenario of the appropriate kind (i.e. which would guarantee a universe suitable for life). If any evidence for such a multiverse is ever forthcoming, it is likely to be much more modest than that. Assuming P(F|¬M) is not close to zero, the second term in the denominator of (19) becomes important. Ignoring Collins’ argument and setting P(M|D) = P(M|¬D) = P(M), the value of L depends crucially on P(M), the prior probability of the multiverse scenario. It certainly seems reasonable to set this to a very low value and so the second term in the denominator becomes dominant. This means that L is much less than 1 and so there would be a substantial degree of residual confirmation. Indeed, as Monton points out, ‘as long as one assigns a probability < 1 to the hypothesis that there are many universes, the theistic fine-tuning argument will still have force’ (Monton 2006: 422). In a similar way to Example 3, there are two evidential pathways by which F could influence the probability of D. First, F would raise the probability of M. By one pathway, this would have a negative influence due to partial undermining via explaining away that would tend to lower the probability of D. (This is what would happen if D and M were marginally independent.) Via a second pathway, the impact of F on M might have a positive influence if there is a positive dependence between D and M that would tend to raise the probability of D. Even if there is no such positive influence via this second pathway, appeals to a multiverse scenario do not seem good enough to undermine the design argument from fine-tuning. The mere possibility that a multiverse exists would not undermine the design argument and even if some evidence F confirming a multiverse scenario were to be obtained, there would still be significant residual confirmation of design.

V. Conclusions Two types of explaining away, partial and complete, have been defined and the conditions under which they occur explored when there are two hypotheses that are a) marginally independent and b) marginally dependent. The case where there

98 p robab i l ity i n th e ph i lo sophy of re l i g i on is some independent but inconclusive evidence for one of the hypotheses has also been considered. The formal results have been applied to design arguments based on biological complexity and fine-tuning in cosmology in order to investigate whether the evidence for design can be explained away. It has been assumed that in the absence of an alternative explanation, the evidence would confirm design in both of these contexts; the only question considered is whether this evidence can be explained away. Closely related to this is the assumption that it is possible to say enough about what a designer would be likely to do, so as to motivate the relevant likelihoods in the case of design. Although this has been questioned (see Sober 2005), it is far from clear that this objection is fatal to design arguments (see Swinburne 2004; McGrew 2004). It turns out that explaining away operates in a different way in the cases of biological complexity and fine-tuning in cosmology. If an appropriate multiverse hypothesis were known to be true this would completely explain away evidence for design if the design and multiverse hypotheses were marginally independent. However, in the absence of very convincing evidence for such a multiverse, fine-tuning does provide significant confirmation of design. In the biological case, evolutionary (and other natural) processes may well partially explain away biological complexity as evidence for design, but it has been argued that they do not completely explain it away. Whether there is significant residual confirmation of design in this case depends mainly on the probability of complex organisms evolving from simple organisms and the development of life in the first place without design. Since there is reason to believe that these probabilities are low and that design is relevant to the existence of complex life, there is reason to believe that there is significant residual confirmation of design in the biological case as well. Much more could be said both for and against the comparative probability judgements presented in this paper. Nevertheless, the approach described here in terms of explaining away is proposed as a general framework for evaluating design arguments. In particular, if it is granted that there is some evidence in favour of design in a given context, defeating the design argument requires not only providing an alternative explanation of the evidence but showing that this alternative explanation explains away the evidence for design so that there is little or no residual confirmation of design.14

14 This paper was presented at the Formal Methods in Epistemology of Religion conference at the Katholieke Universiteit Leuven in June 2009. I would like to thank the participants for a number of questions and suggestions that have helped to improve the chapter. I would especially like to thank Dr Bruce Langtry for extremely detailed feedback on an earlier draft of the paper and Dr Jake Chandler, Dr Victoria Harrison, Dr Joshua Thurow, and anonymous reviewers for very helpful comments and corrections. Finally, I am very grateful to Prof. Richard Swinburne, Prof. Timothy McGrew, Dr Lydia McGrew, Graham Veale, and Richard Smith for interesting and profitable discussions on design arguments.

can evi de nce for de si gn b e e x p lai ne d away ? 99

Appendix: Proof of Theorems Proof of Theorem 1. From Bayes’ theorem, P(D|E, A) =

P(A|E, D) × P(D|E). P(A|E)

Hence, P(D|E, A) < P(D|E) ⇔ P(A|E, D) < P(A|E) P(E|A)P(A) P(E|D, A)P(A|D) < P(E|D) P(E)  ⇔ P(E|D, A)P(A|D) P(E|D)P(D) + P(E|¬D)P(¬D)  < P(E|D) P(E|D, A)P(A|D)P(D) + P(E|¬D, A)P(A|¬D)P(¬D) ⇔

⇔ P(E|D, A)P(A|D)P(E|¬D) < P(E|D)P(E|¬D, A)P(A|¬D)  ⇔ P(E|D, A)P(A|D) P(E|¬D, A)P(A|¬D) + P(E|¬D, ¬A)P(¬A|¬D)  < P(E|¬D, A)P(A|¬D) P(E|D, A)P(A|D) + P(E|D, ¬A)P(¬A|D) ⇔ P(E|D, A)P(E|¬D, ¬A)P(A|D)P(¬A|¬D) < P(E|¬D, A)P(E|D, ¬A)P(A|¬D)P(¬A|D).



Proof of Corollary 1. Since D and A are marginally independent P(A|D) = P(A|¬D) and so P(¬A|¬D) = P(¬A|D). From these equalities and Theorem 1, the result follows immediately.  Proof of Theorem 2. From Bayes’ theorem, P(D|E, A) =

P(E, A|D) × P(D). P(E, A)

Hence, P(D|E, A) ≤ P(D) ⇔ P(E, A|D) ≤ P(E, A)  ⇔ P(E, A|D) ≤ P(E, A|D)P(D) + P(E, A|¬D)P(¬D)  ⇔ P(E, A|D) 1 − P(D) ≤ P(E, A|¬D)P(¬D) ⇔ P(E, A|D) ≤ P(E, A|¬D) ⇔ P(E|D, A)P(A|D) ≤ P(E|¬D, A)P(A|¬D).



Proof of Corollary 2. Since D and A are marginally independent P(A|D) = P(A|¬D). From this equality and Theorem 2, the result follows immediately. 

100 probabi lity in the phi lo sophy of re lig ion Proof of Theorem 3. From Bayes’ theorem, P(D|E, F) =

P(F|D, E) × P(D|E). P(F|E)

Hence, P(D|E, F) < P(D|E) ⇔ P(F|D, E) < P(F|E) ⇔ P(F|D, E, A)P(A|D, E) + P(F|D, E, ¬A)P(¬A|D, E) < P(F|E, A)P(A|E) + P(F|E, ¬A)P(¬A|E) ⇔ P(F|A)P(A|D, E) + P(F|¬A)P(¬A|D, E) < P(F|A)P(A|E) + P(F|¬A)P(¬A|E)  ⇔ P(F|A) P(A|D, E) − P(A|E)  < P(F|¬A) P(¬A|E) − P(¬A|D, E)   ⇔ P(F|A) − P(F|¬A) × P(A|D, E) − P(A|E) < 0 ⇔ P(A|D, E) < P(A|E),

since P(F|A) > P(F|¬A)

⇔ P(D|E, A) < P(D|E),

using Bayes’ theorem. 

The result then follows from Theorem 1.

Proof of Corollary 3. The result follows immediately from Theorem 3 and the marginal independence of D and A.  Proof of Theorem 4. From Bayes’ theorem, P(D|E, F) =

P(E, F|D) × P(D). P(E, F)

Hence,

P(D|E, F) ≤ P(D) ⇔ P(E, F|D) ≤ P(E, F). P(E, F|D) = P(E, F|D, A)P(A|D) + P(E, F|D, ¬A)P(¬A|D) = P(E|D, A)P(F|A)P(A|D) + P(E|D, ¬A)P(F|¬A)P(¬A|D) P(E, F) = P(E, F|D, A)P(A|D)P(D) + P(E, F|D, ¬A)P(¬A|D)P(D) + P(E, F|¬D, A)P(A|¬D)P(¬D) + P(E, F|¬D, ¬A)P(¬A|¬D)P(¬D) = P(E|D, A)P(F|A)P(A|D)P(D) + P(E|D, ¬A)P(F|¬A)P(¬A|D)P(D) + P(E|¬D, A)P(F|A)P(A|¬D)P(¬D) + P(E|¬D, ¬A)P(F|¬A)P(¬A|¬D)P(¬D)

can evi de nce for de si gn b e e xp laine d away ? 101 Hence,

P(D|E, F) ≤ P(D) ⇔ P(E|D, A)P(F|A)P(A|D)(1 − P(D)) + P(E|D, ¬A)P(F|¬A)P(¬A|D)(1 − P(D)) < P(E|¬D, A)P(F|A)P(A|¬D)P(¬D) + P(E|¬D, ¬A)P(F|¬A)P(¬A|¬D)P(¬D)  ⇔ P(F|A) P(E|D, A)P(A|D) − P(E|¬D, A)P(A|¬D)  < P(F|¬A) P(E|¬D, ¬A)P(¬A|¬D) − P(E|D, ¬A)P(¬A|D) .  Proof of Corollary 4. The result follows immediately from Theorem 4 and the marginal independence of D and A. 

References Carr, B. (2007) ‘The Anthropic Principle Revisited’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge Univerity Press, 77–89. Collins, R. (2007) ‘The Multiverse Hypothesis: a Theistic Perspective’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge Univerity Press, 459–80. Conway Morris, S. (2003) Life’s Solution: Inevitable Humans in a Lonely Universe. Cambridge: Cambridge University Press. Cowell, R. G., A. P. Dawid, S. L. Lauritzen, and D. J. Spiegelhalter (1999) Probabilistic Networks and Expert Systems. Berlin, Heidelberg, New York: Springer. Dawkins, R. (2006) The God Delusion. London: Bantam Press. Eells, E. and B. Fitelson (2002) ‘Symmetries and Asymmetries in Evidential Support’, Philosophical Studies, 107: 129–42. Fitelson, B. (1999) ‘The Plurality of Bayesian Measures of Confirmation and the Problem of Measure Sensitivity’, Philosophy of Science, 66: S362–S378. Gould, S. J. (1989) Wonderful Life: The Burgess Shale and the Nature of History. New York: W. W. Norton & Co. Koperski, J. (2005) ‘Should We Care about Fine-tuning’, British Journal for the Philosophy of Science, 56: 303–19. Lipton, P. (2004) Inference to the Best Explanation. 2nd edition. London: Routledge. McGrew, L. (2004) ‘Testability, Likelihoods and Design’, Philo, 7, 1: 5–21. McGrew, T., L. McGrew, and E. Vestrup (2001) ‘Probabilities and the Fine-tuning Argument: A Sceptical View’, Mind, 110: 1027–37. Monton, B. (2006) ‘God, Fine-tuning, and the Problem of Old Evidence’, British Journal for the Philosophy of Science, 57: 405–24. Paley, W. (1802) Natural Theology, or Evidences of the Existence and Attributes of the Deity Collected from the Appearances of Nature. London: Gould of Lincoln. Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems. San Francisco: Morgan Kaufman. Sober, E. (2002) ‘Intelligent Design and Probability Reasoning’, International Journal for the Philosophy of Religion, 52: 65–80.

102 probabi lity in the phi lo sophy of re lig ion Sober, E. (2005) ‘The Design Argument’. In W. Mann (ed.), The Blackwell Guide to the Philosophy of Religion. Oxford: Blackwell: 117–47. Swinburne, R. (2004) The Existence of God. 2nd edition. Oxford: Oxford University Press. Weisberg, J. (2005) ‘Firing Squads and Fine-tuning: Sober on the Design Argument’, British Journal for the Philosophy of Science, 56: 809–21.

6 Bayes, God, and the Multiverse Richard Swinburne

I. Introduction A major probabilistic argument of natural theology claims that the ‘fine-tuning’ of the laws of nature and of the initial conditions of the universe (at the time of the Big Bang) which was necessary for the evolution of embodied beings of our level of intelligence (and made it physically very probable that they would evolve somewhere sometime1 ) makes it (inductively) probable that the universe was created by God.2 This is because, the argument claims, it is most improbable that the universe could have such intelligent life-producing features by mere chance, whereas a God would seek to produce intelligent life. (The normal and in their general effect uncontroversial aspects of ‘finetuning’ to which scientists draw our attention are the facts that the density, velocity, and ratios of the different components of the Big Bang, and the constants of laws of nature all lie within the extremely narrow limits required in the above respect for the evolution of intelligent life.3 ) The objection to such arguments which has appealed most to 1 I shall assume, in order not to complicate my exposition, that a physical theory which explained the evolution of the bodies of intelligent beings would thereby explain the evolution of embodied intelligent beings, understood—as I understand it here—in the sense of conscious intelligent beings. In fact in my view no physical theory is in the least likely to be able to explain the evolution of consciousness, and so for example why human bodies are the bodies of conscious beings. The argument from fine-tuning is really an argument from the universe being fine-tuned so as to produce bodies of intelligent beings. The occurrence of consciousness connected to these bodies provides a further argument for the existence of God. See Swinburne (2004: 192–212). 2 See Appendix A. 3 Most expositions of the ‘fine-tuning’ assume the standard model of particle physics (including the ‘four forces’) and then show that only for one narrow range (or a few narrow ranges) of the constants involved in this model (and of values of the variables of the properties with which this model deals) would intelligent embodied beings evolve. Some expositions assume a somewhat wider class of possible models, for example, those of seriously considered scientific theories (some of which do not include the standard model) ‘and/or the ones we can perform calculations for’ (Collins 2009: 240). This latter phrase is that of Robin Collins who gives by far the fullest and most up-to-date detailed account of the extent of the fine-tuning, that is of just how much different constants and variables could vary if the universe was still to be productive of intelligent life. (See his very long article Collins 2009.) I have argued that in order to show the improbability of fine-tuning if there is no God, one needs to show more than is shown in these expositions. One needs to show that it is (if there is no God) improbable that any scientific theory would have the consequence that intelligent embodied beings would evolve, and that involves showing that

104 p robab i l ity i n th e ph i lo s ophy of re l i g i on scientists of recent years is the ‘multiverse’ argument. Science gives reason to believe, it is claimed, that there are innumerable different universes (together constituting a ‘multiverse’), each with different initial conditions and/or laws of nature; and so it is not improbable that one of these universes would be an intelligent-life-producing universe; and—as we are intelligent beings—it is inevitable that we will find ourselves in such a universe, and so there is no need to postulate a God to explain the characteristics of our universe. The purpose of this paper is to investigate this objection. In order to do so in a short paper, I need to summarize certain results which I claim to have established elsewhere. I need to begin by articulating the structure of any probabilistic argument to a causal explanation, and then go on to outline the structure of any such argument to the existence of God from the general features of the universe (including fine-tuning). After that I shall note the various kinds of multiverse which have been postulated and the evidence for them; and go on to claim that, while there might well be a multiverse of a limited kind, that would not make a great difference to the strength of a probabilistic argument from fine-tuning to the existence of God.

II. Five Criteria A hypothesis h purporting to provide a causal explanation of evidence e is, I suggest, for background evidence k, probable in so far as: 1. 2. 3. 4. 5.

If h, then given k, probably e; If ∼h, then given k, probably ∼e; h is a simple hypothesis; h ‘fits with’ with k; h has small scope.4

By ‘background evidence’ I mean evidence about how things behave in areas outside the range that h purports to explain. I claim that these criteria apply whether h is an explanation of an inanimate (or scientific) kind in terms of initial conditions and laws of nature, or one of a personal kind in terms of persons, their powers, beliefs, and it is improbable that any scientific theories other than those currently seriously considered would have this consequence, for example, because they would need to be very complicated. For my attempt to argue for this improbability see Swinburne (2004: 181–8). 4 I have expounded these criteria in many other places, normally—by combining what I am listing as criteria (1) and (2) as one criterion—as ‘four criteria’. See for example Swinburne (2001: 80–3); I did however there mistakenly confuse the ‘scope’ of a hypothesis with the size of the field which it purports to explain. The latter is only one of the two factors which determine the scope of a hypothesis, the other being the precision of its predictions within the field. What limits the need for a fit with background evidence (criterion 4) is the size of the field; the precision of predictions is irrelevant. I am grateful to a referee for drawing my attention to the confusion.

baye s, g od, and the multive r se 105 purposes. I assume that it is irrelevant to the evidential force of e whether e is known before or after the formulation of h; and so I shall use ‘predict’ in the sense that in so far as h makes e (for given k) probably true, h ‘predicts’ e with that degree of probability. So understood, (1) is simply the criterion that a hypothesis is more probable, the more probable it makes its observed predictions (the more probable, the more of them and the more accurate they are); (2) is simply the criterion that h is more probable if rival hypotheses of any significant probability (on criteria other than (2)) predict not-e. If k is itself a causal hypothesis about other fields, then h ‘fits with’ k in so far as (h & k) is simpler than any (h∗ & k), where h∗ is a rival to h in explaining e. In such a case, I shall say that h ‘meshes with’ k. If k are pieces of evidence of the same level of generality as e, then h ‘fits with’ k in so far as k makes a theory t more probable than any other theory of its field and (h & t) is simpler than any (h∗ & t) where h∗ is a rival of the above kind. So (4) claims that a hypothesis purporting to explain evidence in a named field (for example, about whether John committed a particular crime) is more probably true in so far as it fits (in either of these ways) with other things we know, for instance, whether John has committed other such crimes in the past. But the larger the field covered by the hypothesis, the less is the role for (4). A large-scale theory of physics purports to explain so much that there is little else for it to fit ‘with’. The ‘scope’ of h is greater, as I shall understand this notion, the more and the more precise are its predictions (true or false, observed or unobserved). The more predictions a hypothesis makes and the more precise they are, the greater the probability that the hypothesis contains some error. I suggest that the practice of science shows that scientists give (5) less weight than the other criteria, since they regard large-scale theories which make predictions about a large field and which satisfy the other criteria well as very probably true. There will always be an infinite number of theories (some of them making totally different predictions for the future from others) satisfying criteria (1), (2), and (5) for any value of the ‘probably’ in (1) and (2). In a situation where h covers such a large field that there is no significant contingent k, everything will depend on criterion (3), simplicity. And if the field covered by h is smaller and there is significant contingent k, all will depend on criteria (3) and (4); but as simplicity is crucial for assessing ‘fit with background evidence’, everything will again depend on simplicity. So simplicity is an all-important criterion without which we can make no inferences beyond our evidence. It follows that without an objective understanding of simplicity it will not be an objective matter whether evidence makes this or that hypothesis probable or improbable. I suggest that our ordinary practice, and that of scientists, historians and detectives, shows that we understand a hypothesis as simple to the extent to which it postulates few substances (entities), few kinds of substances, few properties, few kinds of properties, more readily observable properties, few relations between properties (in other words, few laws), and simpler kinds of relations between properties (that is, ones involving simpler mathematics—a notion which can to a considerable extent be

106 probabi lity in the phi lo sophy of re lig ion defined objectively).5 If we think of scientific explanation as explanation by initial conditions (of substances and their properties) and laws, the principle of simplicity requires us to postulate the simplest combination of these. If we think of ‘laws’ as generalizations about the powers and liabilities of individual substances (liabilities to exercise powers in certain circumstances, that is) then the distinction between laws and initial conditions disappears, and the principle tells us just to postulate the fewest substances with the fewest and simplest properties (including powers and liabilities). For personal explanation, the principle of simplicity requires us to postulate the fewest persons with the fewest and simplest powers (for instance, to move limbs), beliefs and purposes. Since spatially extended substances consist of a number of smaller substances which stick together or act together (even if they cannot be separated into those small substances), the ‘few substances’ sub-criterion requires us to postulate substances no larger in spatial extent than are necessary for explanatory purposes. The various sub-criteria of simplicity often need to be weighed against each other, for example, postulating few laws against postulating more complicated laws. It is not always clear which postulation is overall the simplest, but in many paradigm cases it is clear. These criteria are captured by the probability calculus and in particular by Bayes’ theorem: P(h|e&k) =

P(e|h&k)P(h&k) P(e|k)

I read this theorem as claiming that for any propositions h and e, in so far as the probabilities occurring in the theorem can be given a numerical value, it correctly states the numerical relationships that hold between them. In so far as they cannot be given precise numerical values, Bayes’ theorem simply claims that all propositions of comparative probability, that is about one probability being greater than (or much greater than) or equal to or less than (or much less than) another probability, which can be deduced from the theorem are true. For example it follows that if P(e|h1 &k) = P(e|h2 &k), then P(h1 |e&k) > P(h2 |e&k) if and only if P(h1 |k) > P(h2 |k). In the special case where h is a causal hypothesis purporting to explain e with background evidence k, our five criteria tell us how to assess the probabilities on the right-hand side. P(e|h&k) measures the degree to which criterion (1) is satisfied. P(h|k) is large to the extent to which h fits with k (criterion 4) and to the extent to which its intrinsic features (simplicity—criterion 3; and scope—criterion 5) make it probable. When there is no contingent background evidence, k is any tautology and P(h|k) is then called the ‘intrinsic probability’ of h, and is a function solely of its simplicity and scope. By another theorem of the calculus P(e|k) = P(e|h&k)P(h|k) + P(e|h1 &k)P(h1 |k) + P(e|h2 &k)P(h2 |k) . . . and so on for all hypotheses hn of the same field and of equal

5 For analysis and justification of this understanding of simplicity, see Swinburne (2001: 83–102).

baye s, god, and the multive r se 107 scope which are mutually exclusive and exhaustive. So P(e|k) is large in so far as one or more rival simple hypotheses which fit with any contingent k make e probable (which spells out criterion 2).6 We have grounds to believe in states of affairs other than those which provide a probable causal explanation of our evidence if and only if our evidence makes it probable that they explain or are explained by (that is, cause or are caused by) those states which explain our evidence, or are connected by causal relations to the latter states. Thus I can predict rain tomorrow if I have a probable meteorological theory which provides a causal explanation of past weather patterns (or more general phenomena), and which predicts that whatever air currents caused the most recent weather patterns will also cause rain tomorrow. The probability calculus gives us a precise formula for prediction and retrodiction: the probability of a future event, such as rain tomorrow (r) on evidence (e) is the sum of probabilities of different theories (hn ) on that evidence, each multiplied by the probability of the event on that theory and the evidence. P(r|e&k) = n P(r|hn &e&k)P(hn |e&k) A similar formula governs retrodiction, for example, to the probability that it rained in Oxford on 1st January of the year 2 million bc. And we can use joint applications of retrodiction and prediction to infer further events. For example, we can retrodict from a crater that there was a large meteor impact in 60 million bc, and then predict that this would have led to much of the surrounding country being covered by a dense dust cloud which would have led to the extinction of the dinosaurs. But our recognized ways of acquiring knowledge of things past or future or unobserved give us no licence to ascribe probabilities to states of affairs different from their intrinsic probabilities unless our evidence suggests those states of affairs are causally connected to that evidence. The intrinsic probability of a state is the probability on mere tautological evidence that it would occur—which intuitively is a very small probability.

III. Probabalistic Natural Theology Natural theology claims that the most general features of the universe show that there is a God. I will construe a probabilistic natural theology as claiming that those general features make it probable that God causally explains the existence of the universe with these general features, by bringing it into existence (if it had a beginning) and by conserving it in existence as long as it has existed (whether for a finite time or forever).7 6 See Appendix B. 7 If our universe or (see later) the multiverse of which it is a member were to prove everlasting, then

the argument from its being fine-tuned so as to produce intelligent life at some time or place would be an argument, as well as from its laws of nature, not now from its initial conditions, but from its boundary conditions in the sense of those general features which it has at every moment of past time. The argument would be an argument from the boundary conditions being such and every physical object having the powers and liabilities codified in the ‘laws’ being such as to make it probable that God would have been sustaining

108 probabi lity in the phi lo sophy of re lig ion Here I shall be concerned solely with the general feature of its fine-tuning and its contribution to a cumulative natural theology. I understand by ‘God’ an essentially everlasting omnipotent, omniscient, perfectly free person. The natural theologian must claim that such a being is a very simple substance of such a kind as quite probably to bring about a universe with the general characteristics of our universe. I have argued elsewhere (Swinburne 2004: 93–109, and most recently and fully Swinburne 2010) that God so defined is a very simple substance, and that his other properties (including essential perfect goodness and not having a spatial extension) follow from this definition. For reasons of space I cannot repeat those arguments here. God’s perfect goodness means that he will be as good as it is logically possible to be; and this, I suggest, is naturally construed as follows. In circumstances where there is a best possible action to do, he will do it, in circumstances where there are two or more equal best actions, he will do one of them; but in circumstances where there are available to God an infinite number of incompatible actions, each better than some other one, he cannot do a best action but he can and will do one of these actions. But if these latter actions can be divided into kinds, such that there is a best or equal best kind of action (but no best kind of the kind), he will do some action of that or those kinds. Intelligent beings are as such a good thing, and so God has reason to bring them about. But humans are a special kind of intelligent being. We can choose between doing (limited) moral good or evil to ourselves and each other; we can discover by rational inquiry which actions are good and which are evil, and discover how to extend our power over the universe, and discover deep truths about the nature and origin of the universe; and by our choices to do good or evil, over time we can form our own characters—so that doing good or evil (as the case may be) comes naturally to us. We thus have a special kind of goodness, not possessed (as far as we know) by any other kind of conscious being including God himself (who can do no evil, and who is essentially omniscient and perfectly good). So God has substantial reason to bring about humans, even though he also has substantial reason not to bring them about in view of the evil they may well do (even if that evil serves some greater good). So I suggest that it would be an equal best kind of action for God to bring about humans as not to bring about humans, and so there would be a probability of half that God will bring about some humans (see Swinburne 2004: 110–32). From this it follows that there is at least that probability that God will bring about intelligent beings, and so the fine-tuning of the universe in the respects referred to, which, I am assuming, is necessary for the evolution of intelligent life and makes it probable. Now of course we cannot really

these conditions at every moment of past time. An analogy would be if we were to find a hall where a very large number of puppets are dancing in unison. If we learn that this has been going on forever, that would not remove the need for an explanation in terms of a common cause; but the common cause would have to have been a cause operating throughout all past time, for example some puppet master who had been controlling the puppets everlastingly by invisible strings.

baye s, god, and the multive r se 109 give exact probabilities to God’s actions, but the unique goodness of humans in the respects I have described does make it quite probable that God will create humans. (I shall understand in future by ‘humans’ any beings with the characteristics of humans described above.) The argument from fine-tuning can only make a positive contribution to natural theology if the evil (the suffering and wrongdoing) which accompanies humans (and animals) on earth, and is presumably a probable consequence of the particular tuning of the laws and initial conditions of the universe, does not totally annihilate the positive force of the argument as so far stated. It must be the case that, despite the evil, the existence of humans, and so the fine-tuning which produces them, increases the probability of the existence of God (from whatever is shown by other evidence). To show this requires a theodicy. The theodicy must show that there is a significant probability that any evil on earth is such that by allowing it to occur, a perfectly good God forwards some good purpose which could not be forwarded in any better way. Hence it will have to show that there is a significant probability that all such evil is caused by the free choice of an agent, or has an effect which provides an opportunity (not provideable in any better way) for some agent to do a good action or to acquire knowledge which will allow an agent to do a good action; and that God has the right to allow the amount and distribution of evil which occurs. If this cannot be shown, the argument will not have any force. But if an adequate theodicy can be provided, it will only justify kinds of evil for which it can be shown that they forward a good purpose which could not be forwarded in any better way; and which are of limited length and intensity (for example, any human suffering for no more than 80 years or so). I am going to assume what I have argued elsewhere, that this can be shown for the evil which we find on earth.8 Nevertheless God, like a good parent who gives so much to his children, has the right to impose only a limited amount of suffering on his children for their sake or the sake of others. It follows that there is a probability of 0 that God will bring about beings who suffer pointlessly (that is, their being allowed to suffer provides no benefit to themselves or anyone else), or endlessly; or do evil pointlessly (that is, their being allowed to do evil provides no benefit to anyone). It is also much more probable that God would bring about humans (who have a kind of goodness God lacks) than that he would bring about other finite intelligent beings who have a kind of goodness (for example, always to choose the good) already exemplified in God himself—for the reason that this kind of goodness is already exemplified (although of course, in view of the goodness of the latter beings, it is by no means very improbable that God would create more of them). So, I claim, as must any probabilistic natural theologian, that—given an adequate theodicy for the evil on earth—theism is a very simple hypothesis (criterion 3) which

8 For my theodicy see Swinburne (2004: 219–72), and—more fully—Swinburne (1998).

110 probabi lity in the phi lo sophy of re lig ion also satisfies criterion (1). Since it purports to explain almost everything else, criterion (4) is irrelevant; and given its great simplicity and the relative imprecision of its predictions, (5) will not detract very much from its probability. All turns on criterion (2) which in turn depends on whether there are rival hypotheses which satisfy criteria (1) or (3) better. Rival hypotheses which postulate more or weaker godlike persons as the ultimate causes of the universe are clearly going to be less simple than theism (given that, for example, being omnipotent is a simpler property than having such-and-such a large but limited degree of power). A scientific (inanimate) hypothesis which satisfied criterion (1) well would postulate some first substance, or a backwardly infinite succession of substances, either ones which have normal physical properties or the property of a liability to produce good states. Given that laws of nature cannot affect the world unless there is some substance on which they operate, postulation of the existence of a substance with a liability to produce good states is the way we should construe the hypothesis put forward by John Leslie (1979) and Hugh Rice (2000), and considered by Derek Parfit (1992), that there is a propensity in nature to produce the good. I argue elsewhere (Swinburne 2010) that this hypothesis is not as simple a hypothesis as theism. What I want to consider in this paper is the possibility of a multiverse hypothesis which would be as probable as, or more probable than, theism. I shall understand a ‘universe’ in the rather loose way in which physicists use this term as (in effect) a system of physical objects (for example, stars and planets) spatially related to (that is, at some distance in some direction from) each other, but either a long way away (relative to their distance from other members of the same system) from any other such system, or not spatially related to any other such system. It seems to me fairly evident that any ordinary non-theistic one-universe hypothesis (for example, the hypothesis that the Big Bang was the explosion of a first physical substance without itself having a cause) which predicts the fine-tuning of the universe which theism also purports to explain will not be nearly as simple as theism; and so the disjunction of all such hypotheses will not be as simple as theism.9 A first physical substance, however small, would be an extended substance and so less simple than God; it needs to be governed by such general laws as the laws of Quantum Theory and of the four forces (or perhaps a unified ‘theory of everything’); and the constants of its laws and the variables of its initial conditions would need to be very fine-tuned to cause the evolution of intelligent beings and so very un-simple in comparison to theism. So, in Bayesian terms, on the assumption of only one universe, with h as theism, h∗ as the disjunction of all possible physical one-universe theories conjoined with the non-existence of God or gods, e1 as the fine-tuning of the universe so as to produce intelligent life and k as tautological background evidence, 9 Although a disjunction of hypotheses will always have smaller scope than any one of its disjuncts (and for that reason be more probably true), the single disjunct will always be simpler. We saw earlier that great simplicity (together with some satisfaction of the other criteria) suffices to make a hypothesis probably true.

baye s, god, and the multive r se 111 P(h|k) is so much greater than P(h∗ |k), that even though P(e1 |h&k) = 1/2 and even if P(e1 |h∗ &k) = 1, P(h|e1 &k) P(h∗ |e1 &k).

IV. Interpretations of ‘Multiverse’ Hence the suggestion of a universe-generating mechanism which produces many universes (a multiverse) each differing from the others in its laws and/or initial (or boundary) conditions, in such a way that it is probable that there will be at least one universe with the most general characteristics of our universe, including producing intelligent beings in which, inevitably, if we exist, we will find ourselves. To have reason to believe that there is such a multiverse, by my earlier argument we would need reason to postulate a substance, either our universe at some early stage or another universe or some other physical state such as a vacuum field, which would have caused the existence of our universe fine-tuned to produce intelligent life, and which would have caused or been caused by other universes. By my earlier argument that reason would have to consist in the fact that the postulated universe or other physical state satisfies criteria (1) or (3) better than does the one-universe hypothesis. And if we are to have reason to postulate it as the uncaused cause of the fine-tuning of the universe, it would have to satisfy these criteria better than does theism. In the light of these considerations let us assess the various multiverses on offer. Max Tegmark has distinguished four ‘levels’ of multiverse. A first-level multiverse is one consisting of many universes, all governed by the same laws as operate in our universe but with different initial conditions (see Tegmark 2007). There are (or could be) two types of physical theory which predict a level-1 multiverse. One is a theory according to which universes are generated by something else physical (for example, ‘the vacuum field’). The other type of theory is one according to which universes are generated by other universes. The obvious example of the first type is an inflation theory, according to which the fluctuating ‘vacuum field’ is continually expanding due to its internal energy; however inflation comes to an end in a particular region when fluctuations lead to a potential minimum, and that provides initial conditions for a universe. If the vacuum field is everlasting in time and infinite in space, ‘inflation . . . generates all possible initial conditions with non-zero probability’ (Tegmark 2007: 104). A second-level multiverse is a multiverse of universes which differ not merely in their initial conditions but in the constants of the fundamental laws, leading to different lower-level laws in different universes. The mechanism of production will lead to ‘breaking the underlying symmetries of particle physics’ which ‘will change the line-up of elementary particles and the effective equations that describe them’ (Tegmark 2007: 107). Many theories of the inflation type also have this feature. An example of a theory of the universes-generated-by-universes type which has this feature is Lee Smolin’s CNS (Cosmological Natural Selection) theory that new universes with different initial

112 probabi lity in the phi lo sophy of re lig ion conditions and lower-level laws are generated from black holes in an old universe (see Smolin 2007). Now both level-1 and level-2 multiverses seem to me to constitute serious physical theories which should each be assessed by the criteria described earlier. Two obvious questions arise: are they in any way simpler than the one-universe theory, and do they lead to predictions which differ from those of the one-universe theory? A level-1 inflation theory postulates a far larger initial substance (a vacuum field) than our universe at its beginning, and for that reason is less simple than the one-universe theory. The advocates of such a theory claim as its great merit that it does not require the kind of ‘fine-tuned’ initial conditions needed by a one-universe theory in order to bring about somewhere at some time a universe of our kind. But if we suppose that the vacuum field has been inflating for only a finite time, it would require special initial conditions to get the inflation started.10 Yet to suppose that it has been inflating forever and so has always been of infinite extension does seem to involve postulating a very unsimple first substance. These considerations put in doubt whether a level-1 inflation theory is any simpler than a one-universe Big Bang theory. Perhaps the simplest kind of level-2 inflation theory is one based on ‘a Grand Unified’ theory of physics which derives three of the four forces from one more general law and allows different sets of values for the fundamental constants in the different universes produced when the vacuum field reaches a potential minimum. In this respect its laws might well seem simpler than those of a one-universe theory, but it has been questioned whether it allows enough variations in the constants to make probable the occurrence of a universe with the laws of our universe. If it is to generate enough different systems of lower-level laws (to make it probable that our universe would be produced), it may need to be backed up by string theory which has a very large number of unstable states (see Collins 2009: 264); this instability ‘allows the Universe [i.e., multiverse] to sample all of a large part of the landscape’ (Susskind 2007: 262). String theory, in its now generalized form of ‘M-theory’, however, might seem very complex, and is certainly not an established part of physics.11 A Smolin-type level-2 multiverse needs

10 ‘The onset of inflation seems to require very special initial conditions’ (Stoeger 2007: 454). See Stoeger’s references to Penrose, Ellis and others. See also Smolin: ‘On several plausible hypotheses about the initial state, the conditions required for a region to begin inflating are improbable’ (Smolin 2007: 334). 11 In his recent ‘popular’ book Stephen Hawking claims that M-theory ‘is the only candidate for a complete theory of the universe. There is no other consistent model’ (Hawking and Mlodinow 2010: 181). Presumably what he means is that it is the simplest theory consistent with the data (where ‘consistent with the data’ is understood in terms of having the relation to the data set out in my first two criteria in Section II). But, given that ‘nobody has yet written down the equations that govern the full M-theory, let alone solved them’ (Davies 2007: 129), that seems an ill-justified claim. My response to the much-publicized claim of Hawking’s book that because the multiple universes ‘arise naturally from physical law’, ‘their creation does not require the intervention of some supernatural being or god’ (Hawking and Mlodinow 2010: 8–9) is that certainly it would not require the ‘intervention’ of such a being, but the all-important question remains whether the operation of the relevant physical laws themselves would be best explained by the agency of God. I claim that my arguments here, together with other arguments contained in Swinburne (2004), show that it would, and that these arguments make the existence of God significantly more probable than not.

baye s, god, and the multive r se 113 only a universe of our size to generate the multiverse, and that first-universe also does not need to have any very special features; but CNS postulates some ‘very speculative physics’ (Carr 2007: 84), additional to normal physics governing the formation of black holes and their ‘bouncing back’ into new universes. Any level-2 universe needs general laws with far fewer constants than ours; but it still needs the laws of Quantum Theory, the Pauli exclusion principle, forces both of attraction and repulsion, and the laws of Relativity Theory. Overall, while it is immensely difficult to assess the comparative simplicity of multiverse theories, I am going to assume (in order to give the multiverse objector as much as possible) that on the whole one or more multiverse theories are simpler than our one-universe theory, and that in consequence their disjunction is in this respect significantly more likely to be true than the disjunction of one-universe theories. What about predictions (in my wide sense)? Since other universes cannot be observed (being too far distant for any light from them to reach us), these must be of phenomena in our universe which are consequences of a particular multiverse theory, and which would otherwise be arbitrary features of initial conditions or other inexplicable features of our universe. The great merit of inflation theory was its ability to predict the ‘smoothness’ of the universe—that it has (to a high degree of approximation) the same density and rate of expansion in all regions—which is an arbitrary feature of the initial conditions in a one-universe theory. Inflation evens out the bumps in the vacuum field. A further merit of inflation theory was its ability to explain the approximations, in the form of the exact values of the tiny temperature fluctuations in the cosmic background radiation. There is much discussion about whether there are any other obvious known features of our universe predicted by a particular multiverse theory but otherwise inexplicable, and what other tests might reveal new such features; and what observations would count against any particular multiverse theory. One obvious relevant type of observation concerns whether the actual values of the human-life-producing features lie in the middle of the range predicted by a multiverse theory, or whether they are on the edge of those allowed by the theory—which would be improbable if the theory is true.12 It has however been argued that inflation theory (even in its level-1 variety) makes one prediction about our universe whose evident falsity makes the theory itself very improbable: Boltzmann argued that our entire universe was an immensely rare ‘fluctuation’ within an infinite and eternal time-symmetric domain . . . If Boltzmann were right, we would be in the smallest fluctuation compatible with our awareness—indeed, the overwhelmingly most likely configuration would be a universe containing nothing but a single brain with external sensations fed into it. (Rees 2007: 67)

12 See for example Aguirre (2007). A number of the articles in Carr (2007) discuss possible observations which might confirm or disconfirm various multiverse theories.

114 p robab i l ity i n th e ph i lo s ophy of re l i g i on These ‘external sensations’ would most likely give the brain a totally false picture of the universe in which it lived, and the brain would continue to exist only for a very short time. Modern inflation theory differs from Boltzmann’s theory mainly in holding that the vacuum field is continually inflating, but Collins has argued plausibly that inflationary cosmology gives rise to the same problem (Collins 2009: 265–71). The proportion of minimum potential states which give rise to Boltzmann-brain universes (as they are called) is far greater than those which give rise to a vast universe like ours containing a planet on which there are many intelligent beings who, we suppose, have a roughly correct picture of what that universe is like. (If we assume that the apparent universe is a delusion produced by our brains, we could have no reason to believe in other universes, or much else beside our own existence over a very short period of time.) So, if this argument is correct, the evidence that we do live in a vast universe with other intelligent beings on earth and have a roughly correct picture of the world counts massively against inflationary theory, which is generally recognized as by far the most probable kind of level-1 or level-2 theory. Tegmark sees the Many-Worlds-Interpretation (MWI) of Quantum Theory as constituting a separate level of multiverse: level-3. But it is really, as he acknowledges, a peculiar variant of either level-1 or level-2. The ‘peculiarity’ consists in the peculiar interpretation given to the ψ-wave function of Quantum Theory. This function is the basic underlying process (of the Universe, or parts of it); it develops in a deterministic way until an observation is made. An observation ‘collapses the wave packet’ by yielding a particular result (for example, the value of some variable such as the position of a particle within a narrow range). The ψ-function however only allows us to predict in advance different results of an observation with different degrees of probability. The various ‘interpretations’ of Quantum Theory try to explain what makes it the case that an observation yields one particular result rather than another—for example, on one interpretation the observation creates the result, whereas on a different interpretation the observation reveals the result which the ψ-function has already produced. Most of the interpretations have the problem that Quantum Theory cannot be complete because, whether the ‘collapse’ creates or reveals the result, it cannot explain why observation collapses the deterministic wave packet at all, so as to yield one result rather than another. MWI claims to solve this problem by claiming that observation produces all possible results. An observation with n possible outcomes divides the world into n branches (n universes), all of which are actual universes. There are very considerable problems in giving an intelligible and logically coherent account of MWI. The first major problem is: what happens to the observer? MWI cannot say that he will be found only in one branch, since then Quantum Theory would be incomplete in not being able to explain why he is found in this branch rather than that branch. The answer usually given is that the universe-division splits the observer (as well as the rest of the universe) into a number of successor-persons (each of whom is partly identical to the original observer), one in each branch. There are to my mind insuperable philosophical difficulties in supposing that persons, unlike inanimate things, can be split. If I were to be split into two persons and one of them

baye s, god, and the multive r se 115 loses $1m and the other gains $1m, then presumably I neither gain money nor lose money; but no subsequent person in either world neither gains nor loses. Yet even if sense could be given to the notion of a person being split, this account then seems to provide no answer to what it means to say that one result has a probability (or ‘weight’ as MWI theorists sometimes call it) (say, 2/3) greater than that of another result (say, 1/3), since both results inevitably occur. An alternative for MWI to the split-observer answer is to say that what we call ‘one observer’ before the observation really consists of a collection of innumerable persons (presumably an infinite number of such), each with a predetermined but unknown future history; and what the probability of an observation measures is the probability that a given already existing observer is one of the innumerable observers predestined to observe that result. But of course it seems immensely implausible to suppose that what we normally take to be one observer really consists of an infinite number of distinct persons. And even if all these philosophical difficulties could be overcome, and a logically coherent account of MWI be produced, there would still be a need to show why it should be believed in preference to an alternative interpretation of Quantum Theory.13 The reason that MWI alone allows Quantum Theory to be a complete deterministic theory of the universe doesn’t seem much of a reason unless it could be supplemented by MWI making predictions not made by other interpretations. Yet even if MWI were shown to be probably true, it would not raise any new problems for theism beyond those produced by a level-2 multiverse. For a level-3 multiverse would contain no worlds of a kind other than would occur in a level-2 multiverse; the most general laws of Quantum Theory and the four forces would be the same, and so the kinds of universe produced from any particular initial state (or over infinite time) would be the same. Finally there is the level-4 multiverse, which Tegmark states as the claim that ‘all mathematical structures exist’; every possible system of laws of nature and initial conditions is instantiated in one and only one universe, not because of any causal process which brings about these universes—but because that is just how things are (Tegmark 2007: 118). But why stop there? Why not suppose that every possible law-like and chaotic universe exists, as David Lewis (1986) has suggested? And finally, even beyond Lewis, why suppose that each universe described in full detail in terms of the pattern of substances and their properties (and relations) is unique; maybe there are innumerable qualitatively identical universes differing in the physical matter of which they are made? But, as seen above, we have grounds to postulate a universe only if doing so provides a probable explanation of phenomena, or is a consequence of such an explanation (that is, the postulated universe is postulated to be causally connected to our universe). We could have such grounds only for universes of level-1 or level-2 (and perhaps level-3) multiverses. Universes of level-4 multiverses, which do not belong to level-1, level-2, or level-3 multiverses, are not caused to exist by our universe, or by any universe which causes our universe, or by any universe which is caused by any such

13 For these problems with MWI, see the various papers in Saunders et al. (2010).

116 probabi lity in the phi lo sophy of re lig ion universe, and so on. The criterion of simplicity insists that we should not postulate many entities of a kind, when few entities of that kind will explain our evidence. We can have no good reason to believe in level-4 (or higher level) universes. So there could only be good reason (of a normal scientific kind) to believe in a level-1 or level-2 multiverse (or perhaps also level-3, though, as we have seen, that would make no difference to the kinds of universes produced).

V. Multiverse and Intelligent Life But what if there is a God? Does God who, as we have seen, has substantial reason to bring about humans and so one universe productive of humans, have reason for bringing about any more universes? However many civilizations of humans God makes (that is, intelligent beings similar to humans in their general characteristics, as described above), and so planets containing them, it would be better if he made more of them since they are a good thing. So, although God cannot make the best number of such planets, it is not very improbable that he would make quite a few. The initial conditions and laws of our universe may well be such as to lead to more than one planet containing a human civilization. Or maybe not, because we do not know how big that universe is and—even given the normally discussed initial conditions necessary for intelligent life—how rare are the initial conditions (among the conditions necessary for intelligent life) sufficient to produce human life. Most universes in a multiverse will not be conducive to the emergence of intelligent life, let alone human life. So does God have any reason for making universes not so conducive? I suggest that many universes are beautiful things in themselves, great works of art, even if bereft of any life. So certainly God has some reason to produce all those other universes. Does he have any reason to bring about another universe containing planets productive of intelligent life, if there are anyway other such planets in our universe? I suggest that he does, because variety is a good thing, but only so long as the other intelligent life is a good form of life, whether human or non-human. It follows that any multiverse God makes will have such laws and initial conditions as necessary to bring about only good intelligent life (human or of some other kind, for example, intelligent beings with no propensity to do evil); or laws and initial conditions such that this will be (physically) very probable and, if necessary, God will intervene to stop the improbable occurring, that is to stop the evolution of bad intelligent life. The argument from fine-tuning as analysed so far has ignored the fact that the universe is such that—either through an initial fine-tuning, or as a result of the indeterministic laws of Quantum Theory—not merely does it produce intelligent life, but that it produces human life.14 I gave reason to suppose that it was quite probable that God would bring about not just intelligent life, but human life. And in presenting

14 I discuss the good features of humans at greater length in Swinburne (2004: 219–35).

baye s, god, and the multive r se 117 the argument to God I made the assumption that there is a theodicy adequate to show that the amount of evil in the world does not cancel out the positive force of the argument. I pointed out that such a theodicy could explain only certain limited kinds of evil, for example, any humans suffering for no more than 80 or so years, which I assumed were the only kinds of evil on earth. But there could be universes which contained bad intelligent life. There could be intelligent beings who suffer much more intensely or for much longer than they do on earth (and not by their own choice). There could be a race of intelligent beings who (not as a result of their own choice or the influences of the choices of others) love to cause suffering to those of other races, having no natural compassion for them and no moral sense.15 There could be a race of intelligent beings (for example, conscious computers or Boltzmann brains) with hard exterior skeletons making them unable to do any harm or provide any help to each other or to themselves (or to be helped by other beings); and yet there could be just as much total suffering as there is in our world with none of it resulting from free choice or providing information for the sufferers or others which they could use to avoid future suffering. And so on. I see no reason to suppose that natural selection would soon weed out such beings. And anyway natural selection can only select from variants thrown up by genetic recombinations and mutations, and there could be universes in which genes for the above characteristics were linked to ones which gave their possessors selective advantages, or ones in which most combinations and most mutations still produced genes of the same evil kind, or universes in which there was such abundance of food and space that natural selection did not operate. But, as far as we know (and given my assumption about there being a theodicy adequate to explain the limited evil on earth), our universe produces only good intelligent life; and in particular it produces humans, the kind of good intelligent life which God is most likely to produce. I have suggested that there is a probability of 0 that God would create bad intelligent life, but a considerable probability that he would create humans. Just how probable it is that (if there is no God) the initial conditions and laws which would very probably produce intelligent life would produce humans and no bad intelligent life, it is not easy to say until physicists, biologists, etc., have worked out the consequences of different laws and initial conditions in immensely more detail than they have been able to do so far. But it does look not too improbable that circumstances fairly similar to those produced by the Big Bang which produced the human genotype would have produced one in which there was bad intelligent life. (Scientists have recently drawn attention to the risks involved in making contact with any alien beings discovered by SETI.) And plausibly there could be universes of some 15 While some natural altruism for those of one’s own group may well give a selective advantage, altruism

which extends to care for the old and sick and even to competing groups is surely disadvantageous. And so too is a ‘moral sense’, leading one to believe that one ought to help the old and sick, and so on, even where one does not have any natural altruism. As T. H. Huxley put it, ‘the practice of that which is ethically best. . . involves a course of conduct which in all respects, is opposed to that which leads to success in the cosmic struggle for existence’ (Huxley 1894: 81–2).

118 probabi lity in the phi lo sophy of re lig ion level-2 multiverse which produced bad intelligent beings by a mechanism other than the genetic mechanism which operates on earth. The fact that, as far as we know, our universe contains no bad intelligent life (and in particular that we are not bad beings of this kind) but that it contains humans, is evidence that if there is a multiverse, it is a God-produced multiverse. And of course it is also evidence that if there is only one universe, it is God-produced. Even those who doubt my theodicy (that is, doubt whether all suffering and evil-doing on earth is such that there is a point in it being allowed to occur) must admit that there could be universes a great deal worse than ours, universes which much more obviously contained bad intelligent life, or ones which contained less good than ours (for example, ones in which the good intelligent beings were not humans). The fact that the situation is not like this, but could well be if there were a Godless multiverse, increases the probability of the existence of God.

VI. A Bayesian Context Now let’s put all this in Bayesian terms. As before, let h be theism. In order to simplify the formalization I omit k and read expressions such as P(h) as the probability of h on tautological background evidence (that is, the intrinsic probability of h). Let e1 be the currently available evidence of the fine-tuning of a universe (ours), in the kind of respects to which physics has drawn our attention, to produce intelligent life, previously denoted by e. Assume that we have no further evidence relevant to the existence or non-existence of a multiverse; but suspect that our assessment of the probability of h on e is mistaken through not taking into account this possibility. So we may rephrase Bayes’ theorem so as to take this into account. Let m be the hypothesis of the existence of a multiverse of some kind (that is, the disjunction of possible multiverses) which will produce at some time a fine-tuned universe; and let u be the hypothesis that there is only one universe and it is fine-tuned. All actual one-universe or multiverse theories merely give a high value less than 1 to P(e1 |u) and P(e1 |m); but to make the argument less cumbersome without making any crucial difference to it, I shall assume P(e1 |u) = P(e1| m) = 1; and then of course e1 = (u v m). Then: P(h|e1 ) =

P(h&m) + P(h&u) P(h&m) + P(h&u) + P(∼h&m) + P(∼h&u).

Humans are such good things that I have attributed the value of 1/2 to P(e1 |h). So P(h&m) + P(h&u) = 1/2P(h). I have suggested that God has some reason to produce a multiverse rather than just one universe, because of the goodness of the variety of different kinds of universes which would result—although not of course a multiverse which would bring about the existence of bad intelligent life. So I suggest that P(h&m) ≥ P(h&u). I have already suggested that P(∼h&u) is very small. So all turns on whether P(∼h&m) is much bigger than P(∼h&u); and that turns on whether the initial conditions and laws of possible multiverses are much simpler and so a priori more probable than those of a single universe. I said earlier that (in order to give the multiverse objector as much as possible) I was going to assume that P(∼h&m) is quite

baye s, god, and the multive r se 119 a bit greater than P(∼h&u). However P(∼h&m) is still going to be vastly smaller than P(h&m) = P(m|h)P(h) > 1/4P(h). This is because only a multiverse (of level-1 or level-2; or, if it is coherent, level-3) with a moderately small range of initial conditions (an extended state of matter-energy such as a vacuum field of a certain size with a certain range of fluctuations) and some complicated laws (for example, laws of Quantum Theory) can lead to a universe having e1 . This is a vastly complicated hypothesis in comparison with the hypothesis of one unextended essentially omnipotent, omniscient and perfectly free substance. So, given no new evidence, my conclusion is P(h|e1 ) is smaller than it would be if we ignored the possibility of a multiverse but still much greater than P(∼h|e1 ). Now suppose that there is evidence e2 of the kind considered by physicists confirming or disconfirming m. This may be evidence already known (for example, the smoothness of the universe, or the fact that we are not Boltzmann brains) or something newly discovered. I see every reason to suppose that a perfectly good God will not give us deliberately misleading evidence on this matter. So P(e2 |m&h) = P(e2 |m) = P(e2 |m&∼h); P(e2 |u&h) = P(e2 |u) = P(e2 |u&∼h). Then P(h|e1 &e2 ) =

P(e1 &e2 |h)P(h) P(e1 &e2 |h)P(h) + P(e1 &e2 |∼h)P(∼h)

=

P(e1 &e2 &h) P(e1 &e2 &h) + P(e1 &e2 &∼h)

=

P((m v u)&e2 &h) P((m v u)&e2 &h) + P((m v u)&e2 &∼h)

=

P(m&e2 &h) + P(u&e2 &h) P(m&e2 &h) + P(u&e2 &h) + P(m&e2 &∼h) + P(u&e2 &∼h)

=

P(e2 |h&m)P(h&m) + P(e2 |u&h)P(u&h) P(e2 |h&m)P(h&m) + P(e2 |u&h)P(u&h) + P(e2 |m&∼h)P(m&∼h) + P(e2 |u&∼h)P(u&∼h)

=

P(e2 |m)P(h&m) + P(e2 |u)P(h&u) P(e2 |m)P(h&m) + P(e2 |u)P(h&u) + P(e2 |m)P(m&∼h) + P(e2 |u)P(u&∼h)

The effect of e2 has been that extra terms P(e2 |m) and P(e2 |u) have been inserted so as to increase or decrease the relative value of conjunctions involving m as against those involving u. Consider the two extreme cases. The first is that e2 is incompatible with u. Then the equation reduces to: P(h|e1 &e2 ) =

P(h&m) P(h&m) + P(∼h&m)

Given that P(h&m) is not very different from P(h&u), the effect of e2 on h will depend once again on whether m is much simpler than u. On the assumption that it is quite a bit simpler but not too much simpler, e2 will diminish the probability of h but not by very much: P(h|e1 &e2 ) < P(h|e1 ), but still P(h|e1 &e2 ) P(∼h|e1 & e2 ). The other extreme case is that e2 is incompatible with m. Then the equation reduces to:

120 probabi lity in the phi lo sophy of re lig ion P(h|e1 &e2 ) =

P(h&u) P(h&u) + P(∼h&u)

Given that m is simpler than u, and that P(h&u) is not very different from P(h&m), this has the consequence that e2 increases the probability of h. If e2 only raises or lowers the probability of m or u as the case may be without being incompatible with one of them, then the resulting value of P(h|e1 & e2 ) will lie between the two extreme values. (There is a near-extreme case if m makes it very probable that humans would be Boltzmann brains, and e2 includes the evidence that we are not Boltzmann brains.) However, we must now bring in e3 : that the universe is fine-tuned in the further respect that it produces not merely any intelligent life, but humans, and no bad intelligent life on earth or on any other planet which we have been able to study. To repeat the earlier formula: P(h|e1 &e2 ) = Now

P(e1 &e2 |h)P(h) P(e1 &e2 |h)P(h) + P(e1 &e2 |∼h)P(∼ h)

P(e1 &e2 &e3 |h) P(e1 &e2 &e3 |∼h) > P(e1 &e2 |h) P(e2 &e2 |∼h)

since e3 is more to be expected (given e1 & e2 ) if h then if ∼h. The multiverse hypothesis makes no contribution towards explaining why intelligent life (e1 ) takes a human form (e3 ). So the effect of adding e3 to the evidence will be to raise the probability of h significantly (whether combined with m or u), even if e2 lowered it significantly by constituting conclusive evidence for a multiverse hypothesis. All told, the result of this complicated argument is that recognizing the possible existence of a multiverse does not make a great difference to the strength of probabilistic arguments from the fine-tuning of the universe to the existence of God. Of course, this result could be overturned if someone produces an argument from physics to show that some of the values which I have allocated to probabilities on the strength of my limited physical intuition are badly mistaken; but in the absence of that I conclude that the possible existence of a multiverse does not greatly diminish the powerful force of the argument from fine-tuning to the existence of God.16

Appendix A Inductive probability is a measure of the probability of a hypothesis on some body of evidence; whereas physical probability is a measure of the degree to which an event is predetermined by its causes. I assume that (very roughly) there is a correct measure of inductive probability, 16 I am most grateful to Robin Collins who commented on an earlier version of this paper, and provided me with much help in understanding the consequences of an inflationary multiverse for the prevalence of Boltzmann brains. Many thanks also to two referees who provided very helpful detailed comments on the penultimate version of the paper.

baye s, god, and the multive r se 121 which I call ‘logical probability’ (see Appendix B). The argument is normally presented merely as an argument from the existence of ‘fine-tuned’ necessary conditions for the evolution of intelligent life. But the argument would not have so much force if despite the initial conditions (at the time of the Big Bang) and laws providing such necessary conditions, it was still physically very improbable (because of the indeterministic character of the laws) that intelligent life would evolve. For the argument depends on the supposition that God would seek to bring about intelligent life, and would have some reason to do so by an evolutionary process. In that case he would have to have made the initial conditions such that it was (physically) very probable that intelligent life would evolve. Otherwise it is (logically) very probable that he would have needed to intervene at a later stage in the natural order to produce the desired effect; and the argument from fine-tuning would need to be backed up by a further argument to show that it was very probable that he had done this, if it was to be as strong as an argument from the initial conditions being such that intelligent life would very probably evolve. Some biologists do of course offer arguments to show divine intervention at a later stage. (See the discussion of such arguments in Swinburne 2004: 346–9.) Such arguments would need to suggest a reason why God would have made a universe which needed such intervention at a later stage instead of being wound up at the beginning so as very probably to produce intelligent life. But I read the evidence of fine-tuning as showing that the initial conditions of the universe at the time of the Big Bang were such that, given the vast size of the universe and despite the indeterministic character of the laws of Quantum Theory, it is physically very probable that intelligent life would evolve at some time somewhere. So I am going to understand the argument in this stronger sense, even if many of its exponents do not so read it. (For a discussion of the relation to each other of the two different kinds of argument, see Dougherty and Poston (2008).) Why would an argument for God having created the universe to be such as to produce intelligent life not need to show that at its beginning necessarily (not merely ‘very probably’) intelligent life would evolve? A major reason why God would create intelligent beings would be for them to exercise free will in a sense which involves their actions not being predetermined by physical causes (see the discussion in the body of this paper), and for this reason a small degree of indeterminism in the physical laws is necessary. Unless God was to intervene to change the laws of nature, that indeterminism would need to be there from the beginning. Only if (very improbably) the universe began to evolve in such a way that it would not produce human life anywhere, would God need to intervene to redirect its development.

Appendix B As I am using Bayes’ theorem, it concerns ‘logical probability’ in the sense of the objectively correct probability on evidence determinable a priori. Since, however, intrinsic probability is crucial for determining any prior probability, the theorem can only be applied to determine the logical probability of some explanatory theory h on evidence e(P(h|e&k), if we know the intrinsic probability of h (in order to calculate P(h|k)) and the intrinsic probabilities of all possible rival theories and can calculate all their consequences (in order to calculate P(e|k)). So if we have no idea of the range of possible theories in some field and so of the range of their intrinsic probabilities, we cannot have any idea of the logical probability of any particular explanatory theory on any evidence, and so we will have to make do with a kind of inductive probability which I call ‘epistemic probability’. This is probability relative to our knowledge of possibilities

122 probabi lity in the phi lo sophy of re lig ion and our abilities to calculate consequences. For this distinction between ‘logical’ and ‘epistemic’ probability see Swinburne (2001: 56–73). If, still more sceptically, we do not think that there are any intrinsic probabilities, we cannot apply Bayes’ theorem to determine P(h|e&k). For this reason many theorists try to measure the probability of a theory only by its relative ‘likelihood’, that is the extent to which the observed evidence is (logically) more probable given that theory than given other theories. Collins has a concept which he calls ‘epistemic probability’, which depends on relative likelihood, which leads him to hold that we can only reach judgements of the probability of some theory within a certain comparison range, which he calls the ‘epistemically illuminated range’, that is within the range of theories which are taken seriously by scientists and the consequences of which we can calculate (Collins 2009: 244). Those theories which yield the most probable conclusions are the epistemically most probable ones. Hence, according to Collins, we can only argue from fine-tuning to God on the (provisional) assumption that the only relevant scientific theories are the ones currently discussed whose consequences we can calculate and judge their epistemic probability on the basis of their relative likelihood. By contrast I hold that their simplicity and scope are crucial criteria for assessing the probability of scientific theories, and they enable us to judge the relative intrinsic probabilities of theories; and so that Collins’ attempt to assess the force of the argument from fine-tuning without taking account of intrinsic probabilities must fail. I also argue (in Swinburne 2004) that we can make a very imprecise judgement of the range of intrinsic probabilities which would be possessed by all possible scientific theories of universes (not yet articulated) and that we can compare the total intrinsic probability of theories of that range with the intrinsic probability of the existence of God. This is because the postulation of the existence of God is a very simple postulation, far simpler than any scientific theory could be. Hence we can make judgements of the logical probability of the existence of God on the evidence of fine-tuning.

References Aguirre, A. (2007) ‘Making Predictions in a Multiverse’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 367–87. Carr, B. (ed.) (2007a) Universe or Multiverse? Cambridge: Cambridge University Press. (2007b) ‘The Anthropic Principle Revisited’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 77–91. Collins, R. (2009) ‘The Teleological Argument: An Exploration of the Fine-tuning of the Universe’. In W. L Craig and J. P. Moreland (eds), The Blackwell Companion to Natural Theology. Chichester: Wiley-Blackwell, 202–82. Davies, P. (2007) The Goldilocks Enigma. London: Penguin Books. Dougherty, T. and T. Poston (2008) ‘A User’s Guide to Design Arguments’, Religious Studies, 44: 99–110. Hawking, S. and L. Mlodinow (2010) The Grand Design. London: Bantam Books. Huxley, T. H. (1894) Evolution and Other Essays. London: Macmillan & Co. Leslie, J. (1979) Value and Existence. Oxford: Blackwell. Lewis, D. (1986) On The Plurality of Worlds. Oxford: Blackwell. Parfit, D. (1992) ‘The Puzzle of Reality: Why does the Universe Exist?’, The Times Literary Supplement, 3 July: 3–5. Reprinted in P. van Inwagen and D. W. Zimmerman (eds), Metaphysics: The Big Questions. Oxford: Blackwell, 1998, 418–27.

baye s, god, and the multive r se 123 Rees, M. (2007) ‘Cosmology and the Multiverse’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 57–77. Rice, H. (2000) God and Goodness. Oxford: Oxford University Press. Saunders, S., J. Barrett, A. Kent, and D. Wallace (eds.) (2010) Many Worlds? Everett, Quantum Theory, and Reality Oxford: Oxford University Press. Smolin, L. (2007) ‘Scientific Alternatives to the Anthropic Principle’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 323–67. Stoeger, W. R. (2007) ‘Are Anthropic Arguments, Involving Multiverses and Beyond, Legitimate?’ In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 445–59. Susskind, L. (2007) ‘The Anthropic Landscape of String Theory’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press: 247–67. Swinburne, R. (1998) Providence and the Problem of Evil. Oxford: Oxford University Press. (2001) Epistemic Justification. Oxford: Oxford University Press. (2004) The Existence of God. 2nd edition. Oxford: Oxford University Press. (2010) ‘God as the Simplest Explanation of the Universe’, European Journal for Philosophy of Religion, 2: 1–24. Also published in A. O’Hear (ed.), Philosophy and Religion, Royal Institute of Philosophy Supplement 68. Cambridge: Cambridge University Press, 2011. Tegmark, M. (2007) ‘The Multiverse Hierarchy’. In B. Carr (ed.), Universe or Multiverse? Cambridge: Cambridge University Press, 99–127.

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PART III

Evil

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7 Comparative Confirmation and the Problem of Evil Richard Otte

I. Introduction One of the most enduring problems for religious belief has been the problem of evil. The problem of evil can take many forms; it can be a spiritual or psychological problem, and has often been construed as a philosophical problem that calls the rationality of religious belief into question. Philosophical problems of evil usually deal with whether the existence of evil in the world makes it irrational to believe in an omniscient, omnipotent, and benevolent God. These philosophical arguments can generally be classified as either deductive arguments from evil or as inductive arguments from evil. The deductive argument purported to show that it was irrational to believe in God because evil was logically inconsistent with the existence of God. More recent discussion of the philosophical problem of evil has moved away from the deductive argument from evil and has focused on whether evil is evidence against the existence of God. Much of this discussion centres around whether our ignorance of a good reason for God to permit evil is evidence that there is no such reason, and thus is evidence against the existence of God. In the following I will look at whether our ignorance of a good reason for God to permit evil provides a problem for the theist in virtue of being evidence against the religious beliefs of a typical theist. Many theists have admitted that evil is evidence against the existence of God, but respond that belief in God is not irrational because there is enough other evidence that supports the existence of God and religious belief. I will take a different approach and will instead argue that theists should not believe evil, or our ignorance of a good reason for God to permit evil, is evidence against religious belief or the existence of God, at all. Philosophers do not agree about the nature of evidence, but I will look at two leading views of evidence and argue that on these views, theists should hold either that our ignorance of a good reason for God to permit evil is irrelevant to typical religious belief or that it confirms these religious beliefs. This surprising result follows from our most popular views about using evidence to compare theories.

128 probabi lity in the phi lo sophy of re lig ion It will be instructive to begin with a quick look at the deductive argument from evil. The main point of an argument from evil is to demonstrate that typical Christians, Muslims, and Jews should consider evil to provide a serious problem for the rationality of their beliefs. According to the deductive argument from evil, the existence of evil is logically inconsistent with the existence of God, who is an omnipotent, omniscient, and benevolent being. Such a being would know about the evil in the world, would want to eliminate it, and would be able to do what he wants. It is now generally recognized that the deductive argument from evil failed because the existence of evil is not logically inconsistent with the existence of God.1 The deductive argument from evil assumes that God would want to eliminate all evil in the world, but theists respond that some evil in the world may bring about a greater good; thus a totally good being may have a good reason for permitting certain evils. Given this, the most that can be concluded from the deductive argument from evil is that God would eliminate all evil that he could eliminate, without eliminating a greater good (Plantinga 1977). More generally, a totally good God could permit any evil which he has a good reason for permitting. Thus the conclusion of the deductive argument from evil must be modified to claim that the existence of God is inconsistent with evils that God does not have a good reason to permit. From this it is clear that evil which God has a good reason to permit does not provide a problem for religious belief; only evil that God has no good reason to permit is relevant to the deductive problem of evil. So we can see that the deductive argument from evil revolves around the issue of whether God could have a good reason to permit the evil in the world. We can view the inductive or evidential argument from evil as taking up the argument from evil at this point. A widely discussed version of the evidential argument from evil, due to William Rowe, attempts to show that there are some evils that God does not have a good reason to permit. Rowe argued that we do not know of any good reason for God to permit certain evils, and this is reason to think that God has no reason to permit those evils. Rowe’s argument begins with the premise: No good state of affairs that we know of justifies an omnipotent, omniscient being in permitting certain specific horrible evils.

From this he concludes No good at all justifies an omnipotent, omniscient, perfectly good being in permitting these specific evils. (Rowe 1986: 263)

1 This assessment is primarily due to Alvin Plantinga’s (1974) free will defense. William Alston writes: ‘Plantinga . . . has established the possibility that God could not actualize a world containing free creatures that always do the right thing’ (1991: 49); Robert Adams says ‘it is fair to say that Plantinga has solved this problem. That is, he has argued convincingly for the consistency of [God and evil]’ (1985: 226). See also William Rowe (1986). For a recent discussion, see Otte (2009).

com parative confirmation and the p roble m of evi l 129 This inductive inference is at the core of Rowe’s evidential argument from evil. What is important is whether our evidence supports there being no good reason for God to permit evil. Much of the work on the nature of evidence involves the use of probability, and not surprisingly, the ‘evidential argument from evil’ and the ‘probabilistic argument from evil’ are often used interchangeably. The basic idea is that if evil is evidence against religious beliefs, there will be some probabilistic relation between evil and those religious beliefs. Although most philosophers writing on the evidential argument from evil assume some sort of logical or epistemic interpretation of probability, in this chapter I need not commit to any specific interpretation. The arguments that follow are compatible with viewing probability as an objective logical relation, but they are also compatible with viewing it as the logic of rational belief. We make many probabilistic judgements, and for the purposes of this chapter it will not matter whether we take these to be subjective probabilities or to be estimates of objective probabilities. Most discussions of the problem of evil look at the relation between evil and some designated core or essential part of theistic belief, such as that there exists an omnipotent, omniscient, and benevolent being. The basic idea is that any problem that evil raises for this core is supposed to transfer to a typical theist’s religious beliefs. With regard to the problem of evil, the idea is that if evil is a problem for these core beliefs, then evil is also a problem for traditional religious beliefs. For example, if the deductive argument from evil were correct, the existence of an omniscient, omnipotent, benevolent being would be logically inconsistent with the existence of evil. This logical inconsistency would transfer to any set of religious beliefs that include evil and an omnipotent, omniscient, and benevolent being. However this methodology will not work when we leave the deductive argument from evil and focus on the evidential argument from evil. Evidence against an omnipotent, omniscient, and benevolent being need not be evidence against typical religious beliefs, even though those religious beliefs include belief in an omnipotent, omniscient, and benevolent being. As Rudolf Carnap and others have noted, some evidence might lower the probability of H1 and lower the probability of H2 , but raise the probability of H1 & H2 .2 Evidence that disconfirms the core of a set of beliefs may not disconfirm the larger set of beliefs. Because of this, in discussing the evidential argument from evil we cannot abstract a core from typical religious beliefs and look at the whether evil is evidence against that core. We need to look at whether evil is evidence against the religious beliefs as a whole, and whether evil is evidence against the existence of an omniscient, omnipotent, and benevolent being is irrelevant. Christianity, Islam, and Judaism say more that is relevant to evil than that an omnipotent, omniscient, and benevolent being exists. We need to include these other beliefs when investigating whether evil and related facts are evidence against those religions.

2 See R. Carnap (1950); W. C. Salmon (1975); and R. Otte (2000).

130 probabi lity in the phi lo sophy of re lig ion One doctrine that has a long history in theistic religions is that we should not expect to know or understand God’s reasons for acting, including his reason to permit evil. Various explanations are given for this epistemic distance between God and humans, but the basic idea is that God’s knowledge and understanding is far beyond ours and we should not expect to understand his thoughts. This is illustrated in the story of Job, in which the problem of evil plays the central role. After several chapters of discussion of possible explanations for what befell Job, we get to God’s response in Chapter 38. At this point we might expect God to give Job some sort of theodicy or explanation of the evils he suffered, but we find no such thing. Instead, we get a lengthy statement about how God’s ways are beyond our ways, and how his knowledge far surpasses ours. The idea that we are unable to understand God’s reasons is widespread, and this scepticism about our ability to understand God’s reasons is also at the heart of what has recently become known as ‘sceptical theism’. According to sceptical theism we should not infer there is no good reason for God to permit evil from our ignorance of any such reason.3 In this chapter I will focus on the views of a typical theist who believes we are unable to understand God’s reason for permitting evil. Although some theists have presented theodicies that purport to explain God’s reasons for permitting evil, the argument of this chapter is not directed towards them; instead I will focus on theists who do not think any known theodicy is adequate. I will call these theists ‘sceptical theists’ and will use ‘UNKNOWABLE GOOD’ to abbreviate their views: UNKNOWABLE GOOD: Evil exists, and God has a good reason to permit it that we are not capable of understanding.

Certainly there is more relevant to theists’ beliefs than what is contained in UNKNOWABLE GOOD, and UNKNOWABLE GOOD indicates a bare minimum of what the theistic views I am discussing hold that is relevant to our argument. In this paper I will focus on what Peter van Inwagen (2006) calls the global argument from evil, and thus I formulated UNKNOWABLE GOOD to not imply any particular evils. My argument is also applicable to the local argument from evil, and we could easily have construed UNKNOWABLE GOOD to contain a description of some evils that exist or a description of types of evils that exist. In managing our beliefs we need to look not only at the evidence and hypothesis we are interested in, but also at the competing hypotheses. Elliot Sober says: When scientists wish to assess the credentials of an explanatory hypothesis, a fundamental question will be: What are the alternative hypotheses that compete with the one in which you are really interested? This is the idea that theory testing is a contrastive activity. To test a theory T is to test it against at least one competing theory T . (Sober 1994: 120)

3 There are numerous articles on sceptical theism, but see S. J. Wykstra (1984), M. Bergmann (2001), and W. Rowe (2001).

com parative confirmation and the p roble m of evi l 131 So in order to know whether our ignorance of a good reason for God to permit evil is evidence for God having no good reason to permit evil, we must look at the alternative hypotheses or possibilities. For our purposes we can take the major competitor to UNKNOWABLE GOOD to be NO GOOD: NO GOOD: Evil exists and there is no good reason for God to permit it (and thus God does not exist).

Another competitor to UNKNOWABLE GOOD is KNOWABLE GOOD, which is held by those who accept some theodicy: KNOWABLE GOOD: Evil exists, and God has a good reason to permit it that we are able to understand.

These three positions are pairwise inconsistent; any two of them are inconsistent with each other. In the following I will make use of contemporary discussions of evidence and confirmation to discuss whether our ignorance of a good reason for God to permit evil favours one of the above hypotheses over the others. I am interested in whether our ignorance of a good reason should give a typical theist who holds UNKNOWABLE GOOD a reason to think that theism is disconfirmed compared to NO GOOD. In other words, I am interested in whether our evidence supports or favours NO GOOD or UNKNOWABLE GOOD. I will argue that a typical sceptical theist should not regard ignorance of a good reason for God to permit evil to be evidence that God does not have a good reason to permit evil. On the contrary, a theist should hold that our ignorance is either irrelevant to there being a good reason, or is evidence for God having a good reason to permit evil. A related argument would show that the evil in the world is not evidence that God does not exist; at best, the theist should refrain from judgement about whether evil is evidence against the existence of God. Our ignorance of a good reason for God to permit evil does not generate a philosophical problem of evil for typical Christians, Muslims, and Jews. The philosophical literature on probability and evidence contains two main views about using probability to compare the effect of evidence on competing hypotheses: Likelihoodism and Bayesianism. I will begin my discussion by looking at this issue from the perspective of Likelihoodism, and I will then discuss whether some Bayesian accounts of confirmation give the same results. We will see that neither supports the view that our ignorance of a good reason for God to permit evil is evidence that there is no good reason. As a result, neither Likelihoodism nor Bayesian accounts of evidence provide the basis for a probabilistic argument from evil.

II. Likelihoodism Likelihoodism is a theory of comparative or contrastive confirmation; the notion that confirmation involves comparing hypotheses is built into Likelihoodism. A central

132 probabi lity in the phi lo sophy of re lig ion idea of Likelihoodism is that in using evidence to compare theories we look at the likelihoods, the probability of the evidence on the hypotheses. An advantage of Likelihoodism is that prior probabilities play no role in comparing evidential strength; this makes Likelihoodism especially useful in discussions in which there is widespread disagreement about prior probabilities, such as in philosophy of religion. Likelihoodists generally accept what has come to be known as the ‘Law of Likelihood’: L: Evidence E favours hypothesis H1 over H2 iff P(E/H1 ) > P(E/H2 )4 Although some Likelihoodists endorse a quantitative measure of evidence favouring one hypothesis over another, in this paper we will work only with the qualitative concept. An immediate corollary of the Law of Likelihood is the following: C: If P(E/H1 ) = P(E/H2 ), then E does not favour H1 over H2 . According to Likelihoodism, if we have two hypotheses and some evidence, the hypothesis upon which the evidence is most likely is the one that is favoured by the evidence, and if the evidence is equally likely on both hypotheses, the evidence does not support one hypothesis over the other. The hypotheses we are comparing are UNKNOWABLE GOOD and NO GOOD. Since we are focusing on a version of the global argument from evil, for our evidence, consider: NO KNOWN: Evil exists and we do not know of a good reason for God to permit it.

We are interested in how likely NO KNOWN is on these hypotheses. Even though NO GOOD and UNKNOWABLE GOOD are competing hypotheses, both imply that we do not know of a good reason for God to permit evil. NO GOOD implies NO KNOWN because we cannot know of a good reason if there is none. UNKNOWABLE GOOD implies NO KNOWN because we do not know a good reason for God to permit evil if we are not capable of understanding it. From this it follows that P(NO KNOWN/NO GOOD) and P(NO KNOWN/UNKNOWABLE GOOD) are both equal to 1. Given this, corollary C implies that NO KNOWN does not support NO GOOD over UNKNOWABLE GOOD. In the introduction to this chapter I pointed out that the problem of evil comes down to whether or not God has a good reason to permit evil. We see that on the Likelihoodist conception of evidential support, our not knowing of a good reason for God to permit evil does not support God not having a good reason to permit evil over typical theism; we need to look elsewhere for evidence against the existence of God. However NO KNOWN is evidence against KNOWABLE GOOD as compared to NO GOOD and UNKNOWABLE GOOD. P(NO KNOWN/KNOWABLE GOOD) is certainly less than 1, and thus P(NO KNOWN/KNOWABLE GOOD) < P(NO KNOWN/UNKNOWABLE GOOD) and P(NO KNOWN/KNOWABLE 4 See B. Fitelson (2007) ; R. Royall (1997); E. Sober (1994); and I. Hacking (1976).

com parative confirmation and the p roble m of evi l 133 GOOD) < P(NO KNOWN/NO GOOD). From this it follows from Likelihoodism that NO KNOWN supports UNKNOWABLE GOOD and NO GOOD over KNOWABLE GOOD. Although NO KNOWN is irrelevant to the choice between UNKNOWABLE GOOD and NO GOOD, it is not irrelevant when considering KNOWABLE GOOD. And if we let GOOD be the disjunction of UNKNOWABLE GOOD and KNOWABLE GOOD, it also follows that NO KNOWN favours NO GOOD over GOOD. Many attempts at formulating a global problem of evil differ from the above and take our evidence to be only that evil exists, or perhaps rely on a description of some particularly bad evils. However one might argue that if our evidence were simply the existence of evil instead of NO KNOWN, then according to Likelihoodism this evidence would support traditional theistic religions over naturalism. The major theistic religions, Christianity, Islam, and Judaism, all imply that evil exists; certainly these religions would be false if there were no evil, because they make historical claims about specific evils. It is difficult, perhaps impossible, to define naturalism, but it is plausible to think naturalism does not imply that evil exists. If so, the existence of evil is more likely given theistic religions than given naturalism, and by Likelihoodism evil confirms theism over naturalism. It is important to look at all of our relevant evidence, which goes beyond the existence of evil, and to look at actual competing beliefs instead of core ideas implied by them. If we were focusing on the local problem of evil and had taken our evidence to be the specific evils we know of instead of the existence of evil, then neither UNKNOWABLE GOOD nor NO GOOD would imply our evidence. However, it is difficult to see why this evidence should be more likely on naturalism than on one of the theistic religions. The theistic religions all have accounts of evil in the world, and there is no reason to think that any particular evil or the distribution of evils in the world is less likely on the theistic religions than on naturalism. Although much more could be said about the local problem of evil, in this paper I will not focus on how likely specific evils are given UNKNOWABLE GOOD and NO GOOD. Instead I focus on the global problem of evil and will look at our ignorance of a good reason for God to permit evil. At this point one might object that our evidence is not that we do not know of a good reason for God to permit evil, but rather that we are unaware of a good reason. This can be expressed by: UNAWARE: Evil exists and we are not aware of a good reason for God to permit it.

With UNAWARE we are saying that we do not believe we know of a good reason for God to permit evil, or we do not think we see a good reason for God to permit evil; this is consistent with us believing we are ignorant of any good reason God may have to permit evil. UNAWARE is different than NO KNOWN, because UNAWARE says we do not accept any theodicy as being adequate, but NO KNOWN does not imply this. UNAWARE is stronger than NO KNOWN; although UNAWARE implies NO KNOWN, NO KNOWN does not imply UNAWARE. It could be the case that we

134 p robab i l ity i n th e ph i lo s ophy of re l i g i on think we know of a good reason even though we do not know a good reason; if so, NO KNOWN would be true but UNAWARE would be false. For example, one might accept a theodicy as adequate, but be mistaken in that the theodicy is not successful. Our above argument relied on the fact that NO GOOD and UNKNOWABLE GOOD both implied NO KNOWN, but neither NO GOOD nor UNKNOWABLE GOOD imply UNAWARE; it is possible that we could believe we are aware of a good reason for God to permit evil, even if NO GOOD or UNKNOWABLE GOOD were true. Thus we cannot reason as we did above if our evidence is UNAWARE instead of NO KNOWN; whether UNAWARE is evidence for NO GOOD over UNKNOWABLE GOOD depends on the values of P(UNAWARE/NO GOOD) and P(UNAWARE/UNKNOWABLE GOOD). These probabilities are not equal to 1, because we need to assign some positive probability to (NO GOOD and not-UNAWARE) as well as (UNKNOWABLE GOOD and not-UNAWARE). Even if we think we know of a good reason for God to permit evil, it is possible that we could be mistaken about that reason because either there is no good reason or there is a good reason that we cannot comprehend. Unfortunately, even though these probabilities are crucial to the problem of evil there is almost no discussion of them in the literature.5 According to Likelihoodism, the relevance of these probabilities to the evidential argument from evil is clear. If we think P(UNAWARE/NO GOOD) is equal to P(UNAWARE/UNKNOWABLE GOOD), then the corollary to the Law of Likelihood tells us that UNAWARE does not favour one over the other. If we think P(UNAWARE/NO GOOD) is greater than P(UNAWARE/UNKNOWABLE GOOD), then the likelihood principle L tells us that our evidence UNAWARE supports NO GOOD over UNKNOWABLE GOOD. And if we think P(UNAWARE/ UNKNOWABLE GOOD) is greater than P(UNAWARE/NO GOOD), then the likelihood principle L tells us that UNAWARE supports UNKNOWABLE GOOD over NO GOOD. From this it is clear that a defender of the evidential argument from evil is committed to showing P(UNAWARE/NO GOOD) > P(UNAWARE/UNKNOWABLE GOOD), because a theist who holds this will be committed to evil being evidence against God existing. However, I am not aware of any plausible argument for this claim in the literature. On the contrary it appears rational for a theist to hold that P(UNAWARE/UNKNOWABLE GOOD) is greater than P(UNAWARE/NO GOOD), that the probabilities are equal, or even to withhold judgement on which probability is greater. Sceptical theists who do not believe they have a theodicy and accept UNKNOWABLE GOOD, generally believe the reason they are unaware of an acceptable theodicy is because their minds are working well and they would be in error if they were to accept a theodicy when there was no acceptable one that we can understand. In other

5 But see R. Otte (2002) and W. Rowe (1998: 550) for a discussion of some related probabilities.

com parative confirmation and the p roble m of evi l 135 words, these theists believe their minds are fairly reliable when reasoning about matters of theodicy, and since there is no acceptable theodicy that we can understand, we would be mistaken to accept one. These theists who accept UNKNOWABLE GOOD may naturally think it unlikely that they would wrongly accept a theodicy, and thus they hold that P(UNAWARE/UNKNOWABLE GOOD) is high. These sceptical theists may also think that P(UNAWARE/NO GOOD) is low, or at least lower than P(UNAWARE/UNKNOWABLE GOOD). Plantinga (1993) argued that on the assumption of naturalism and evolutionary theory, the probability of our cognitive faculties being reliable was either low or inscrutable. In response many philosophers questioned whether we could really survive if many of our basic cognitive faculties were not reliable. For our purposes we can ignore most of that debate, because we are here only interested in our being unaware of a good reason for God to permit evil. UNAWARE is clearly different from the types of beliefs that many think are necessary for survival, such as perceptual beliefs, and it is more akin to abstract scientific or philosophical beliefs. Even if we were to grant that one’s survival would be in danger if one did not form reliable beliefs about predators when they were present, this reasoning does not automatically carry over to UNAWARE. The truth of our abstract scientific, philosophical, or theological beliefs is basically irrelevant to survival by natural selection. Natural selection only operates on the actions resulting from our beliefs, and our abstract beliefs could be false, and yet easily give rise to actions that have survival value. For example, suppose we have two qualitatively identical worlds, W1 and W2 , in which we hold the same abstract scientific, philosophical, and theological beliefs. Let most of these abstract beliefs be true in W1 but false in W2 . That is, the worlds W1 and W2 appear identical, but our abstract beliefs are mostly true in one but not the other. Our ability to survive would be the same in these two worlds; the truth or falsity of our abstract beliefs would have no effect on our survival. Naturalists even hold that humans have a widespread tendency to form religious and supernatural beliefs, even though these beliefs are all false. Given this, it is difficult to see why humans would not mistakenly hold theodicies if naturalism were correct; naturalism gives no reason to think our minds would be reliable on topics such as UNAWARE, and one can reasonably hold that P(UNAWARE/NO GOOD) is not high. Thus sceptical theists can hold that P(UNAWARE/UNKNOWABLE GOOD) is high and P(UNAWARE/NO GOOD) is not high, which has the consequence that UNAWARE favours UNKNOWABLE GOOD over NO GOOD. Theists can rationally hold that our ignorance of a good reason for God to permit evil supports the existence of God having a reason we can’t comprehend over there being no such reason (and God not existing). In other words, many theists’ beliefs are such that UNAWARE creates a problem of evil for naturalism, not for traditional theism. However, the above reasoning is not uncontroversial, and a rational theist may reject it and think P(UNAWARE/NO GOOD) is high. Since P(UNAWARE/ UNKNOWABLE GOOD) is also high, corollary C to the Likelihood principle tells us that UNAWARE supports NO GOOD and UNKNOWABLE GOOD

136 p robab i l ity i n th e ph i lo sophy of re l i g i on equally. Or a theist may think P(UNAWARE/UNKNOWABLE GOOD) is as low as P(UNAWARE/NO GOOD). Theists who hold that P(UNAWARE/ UNKNOWABLE GOOD) is about the same as P(UNAWARE/NO GOOD) will hold that UNAWARE is neutral evidence for NO GOOD and their own beliefs. Although I have argued that a theist may rationally believe P(UNAWARE/ UNKNOWABLE GOOD) is at least as high as P(UNAWARE/NO GOOD), puzzlement about these probabilities is completely understandable. Perhaps one reason these probabilities have seen such little discussion in the literature is that most people simply aren’t sure about them; it is very difficult to come up with any obvious estimates of their values. For both of these probabilities, one can think of arguments that the value is high and arguments that the value is low. Because of this we may want to withhold judgement on the probabilities and on whether P(UNAWARE/NO GOOD) is greater than P(UNAWARE/UNKNOWABLE GOOD).6 If so, we should also withhold judgement on whether UNAWARE supports NO GOOD over UNKNOWABLE GOOD. I am here appealing to the following general principle: If we know that C is contingent and is a necessary condition of some belief B, and it is rational to withhold belief in C, then it is rational to withhold belief in B.7

Since we know (assuming Likelihoodism) that P(UNAWARE/NO GOOD) being greater than P(UNAWARE/UNKNOWABLE GOOD) is a necessary condition of UNAWARE favouring NO GOOD over UNKNOWABLE GOOD, and since it is rational to withhold judgement on whether P(UNAWARE/NO GOOD) is greater than P(UNAWARE/UNKNOWABLE GOOD), the above principle tells us it is also rational to withhold judgement on whether UNAWARE supports NO GOOD over UNKNOWABLE GOOD. If theists are unsure about the relation of P(UNAWARE/NO GOOD) and P(UNAWARE/UNKNOWABLE GOOD) they are not obligated to hold that UNAWARE supports NO GOOD over UNKNOWABLE GOOD; for all they know, UNAWARE may support UNKNOWABLE GOOD over NO GOOD. UNAWARE does not provide the basis for an evidential problem of evil for these theists. We have seen that sceptical theists can hold either that P(UNAWARE/ UNKNOWABLE GOOD) is greater than P(UNAWARE/NO GOOD), or that the probabilities are equal, or they can withhold judgement about the relation between those probabilities. In each of these cases we saw that UNAWARE does not favour NO GOOD over UNKNOWABLE GOOD. Thus the evidential argument from evil fails from a Likelihoodist perspective.

6 Standard Bayesian accounts require probability be precise values, although vague belief can be modelled in various ways. See P. Walley (1991); B. van Fraassen (1990) and (1995). 7 The requirement that C be contingent is important. Logical truths are necessary conditions of everything and it is also rational to withhold judgement on whether certain sentences are logical truths. If we did not require C to be contingent, then the principle would require us to withhold judgement on everything.

com parative confirmation and the p roble m of evi l 137

III. Bayesianism The above argument shows that Likelihoodism does not provide a foundation for a successful evidential argument from evil. In response, the defender of the evidential argument from evil may appeal to a fundamentally different approach to comparative confirmation based on Bayesian confirmation theory. For the Likelihoodist, confirmation is fundamentally a comparative notion. In contrast to this, Bayesian accounts of comparative confirmation are defined in terms of a more fundamental degree of confirmation relation that holds between evidence and hypothesis. There are many different Bayesian measures of confirmation. Fitelson (2007) proposes the following principle which defines contrastive confirmation in terms of degree of confirmation: † : E favours H1 over H2 iff E confirms H1 more strongly than E confirms H2 . The general strategy Fitelson proposes is that in order to determine whether some evidence E supports H1 over H2 , we should first look at how strongly E confirms H1 and E confirms H2 , and then compare those values.8 There are several Bayesian proposals for measuring degree of confirmation, and each of these will generate a different account of the comparative favouring relation. The three most popular Bayesian accounts of the degree of confirmation of H by E are: a. The degree of confirmation of H by E is given by the difference between P(H/E) and P(H): P(H/E) – P(H). b. The degree of confirmation of H by E is given by the ratio of P(H/E) to P(H): P(H/E)/P(H).9 c. The degree of confirmation of H by E is given by the likelihood ratio of P(E/H) to P(E/not-H): P(E/H)/P(E/not-H). (See Fitelson 2007) A discussion of the details of these Bayesian accounts would take us too far afield from our goal of looking at whether Bayesian measures of confirmation lend support to an evidential argument from evil. In what follows I will not argue for one of these accounts of confirmation over another, but will only consider whether they give a sceptical theist reason to think that UNAWARE supports NO GOOD over UNKNOWABLE GOOD. To begin, it is worth noting that each of the above measures of confirmation implies the following sufficient condition for E supporting H1 over H2 , which Fitelson calls the ‘Weak Law of Likelihood’:

8 Principle † is not uncontroversial. See J. Chandler (2010) for an interesting discussion of † and a defence of the Law of Likelihood, and B. Fitelson (forthcoming) for a response. 9 Using this measure of confirmation with † generates a favouring relation that is equivalent to the Law of Likelihood L.

138 probabi lity in the phi lo sophy of re lig ion WLL: Evidence E favours hypothesis H1 more than hypothesis H2 if P(E/H1 ) > P(E/H2 ) and P(E/not-H1 ) ≤ P(E/not-H2 ). (See Fitelson 2007)10

According to Fitelson, ‘[d]ozens of relevance measures . . . have been proposed and defended in the literature . . . [a]nd all of these are such that [the measures of non-relational confirmation based on them] entails (WLL)’ (2007: 479). Applied to the problem of evil, this tells us that UNAWARE will favour UNKNOWABLE GOOD over NO GOOD if P(UNAWARE/UNKNOWABLE GOOD) > P(UNAWARE/NO GOOD) and P(UNAWARE/not-UNKNOWABLE GOOD) ≤ P(UNAWARE/not-NO GOOD). Recall that a goal of the evidential argument from evil is to show that a theist should hold that evil or some related evidence is evidence against the theist’s beliefs. In response, for the reason given above, I argued that sceptical theists may hold that P(UNAWARE/UNKNOWABLE GOOD) > P(UNAWARE/NO GOOD). But unlike Likelihoodism, not all Bayesian positions imply that this is sufficient for UNAWARE to support UNKNOWABLE GOOD over NO GOOD. In applying WLL to this situation, we need to also look at P(UNAWARE/not-NO GOOD) and P(UNAWARE/not-UNKNOWABLE GOOD). It will be easiest to compare these probabilities if we become clearer about what NO GOOD, not-NO GOOD, UNKNOWABLE GOOD, and not-UNKNOWABLE GOOD are claiming. This can be more easily seen if we break NO GOOD and UNKNOWABLE GOOD down to more basic components. Since both NO GOOD and UNKNOWABLE GOOD imply that evil exists, for heuristic reasons we will move the existence of evil into our background knowledge and will not explicitly state it when discussing NO GOOD and UNKNOWABLE GOOD. This makes the formal argument easier to follow, although the results are unaffected if we retain the existence of evil in the two hypotheses.11 Let us use the following abbreviations: G: God exists R: there is a good reason for God to permit evil RC: there is a good reason for God to permit evil that is comprehensible by humans. Using these abbreviations we can see that NO GOOD is equivalent to not-R and UNKNOWABLE GOOD is equivalent to (G & R & not-RC). Conversely, not-NO GOOD is equivalent to R, and not-UNKNOWABLE GOOD is equivalent to (not-G or not-R or RC). Now consider the following chart which lists the possible combinations of these three propositions:

10 See J. Joyce (2008) for further discussion of this principle. 11 See the next note for details.

com parative confirmation and the p roble m of evi l 139

a) b) c) d) e) f) g) h)

G T T T T F F F F

R T T F F T T F F

RC T F T F T F T F

Lines c and g are not possible because RC implies R; there being a good reason that we can understand implies that there is a good reason. Assuming God does not exist if he does not have a good reason to permit evil, line d is also not possible. We are left with 5 mutually exclusive possibilities: a, b, e, f, and h. Since NO GOOD is equivalent to not-R, given our abbreviations it is easy to see that NO GOOD is equivalent to line h; h is the only possibility in which not-R is true. UNKNOWABLE GOOD is equivalent to (G & R & not-RC); given our abbreviations we see this holds only in line b. Since the possibilities are exhausted by lines a, b, e, f, and h, not-NO GOOD is equivalent to the disjunction of lines a, b, e, and f (a or b or e or f ) and not-UNKNOWABLE GOOD is equivalent to the disjunction of lines a, e, f, and h (a or e or f or h). Thus we have P(UNAWARE/ not-NO GOOD) = P(UNAWARE/ a or b or e or f ) and P(UNAWARE/ not-UNKNOWABLE GOOD) = P(UNAWARE/ a or e or f or h). Notice that not-NO GOOD and not-UNKNOWABLE GOOD have the disjunction (a or e or f ) in common; they differ only on b and h. It is clear that the relation between P(UNAWARE/a or e or f or h) and P(UNAWARE/ a or b or e or f ) depends on how likely UNAWARE is given b and given h. From this it follows that the relation between P(UNAWARE/not-NO GOOD) and P(UNAWARE/not-UNKNOWABLE GOOD) depends on P(UNAWARE/b) and P(UNAWARE/h).12 If P(UNAWARE/ h) ≤ P(UNAWARE/b), then P(UNAWARE/a or e or f or h) ≤ P(UNAWARE/ a or b or e or f ), which immediately implies that P(UNAWARE/not-UNKNOWABLE GOOD) ≤ P(UNAWARE/not-NO GOOD). 12 This same argument goes through if we keep the existence of evil in NO GOOD and UNKNOWABLE GOOD instead of putting it in our background knowledge. Let E be that evil exists; in this case UNKNOWABLE GOOD is represented by (G & R & not-RC & E), and NO GOOD is represented by (not-R & E). We also know not-NO GOOD is equivalent to (R v not-E), and not-UNKNOWABLE GOOD is equivalent to (not-G v not-R v RC v not-E). We now have 16 possibilities instead of 8; let possibilities a–h include E, and let possibilities i–p include not-E. Excluding impossible situations, we have the real possibilities are: a, b, e, f, h, i, j, m, n, p. As before, NO GOOD is equivalent to h and UNKNOWABLE GOOD is equivalent to b. And not-NO GOOD is equivalent to the disjunction (a v b v e v f v i v j v m v n v p) and not-UNKNOWABLE GOOD is equivalent to the disjunction (a v e v f v h v i v j v m v n v p). P(UNAWARE/not-NO GOOD) = P(UNAWARE/ a v b v e v f v i v j v m v n v p) and P(UNAWARE/not-UNKNOWABLE GOOD) = P(UNAWARE/ a v e v f v h v i v j v m v n v p). As before, the only difference in what is conditionalized on is b and h, and thus the same result follows.

140 probabi lity in the phi lo sophy of re lig ion So let us reconsider the case of a sceptical theist who holds that God’s reason for permitting evil is incomprehensible to us, sees no reason to doubt the reliability of our faculties that produce UNAWARE given theism, but has doubts about the reliability of these faculties given naturalism. This theist holds that P(UNAWARE/UNKNOWABLE GOOD) > P(UNAWARE/NO GOOD), which is the first condition in WLL for UNAWARE to support UNKNOWABLE GOOD over NO GOOD. This is equivalent to P(UNAWARE/b) > P(UNAWARE/h), which implies P(UNAWARE/h) ≤ P(UNAWARE/b). We saw above that this implies P(UNAWARE/not-UNKNOWABLE GOOD) ≤ P(UNAWARE/not-NO GOOD), which is the second condition in WLL for UNAWARE to favour UNKNOWABLE GOOD over NO GOOD.13 Since this sceptical theist holds both P(UNAWARE /UNKNOWABLE GOOD) > P(UNAWARE/NO GOOD) and P(UNAWARE/not-UNKNOWABLE GOOD) ≤ P(UNAWARE/not-NO GOOD) both conditions of WLL are met, and WLL tells us that UNAWARE supports UNKNOWABLE GOOD over NO GOOD. Bayesian approaches to comparative or contrastive confirmation have the result that these sceptical theists should not think that our ignorance of a good reason for God to permit evil supports there actually being no good reason over God having a good reason we cannot understand. As a result, according to Bayesian accounts of contrastive confirmation the standard evidential argument from evil does not arise for these typical theists. On the contrary, Likelihoodism and Bayesian accounts of comparative confirmation have the result that some sceptical theists are entitled to hold that the evidential argument from evil is a problem for naturalists.

IV. Objection In this chapter I have examined the problem of evil from the perspective of someone who believes that humans are incapable of understanding God’s reason for permitting evil. This view is in fact widely held by theists. Nevertheless one might object to an argument based on this perspective on methodological grounds. I focused on whether these typical theists were susceptible to a problem of evil by looking at whether UNAWARE supported NO GOOD over UNKNOWABLE GOOD; I did not discuss whether UNAWARE supported NO GOOD over GOOD, where GOOD is the disjunction of UNKNOWABLE GOOD and KNOWABLE GOOD. One might object that I should have looked at whether evil and UNAWARE favour GOOD over NO GOOD, instead of looking at NO GOOD and UNKNOWABLE GOOD. 13 The same argument will go through if we let UNKNOWABLE GOOD not imply that God

exists and instead be equivalent to R & not-RC, in which case UNKNOWABLE GOOD does not imply that God exists. UNKNOWABLE GOOD will then be equivalent to (b v f) and notUNKNOWABLE GOOD will be (a v e v h). In this case P(UNAWARE/not-UNKNOWABLE GOOD) ≤ P(UNAWARE/not-NO GOOD) iff P(UNAWARE/b v f ) ≤ P(UNAWARE/h). But this is equivalent to P(UNAWARE/UNKNOWABLE GOOD) ≤ P(UNAWARE/NO GOOD).

com parative confirmation and the p roble m of evi l 141 It might be claimed that although UNAWARE may not be evidence for NO GOOD over UNKNOWABLE GOOD, it is evidence for NO GOOD over GOOD, and this does generate a problem of evil for the rationality of typical theists’ beliefs. This objection assumes that UNAWARE favours NO GOOD over GOOD, but this assumption may be rejected by sceptical theists. GOOD is equivalent to the disjunction of KNOWABLE GOOD and UNKNOWABLE GOOD, and thus the probability of UNAWARE given GOOD will be: P(UNAWARE/GOOD) = P(UNAWARE/UNKNOWABLE GOOD) P(UNKNOWABLE GOOD) + P(UNAWARE/KNOWABLE GOOD) P(KNOWABLE GOOD)

Sceptical theists hold UNKNOWABLE GOOD, and thus assign KNOWABLE GOOD a very low probability. This implies P(UNAWARE/GOOD) is very close to P(UNAWARE/UNKNOWABLE GOOD). Because of this, looking at GOOD instead of UNKNOWABLE GOOD does not make a significant difference to the argument. The basic assumption of this objection is false, according to sceptical theists. But even if we did grant the assumption this objection would still fail. In the introduction I mentioned that it is common in philosophy of religion to discuss the rationality of religious beliefs not by looking at those beliefs themselves, but at a ‘core’ of beliefs entailed by those beliefs. Elsewhere, I discussed this methodology in detail and argued this methodological assumption is problematic (see Otte (2011)). The reason I did not discuss the relation between UNAWARE and GOOD here is because it is irrelevant to the rationality of typical theistic belief, since most theists hold UNKNOWABLE GOOD. Even though UNKNOWABLE GOOD implies GOOD, evidence that favours NO GOOD over GOOD may also favour UNKNOWABLE GOOD over NO GOOD. We cannot infer anything about whether some evidence favours NO GOOD or UNKNOWABLE GOOD by looking at some ‘core’ implied by UNKNOWABLE GOOD, and arguing that the evidence supports NO GOOD over the core implied by UNKNOWABLE GOOD. UNAWARE can favour NO GOOD over GOOD, and yet also favour UNKNOWABLE GOOD over NO GOOD. Comparing GOOD and NO GOOD is not useful if we are interested in the rationality of theists who hold UNKNOWABLE GOOD.

V. Conclusion We have been investigating whether a typical theist should think that the probabilistic argument from evil actually poses a problem for religious belief. We saw that neither Likelihoodism nor Bayesian accounts of contrastive confirmation give a typical theist reason to think that our ignorance of a good reason for God to permit evil supports there being no good reason over the reason being incomprehensible by us. But if theists are rational in holding that there is a good reason for God to permit evil, then evil is not problematic for belief in God. On the contrary, theists may reasonably hold that

142 probabi lity in the phi lo sophy of re lig ion our ignorance of a good reason actually supports their religious beliefs over naturalism. From this we see just how difficult it is to formulate any problem of evil that should be convincing to most theists.14

References Adams, R. (1985) ‘Plantinga on the Problem of Evil’. In J. E. Tomberlin and P. van Inwagen (eds), Alvin Plantinga. Dordrecht: D. Reidel, 225–55. Alston, W. P. (1991) ‘The Inductive Argument from Evil and the Human Cognitive Condition’, Philosophical Perspectives, 5: 29–67. Bergmann, M. (2001) ‘Sceptical Theism and Rowe’s New Evidential Argument from Evil’, Noûs, 35, 2: 278–96. Carnap, R. (1950) Logical Foundations of Probability. Chicago: University of Chicago Press. Chandler, J. (2010) ‘Contrastive Confirmation: Some Competing Accounts’, Synthese, 1–10. Fitelson, B. (2007) ‘Likelihoodism, Bayesianism, and Relational Confirmation’, Synthese, 156, 3: 473–89. (forthcoming) ‘Contrastive Bayesianism’. In M. Blauw (ed.), Contrastivism in Philosophy. London: Routledge. Hacking, I. (1976) Logic of Statistical Inference. Cambridge: Cambridge University Press. Howard-Snyder, D. (1996) The Evidential Argument from Evil. Bloomington: Indiana University Press. Joyce, J. (2008) ‘Bayes’ Theorem’. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2008 Edition). Otte, R. (2000) ‘Evidential Arguments from Evil’, International Journal for Philosophy of Religion, 48: 1–10. (2002) ‘Rowe’s Probabilistic Argument from Evil’, Faith and Philosophy, 19, 2: 147–71. (2009) ‘Transworld Depravity and Unobtainable Worlds’, Philosophy and Phenomenological Research, 78, 1: 165–77. (2011) ‘Theory Comparison in Science and Religion’. In K. J. Clark and M. Rea (eds), Science, Religion, and Metaphysics: New Essays on the Philosophy of Alvin Plantinga. Oxford: Oxford University Press. Plantinga, A. (1974) The Nature of Necessity. Oxford: Oxford University Press. (1977) God, Freedom, and Evil. Grand Rapids, MI: Eerdmans. (1993) Warrant and Proper Function. Oxford: Oxford University Press. Rowe, W. (1986) ‘The Problem of Evil and Some Varieties of Atheism’. In D. Howard-Snyder (ed.), The Evidential Argument from Evil. Bloomington: Indiana University Press. (2001) ‘Sceptical Theism: A Response to Bergmann’, Noûs 35, 2: 297–303. Rowe, W. L. (1998) ‘Reply to Plantinga’, Noûs 32, 4: 545–52. Royall, R. (1997) Statistical Evidence. London: Chapman and Hall.

14 I would like to thank the anonymous referees and especially Jake Chandler for comments that made this a much better chapter.

com parative confirmation and the p roble m of evi l 143 Salmon, W. C. (1975) ‘Confirmation and Relevance’. In G. Maxwell and R. Anderson (eds), Induction, Probability, and Confirmation: Minnesota Studies in the Philosophy of Science. Volume 6. Minneapolis: University of Minnesota Press, 3–36. Sober, E. (1994) From a Biological Point of View. Cambridge: Cambridge University Press. van Fraassen, B. C. (1990) ‘Figures in a Probability Landscape’. In J. M. Dunn and A. Gupta (eds), Truth or Consequences. Dordrecht: Kluwer Academic Publishers, 345–56. (1995) ‘Belief and the Problem of Ulysses and the Sirens’, Philosophical Studies, 77, 1: 7–37. van Inwagen, P. (2006) The Problem of Evil. Oxford: Clarendon Press. Walley, P. (1991) Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall. Wykstra, S. J. (1984) ‘The Humean Obstacle to Evidential Arguments from Suffering: On Avoiding the Evils of “Appearance”’, International Journal for Philosophy of Religion, 16, 2: 73–94.

8 Inductive Logic and the Probability that God Exists: Farewell to Sceptical Theism Michael Tooley

I. Introduction and Overview The idea that at least some of the evils that are present in the world constitute a problem for belief in the existence of God is both an ancient idea—going back at least to Job, and presumably beyond—and a very natural one. But how best to develop that basic idea, and to convert it into an explicitly formulated argument in support of the non-existence of God, has remained unclear. Most initial attempts to do this involved arguments designed to show that the existence of God—understood as an omnipotent, omniscient, and morally perfect person—is logically incompatible either with the existence of any evil at all, or, at least, with the existence of certain types of evil, such as natural evils, or especially horrendous evils. It is now widely felt, however, that such incompatibility claims cannot be sustained, and, as a result, the focus of discussion has shifted to evidential versions of the argument from evil, where the thesis is, instead, that there are facts about the evils to be found in the world that render it unlikely—and perhaps very unlikely—that God exists. Formulating a satisfactory evidential argument from evil, however, has also proved to be very difficult, and so the question arises whether there is, in the end, any satisfactory version of the argument from evil. To show that there is, three tasks need to be carried out. First of all, one needs to argue that that there are states of affairs in the world that, no matter how carefully one considers them, are morally problematic in the following way: when one considers the moral status of allowing any of the states of affairs in question, and reflects upon the rightmaking and wrongmaking properties that there is good reason to believe such an action would have, the wrongmaking properties outweigh the rightmaking properties. So the claim here is that if one considers, for example, the actions of not preventing the Lisbon earthquake, or the Holocaust, when one could have done so, any rightmaking properties that one has good reason to believe such actions would possess

inductive log ic 145 are not sufficiently significant to outweigh the known wrongmaking properties, such as allowing ordinary people to undergo enormous suffering or death. Secondly, one needs to show that facts of the preceding sort about the evils to be found in the world do indeed make it more likely than not that God does not exist. Finally, it also needs to be shown that neither an appeal to the idea that belief in the existence of God can be non-inferentially justified, nor an appeal to positive evidence or arguments in support of the existence of God, can overcome the prima facie case against the existence of God provided by the existence of such evils, so that, as a consequence, even when all the ways in which one might attempt to show that belief in the existence of God is justified are taken into account, it still remains the case that the non-existence of God is more likely—and perhaps much more likely—than the existence of God. In this essay, I shall not address either the first or the third of these tasks in any detail. As regards the former, I think that the vast majority of present-day philosophers of religion would agree that if one considers the known rightmaking and wrongmaking properties, for example, of allowing the Lisbon earthquake, the wrongmaking properties outweigh the rightmaking. Nevertheless, there are serious philosophers, such as John Hick (1966) and Richard Swinburne (1979, 1988, 1996a, and 1996b) who have argued that it is possible to set out what I would refer to as a complete theodicy, where, as I shall use that term, this involves being able to describe, for every actual evil found in the world, some state of affairs that it is reasonable to believe exists, and that would provide an omnipotent and omniscient being with a morally sufficient reason for allowing the evil in question.1 The basic point that needs to be made about all theodicies, however, is simply this.2 The theodicist’s approach is to point to some feature that is clearly and uncontroversially present in what initially appears to be a great evil, and then to claim that that feature not only is morally significant, but also is so to such a degree that it outweighs the known wrongmaking properties. Thus, for example, in the case of the Holocaust, the theodicist might, first of all, point out that if an omnipotent and omniscient being exists, his not having intervened, say, to enable an attempt to assassinate Hitler to succeed, has the rightmaking feature of leaving the world one where agents can freely bring about very great evils, and then, secondly, claim that this rightmaking feature is so significant that it outweighs the suffering and deaths of six million people. The latter 1 It should be noted that the term ‘theodicy’ is sometimes used in a stronger sense, according to which a person who offers a theodicy is attempting to show not only that such morally sufficient reasons exist, but that the reasons cited are in fact God’s reasons. Alvin Plantinga (1974: 10; 1985: 35) and Robert Adams (1985: 242) have used the term in that way, but, as has been pointed out by a number of writers, including Richard Swinburne (1988: 298), and William Hasker (1988: 5), that is to saddle the theodicist with an overly ambitious programme, and one that is completely unnecessary for rebutting the argument from evil. 2 Quite a number of different theodicies have been advanced, and a thorough discussion of them would be very lengthy indeed. I have, however, offered a brief critical survey in my online essay ‘The Problem of Evil’ (2010), where I consider, in Section 7, theodicies based on the ideas of (a) soul-making, (b) the value of free will, (c) the importance of the freedom to do great evil, and (d) the need for natural laws.

146 probabi lity in the phi lo sophy of re lig ion claim, however, is not one, it seems, that most people accept. The theodicist’s moral intuitions, then, differ from the moral intuitions of most people. Epistemologically, that would seem to be a very strong prima facie reason for concluding that if there are objective moral values, the theodicist’s moral claim is in all probability mistaken. The theodicist needs to argue, then, that this is not the case. Theodicists have not, I think, seriously attempted to meet this epistemological challenge. The third task—that of showing that no positive consideration in support of the existence of God can overcome the evidential argument from evil—would also require extended discussion, although many arguments that are offered in support of the existence of God can be quickly set aside on the ground that they provide no reason for attributing moral goodness, let alone moral perfection, to the being in question.3 The upshot is that my focus here will be confined to the second of the above tasks—that of showing that, and how, one can establish that, relative to evidence that consists simply of facts about the evils to be found in our world, the existence of God is unlikely—indeed, extremely unlikely. But even here, my discussion will be incomplete, since I shall not repeat the detailed argument that I set out in my opening statement in the Knowledge of God volume (Plantinga and Tooley 2008: 117–21). What I shall do is simply to focus on the crucial inductive step in the argument, with which the argument as a whole stands or falls. Why am I returning to this issue, given that I discussed it quite fully in Knowledge of God (2008: 126–50)? There are two reasons. First of all, after Plantinga and I had exchanged our opening statements, I discovered an idea that was relevant to my basic argument, and that allows one to recast the mathematical argument in a way that is both simpler and more accessible. At that point it was, of course, too late to make use of the new formulation—which I am setting out here. Secondly, although there have been surprisingly few reviews of the Knowledge of God volume, and so virtually nothing by way of criticism to respond to, there is an important issue that has been raised in a discussion between Alexander Pruss (2010) and Branden Fitelson (2010). It concerns the question of what system of inductive logic should be used in formulating the argument from evil, and this is a question that I think it is important to address.

II. The Use of Inductive Logic On my formulation of the argument from evil, the relevant evidence consists of facts concerning the existence of evils whose known wrongmaking properties outweigh their known rightmaking properties. How can one show that such evidence makes it more likely that God does not exist than that God does exist? The only way, it seems to me, is by bringing inductive logic to bear upon the question. Many present-day 3 A slightly more detailed discussion can be found, however, in my closing statement in the debate volume Knowledge of God, which I co-authored with Alvin Plantinga (Plantinga and Tooley 2008: 241–7).

inductive log ic 147 philosophers, of course, have serious doubts about whether there is any objectively correct, inductive logic. If those doubts are justified, and if there is, in general, no unique, correct value for the a priori probability that a given proposition is true, the question arises whether the assignment of an a priori probability to a given proposition is constrained in any way. Given some particular proposition, is there anything that prevents one from selecting any value from zero up to and including one as the a priori probability that that proposition is true? Strong arguments can be offered for holding that one’s assignments of probabilities to different propositions should respect the axioms of the theory of probability. So, for example, the probabilities that one assigns to a proposition and to its negation should add up to one, and the probability that one assigns to a disjunction should not be greater than the sum of the probabilities one assigns to its disjuncts. But these consistency constraints allow one to choose any particular contingent proposition, and assign to that proposition any probability that one wants. It seems to me that such a view entails scepticism about induction, and that such scepticism is epistemologically disastrous. This is not, however, an issue that I shall pursue here. Instead, I shall simply assume that a certain approach to inductive logic is sufficiently close to the right account for our present purposes, and then offer an argument for the conclusion that, given that approach, it follows that the existence of God is very unlikely relative to propositions concerning the evils to be found in the world. The fact that my argument is based upon a certain approach to inductive logic does not mean, however, that the argument will be of interest only to those who accept the approach in question. For, first of all, philosophers who favour a different approach to inductive logic may very well be able to recast my argument within the framework of their own preferred approach. Secondly, readers who do not share my optimism about the existence of an objectively correct inductive logic can approach the argument from the point of view of their own, purely subjective probabilities. The question, in that case, will be whether the equiprobability assumptions that are part of the approach to inductive logic that I favour agree with their own subjective probabilities—or, at least, whether the relation is sufficiently close that very similar results can be derived.

III. Rudolf Carnap’s Structure-description Approach to Inductive Logic The inductive logic that I shall use here is essentially that set out by Rudolf Carnap in the Appendix, ‘Outline of a Quantitative System of Inductive Logic’, to his book Logical Foundations of Probability (1962). A detailed consideration of Carnap’s approach will not, however, be necessary, since everything that is crucial for our present purposes can

148 probabi lity in the phi lo sophy of re lig ion be explained in terms of three fundamental concepts: the concept of a state-description, the concept of a structure-description, and the concept of a predicate that is maximal with respect to a set of properties. Consider, then, a very simple world that contains only three individuals—a, b, and c—and only one basic property—P. (The idea of basic properties is crucial, since otherwise there is, arguably, no satisfactory answer to Goodman’s ‘new riddle of induction’ (1955: 59–83).) A state-description will then be a proposition that specifies, for each of the three individuals, whether that individual has property P or not. Given that, in the case of each individual, the individual may either possess property P or not, there are, for a world that contains only three individuals, (2 x 2 x 2) possible state-descriptions, which are as follows: (1) (2) (3) (4) (5) (6) (7) (8)

Pa & Pb & Pc Pa & Pb & ∼Pc Pa & ∼Pb & Pc ∼Pa & Pb & Pc Pa & ∼Pb & ∼Pc ∼Pa & Pb & ∼Pc ∼Pa & ∼Pb & Pc ∼Pa & ∼Pb & ∼Pc

To define the logical probability that one proposition has, given another, one needs to determine the weights to be assigned to the propositions belonging to some fundamental set of propositions. One very natural idea is to assign equal weight to all state-descriptions, so that each of the above eight state-descriptions would be assigned the weight 1/8. But suppose that one does that, and that one learns that Pa and Pb are the case. Given that, what is the probability that Pc? If Pa and Pb are the case, only two state-descriptions can obtain, namely: (1) Pa & Pb & Pc (2) Pa & Pb & ∼Pc In the first of these, Pc is the case, while in the second it is not. If both of these state-descriptions are assigned the same weight—namely, 1/8—it follows that the probability that Pc, given Pa and Pb, is equal to 1/2. But this is precisely the probability that Pc initially had, before one learned that Pa and Pb were the case. Moreover, this would be so regardless of whether there were three individuals, or a million: the probability that the next individual would have property P would still be equal to 1/2, regardless of how many individuals had been examined and found to have property P. Assigning equal weight to state-descriptions appears to have, therefore, the unhappy consequence that one can never learn from experience via induction, as Carnap argued (1962: 564–5). To solve this problem, Carnap proposed that equal

inductive log ic 149 weight should be assigned, not to state-descriptions, but to what he referred to as ‘structure-descriptions’. What is a structure-description? The basic idea is that a structure-description provides one only with statistical information about how many individuals have a given property. Thus, in the case of the mini world mentioned above, one has the following four structure-descriptions: (1) (2) (3) (4)

All three individuals, a, b, and c, have property P. Two of the three individuals have property P, and one does not. One of the three individuals has property P, and two do not. None of the three individuals has property P.

Two of those structure-descriptions each correspond to a single state-description, while the other two each correspond to three state-descriptions, as follows: Structure-description 1: Structure-description 2:

Structure-description 3:

Structure-description 4:

(1) Pa & Pb & Pc (2) Pa & Pb & ∼Pc (3) Pa & ∼Pb & Pc (4) ∼Pa & Pb & Pc (5) Pa & ∼Pb & ∼Pc (6) ∼Pa & Pb & ∼Pc (7) ∼Pa & ∼Pb & Pc (8) ∼Pa & ∼Pb & ∼Pc

On Carnap’s proposal, then, each of these four structure-descriptions is assigned the weight 1/4, and that weight is then distributed equally over the state-descriptions that belong to the structure-description in question. Thus the state-description Pa & Pb & Pc, since it is the only state-description that belongs to the first structure-description, is assigned the weight 1/4, whereas the state-description Pa & Pb & ∼Pc, since it is one of three state-descriptions that belong to the second structure-description, is assigned the weight 1/12. How does this change affect things? Assume, once again, that Pa and Pb obtain. As before, the probability that Pc is the case will be equal to the ratio of the weight of the one remaining state-description where Pc is true—namely, (1) Pa & Pb & Pc—to the total weight of all of the state-descriptions where Pa and Pb are true—namely, (1) Pa & Pb & Pc and (2) Pa & Pb & ∼Pc. When equal weights are assigned to structure-descriptions, rather than to state-descriptions, that ratio is then equal to (1/4) = 3/4. (1/4 + 1/12) The upshot is that when equal weight is assigned to structure-descriptions, rather than state-descriptions, the probability that the next individual has a certain property

150 p robab i l ity i n th e ph i lo sophy of re l i g i on does depend upon how many previously observed individuals have had that property. One can, therefore, learn from experience, via induction. Finally, there is the concept of predicates that are maximal with respect to some set, S, of properties, where these are predicates that, when applied to an individual, indicate, either explicitly or implicitly, for every property in S, whether the individual in question has the property or not. So, for example, if S is a set of three mutually compatible properties, with associated predicates ‘P’, ‘Q’, and ‘R’, the following eight predicates will be maximal with regard to that set: ‘P & Q & R’, ‘P & Q & ∼R’, ‘P & ∼Q & R’, ‘∼P & Q & R’, ‘P & ∼Q & ∼R’, ‘∼P & Q & ∼R’, ‘∼P & ∼Q & R’, and ‘∼P & ∼Q & ∼R’. Alternatively, if S is instead a set of three mutually incompatible properties, with associated predicates ‘P’, ‘Q’, and ‘R’, then those predicates themselves are maximal with respect to S, since each logically entails the presence of one of the three properties, and the absence of the other two.

IV. The Application of Inductive Logic to the Argument from Evil: An Overview The obstacles in the way of an exact calculation Given these ideas, and a Carnapian-style, structure-description approach to inductive logic, let us now turn to the problem at hand. Suppose that there are n events, each of which is such that, judged in the light of the totality of known rightmaking and wrongmaking properties, it would be morally wrong to allow. What is the probability that there are unknown rightmaking and wrongmaking properties such that, given the totality of rightmaking and wrongmaking properties, both known and unknown, it would not be morally wrong, all things considered, to allow any of the n events? The calculation of an exact answer to this question is a complicated matter for a number of reasons. First, there is no logical limit upon the number of unknown morally significant properties that there may be, so in the absence of some substantive moral theory that can be shown to be correct and that entails such a limit, one needs a calculation that sums over an infinite number of possibilities. Secondly, unknown rightmaking and wrongmaking properties can vary in significance from ones that are quite trivial to ones whose significance is very great indeed. Thus there might, for example, be an unknown rightmaking property that could have been possessed by an act of allowing the Lisbon earthquake, but that would not have been sufficiently weighty to make it the case that that act would not have been wrong, all things considered. So one also needs a calculation that sums over the nondenumerable number of possibilities with respect to strengths of unknown rightmaking and wrongmaking properties. Thirdly, the n events that one is focusing upon may vary enormously with regard to the extent to which, judged by known rightmaking and wrongmaking properties, it would have been morally wrong to allow the event in question, so that unknown rightmaking properties that would render it morally permissible

inductive log ic 151 to allow one event might not suffice at all in the case of some other event. Consequently, the calculation would also need to be geared to the specific events that one is considering. Finessing the problem: calculating an upper bound Clearly, then, there are a number of serious difficulties that stand in the way of calculating an exact answer to our question. Because of these obstacles, I shall not attempt to carry out such a calculation. My approach, instead, will be to argue that, given a set of n events, each of which, judged by known rightmaking and wrongmaking properties, it would be morally wrong to allow, one can place an upper bound upon the probability that, judged in the light of the totality of rightmaking and wrongmaking properties, both known and unknown, it is not morally wrong to allow any of those n events. How can this be done? The basic ideas are as follows. Let E be some set of n events, each of which is such that, judged only by the rightmaking and wrongmaking properties of which one is aware, it would be morally wrong to allow that event. Let S be some structure-description involving an attribution of unknown rightmaking and wrongmaking properties to the n actions of allowing one of the events in E. A structure-description S is positive if, judged only by the unknown rightmaking and wrongmaking properties, none of the n actions would be either a morally neutral action or a morally wrong action. A structure-description is favourable if, judged by the totality of rightmaking and wrongmaking properties, both known and unknown, none of the n actions is morally wrong. Any favourable structure-description must be a positive structure-description, but the converse need not be the case, since a given unknown rightmaking property of an action may not be sufficiently weighty to outweigh that action’s known wrongmaking property or properties. Now what one would really like to determine is the probability that none of the n actions is morally wrong. But as I noted above, doing that is a very complicated matter. What is much less complicated is determining the probability that a structure-description involving an attribution of unknown rightmaking and wrongmaking properties to the n actions is a positive structure-description. Then, since a structure-description cannot be a favourable structure-description without being a positive one, the likelihood that a structure-description involving an attribution of unknown rightmaking and wrongmaking properties to the n actions is a positive one will provide an upper bound on the probability that none of the n actions is morally wrong, all things considered. The method But how can one calculate the probability that a structure-description will be a positive one? My way of proceeding will be as follows. First, I shall show how one can establish a

152 probabi lity in the phi lo sophy of re lig ion formula that gives the probability that, if there are precisely k unknown, morally significant properties, of which r are rightmaking properties, a structure-description involving an attribution of those properties to n actions is a positive structure-description. This formula will then be an upper bound upon the probability that such a structuredescription will be, not only a positive structure-description, but also a favourable one. Secondly, I shall then show how one can establish a formula stating the probability, given only that there are precisely k unknown, morally significant properties, and where the number of rightmaking and wrongmaking properties is not specified, that a structure-description involving an attribution of those properties to n actions is a positive structure-description. Then, thirdly, we shall see that that formula enables one to arrive very quickly at another formula that places an upper bound upon the probability that a structure-description involving the attribution of unknown rightmaking and wrongmaking properties to n actions is a favourable one, and an upper bound that holds regardless of the number of unknown, morally significant properties.

V. Step One: The Case of k Unknown Morally Significant Properties, where r are Rightmaking Properties, and w are Wrongmaking First of all, then, what is the probability that a structure-description involving n actions and k unknown, morally significant properties, of which exactly r are rightmaking and w are wrongmaking, is a positive structure-description? Given the approach to inductive logic discussed above, we need to determine the ratio of the number of positive structure-descriptions involving the r rightmaking and w wrongmaking properties, and the n actions, to the total number of relevant structure-descriptions. So we need a formula that will enable us to determine the number of structure-descriptions in each case. It turns out that, if we have n things, and m maximal predicates, the number of structure-descriptions is equal to the number of ways of choosing (m − 1) things from a set of (n + m − 1) things, as can be shown by the following simple, but elegant argument, from Carnap’s Logical Foundations of Probability (1962: 159–60): The structure-descriptions in question may be represented by serial patterns consisting of n dots and m − 1 strokes, as follows: the number of individuals which have the first property is indicated by the number of dots preceding the first stroke; that of the second property by the dots between the first and second stroke, etc.; finally, that of the mth property by the dots following the last stroke. (For example, the pattern ‘. . . //./’ indicates the numbers 3, 0, 1, 0 for the four properties.) Therefore the number sought is equal to the number of possible patterns with n dots and m − 1 strokes. These patterns may be produced by starting with a series of n + m − 1

inductive log ic 153 dots and then replacing a subclass of m − 1 of them by strokes. The number of these subclasses is [n+m−1 Cm−1 ] . . . ; therefore this is also the number of possible patterns.4 j

The expression j Ck is one among a number of expressions, including j Ck , Ck , and   j , that are used to represent the number of ways in which one can choose k things k from a set containing j things. It turns out that j Ck is equal to the following:

j × (j − 1) × (j − 2) × · · · × 3 × 2 × 1 [(j − k) × (j − k − 1) × (j − k − 2) × · · · × 3 × 2 × 1] × [k × (k − 1) × (k − 2) × · · · × 3 × 2 × 1]

This is a rather unwieldy expression. But the product of the natural numbers ranging from j down to 1, that is, j × (j − 1) × (j − 2) × · · · × 3 × 2 × 1 – which is referred to as j factorial – is typically written as j! This allows one to express the value of j Ck j! much more compactly as (j−k)!×k! , or, dropping the multiplication sign in favour of j! juxtaposition, (j−k)!k! Since the number of structure-descriptions that involve m maximal predicates, together with n events, is equal to n+m−1 Cm−1 , that can now be written as (n+m−1)! (n+m−1)! ((n+m−1)−(m−1))!(m−1)! —that is, as n!(m−1)! .

There is one more preliminary that must be discussed before we proceed. Consider a case where there are only two unknown, morally significant properties, one rightmaking and the other wrongmaking, and where the two properties are compatible. Then for any action there are four possibilities: it may have both properties, or only the rightmaking property, or only the wrongmaking property, or neither property. If the action has both properties, the question arises as to what the overall result is. Is the combination of the two properties rightmaking, or is it wrongmaking, or do the two properties cancel one another out? The natural way of dealing with this problem is to formulate things not in terms of the fundamental properties themselves, but in terms of conjunctions of properties and absences of properties. Thus, rather than referring to properties R and W , one can refer to the conjunction of R and W , the conjunction of R with the absence of W , the conjunction of W with the absence of R, and the conjunction of the absence of both. The result is then a family containing four incompatible properties—or quasi-properties. If R is a weightier property than W , the family will contain two rightmaking combinations, one wrongmaking combination, and one neutral combination. If, on the other hand, W is a morally more significant property than R, the family will contain one rightmaking combination, two wrongmaking combinations, and one neutral combination. Finally, if R and W are equally weighty, the family will contain one rightmaking combination, one wrongmaking combination, and two morally neutral combinations.  4 Carnap uses the expression n+m−1 Cm−1 .

 n+m−1 , where I have inserted the alternative expression m−1

154 p robab i l ity i n th e ph i lo s ophy of re l i g i on The upshot, in short, is that regardless of whether all, some, or none of one’s basic properties are mutually incompatible, one can formulate things in terms of a family of incompatible properties, or quasi-properties. This, then, is the approach that I shall be using: the properties (or, if one prefers, quasi-properties) being considered will always belong to a family of properties, where this means that all of the properties, or quasi-properties, are mutually incompatible. This way of proceeding will, without sacrificing any generality, greatly simplify the discussion. Let us consider, then, a world where there are k unknown morally significant properties, all of them mutually incompatible, of which r are rightmaking and w are wrongmaking. It is now a simple matter to calculate the total number of structuredescriptions involving those properties and the n actions, for in this case we have a total of (k + 1) maximal predicates, corresponding to the k mutually incompatible properties, together with the predicate representing the absence of all of them. So, by Carnap’s formula, the total number of structure-descriptions is equal to n+k Ck .

Next, what is the number of positive structure-descriptions? To arrive at the answer, recall Carnap’s idea: one starts out, if there are n actions, and (k + 1) maximal predicates, with (n + k) dots, and changes k of them to strokes. So one starts with the following: ...................... [

.......................

(n + k) dots

]

It does not matter, of course, which property one starts with in replacing dots with strokes to represent the number of things that have that property, nor does it matter whether one starts on the right-hand side or the left-hand side of the series of dots. So let us start on the right-hand side, and, for vividness, let us replace a dot with the letter ‘q’ to represent the quasi-property of not having any of the k unknown morally significant properties. Clearly, it is the dot at the right-hand end of the series that we must replace with that letter, for if there were any dots to the right of the letter, those would represent actions that lacked all of the unknown morally significant properties, and therefore they would be actions that were morally wrong, all things considered. So what we now have is this: ...................... [

.......................q

(n + k − 1) dots

]

Next, let us replace w dots with ‘*’s to represent the w unknown wrongmaking properties. Once again, none of those symbols can have any dots to the right of it, since any such dots would then represent actions that had the unknown wrongmaking property in question, and therefore they would be actions that were morally wrong, all things considered. What we now have is this:

inductive log ic 155 ............. [

. . . . . . . . . . . . . . .∗∗∗∗

(n + k − 1 − w) dots

∗∗∗∗ q

][w occurrences of ∗ ]

But (k–w) is equal to r, the number of unknown rightmaking properties. So we can rewrite this as: ............. [

. . . . . . . . . . . . . . .∗∗∗∗

(n + r − 1) dots

∗∗∗∗ q

][w occurrences of ∗ ]

Finally, we need to replace (r − 1) of the remaining (n + r − 1) dots by strokes to divide the n dots that we shall then be left with up into r groups that will represent how many things have each of the r rightmaking properties. The upshot is that the number of positive structure-descriptions involving r rightmaking properties and w wrongmaking properties, along with n actions, is the number of ways of choosing (r − 1) dots to be replaced with strokes out of the (n + r − 1) dots that remain after we have represented the number of actions—namely, zero—that have either one of the wrongmaking properties or the quasi-property of not having any of the unknown morally significant properties. Accordingly, the number of positive structure-descriptions involving r rightmaking properties and w wrongmaking properties, along with n actions, is equal to n+r−1 Cr−1 .

But what if r = 0? The expression n+r−1 Cr−1 makes no sense in that case. But that is not a problem, since if all of the unknown, morally significant properties are wrongmaking properties, the number of positive structure-descriptions is simply 0. As is clear from the expression n+r−1 Cr−1 , the number of positive structure-descriptions depends only on n, the number of actions, and on r, the number of rightmaking properties. The number of wrongmaking properties—w—is irrelevant, since none of the actions can have a wrongmaking property. The upshot is that one can now set out an expression giving the probability that a structure-description involving n actions, along with a total of k unknown, morally significant properties, of which r are rightmaking properties, will be a positive structure-description. The probability is equal to the ratio of the number of positive structure-descriptions to the total number of structure-descriptions, and so it is given by the expression n+r−1 Cr−1 n+k Ck

—except for the case where r = 0, when it will be simply 0.

156 probabi lity in the phi lo sophy of re lig ion

VI. Step Two: The Case of k Unknown Morally Significant Properties, Given No Further Information The initial equation We now have representations of the probability that a structure-description is positive for the cases where r ranges in value from 0 up to k, and this allows us to move on to the second stage of the argument. The central point here is that each of the (k + 1) possibilities with regard to the number of unknown properties that are rightmaking rather than wrongmaking must be treated as equally likely. For, just as, in the case of particulars and first-order properties, if there are two particulars, and one property, the method of structure-descriptions involves treating it as equally likely that none of the particulars has the property, that one particular does, and that both particulars do, so when the method of structure-descriptions is applied to properties and properties of properties, if one has two first-order properties, and a second-order property, it must be taken as equally likely that none of the first-order properties has the second-order property, that one first-order property does, and that both first-order properties do. Accordingly, to arrive at the probability that a structure-description involving k unknown, morally significant properties, along with n actions, is a positive structure-description, we must average the probabilities associated with those (k + 1) possibilities concerning the number of the k properties that are rightmaking properties. Therefore, if we now introduce the expression ‘P(k, n)’ to represent the probability, if there are k unknown, morally significant properties, that a structure-description involving those properties along with n actions will be a positive structure-description, we have the following equation:   1 P(k, n) = k+1   n+k−1 Ck−1 n+k−2 Ck−2 n+2 C2 n+1 C1 n C0 × + + ... + + + +0 n+k Ck n+k Ck n+k Ck n+k Ck n+k Ck Or, more simply:   1 P(k, n) = k+1   n+k−1 Ck−1 + n+k−2 Ck−2 + . . . . + n+2 C2 + n+1 C1 + n C0 + 0 × n+k Ck A plausible hypothesis concerning a simple expression for the value of P(k, n) The next step is this. We want to find an expression that will enable one to work out the value of P(k, n) much more quickly than the expression we now have—an expression that could be very long indeed. A natural way of

inductive log ic 157   1 attempting to arrive at such an expression is by calculating the value of k+1   n+k−1 Ck−1 +n+k−2 Ck−2 +....+n+2 C2 +n+1 C1 +n C0 +0 for a few small values of k, to see if n+k Ck any pattern is evident.    C0 +0  Suppose k = 1. Then our expression reduces to 12 nn+1 C1 , and this in turn is 1  1  equal to 2 n+1 . So we have:    1 1 . 2 n+1   

P(1, n) =

Next, let k = 2. Then we have 13 n + 1 Cn1++2 Cn C2 0 + 0 , which is equal to   2  1  (n + 1) + 1 + 0 , which reduces to (n + 2)(n + 1) 3 n + 1 . So this gives us: 2

P(2, n) =

1 3

   1 2 . 3 n+1

This looks promising, but let us consider one more value: Then we have   k = 3. + 1)  1  ( (n + 2)(n  1   n + 2 C2 +n + 1 C1 +n C0 +0  )+(n + 1) + 1 + 0 2 , which is equal to 4 , (n + 3)(n + 2)(n + 1) 4 n + 3 C3  (n + 2)(n + 1)+2(n + 2)   3! (n + 2)(n + 3)      ( ) ( ) 2 2 , and thus to 14 , which in turn is equal to 14 (n + 3)(n + 2)(n + 1) (n + 3)(n + 2)(n + 1) 3×2 3! 3  1  which is equal to = 4 n + 1 . So we have that    1 3 . P(3, n) = 4 n+1

We could hardly have  hoped   for a simpler pattern, since it appears that the general k rule is that P(k, n) = k + 1 n +1 1 . A proof by mathematical induction But can it be shown, then, that the following is true:       1 1 k n + k − 1 Ck − 1 + . . . . + n + 2 C2 + n + 1 C1 + n C0 + 0 ? = k+1 k+1 n+1 n + k Ck The answer is that this can be done via mathematical induction, where, first, one shows that the equation is true for k = 1, and then, secondly, one shows that, if it is true for any particular value of k, it must also be true for the next value: (k + 1). First of all, then, the base case, where k = 1. We have already seen that when k = 1,    C0 +0  the left-hand side of this equation reduces to 12 nn+1 C1 , and that this is equal to    1  1  k 1 2 n+1 . But this is equal to k+1 n+1 , when k = 1. This brings us to the inductive step, where it has to be shown that if the equation holds for k, then it also holds for (k + 1). So assume that it holds for k: that is, that

158 probabi lity in the phi lo sophy of re lig ion 

1 k+1



n+k−1 Ck−1

+ . . . . + n+2 C2 + n+1 C1 + n C0 + 0 n+k Ck





k = k+1



 1 . n+1

side of the equation becomes  When   we replace k by (k + 1), the left-hand  1 n+k Ck +n+k−1 Ck−1 +....+n+2 C2 +n+1 C1 +n C0 +0 , while the right-hand side of the k+2   Ck+1  n+k+1 1 equation becomes k+1 k+2 n+1 . Thus we now need to show that, if the equation holds for k, these last two quantities are equal, so that the equation also holds for (k + 1). To see that this is so, let us begin, first, by eliminating the combinatorial denominator n+k+1 Ck+1 in favour of an expression involving n+k Ck . Since n+k+1 Ck+1 is equal to (n+k+1)n+k Ck (n+k+1)! (n+k)! . n!(k+1)! , while n+k Ck is equal to n!k! , one can see that n+k+1 Ck+1 = (k+1) When that substitution is made, we have the following expression: 

1 k+2



n+k Ck

+ n+k−1 Ck−1 + . . . . + n+2 C2 + n+1 C1 + n C0 + 0 (n+k+1) (k+1) n+k Ck

 .

This in turn can be rewritten as:    (k + 1) n+k Ck + n+k−1 Ck−1 + . . . . + n+2 C2 + n+1 C1 + n C0 + 0 . (k + 2) (n + k + 1) n+k Ck Next, let us divide the combinatorial part of this expression into a sum of two fractions, as follows:    (k + 1) n+k Ck n+k−1 Ck−1 + . . . . + n+2 C2 + n+1 C1 + n C0 + 0 + . (k + 2) (n + k + 1) n+k Ck n+k Ck In   of the assumption that the equation   holds  for k,  however, we have that  view 1 k 1 n+k−1 Ck−1 +....+n+2 C2 +n+1 C1 +n C0 +0 = k+1 n+1 . Accordingly, we have n+k Ck   k+1   C +....+ C + C + C +0 Ck k n+2 2 n+1 1 n 0 = 1. If we then = n+1 . In addition, n+k that n+k−1 k−1 n+k Ck n+k Ck substitute into the above expression in accordance with these two equations, we then have:    (k+1) k 1 + n+1 , which then simplifies to   (k+2)(n+k+1)   (k+1) n+k+1 (k+2)(n+k+1) n+1 , which in turn is equal to 

k+1 k+2



 1 . n+1

The latter, however, is just the right-hand side of the equation for the case of (k + 1). It has been shown, accordingly, that if the equation holds for k, it holds for (k + 1). This completes the argument by mathematical induction, and it has therefore been shown that the following equation is true:

inductive log ic 159      1 1 k n+k−1 Ck−1 + . . . . + n+2 C2 + n+1 C1 + n C0 + 0 . = k+1 k+1 n+1 n+k Ck    k 1 It has been proven, then, that P(k, n) = k+1 n+1 , where ‘P(k, n)’ has been defined as the probability, if there are k unknown, morally significant properties, that a structure-description involving those properties, together with n actions, will be a positive structure-description. But since a structure-description cannot be a favourable structure-description unless it is a positive one, P(k, n) is an upper bound on the probability, if there are k unknown, morally significant properties, that a structure-description involving those properties, together with n actions, will be not only a positive structure-description, but a favourable one. P(k, n) represents, then, an upper bound upon the probability that none of the n events in question is such that it would be morally wrong to allow that event, given the totality of all rightmaking and wrongmaking properties, known and unknown. 

VII. Step Three: An Upper Bound that Does Not Depend upon Information about the Number of the Unknown, Morally Significant Properties The third and final step involves finding a formula that both places an upper bound upon the probability that a structure-description involving the attribution of unknown rightmaking and wrongmaking properties to n actions is a favourable one, and that does not involve any information about the number of unknown, morally significant properties that there are.  can now be done very quickly, for given the simple expression P(k, n) =  This k 1 k+1 n+1 , it is immediately evident how that upper bound depends upon k and n. First of all, given any fixed value for n, the upper bound increases as k increases. Secondly, given any fixed value for k, the upper  bound decreases as n increases. Thirdly, k for a fixed value of n, since the limit of k+1 , as k increases without limit, is equal to   1 . one, the limit of the upper bound is equal to n+1 We have, in short, the following important conclusions: (1) The upper bound on P(k, n) is a monotonically increasing function of k. (2) The upper bound on P(k, n) is a monotonically decreasing function of n. (3) For a fixed value  of n, the upper bound on P(k, n), for any value of k, cannot be 1 greater than n+1 . The upshot is this. Consider any n events where each event is such that, judged only in terms of the morally significant properties of which we are aware, it would be morally wrong to allow the event in question. Then, regardless of how many unknown morally significant properties the world contains, the probability that none of those

160 p robab i l ity i n th e ph i lo sophy of re l i g i on events is such that it is morally wrong to allow that event, in the light of the totality of all rightmaking  and wrongmaking properties, known and unknown, can never be 1 greater than n+1 . We can construct, then, the following sort of table: Value of n 1 2 3 4 5 100 1000 1,000,000 1,000,000,000

Upper Bound on P( k,n), for any value of k: 1/2 1/3 1/4 1/5 1/6 1/101 1/1001 1/1,000,001 1/1,000,000,001

VIII. The Likelihood that God Exists How many events that have already taken place in the world are such that, judged only by the rightmaking and wrongmaking properties of which we are aware, it would have been morally wrong to have allowed the event in question? It would seem that the number of such events must be very large indeed. For given that the present population of the world is about six billion, an enormous number of people have died in the past, many of whom also experienced the physical and mental deterioration associated with aging, and judging by the rightmaking and wrongmaking properties of which we are aware, allowing either of these things would surely have been morally justified only in a very small proportion of cases. So there are relevant events such that the value of n is very high indeed, with the result that the upper bound on P(k, n) is very, very low. This in turn entails that the probability that God exists is also very, very low, in view of the following argument: (1) Let n be the number of events in the world that are such that, judged only by the rightmaking and wrongmaking properties of which we are aware, it would have been morally wrong to have allowed any of those events. Then n is a very large number.   1 is an upper bound on P(k, n), (2) If n is a very large number, then, since n+1 there is an upper bound on P(k, n) that is a very small number. Therefore, from (1) and (2): (3) There is an upper bound on P(k, n) that is a very small number.

inductive log ic 161 (4) Any upper bound on P(k, n) is an upper bound on the probability that none of the n events in question is such that it would have been morally wrong to have allowed that event. Therefore, from (3) and (4): (5) There is a very small number that is an upper bound on the probability that none of the n events in question is such that it would have been morally wrong to have allowed that event. (6) Given that God is by definition an omnipotent, omniscient, morally perfect, and everlasting creator of the universe, the proposition that God exists entails the proposition that at every past time there was an omnipotent, omniscient, and morally perfect person. (7) The proposition that at every past time there was an omnipotent, omniscient, and morally perfect person entails that none of the n events in question is such that it would have been morally wrong to have allowed that event. Therefore, from (6) and (7): (8) The proposition that God exists entails that none of the n events in question is such that it would have been morally wrong to have allowed that event. (9) For any propositions p and q, if p entails q, then any upper bound on the probability that q is true is also an upper bound on the probability that p is true. Therefore, from (8) and (9): (10) There is a very small number that is an upper bound on the probability that God exists.

IX. Alternative Inductive Logics? As I noted at the beginning of Section III, the inductive logic that I am using is essentially that set out by Rudolf Carnap in his book Logical Foundations of Probability (1962). But Carnap later developed a modified system, set out in his two-part article ‘A Basic System of Inductive Logic’ (1971 and 1980), which is only slightly more complex, and which avoids certain objections. Fitelson (2010) has suggested that Carnap’s later system is to be preferred, and other writers, such as Patrick Maher in his article ‘Explication of Inductive Probability’ (2010), which contains a very concise and accessible exposition of Carnap’s later approach, have defended that alternative approach to inductive logic. Though I have not done so here, I think that the idea of recasting the above argument in terms of a slightly different system of inductive logic is well worth pursuing. But I favour a system that diverges slightly from both Carnap’s earlier system, and also his later one. My reason is this. Carnap’s later approach differs from his earlier approach in the following way. His earlier approach involved a single system that generated definite logical probabilities for one proposition relative to another; the later approach, by contrast, does not do so. Instead, it involves a general framework that covers a variety

162 probabi lity in the phi lo sophy of re lig ion of possible systems of inductive logic. Part of the reason for this is that Carnap thought that the probability that something has a given property depends not only upon how many other things have precisely that property, but also upon the extent to which things have properties that resemble, to a greater or lesser extent, the property in question (1980: 32–49). But even when this ‘analogy influence’ was set aside, Carnap, on his later approach, was still left with a range of different inductive systems. Those different systems, in turn, could be expressed via a formula that involved a variable, λ, whose values could range from 0 to ∞. One had, then, what was referred to as a λ-continuum of inductive systems. Carnap (1980: 94–5) argued that there were decisive objections to setting λ equal to either of the extreme values 0 or ∞, but in the case of other values there seemed to be nothing that one could appeal to beyond the fact that certain choices—such as the choice of very large values for λ—lead to results that are intuitively implausible (1980: 107–8). But even if such considerations are accepted as epistemically relevant, one is still left with a significant range of possible values for λ, an outcome that Carnap did not find entirely satisfactory (1980: 11–19). The alternative approach that I think is worth exploring is as follows. A key feature of Carnap’s later approach involves the idea of a family of properties, where a family of properties is a set of genuine properties that are mutually exclusive, and where the set is not a proper subset of some larger set of mutually exclusive, genuine properties. Thus the set of phenomenal colour properties, and the set of all mass properties, are families of properties. Given this concept, the key feature is this. Suppose that, using one of Carnap’s later systems, and given a framework that involves n families of properties, the logical probability of hypothesis H relative to evidence E is equal to k. Suppose, further, that H and E do not involve all n families. Then given a framework that involves only the families of properties that are involved either in H or in E, the probability of H given E must also be equal to k. In short, ignoring families for properties that are not involved either in the hypothesis or the evidence does not affect the probability of H upon E. The approach to inductive logic that I have in mind would share this feature, but it would differ from Carnap’s approach in the following way. On Carnap’s approach, if one has a hypothesis H and evidence E that involve only a single family of properties, the logical probability of H given E depends upon the specific system of inductive logic that one chooses. In particular, different values of λ will generate different probabilities. On the approach that I am inclined to favour, by contrast, the value of the logical probability of H given E in such a case would be generated by applying a modified version of the structure-description approach to inductive logic that characterized Carnap’s earlier system of inductive logic. The basic idea is as follows. Consider a possible world that, like the mini world of Section II, contains only three individuals—a, b, and c—but that differs in that it involves many families of properties, one of which contains only property P. Then Pa & Pb & ∼Pc is no longer a state-description, since a state-description specifies all

inductive log ic 163 of the properties that each individual has. But Pa & Pb & ∼Pc does specify, for every individual, whether that individual has or lacks the single positive property P that belongs to the family of properties that contains only property P. So let us refer to Pa & Pb & ∼Pc as a ‘family-relative state-description’. As with state-descriptions, so with structure-descriptions. Consider the proposition that all three individuals have property P. In our original mini world, that was a structure-description, since it contained complete information about how many individuals had the only positive property in that world. In the expanded mini world, where there are many families of properties, the proposition that all three individuals have property P is no longer a structure-description, since it contains no information about the many families of properties other than the one that contains P as its only positive property. But it does give one complete information about how many individuals have the single positive property belonging to that family. Let us therefore refer to it, then, as a ‘family-relative structure-description’. Given these family-relative conceptions of state-descriptions and structure-descriptions, the central idea is simply this. On Carnap’s earlier approach, equal probabilities were assigned to all structure-descriptions, and the sum of those probabilities was equal to one. In addition, the probability assigned to a given structure-description was then distributed equally among the state-descriptions falling under that structure-description. On the approach that I am suggesting, equal probability is assigned to all the family-relative structure-descriptions associated with a given family of properties, and the sum of those probabilities is to be equal to one. The probability assigned to a given family-relative structure-description is then to be distributed equally among the family-relative state-descriptions falling under that family-relative structure-description.

X. Summing Up As I said at the beginning of Section II, it seems to me that there is only one way that progress can be made in the philosophical examination of the argument from evil, namely, by bringing serious inductive logic to bear upon the question. In this essay, I have attempted to do this by using Carnap’s structure-description approach to inductive logic to set out a mathematical formulation of the argument from evil. As I have just indicated, however, it may well be that some alternative system of inductive logic would be preferable. It seems to me very likely, however, that very similar results to those that I have derived here can also be proved, by essentially parallel arguments, in any plausible alterative system of inductive logic. In any case, given the approach that I have followed here, we have the following results. First of all, given n events, each of which is such that, judged simply by known rightmaking and wrongmaking properties, it would have been morally wrong to allow that event, the probability that, judged in the light of all rightmaking and wrongmaking properties, known and unknown, it was not morally wrong to allow any 1 . Secondly, and as we have also seen, additional of those events, must be less than n+1

164 p robab i l ity i n th e ph i lo s ophy of re l i g i on information simply about the number of unknown morally significant properties that 1 . Thirdly, there is good exist could not serve to make the probability greater than n+1 reason for supposing that the number of events, n, in the actual world, each of which is such that, judged simply by known rightmaking and wrongmaking properties, it would be morally wrong to allow that event, is very high, and therefore there is good 1 is very low. Fourthly, since the reason for holding that the corresponding value of n+1 proposition that God exists entails that none of the n events in question is such that it was morally wrong to allow that event, the probability that God exists must also be 1 . The probability that God exists is, therefore, very low indeed relative to less than n+1 facts concerning the evils found in our world.

References Adams, R. M. (1985) ‘Plantinga on the Problem of Evil’. In J. E. Tomberlin and P. van Inwagen (eds), Alvin Plantinga. Dordrecht: D. Reidel, 225–55. Carnap, R. (1962) Logical Foundations of Probability. 2nd edition. Chicago: University of Chicago Press. (1971) ‘A Basic System of Inductive Logic’, Part I. In Rudolf Carnap and Richard C. Jeffrey (eds), Studies in Inductive Logic and Probability, Volume I. Berkeley and Los Angeles: University of California Press, 33–165. (1980) ‘A Basic System of Inductive Logic’, Part II. In Richard C. Jeffrey (ed.), Studies in Inductive Logic and Probability, Volume II. Berkeley and Los Angeles: University of California Press, 7–155. Fitelson, B. (2010) ‘Comments on Alexander Pruss’s “Tooley’s Use of Carnap’s Probability Measure” ’: http://prosblogion.ektopos.com/archives/2010/02/tooleys-use-of.html. Goodman, N. (1955) Fact, Fiction, and Forecast. Cambridge, MA: Harvard University Press. Hasker, W. (1988) ‘Suffering, Soul-Making, and Salvation’, International Philosophical Quarterly, 28: 3–19. Hick, J. (1966) Evil and the God of Love. New York: Harper & Row. Maher, P. (2010) ‘Explication of Inductive Probability’, Journal of Philosophical Logic, 39: 593–616. Plantinga, A. (1974) God, Freedom, and Evil. New York: Harper & Row. (1985) ‘Self-Profile’. In J. E. Tomberlin and P. van Inwagen (eds), Alvin Plantinga. Dordrecht: D. Reidel, 3–97. and M. Tooley (2008) Knowledge of God. Oxford: Blackwell. Pruss, A. (2010) ‘Tooley’s Use of Carnap’s Probability Measure’: http://prosblogion.ektopos. com/archives/2010/02/tooleys-use-of.html Swinburne, R. (1979) The Existence of God. Oxford: Clarendon Press. (1988) ‘Does Theism Need a Theodicy?’, Canadian Journal of Philosophy, 18: 287–312. (1996a) Is There a God? Oxford: Oxford University Press. (1996b) ‘Some Major Strands of Theodicy?’ In D. Howard-Snyder (ed.), The Evidential Argument from Evil. Bloomington: Indiana University Press, 30–48. Tooley, M. (2010) ‘The Problem of Evil’. In E. N. Zalta (ed.), Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/evil/.

PART IV

Pascal’s Wager

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9 Blaise and Bayes Alan Hájek

I. Introduction Pascal’s Wager, like the Ontological Argument, is a watershed in the philosophy of religion. And just as ‘the Ontological Argument’ is something of a misnomer—there are many such arguments—so is ‘Pascal’s Wager’. In a single paragraph of his Pensées, Blaise Pascal in fact presents at least three wagers—at least three arguments for believing, or for cultivating belief, in God.1 Ian Hacking offers three reconstructions of them using the apparatus of Bayesian decision theory (see Hacking 1994). He also makes a point of granting that each of these arguments is valid. While he questions Pascal’s premises, he is emphatic that Pascal’s conclusion really does follow from those premises in each case. I will contend here, equally emphatically, that all three arguments are invalid. Hacking portrays the first wager as an ‘argument from dominance’, although as we’ll see, he is unclear about what that means. He contends that an argument from dominance is automatically valid: if there is a dominating act, then rationality requires it to be performed. I demur, and I show that such an argument is invalid without a further assumption about the relevant probabilities. Edward McLennen characterizes the first wager with a stronger sense of dominance, which he calls ‘superdominance’ (see McLennen 1994). But I argue that even superdominant acts need not be performed, with no offence to rationality. He understates the strength of the first wager: it is stronger than one from superdominance in his sense, twice over. So I strengthen his notion of superdominance twice over, finally doing justice to the first wager. And yet still the argument is invalid, I argue. I offer two ways that the premises of the argument could be strengthened so as to render it valid, although then its premises are considerably less plausible. Nonetheless, I find textual support for attributing the first strengthening to Pascal himself. I next turn to Pascal’s second and third wagers, which famously appeal to considerations of expected utility. The second wager makes the highly implausible assumption that the probability of God’s existence is exactly ½, but the third wager weakens this 1 Indeed, McLennen (1994) finds four such arguments.

168 probabi lity in the phi lo sophy of re lig ion assumption to the probability of God’s existence merely being positive—this is Pascal’s best-known ‘Wager’. Rehearsing and building on points I made in my paper ‘Waging War on Pascal’s Wager’, I argue that these wagers are also invalid (Hájek 2003a). I then attempt some reparation work. In that paper, I offered four valid reformulations of the Wager, although I argued that they did violence to Pascal’s theology. Here I will offer two further valid reformulations, and at least the first of these seems to be more faithful to that theology. In the process I will suggest and defend new decision rules appropriate for the comparison of acts of infinite and indeterminate expected utility.

II. The Argument from Dominance Hacking has done us a service in distinguishing three arguments of Pascal’s, and in showing how each can be given a decision-theoretic gloss. I will follow his naming of the arguments, but I will differ from him on some points of detail, and I will part company with him when it comes to assessing them.2 The first argument is ‘the argument from dominance’. Pascal maintains that we are incapable of knowing whether God exists or not, yet we must ‘wager’ one way or the other. Reason cannot settle which way we should incline, but a consideration of the relevant outcomes supposedly can: You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. . . . Let us weigh the gain and the loss in wagering that God is. . . . If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. (Pascal 1910: 233)3

So we have two possible states of the world: God exists, or God does not exist; and two possible actions: wagering that God exists, and wagering that he does not. It will not be important for our purposes to determine exactly what such ‘wagerings’ involve. Let’s simply accept Pascal’s idea that wagering for God involves living a pious life, ‘taking the holy water, having masses said, etc.’, steps likely to foster (or to sustain) belief in God; and that to avoid taking such steps is to wager against God. We can formulate the argument in terms of the following decision matrix: Wager for God Wager against God

God exists Salvation Misery

God does not exist Status quo Status quo

Under either hypothesis about the way the world is—God exists, or does not—the result of wagering for God is at least as good as the result of wagering against God. Thus, according to Pascal, wagering for God is rationally required. Hacking says 2 Here I will extend considerably my discussion in (1997). 3 All quotes by Pascal are taken from Pascal (1910).

blaise and baye s 169 that ‘the wager “God is” dominates the wager “he is not”. The decision problem is solved. The argument is valid’ (1994: 25). Earlier, Hacking explains that dominance occurs ‘when one course of action is better no matter what the world is like . . . ’ (1994: 22). This is usually called strict dominance. Firstly, note that we do not have a case of such dominance here, for if God does not exist, the outcomes are essentially the same. In Hacking’s words: ‘If God is not, then both courses of action are pretty much on a par’ (1994: 25). In the jargon, we have instead a case of weak dominance.4 Hacking writes: ‘If one act dominates all others, the solution to our decision problem is “perform the dominant act”’ (1994: 22). We see, then, why he regards the argument as valid. But this is too quick: even strictly dominating actions are not necessarily the best ones to perform. Whether they are or not depends on the relevant conditional probabilities of the states, given the actions (according to evidential decision theorists), or probabilities that reflect the causal efficacy of the actions in bringing about the states (according to causal decision theorists). In any case, probabilistic considerations can make all the difference. Consider the following decision problem. There are two possible states of the world: you are offered an attractive job, or you are not. There are two possible actions you can perform: you can apply for the job, or not. Given that you are offered the job, you prefer not to have applied: what an honour to be offered the job out of the blue! Given that you are not offered the job, you would prefer not to have applied: you don’t like being rejected! You want the job, above all, and you have a decent chance of getting it; but you know that in order to get it, you must apply for it. Here is the decision matrix, with the outcomes ranked from best to worst: Not apply Apply

You are offered job best 2nd best

You are not offered job 3rd best worst

Not applying strictly dominates applying; yet applying is clearly the best thing for you to do. Note that we can a fortiori make the same point for a weakly dominating act. Change the story so that you are indifferent between applying and not applying in the case that you are offered the job either way. Despite the fact that not applying then weakly dominates applying, applying is still the best thing to do, as a consideration of the relevant probabilities will bring out. Punch line: neither weak nor strict dominance is sufficient to make an act choice-worthy.5 The states of the world and your actions are not probabilistically or causally independent of each other in this case: it is far more likely that you are offered the job given that you apply than given that you do not, and your applying for the job is 4 Immediately after his explanation of dominance, as quoted in the previous paragraph, Hacking goes on to formulate the notion ‘[s]chematically’. But that turns out to be a formulation of weak dominance, thus something different from what immediately preceded it. 5 This is hardly an original point—for example, one-boxers in Newcomb’s problem and co-operators in the Prisoner’s Dilemma have been saying this for years, although these cases are more controversial.

170 probabi lity in the phi lo sophy of re lig ion causally relevant to your getting it. Now perhaps in Pascal’s argument, the states and actions are probabilistically and causally independent of each other: ‘whether God exists or not does not depend on what I do’, a friend of dominance reasoning might say. And arguably if the states and actions are independent of each other, then even weak dominance is sufficient to make an act choice-worthy. (Soon we will see that even that is too quick.) But it’s not clear that rationality requires you to regard what you do to be independent of whether God exists or not. Maybe by your lights there is positive dependence between wagering for God and his existence: maybe God helps people come to believe in Him, in which case your wagering for God provides some evidence that He exists. To be sure, that sort of dependence would only help the Wager. But maybe by your lights there is negative dependence, which would undermine the dominance reasoning. Pascal would need a further premise that rationality forbids this—a constraint that would sit awkwardly with Bayesian orthodoxy’s permissiveness about prior probability assignments, and perhaps even with his own insistence that ‘[r]eason can decide nothing here’. In any case, this would introduce probabilistic considerations, which are supposed to play absolutely no role in this first argument. The fact that wagering for God weakly dominates, or even strictly dominates, wagering against God is not enough to clinch the case for wagering for God. More germane, perhaps, is the fact that in a certain specific sense, wagering for God (weakly)6 superdominates wagering against God, according to Pascal. McLennen suggests that this is really an argument from a principle of superdominance: each of the outcomes associated with betting on God is at least as good as, or better than, each of the outcomes associated with betting against God. In such a case, we are spared the burdensome business of sorting out notions of probabilistic and causal dependence or independence between our choice of an action and the relevant states. (1994: 118)

If wagering for God superdominates wagering against God like this, one cannot go wrong by wagering for God: even in the worst case scenario, there is no way that wagering against God could do better. More germane, perhaps, but still not good enough: (weakly) superdominant acts in McLennen’s sense are not automatically the best ones to perform, not necessarily rationally mandated. Consider this case: either it is snowing in Siberia, or it is not, a matter of indifference to me; and I am indifferent between receiving a dollar bill and receiving two 50 cent coins: Snow in Siberia No snow in Siberia Receive a dollar bill $1 richer $1 richer Receive two 50 cent coins $1 richer $1 richer

6 I prefix the word ‘weakly’, since the inequality ‘at least as good as’ is weak; the words ‘or better than’ don’t add anything.

blaise and baye s 171 Each of the outcomes associated with receiving a dollar bill is at least as good as each of the outcomes associated with receiving two 50 cent coins. So receiving a dollar bill (weakly) superdominates receiving two 50 cent coins in McLennen’s sense. Yet choosing to receive a dollar bill is not rationally mandated. After all, receiving two 50 cent coins is just as good—indeed, it (weakly) superdominates receiving a dollar bill! I cannot go wrong with receiving a dollar bill; but I cannot go wrong with receiving two 50 cent coins either. More germane still is the fact that wagering for God (weakly) superdominates wagering against God in a stronger sense, according to Pascal: each of the outcomes associated with wagering for God is at least as good as each of the outcomes associated with wagering against God and in at least one state of the world, wagering for God is strictly better than wagering against God. More generally, say that act 1 superdominates+ act 2 if each of the outcomes associated with act 1 is at least as good as each of the outcomes associated with act 2, and in at least one state of the world, act 1 yields an outcome that is strictly better than act 2 does. The case for wagering for God is even better than McLennen makes it out to be, since wagering for God does not merely superdominate wagering against God—the former superdominates+ the latter. More germane still, but still not good enough: even superdominant+ acts are still not automatically the best ones to perform. Consider this case: ordinarily I would rather go to the party than stay home. But I know that a somewhat irritating person will make a point of going if I go: he pays close attention to my movements, and he loves to corner me, regaling me endlessly with his philosophical views. Going to a party free of him is my favourite outcome, but his presence there would make me indifferent between going, and staying home. Here is the ranking of the outcomes, with a three-way tie for second best: I go I stay home

He does not go Best 2nd best

He goes 2nd best 2nd best

Going superdominates+ staying home in this stronger sense. But I am still not rationally compelled to prefer going to staying home. For even though I do not rule out his not going (I assign positive probability to the first column), I do rule out his not going given that I go. I am convinced that I will get a second best outcome either way, so I can just as rationally stay home. Note that we are not spared the burdensome business of sorting out notions of probabilistic and causal dependence or independence between our choice of an action and the relevant states, for it is just such dependence that undercuts the superdominant, and even the superdominant+ act. But we can imagine the case for going being still stronger. Suppose that the scenario in which I stay home and he does not go is strictly worse than all the others—I vividly picture how much fun the party is, all the more so because the annoying guy isn’t there, and I regret the fact that I am missing it!

172 probabi lity in the phi lo sophy of re lig ion

I go I stay home

He does not go Best Worst

He goes 2nd best 2nd best

We have started with the previous example’s superdominance+ of going over staying home, and we have strengthened it by making one of the payoffs associated with staying home strictly worse than before—in particular, worse than any of the payoffs associated with going to the party. More generally, say that act 1 superdominates++ act 2 if each of the outcomes associated with act 1 is at least as good as each of the outcomes associated with act 2, and in at least one state of the world, act 1 is strictly better than act 2, and in that state act 2’s outcome is worse than any of the payoffs associated with act 1. Stronger, but still not strong enough! For consistent with all the assumptions we have made, I might assign probability 1 to his going to the party. In that case, I am certain that column 2 will be realized, and I can rationally realize a second best outcome by staying home: after all, by my lights the best outcome can’t be realized. This case parallels Pascal’s first wager in its utility structure. Obviously the payoffs of the outcomes are ordered, with ‘salvation’ best, ‘status quo’ second best, and ‘misery’ worst: Wager for God Wager against God

God exists Best Worst

God does not exist 2nd best 2nd best

If the probability structures are also parallel, then one can likewise rationally realize a second best outcome in the wager. This will be the case if one assigns probability 0 to God’s existence. Again, ruling this out requires an assumption about the probabilities after all. Let me summarize what I have argued so far. Both Hacking and McLennen have understated Pascal’s case here in favour of wagering for God: it doesn’t merely dominate wagering against God, in Hacking’s sense, and it doesn’t merely superdominate wagering against God, in McLennen’s sense. It doesn’t even merely superdominate+ wagering against God! Wagering for God superdominates++ wagering against God!! Yet still the argument is invalid. An atheist who is probabilistically certain that column 2 is realized can rationally realize a second best outcome by wagering against God: after all, by his lights the best outcome cannot be realized. Notice that I said ‘probabilistically certain’ rather than ‘certain’: probability 1 does not imply certainty. For example, suppose a fair coin is tossed infinitely many times. It is not certain to land heads eventually—it could land tails on every toss. Yet the probability of it landing heads eventually is 1. (If you think it is infinitesimally less than 1, see Williamson 2007.) Probability theorists are not being perverse when they call their various limit theorems ‘almost sure’ results: the strong law of large numbers, the central limit theorem, and so on. This is not misleading, the way that calling ¬(p & ¬p) ‘almost a tautology’ would be. The relevant limiting behaviour is not logically necessary.

blaise and baye s 173 This raises the interesting question of whether an agent who gives probability 1 to God’s non-existence, but who is not certain of it, should be moved by this argument from superdominance++. Standard expected utility theory says no: when multiplying the utility of salvation by probability 0 for God’s existence in the expectation formula, the resulting 0 makes no contribution to the expectation, whether it represents sure-atheism, or merely almost-sure-atheism. Yet offhand, this seems to be a strike against standard expected utility theory, for it seems to make a difference to superdominance++ reasoning whether God’s existence is a doxastic possibility or not: if it is, then one ought to believe (granting Pascal his utility matrix). In that case I should strengthen the point: if our atheist is a strict atheist, with God’s existence not even a doxastic possibility, then he is not rationally required to wager for God, despite its superdominance++ over wagering against God. I said earlier that a friend of dominance reasoning might assume that the states and acts are independent of each other. Ironically, we now have a special case of independence of states from acts! Anything that has probability 1 or 0 is probabilistically independent of everything, according to the usual Kolmogorovian/Bayesian construal of independence. After all, according to that construal, X is independent of Y iff P(X and Y ) = P(X) × P(Y ). And if P(X) = 0, then for any Y we have this equality holding: 0 = 0 × P(Y ). So even granting independence of states from acts—which protects against a certain kind of specious dominance reasoning such as in the initial job-offer example—is not enough: it had better not be the degenerate case of independence that comes from assigning probability 1 to one of the states. Said another way, even if we are spared the burdensome business of sorting out notions of probabilistic and causal dependence or independence between our choice of an action and the relevant states, we are still not done. Probabilistic independence is not sufficient even for a superdominant++ act to be rationally mandated—it isn’t if it is the cheap sort of independence that one gets from extreme probabilities. In sum, we have come some distance from Hacking’s characterization of the argument, every step that we have taken has only improved the case for wagering for God, and still the argument for wagering for God is invalid. I must therefore disagree with Hacking’s assessment of Pascal’s first wager. However we construe the Wager—as an argument from dominance, from superdominance, from superdominance+, or from superdominance++—it is simply invalid. Now to turn it into a valid argument, Pascal could: 1. Insist that each of the outcomes associated with wagering for God is strictly better than all of the outcomes associated with wagering against God: the worst outcome associated with wagering for God is strictly better than the best outcome associated

174 p robab i l ity i n th e ph i lo sophy of re l i g i on with wagering against God. We might call this superduperdominance of the former over the latter. Then, finally, considerations of utilities alone would carry the day for wagering for God. This modification would require some further argument, however, for we would need to be convinced that one benefits from having a false belief in God. Offhand, this seems less plausible than the original Wager’s assumption about the utilities. As if anticipating this concern, Pascal adds the following coda at the end of all the wagers: Now, what harm will befall you in taking this side [wagering for God]? You will be faithful, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognise that you have wagered for something certain and infinite, for which you have given nothing. [My italics.]

In my terminology, Pascal seems to be claiming that wagering for God superduperdominates wagering against God after all. If he is right, then there is a good sense in which you do not really face a wager at all: come what may, you are better off wagering for God, period. In that sense, the foregoing wagers might be regarded as otiose. But they nonetheless serve an important dialectical role, seeking to convince someone who does not grant Pascal this strong claim. Furthermore, while it is hardly a criticism of Pascal’s reasoning, an argument from superduperdominance is of little decision-theoretic interest. In his classic work on the history of probability, Hacking describes Pascal as giving us ‘the first well-understood contribution to decision theory’ (1975: viii). As such, we should be grateful that Pascal also offered wagers for God whose resolutions were not so trivial! 2. Deny that rationality permits extreme atheism: forbid an assignment of probability 0 to God’s existence. But one wonders whether this is more rational theology than Pascal should allow himself. How does one square this with his claim that ‘[r]eason can decide nothing here’? For now it seems that reason can decide at least something here: namely, that certain probability assignments, consistent with the probability calculus and orthodox Bayesianism, are rationally impermissible. (See Appendix A.) Pascal’s reasoning appears to be powerful because it apparently makes minimal assumptions about probabilities—most of the work is done by the utilities. This is a familiar point regarding the third, and most famous, of his ‘wagers’, to which I will turn shortly: it only seems to assume that your probability for God’s existence is positive, a very weak assumption. (To be sure, assigning infinite utility to salvation is a very strong assumption.) It is tempting to think that his first wager makes even weaker probabilistic assumptions: namely, none at all! I have argued that this is not the case. Holding fixed the original payoffs, to render the argument valid, a probabilistic assumption is needed: that rationality forbids the assignment of probability 0 to God’s existence. Whether or

blaise and baye s 175 not this assumption is plausible in the end, the fact remains that Pascal’s first argument is invalid as it stands. We have introduced probabilities into the discussion. Pascal does so explicitly in his second and third wagers. Let us move on, then, to further invalid arguments.

III. The Argument from Expectation Pascal is aware of the objection that wagering for God may require giving something up, so that if God does not exist, the wagerer-for is worse off than the wagerer-against. No form of dominance argument could then apply. So Pascal now makes two pivotal moves: he asserts that salvation brings infinite reward (‘an infinity of an infinitely happy life’); and that the probability of God’s existence is 1/2. He then essentially argues that the expectation of wagering for God exceeds that of wagering against God, and hence wagering for God should be preferred. Hacking puts it this way: In the agnostic’s existential situation, the optimal payoff if there is no God is a worldly life. The optimal payoff if there is a God is salvation, of incomparably greater value. Hence, if there is an equal chance of God’s existence or nonexistence, the expectation of choosing the pious life exceeds that of choosing the worldly one. The argument from expectation concludes, act so that you will come to believe in God. . . . [The argument] is valid . . . (1994: 27).

Hacking must be tacitly assuming that salvation is also of incomparably greater value than the result of wagering against God if God exists—otherwise the argument is clearly invalid. For it is consistent with what he says here that if there is a God, wagerers-against also enjoy the optimal payoff: God bestows salvation upon wagerers-for and wagerers-against alike. In that case, the expectation of choosing the pious life need not exceed that of not choosing the pious life.7 Indeed, if the ‘worldly life’ of a wagerer-for is worse than that of a wagerer-against (which was the point of the objection that Pascal responds to, after all), then by a (weak) superdominance argument it would seem that wagering-against is preferable. And if, as in the first argument, the worldly lives of wagerer-for and wagerer-against, in the case that God does not exist, are the same (both ‘status quo’), then the two expectations are the same, so again there is no reason to prefer wagering for God. Pascal is frustratingly obscure at this crucial point, and he almost deserves misinterpretation here. But I will grant that he makes the assumption that wagering against God results in some incomparably lesser reward than salvation, presumably misery once again (although a worldly life would also do):

7 A little pedantically, I avoid Hacking’s phrase ‘choosing the worldly [life]’, since ‘the worldly life’ is here understood as an outcome, not an action. Suppose the wagerer-against gets something much worse than a worldly life if God exists. Then if God exists, one can’t choose a worldly life at all: one either gets something much better, or something much worse.

176 probabi lity in the phi lo sophy of re lig ion

Wager for God Wager against God

p = 1/2 God exists Salvation Misery

1 − p = 1/2 God does not exist Worldly life1 Worldly life2

I distinguish here the two worldly-life outcomes, although this is not important. What is important is that salvation is sufficiently better than all the other outcomes.8 We need not assume that salvation constitutes an infinite reward in order to make the expectation of wagering for God exceed that of wagering against God. But Pascal unequivocally does make that assumption. So he attaches utilities to the outcomes as follows (where the fi are finite numbers):9

Wager for God Wager against God

p = 1/2 God exists ∞ f2

1 − p = 1/2 God does not exist f1 f3

Now the argument has at least the appearance of validity. The expectation of wagering for God is ∞.1/2 + f1 .1/2 = ∞. This exceeds the expectation of wagering against God, namely, f2 .1/2 + f3 .1/2 = some finite value.10 So far, so good. Still, I contend that Pascal’s conclusion that one must wager for God does not follow. In a nutshell, the main point is that the actions {wager for God, wager against God} do not exhaust the set of possible strategies that one might adopt. Moreover, there are other strategies besides wagering for God that have infinite expectation, and Pascal must rule these out if he wants validly to conclude that 8 Note that Hacking makes a mistake here: ‘The argument from expectation with an equal probability distribution requires only that salvation, if God is, is more valuable than sinful pleasures, when there is no God’ (1994: 27). No—to see that this is not so, consider:

Wager for God Wager against God

p = 1/2 God exists 1 0

1 − p = 1/2 God does not exist −3 0

The utility of salvation is 1 (there is no question of salvation ‘if God is not’, so I don’t understand Hacking’s use of a conditional here). This is greater than the 0 utility of sinful pleasures (pursued by the wagerer-against, presumably) when there is no God. Yet the expectation of wagering for God is –1, which is less than the expectation of wagering against God, 0. 9 In assigning a finite value to ‘misery’, I am assuming that the misery is not infinitely bad, having utility −∞. More on this later. 10 The calculations will work even if you think that what you do is not independent of whether God exists—just replace the unconditional probabilities by the appropriate conditional probabilities.

blaise and baye s 177 wagering for God is the unique optimal strategy. I develop this point in the next section.

IV. The Argument from Dominating Expectation Hacking rejects the argument from expectation because of its ‘monstrous premiss of equal chance’ (1994: 27). Said more precisely, what is implausible—yes, perhaps even monstrous—is the assumption that rationality requires one to assign equal probability to God’s existence and His non-existence. But note that the probability assignment of 1/2 really plays no role in the argument. Any positive (finite)11 assignment would work equally well. So for the third argument, Pascal keeps the same decision matrix, but now essentially allows p to be a variable, ranging over all positive values:

Wager for God Wager against God

p>0 God exists ∞ f2

1−p pJ , and wager for the jealous god if pJ > pG . Example 6

Wager for jealous god Wager for grouchy god Wager against all

Jealous god exists

Grouchy god exists

No god exists

1 0 0

0 0 1

0 0 0

Example 7 ( jealous and nice) Next, let’s compare Pascal’s jealous god with a very nice god who rewards everybody. We go straight to the relative decision matrix. The obvious calculations show that wagering for the jealous god is the dominant choice. Plainly, very nice gods finish last. There is just no point in wagering for them.

15 Mougin and Sober accomplish the same result by replacing the infinite reward of salvation with a large finite reward, but employing relative utilities makes this step unnecessary.

196 probabi lity in the phi lo sophy of re lig ion Example 7

For jealous god For nice god Wager against all

Jealous god exists

Nice god exists

No god exists

1 0 0

1 1 1

0 0 0

As a variation (call it Example 7A), substitute 0 for 1 at the top of the second column, in order to represent a deity who likes everyone except followers of the jealous god. In this case, the decision about how to wager depends upon a comparison of the relevant subjective probabilities. You should wager for the jealous god if pJ > pN . Let us take stock. The examples show that there exist subjective probability assignments that could justify the decision to wager against a particular god, or even against all gods. This observation is legitimate, but perhaps it is not very troublesome for the Pascalian. We might say: there are no difficulties here for somebody who possesses subjective probabilities favourable to religious belief. Indeed, it looks as though we have peaceful co-existence of all types of believers and non-believers. If we dig a little deeper, however, the odd results above might be interpreted not as justifying bland pluralism, but rather as indicating the need to justify our initial credences for various theological possibilities. In fact, relative utilities open up new avenues to evaluate these subjective probabilities. That is the point of the Many-Wagers Model, which I shall now explain. By way of motivation, consider once again assumption (3) of the opening section: the decision about how to wager can influence one’s subjective probabilities, either positively or negatively. In light of this assumption, the analysis of Example 5 has a puzzling feature. Given positive probabilities pG and pJ , the atheist option, ‘wager against all’, is competitive if and only if pG > pJ . The basis for your decision to wager against all, at least at first, is that you take the grouchy god more seriously than the jealous god. Yet to take the atheist line is to take steps that will lower your credence in the grouchy god—albeit in the jealous god as well. So long as the inequality is maintained, of course, your action is justified in the same way. Still, persistence appears to lead to a situation where pJ = pG = 0 and pA = 1, removing your original reason for wagering against all.16 Furthermore, the details of the convergence are important, since if the inequality pG > pJ ceases to hold at any point, you should change your wager. In short, ‘wagering against all’ may be self-undermining, whereas wagering for the jealous god is not self-undermining. You assign an appreciable probability to some state of affairs: the existence of a grouchy god. The fact that this probability value is larger

16 With these limit probability values, the different wagers come out with equal expected relative utility. The decision will then depend upon lower-level decision matrices, as pointed out in note 12. It is certainly possible that these matrices will be consistent with a preference to wager against all gods.

many gods, many wag e r s 197 than another is the basis for a decision that leads you to lower your probability for (and, in the long run, to reject) that state of affairs. The crucial point here is that you can foresee this change when you begin your deliberations, and there is no mysterious process involved besides Pascalian belief revision. Given that you accept Pascalian belief revision, there is some tension, if not outright conflict, between your initial reasoning and the final anticipated epistemic state. At the very least, the tension should give you pause. You should be motivated to explore the details of how your probabilities will evolve. These remarks suggest a shift from a static to a dynamic view of Pascal’s Wager. You need to look not just at the expected payoffs for a single decision, but also at the sustainability of your probability assignments (and of wagering decisions based on those assignments). How should you revise the relevant probabilities? The Many-Wagers Model offers one reasonable method. On the Many-Wagers Model, rather than selecting a single wager, you make them all. More accurately, you assign each available pure wager a weight that matches your current credence in the corresponding deity. Thus, a positive probability assignment to a deity (or against all deities) signifies the degree to which you are presently inclined to make the corresponding wager. You start with a set of initial credences, and you watch how the various wagers fare, bearing in mind that wagering will bring about changes to your subjective probabilities. Probabilities increase if they correspond to wagers with the highest (relative) expectation, and decrease if they correspond to the wagers that do worst. You then repeat the process with the modified probability values. All that is missing is the updating rule. One possible model for thinking about this belief revision process is the Replicator Dynamics of evolutionary game theory.17 A problem in evolutionary game theory begins with a finite number of distinct groups, each defined by a particular strategy. Based on the starting population proportion for each group and its expected success rate or relative fitness, we can determine how the population as a whole may be expected to evolve in successive rounds of an interactive process, and we can calculate what the equilibrium proportions will be. How can this apparatus be applied to Pascalian decision problems? In place of groups, we have distinct theological possibilities. In place of competitive strategies, we have wagers. In place of initial population proportions, we have initial credences. The ‘relative fitness’ of a wager is the ratio of its expected relative utility to the average expected relative utility of all wagers. Finally, corresponding to an equilibrium population distribution, we have an equilibrium probability distribution. It helps to think in terms of a metaphorical competition between your inclinations to make different wagers, each of which has some initial appeal. Based on the relative expectations, the corresponding probabilities rise or fall until they approach equilibrium: a set of

17 See Skyrms (1996) and Weibull (1995).

198 probabi lity in the phi lo sophy of re lig ion probability assignments that is invariant under updating. An equilibrium is stable if it is restored following any small perturbation of the probabilities. The updating rule, modelled on the Replicator Dynamics, will be stated shortly. Before examining this rule or reflecting on the philosophical appropriateness of the model, however, let us see how it applies to two of our simplest examples. Example 1 (original Wager) Employing the relative decision matrix, the relative expectation for rejecting the Wager is 0. So long as your initial probability p for God’s existence is positive, p rises to 1 after a single iteration. The only equilibria are p = 0 or p = 1, and only the latter is a stable equilibrium. If the final distribution must be a stable equilibrium, then p = 1 is the only possibility. We can thus interpret Pascal’s argument as showing that p = 1 is the only possible epistemic limit for anyone who begins with p > 0. Example 5 ( jealous and grouchy) Starting from any set of positive initial probabilities pJ , pG and pA , we see that pG drops to 0 after one iteration because the expectation of wagering for a grouchy god is 0. After two iterations, pA drops to 0 as well. The only equilibrium values for (pJ , pG , pA ) are (1, 0, 0), (0, 0, 1) and (0, 1, 0), and only the first of these is a stable equilibrium.18 This means that if you have any inclination to believe in the jealous god (i.e. pJ > 0 initially), the Many-Wagers Model leads to full belief. This is a much stronger result than in our earlier discussion of this example, where belief in the jealous god is vindicated only if pJ > pG initially. Before turning to the remaining examples, let’s have a clear statement of the updating rule. Many-wagers model: updating rule. Suppose there are finitely many possibilities S1 , . . . , Sn with corresponding wagers W1 , . . . , Wn , relative decision matrix A, and initial subjective probability vector p = (p1 , . . . , pn ), where pi is the initial subjective probability for Si . Let U(Wi ) = (Ap)i represent the expected (relative) utility of wager Wi . Let U¯ = p1 U(W1 ) + . . . + pn U(Wn ) represent the average (relative) expected utility. Then the updated subjective probability pi for Si is given by ¯ i. pi = [U(Wi )/U]p

Wagers with higher-than-average expected relative utility lead to upward revision of the corresponding subjective probability, while wagers with lower-than-average expected relative utility lead to downward revision of the corresponding subjective probability.

18 To make sense of the other equilibria, we need to look at the relative decision matrices that represent finite payoffs. See note 12.

many gods, many wag e r s 199 We also need a clear definition of an equilibrium distribution and a stable equilibrium distribution. Equilibrium distribution A probability distribution {pi } over possibilities S1 , . . . , Sn counts as an equilibrium distribution if pi = pi for all i. In other words, the updating rule produces no changes. Stable equilibrium distribution An equilibrium distribution {pi } over possibilities S1 , . . . , Sn is stable if for any small (and mathematically admissible) set of changes pi , application of the updating rule to the distribution {pi + pi } leads to convergence to {pi }. In other words, the updating rule restores the equilibrium in the event of small changes in one’s probability distribution over the set of possibilities.

I propose a corresponding requirement of evolutionary stability:19 a viable probability distribution for a Pascalian decision problem must be a stable equilibrium.20 Importantly, this is put forward as a necessary (but not sufficient) condition for viability, a point to which I shall return later. To illustrate further how the Many-Wagers Model works, let’s consider our other two examples. Example 2 (many jealous gods) With many jealous gods, the result agrees with the analysis of Section II. If you initially assign highest subjective probability to one god, then that probability converges to 1 on repeated application of the updating rule; all other probabilities drop to 0. If there are initially N gods with equal highest probability, then in the limit all others drop out and those in this initial set come to have probability 1/N. At equilibrium, you retain an equally strong disposition to wager for each of these deities. Example 7 ( jealous and nice) Suppose that the situation is as in Example 7, and you start with positive credences pJ , pN , and pA for a jealous god, a (very) nice god and no god (respectively). The expected relative utilities are as follows: U( J) = pJ + pN U(N) = pN U(A) = pN Then U¯ = pJ (pJ + pN ) + (pN · pN ) + (pN · pA ) = p2J + pN . The revised probabilities are as follows: 19 This requirement will be strengthened in the next section, and then qualified in the conclusion. 20 Throughout, I ignore mixed strategies. The introduction of mixed strategies has no impact on the

equilibrium distributions over pure strategies.

200 probabi lity in the phi lo sophy of re lig ion pJ = pN = pA =

pJ + pN p2J + pN p2J

pJ ,

pN pN , and + pN

pN pA . p2J + pN

The multiplier exceeds 1 for pJ but is less than 1 for pN and pA . In the limit, pJ converges to 1. Although there are multiple equilibria, the only stable equilibrium is pJ = 1 and pN = pA = 0. Thus, if pJ > 0, the model leads to full belief in the jealous god. What about the modified version (Example 7A), where the nice deity, for some reason, does not reward followers of the jealous god? In this version, U( J ) = pJ , U(N) = U(A) = pN , and U¯ = pJ · pJ + pN (1 − pJ ). There is a stable equilibrium if pJ = 1 or pJ = 0; if pJ = pN , we have an unstable equilibrium. So if you begin with pJ > pN , then pJ converges to 1. If you begin with pJ < pN , then pJ converges to 0. In this latter case, any combination with pA + pN = 1 is an equilibrium, so the atheist option (wager against all) survives as a kind of free rider. I’ll return to this case in the next section. Let’s pause now to question the appropriateness of the Many-Wagers Model as a tool for resolving Pascalian decision problems, and specifically as a means to handle versions of the many-gods objection. The heart of the model is not so much the specific mathematical formula for updating probabilities as it is the injunction to boost one’s credence when the corresponding wager is perceived as relatively advantageous, and to lower it when the corresponding wager is relatively disadvantageous. The specific updating formula need not prescribe how one’s credences must evolve, but it does reveal the foreseeable direction in which those credences would evolve if one followed Pascalian injunctions about wagering. Indeed, the equilibrium results above, in all four examples, can be shown to be independent of our particular probability updating rule. Let pi , Si , and Wi and U¯ be as in our statement of the updating rule for the Many-Wagers Model. Let us say that a Pascalian updating rule is evolutionarily robust if it has the following property: ¯ For all i, pi > pi if and only if U(Wi ) > U. (The probability of state Si will increase under the updating rule if and only if the expected relative utility of the corresponding wager is above average.)

It is not difficult to see that the equilibrium results above are the same under any evolutionarily robust updating rule.21 So, it does not much matter whether the formula proposed by the Many-Wagers Model provides the best (or only) way to apply Pascalian norms to the step-by-step evolution of one’s credences. All that matters is that one’s probability distribution will 21 It remains possible that convergence behaviour will vary between different evolutionarily robust updating rules.

many gods, many wag e r s 201 be unstable unless one attains equilibrium, and (if we assume possible perturbations) a stable equilibrium at that. Still, it helps to imagine subjective probabilities as evolving according to the model just presented. The analogy with evolutionary game theory provides enough plausibility for the Many-Wagers Model to serve as a useful idealization for how subjective probabilities might evolve in a Pascalian decision problem.

IV. What Remains of the Many-gods Objection? The original version of the many-gods objection undercuts Pascal’s Wager by showing that incompatible wagers have infinite expectation, and thus an equal claim on our rationality. We circumvented that difficulty by employing relative decision matrices, but then we had to face a second version of the objection: there are scenarios in which the subjective probability distributions justify a decision to wager against any particular god or even against all gods. As we have seen, the Many-Wagers Model provides strategies for handling many such scenarios. We can rule out non-equilibrium distributions, i.e. sets of probabilities that are self-defeating. We can also rule out unstable equilibria. What remains of the many-gods objection is the following: even if we impose the requirement that any acceptable probability distribution must be stable under our revision rule, there still remain problematic cases in which our allegiance can be divided between many gods (or directed against all belief in gods). I shall examine three representative cases, with a view to determining how serious a problem they constitute for the Pascalian. I shall propose a strengthened requirement of stability which takes care of these cases. This strengthened requirement is my best attempt to make good on my initial conjecture, that many-gods problems can be convincingly resolved if we set aside difficulties with Pascal’s original (one-god) argument. Example 8 (grouchy cartel)

Wager for A Wager for B Wager for C

A: grouchy exists

B: grouchy exists

C: no god exists

0 1 0

1 0 0

0 0 0

In this scenario, you assign positive probability to each member of a small cartel of grouchy gods who only reward each other’s followers. We have U(A) = pB , U(B) = pA , U(C) = 0, and U¯ = 2pA pB . The only stable equilibrium is (1/2, 1/2). You do best by retaining an equal disposition to wager for each grouchy god. The pattern illustrated here for two grouchy gods clearly generalizes to larger cartels. While this ‘many-gods’ scenario may appear to represent a threat to the Pascalian, I don’t believe that it is any more problematic than the special case of Example 2 considered earlier involving multiple jealous gods, each of whom receives equal

202 p robab i l ity i n th e ph i lo sophy of re l i g i on subjective probability. Indeed, a grouchy cartel, considered as an ensemble, is formally indistinguishable in the Many-Wagers Model from a single jealous god. Example 9 ( grouchy and nice)

Wager for A Wager for B Wager for C

A: grouchy god exists

B: nice god exists

C: no god exists

0 0 1

1 1 1

0 0 0

In this scenario, every starting configuration with positive probabilities pA , pB , pC for all three outcomes leads (on the Many-Wagers Model) to convergence towards pC = 1. In a sense, the atheist wager carries the day; there is no stable equilibrium besides (0, 0, 1).22 This looks like a serious problem for the Pascalian. Note, however, that this equilibrium is unstable if we broaden the requirement of stability somewhat. Evolutionary game theorists consider whether a strategy is vulnerable to invasion by mutants: competitors employing a novel strategy. Let’s allow for an analogous form of invasion in the form of new theological possibilities (with correspondingly new options for wagering). Suppose that somebody in the scenario under consideration comes to believe that a jealous god has a small positive probability pJ . In this case, the Many-Wagers Model leads to pJ = 1.23 The first equilibrium disappears. It is important here that the new theological possibility is ‘in the neighbourhood’. From the bare fact that a deity appears to be logically possible, one need not (indeed, cannot always) infer positive probability.24 The thought here is that the many-gods objection rests on the view that it is not reasonable to assign positive probability only to one deity. I am generalizing this point to the ‘pantheon of possibilities’ introduced in Section III: anyone who assigns one of these gods a positive probability should be willing to entertain a tiny positive probability for the other types (i.e. jealous, nice, or grouchy gods). These are relevant possibilities for anyone who takes Pascal’s argument and the many-gods objection seriously. Accordingly, we define a stronger notion of stability. Strongly stable equilibrium distribution. An equilibrium distribution {pi }, 1 ≤ i ≤ n, over possibilities S1 , . . . , Sn is strongly stable if for any new possibility Sn+1 , and any (mathematically admissible) set of changes pi , 1 ≤ i ≤ n + 1, application of the updating rule to the distribution {pi + pi } leads to convergence to {pi } (and in particular to pn+1 = 0). In other words, the updating rule restores the equilibrium in the event of small changes in one’s probability distribution over an extended set of possibilities. 22 We make sense of this equilibrium using relative decision matrices involving finite payoffs. 23 I assume that the decision table is expanded in obvious ways. For instance, the nice god rewards those

who wager for the jealous god. 24 Gale’s denumerable infinity of sidewalk-crack deities establishes this point (Gale 1991: 350).

many gods, many wag e r s 203 The ‘new possibility’, as noted, will always be a deity from our elementary classification. Corresponding to this definition, I propose the requirement of strong stability: a viable probability distribution for a Pascalian decision problem must be a strongly stable equilibrium. Once again, this is proposed only as a necessary condition for viability. I’ll examine the justification for this strengthened stability requirement in the concluding section. For now, it’s clear that cases like Example 9 do not represent a problem for the Pascalian if the requirement is accepted. Example 7A (jealous and nice) J: Jealous god exists

N: Nice god exists

A: No god exists

1 0 0

0 1 1

0 0 0

Wager for J Wager for N Wager for A

Let’s reconsider the scenario of Section III, in which you initially assign positive probability to a jealous god ( J ), a moderately nice deity (N) who rewards everyone except followers of J, and the possibility (A) that no god exists. We saw that, under the Many-Wagers Model, the stable equilibria are pJ = 1 and any combination with pN + pA = 1. We noted that wagering against all gods is a free rider in this case. As in the previous example, we show that any equilibrium of the second sort (where pN + pA = 1) is not strongly stable. We introduce a new possibility: a deity N  who rewards only followers of N or N  . This possibility might be motivated by the arbitrariness of deity N, who rewards atheists but not traditional believers, but the motivation is, strictly speaking, irrelevant. The point is that by introducing this new possibility and making plausible assumptions, we undermine the equilibrium. The decision table becomes Example 10. Example 10

Wager for J Wager for N Wager for N  Wager for A

J exists

N exists

N  exists

A: No god exists

1 0 0 0

0 1 1 1

0 1 1 0

0 0 0 0

We modify our original equilibrium where pN + pA = 1 by assuming that pN now takes a small positive value ε > 0, and pN + pA = 1 − ε. When we apply the updating rule, we see that the expected utility of wagering for A is now below average, and the new equilibrium occurs when pA = 0 and pN + pN  = 1. So the original equilibrium is not strongly stable. Once again, we have a satisfactory response to this particular instance of the many-gods objection.

204 probabi lity in the phi lo sophy of re lig ion Of our three problematic examples, we resolved the first by noting that a ‘grouchy cartel’ behaves like a peculiar type of jealous god, and we resolved the other two by introducing the requirement of strong stability. The examples point to the robustness of belief in jealous gods. It’s worth noting that any equilibrium involving a single jealous god, or multiple jealous gods assigned equal probability, or (by extension) grouchy cartels, is strongly stable. At the moment, I’m unsure whether or not other types of deity can participate in a strongly stable equilibrium. That leaves room for a remnant of the many-gods objection (and for doubts about the sufficiency of the requirement of strong stability).

V. Conclusion: Convergence and Rationality In this concluding section, I want to address two concerns. The first is about the significance of the mathematical details of the Many-Wagers Model. The second is a more important concern about whether the evolutionary approach put forward here really tells us anything about the rationality of wagering for God. The first issue is this: beyond the requirements of stability and strong stability, what importance attaches to the mathematical formalism, i.e. to the updating rule in the Many-Wagers Model? I suggested earlier that the updating rule provides a useful way to implement the Pascalian idea that making the wager can raise or lower your subjective probabilities. Beyond this, I believe that there is also an important technical point. The mathematical details determine whether or not the repeated application of the updating rule to a particular starting configuration leads to convergence, and what the final limit will be. Although I have no general results, it is instructive to examine one important special case. We shall see that the updating rule strongly supports belief in jealous gods. The special case is a many-gods problem that (unlike any of our previous examples) involves one god of each of the three types: jealous, grouchy, very nice. We make things interesting by supposing that the grouchy god has it in for followers of the jealous god (as well as his/her own devotees), but rewards everybody else. Example 11 is the relative decision table. Example 11 (an important special case)

Wager for A Wager for B Wager for C Wager for D

A: jealous

B: grouchy

C: nice

D: none

1 0 0 0

0 0 1 1

1 1 1 1

0 0 0 0

To add to the difficulty, let’s stack the initial probabilities against the jealous god. As it turns out, provided that our initial probability for the jealous god is positive,

many gods, many wag e r s 205 the Many-Wagers Model leads to convergence of that probability to a limit of 1. Convergence is extremely slow, but it eventually occurs. Belief in jealous gods is tenacious. Let me turn, in closing, to the second question. I have argued that the Many-Wagers Model, with the requirement of strong stability, provides a promising response to the many-gods objection.25 It does so by providing plausible resolutions for problematic ‘many-gods’ scenarios: it recommends a particular action (or set of credences) in some cases and rules out counter-intuitive actions (or credences) in other cases. Still, it is perfectly natural to raise questions about the link between the model and rationality. I shall focus on three such questions. First, the most general question: does the evolutionary approach put forward tell us anything about the rationality of wagering for God? Second, a more specific question: why is the requirement of strong stability a requirement of rationality? Finally, it is important to address one very specific question: what do the arguments in this chapter imply about the rationality of atheism or agnosticism? In answering these questions, it is important to recall the conditional form of my original conjecture: if (1) we can assign positive probability to the existence of deities, (2) we can make sense of infinite utility, (3) we can justifiably revise our beliefs on pragmatic grounds, and (4) we can provide a valid formulation of Pascal’s original argument, then the many-gods objection poses no additional threat. Provided we grant these assumptions and take the many-gods objections seriously, there is a powerful argument for the requirement of strong stability. The very thought that leads to the many-gods problem—the insistence that if we assign positive probability to one deity, we must consider positive probabilities for others—also provides the key to its solution. The requirement of strong stability rests on a modest version of this principle of open-mindedness: once you admit the positive probability of one type of deity, there is no justification for rejecting the other basic types. You ought to be willing to entertain non-zero probabilities. And if you can foresee that entertaining these probabilities would shift your current beliefs (i.e. that your beliefs are not strongly stable), then you have strong reason to reject those beliefs. It follows that you should reject unstable probability distributions and the associated wagering behaviour. I conclude, in answer to the first two questions, that there is a strong argument for regarding the evolutionary approach and the requirement of strong stability as imposing constraints on Pascalian decisions. In answer to the third question, however, there is no parallel argument that somebody who has no inclination to assign positive probability to the existence of any deity, or any deity offering human beings an infinite reward, should be rationally bound to consider positive probability assignments. The requirement of strong stability is restricted: it only applies if one is prepared to grant the initial premise (1) that we 25 As noted in the introductory section, this claim is subject to two limitations: we consider only a finite number of gods, and we rule out infinitesimal probabilities.

206 p robab i l ity i n th e ph i lo sophy of re l i g i on may assign positive probability to various theological possibilities. The requirement of strong stability also depends upon premise (3), which asserts that there can be pragmatic grounds for belief revision. If wagering is either incapable of influencing one’s beliefs or inherently irrational on independent grounds, then we cannot require that one’s probability distribution over various theological possibilities should be strongly stable. In light of this dependence on two key premises, it is clear that the arguments of Sections III and IV have limited implications about the rationality of atheism and agnosticism. Indeed, the limitations are in keeping with the conjecture stated at the outset: the many-gods problem generates no new problems for Pascal, over and above problems that bedevil the original version of his argument.

References Bartha, P. (2007) ‘Taking Stock of Infinite Value: Pascal’s Wager and Relative Utilities’, Synthese, 154: 5–52. Duff, A. (1986) ‘Pascal’s Wager and Infinite Utilities’, Analysis, 46: 107–9. Gale, R. M. (1991) On the Nature and Existence of God. Cambridge: Cambridge University Press. Hacking, I. (1994) ‘The Logic of Pascal’s Wager’. In J. Jordan (ed.), Gambling on God: Essays on Pascal’s Wager. Savage, MD: Rowman & Littlefield, 21–9. Hájek, A. (2003) ‘Waging War on Pascal’s Wager’, Philosophical Review, 112, 1: 27–56. Jordan, J. (ed.) (1994) Gambling on God: Essays on Pascal’s Wager. Savage, MD: Rowman & Littlefield. (1998) ‘Pascal’s Wager Revisited’, Religious Studies, 34: 419–31. (2006) Pascal’s Wager: Pragmatic Arguments and Belief in God. Oxford: Oxford University Press. Mougin, G. and E. Sober (1994) ‘Betting Against Pascal’s Wager’, Noûs, 28: 382–95. Pascal, B. (1948) Pensées. Trans. W. F. Trotter. London: J. M. Dent & Sons. Schlesinger, G. (1994) ‘A Central Theistic Argument’. In J. Jordan (ed.), Gambling on God: Essays on Pascal’s Wager. Savage, MD: Rowman & Littlefield: 83–99. Skyrms, B. (1996) Evolution of the Social Contract. Cambridge: Cambridge University Press. Sobel, H. (1996) ‘Pascalian Wagers’, Synthese, 108: 11–61. Weibull, J. (1995) Evolutionary Game Theory. Cambridge: MIT Press.

PART V

Faith and Disagreement

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11 Does Religious Disagreement Actually Aid the Case for Theism? Joshua C. Thurow

I. Introduction What should we do when we come across people who seem just as competent and well-informed as ourselves on a given issue, and yet disagree with us about that issue? Should we maintain our confidence and assume that the others must really be messed up this time? Or should we be conciliatory, saying ‘well, we’re both being quite reasonable, aren’t we’? Or, alternatively, should we (and the others) reduce our confidence, perhaps suspending judgement altogether about the issue? The third option has recently been endorsed and defended by several philosophers, including Richard Feldman—who nicely expresses his view as follows: After examining the evidence, I find in myself an inclination, perhaps a strong inclination, to think that this evidence supports P. It may even be that I can’t help but believe P. But I see that another person, every bit as sensibly and serious as I, has an opposing reaction. Perhaps this person has some bit of evidence that cannot be shared or perhaps he takes the evidence differently than I do. It’s difficult to know everything about his mental life and thus difficult to tell exactly why he believes as he does. One of us must be making some kind of mistake or failing to see some truth. But I have no basis for thinking that the one making the mistake is him rather than me. And the same is true of him. And in that case, the right thing for both of us to do is to suspend judgment on P. (Feldman 2007: 212)1

David Christensen’s view is quite similar to Feldman’s: These cases, then, suggest the following (admittedly rough) principles for assessing, and reacting to, explanations for my disagreement with an apparent epistemic peer: (1) I should assess explanations for the disagreement in a way that’s independent of my reasoning on the matter under dispute; and (2) to the extent that this sort of assessment provides reason for me to think that the explanation in terms of my own error is as good as that in terms of my friend’s error, I should move my belief toward my friend’s. (Christensen 2007: 199) 1 For defences of this third option see: R. Feldman (2006, 2007), A. Elga (2007), D. Christensen (2007), and R. White (2005).

210 probabi lity in the phi lo sophy of re lig ion Christensen continues: ‘when I have excellent reason to think that the explanation in terms of my own error is every bit as good as that in terms of my friend’s error, I should come close to “splitting the difference” between my friend’s initial belief and my own’ (Christensen 2007: 203). And he concludes that: evidence that can mandate significant changes in rational degree of belief will often mandate changes in rational all-or-nothing belief-state. So there should certainly be many cases where the discovery of disagreement by peers mandates suspension of belief—or even disbelief—in a proposition that was initially believed rationally. (Christensen 2007: 213–14)

Although there are clearly differences between Feldman and Christensen, they have this central idea in common: when you have good reason to believe that someone is an epistemic peer (in other words, that the person has approximately the same evidence evaluating abilities as you and is aware of the same evidence as you) regarding p, you should weight their judgement about p the same, or almost nearly the same, as your own judgement about p. This view has also been called the ‘equal weight view’, which I shall abbreviate to EWV (see Elga 2007; Jehle and Fitelson 2009). Although the EWV is intuitively plausible, and has been defended with interesting arguments, it seems to have a highly unwelcome consequence: we should suspend judgement about nearly any contentious issue in philosophy, religion, and politics since there fairly clearly are epistemic peers on both sides of contentious issues of these kinds and the peers don’t tend to clump on one side of the issue.2 But, to many philosophers, this just seems crazy, unintuitive, too high a price to pay (contra Christensen, who takes this to be good news). Peter van Inwagen, for example, writes: I think that any philosopher who does not wish to be a philosophical skeptic—I know of no philosopher who is a philosophical skeptic—must agree with me that . . . whatever the reason, it must be possible for one to be justified in accepting a philosophical thesis when there are philosophers who, by all objective and external criteria, are at least equally well qualified to pronounce on that thesis and who reject it. (van Inwagen 1996: 139)

Adam Elga dubs this worry ‘the problem of spinelessness’ (2007: 484). Is there any way around the problem of spinelessness? Some will reject the EWV (as Kelly 2006 seems to do). But, more interestingly, others may try to argue that it does not in fact entail spinelessness. There have been a few attempts to do this: (1) Perhaps we can’t share all of our evidence because some of it is private, such as intuitions and experiences (presented and rebutted by Feldman 2007); (2) We might have different standards for weighing the force of evidence (presented and rebutted by Feldman 2007); (3) Almost no two people are epistemic peers on philosophical, ethical, 2 Perhaps the philosophical experts clump to one side with respect to some philosophical issues (one example: global scepticism is false, or at least unreasonable), but not many. If there were a clumping, then of course the EWV would favour belief in whichever way the average of one’s peers leaned. For some evidence about philosophers’ views on a variety of issues, see the results of the PhilPapers.org survey at http://philpapers.org/surveys/results.pl.

re lig ious disag re eme nt and theism 211 political, and religious issues (Frances 2010); (4) In realistic scenarios, you know that you are alert and sober in assessing p, whereas you are far less certain that your peer was alert and sober in assessing p (Frances forthcoming); (5) Regarding controversial areas like philosophy, ethics, politics, and religion, we don’t have a way of evaluating associates as peers ‘based on reasoning that is independent of the disputed issue’, and so we don’t have sufficient reason to regard those that disagree with us about these issues as epistemic peers (Elga 2007); (6) We might reasonably assign different values to prior probabilities which in turn produces disagreement (Oppy 2006).3 In this chapter I develop a new strategy for arguing that the EWV does not entail spinelessness. At the core of my argument is the claim that in cases where two people disagree about p, but each has arguments supporting their belief about p, applying the EWV may entail that they should rationally come to an agreement that p or that not-p. So, far from the EWV entailing spinelessness, depending upon the nature of the dispute, sometimes it will entail that the disputants should, rationally, come to agree that p (or not-p)! I argue for this claim by (1) considering a specific case, and then (2) offering some more general theoretical reasons that, taken together, strongly support it. I then present and discuss a simple formal model of how to weight the views of peers equally; arguing that, despite some limitations and flaws, the model can stand as a useful first approximation. Using this model, I apply the new strategy to religious disagreement. The results illustrate how it could well turn out that the EWV has the consequence that we all should, rationally, accept theism. Although it could also turn out that we all should, rationally, accept atheism. Which actually is the case is shown to depend upon quite a number of details concerning the nature of the disagreement between atheists and theists that are epistemic peers. After considering several objections to my approach, I conclude that the EWV entails spinelessness neither about beliefs in general nor about religious beliefs in particular.

II. The Disagreeing Detectives Case Imagine that there are two detectives, Gary and Isaac, and that they disagree about whether Kelly is guilty of murder. Gary thinks that Kelly is guilty, but Isaac thinks that Kelly is innocent. Furthermore, imagine that Gary and Isaac agree that body of evidence e incriminates Kelly; that is, considered alone, e makes it quite likely that Kelly is guilty. Let e include things like fingerprint evidence, the possession of a good motive, the fact that her blood was found at the scene of the crime. The difference between Gary and Isaac is this: there is a witness who reports seeing Kelly elsewhere at the time of the crime, and Gary does not trust this witness at all, whereas Isaac trusts this witness quite a bit. Isaac trusts her enough to override the force of e, and so he believes 3 Graham Oppy is not explicitly responding to the spinelessness charge, nevertheless his proposal does come in the context of an explanation of how there can be reasonable disagreement.

212 probabi lity in the phi lo sophy of re lig ion that Kelly is innocent. Suppose furthermore that Isaac and Gary are both normally pretty good judges of trustworthiness (they are experienced detectives), and that they have the same evidence about the witness’ trustworthiness, they just disagree about what to believe about her trustworthiness on the basis of this evidence. Furthermore, they recognize each other to have the same general ability at crime-solving; they are crime-solving peers. What should Gary and Isaac believe about Kelly? Once Gary and Isaac talk things through, they should realize that their disagreement about Kelly’s guilt stems entirely from their disagreement about the trustworthiness of the witness. Given that they regard each other, with good reasons, as epistemic peers regarding evaluating trustworthiness, they should suspend judgement about whether the witness is trustworthy. But, then given the fact that e strongly incriminates Kelly, and that they agree about the evidential force of e, they should now both believe that Kelly is guilty. Both Gary’s and Isaac’s overall evidence, once they have suspended judgement about the trustworthiness of the witness, favours the claim that Kelly is guilty. Perhaps the fact that they have suspended judgement about whether independent evidence supports the witness’ reliability gives them reason to believe that there might be (in a stronger sense than bare possibility) a witness to Kelly being elsewhere, and perhaps this in turn is some evidence for her innocence (or, alternatively, it decreases the level of support e gives for the claim that Kelly is guilty). However, it still seems that the evidence overall favours her guilt, so that’s what Gary and Isaac should both believe (although, for the reasons just given, perhaps their degree of belief should be lower than Gary’s initial degree of belief that Kelly was guilty). Here’s a case, then, in which disagreement between epistemic peers about p rationally requires, not suspension of judgement, but agreement that p.4 Does this case fit well with the EWV? Initially, it might seem as if the EWV entails that Gary and Isaac should suspend judgement about whether Kelly is guilty. After all, they are epistemic peers and they disagree about Kelly’s guilt. However, I don’t think that this is correct.5 Their disagreement about Kelly’s guilt derives entirely from their disagreement about the witness’ trustworthiness; the latter disagreement is the more fundamental disagreement. The EWV should be able to recognize this, and we can certainly understand the view in such a way that it does recognize the importance 4 In conversation, Jake Chandler noted that my example seems to conflict with the preservation criterion, which states that if one believes that A and does not believe that ∼B, then one still believes that A upon revising one’s corpus of beliefs by B (see Gärdenfors 1986). My detectives—post revision in light of disagreement—believe that Kelly is guilty, and do not believe that the witness is unreliable (they suspend judgement about that). But, if they did come to believe that the witness is reliable, they would no longer believe that Kelly was guilty. However, Robert Koons notes that ‘from the perspective that uses defeasible reasoning to define belief revision, there is no good reason to accept Preservation. One can add a belief that is consistent with what one already believes and thereby lose beliefs, since the new information might be an undercutting defeater to some defeasible inference that had been successful’ (Koons 2009, section III). When the detectives begin to believe that the witness is reliable, they then have an undercutting defeater for the evidential force of the other evidence they have that incriminates Kelly. So, there are ways of understanding rational belief revision that reject the preservation criterion, and thus make sense of my example. 5 See Appendix A.

re lig ious disag re eme nt and theism 213 of the more fundamental disagreement. When a belief b is held on the basis of other beliefs, we can say that there is an evidential chain supporting b. The links of the chain supporting b may in turn be supported by other links, and so on. We can then say that a belief is more basic/fundamental in justifying b to the extent that it is further back on the chain supporting b. We can then say that a disagreement is more fundamental with respect to b to the extent that it is a disagreement about beliefs that are more basic/fundamental in justifying b.6 With these distinctions, we can understand the EWV as follows: EWV: When S and R disagree about p and S and R recognize each other as epistemic peers, then S and R should weight their judgements about their most fundamental disagreements with respect to p equally (or nearly equally), and make any changes to their less basic beliefs in justifying p (propositional attitude and level of confidence) that are rationally mandated. If any disagreements remain at the level of their less basic beliefs in justifying p, then S and R should weight their judgements about those disagreements equally (or nearly equally). This process should continue until S’s and R’s evidential chains supporting their beliefs about p support the same attitude towards p.

As mentioned above, there are general theoretic reasons for accepting EWV, aside from any support it receives from the disagreeing detectives case. First, if disagreement about a less basic belief is explained by disagreement about a more basic belief, then it seems quite intuitive to think that the disagreement about the less basic belief should be resolved by first going to the root of the disagreement, namely the disagreement about the more basic belief. Second, if a belief that p is supported by beliefs that q and that r, then any information that affects the justification for p by affecting the justification for q and r should be accounted for by making the appropriate adjustment in justification for q and r, and once that is done, then adjusting p in whatever way is demanded by the adjustments made in the justification for q and r. Thus, disagreements about more basic beliefs, such as q and r, should be resolved (they affect the justification for p, but via affecting the justification for q and r), and after doing that, believers should then make whatever changes in their less basic beliefs (for example, belief that p) are rationally required by the changes made concerning the more basic beliefs. It is interesting to note that EWV entails that the general phenomenon observed when considering the Gary and Isaac case—which I shall henceforth call the ‘disagreement rationally requiring assent’ (DRRA) phenomenon—can occur even if there is some disagreement about the evidential force of e (for instance, the fingerprint evidence, the fact that Kelly has a good motive, the fact that Kelly’s blood was found at the scene of the crime) for the claim that Kelly is guilty of murder. If, once the 6 These distinctions suggest that justification is a chain-like relation that ends somewhere in basically justified beliefs, rather than a coherentist, web-like, relation where beliefs are justified to the extent that they fit well in the web. Nevertheless the distinctions can also be applied to a coherentist view because, on that view, some beliefs are more central to the web of belief, and the EWV can then be understood to say that disagreements on those more central beliefs should be resolved first.

214 p robab i l ity i n th e ph i lo s ophy of re l i g i on detectives’ views about the evidential force of e are weighted equally, they agree that e supports Kelly’s guilt, then they should both agree that Kelly is guilty, since they have rationally suspended judgement about the reliability of the witness. So, complete agreement about the evidential force of some of the evidence isn’t necessary in order for the detectives to be justified in believing Kelly is guilty. All that is required is that, once their views about the evidential force of e and the testimony are each weighted equally, they would agree that the evidential force of e supports Kelly’s guilt to a moderate extent, whereas they would suspend judgement about whether the testimony supports Kelly’s innocence, and e supports Kelly’s guilt to a high enough degree to justify belief that she is guilty. More generally, the following are sufficient conditions for the DRRA phenomenon to occur: (1) epistemic peers disagree about p; (2) there is agreement amongst the epistemic peers about the force of some body of evidence e (or, agreement that e supports p to some degree, after having weighted disagreements about the evidential force of e equally), but disagreement about the force of f sufficient to justify suspension of judgement about the evidential force of f ; and (3) once the peers have suspended judgement about the force of f , e is then strong enough to justify belief in p for each of the peers.

III. Equal Weight Now that the EWV has been clarified to show its consistency with the detectives’ case, it’s time to consider the following question: what is it to weigh two (or more) people’s judgements about p equally? The answer to this question turns out to be significantly harder to find than it might initially seem. We will start by modelling degrees of belief using probability functions that satisfy the probability calculus, and thus Bayes’ theorem. According to the simplest and most intuitive model of equal weight, namely the ‘Straight Averaging Equal Weight’ model (henceforth SAEW), if two people S and R at t0 recognize they are epistemic peers, and at some later time recognize that they disagree about p, then we can express the salient features of their epistemic situation at some yet later time t1 as follows: Pr1S (p) = Pr1R (p) =

Pr0S (p) + Pr0R (p) , 2

where Pr1S (p) represents the degree of belief in p that S should have at t1 . This model can easily be generalized to apply to cases in which we find any number of mutually recognized epistemic peers.7

7 It is worth noting that the EWV—particularly as presented in SAEW—has a response to Graham Oppy’s argument, which I briefly presented above. Why should we use our own personal priors independent of other people’s? Why should I think that my priors are any more rational than anybody else’s? If I lack a reason to think that my priors are more rational, then I should weight my peers’ priors the same as my own.

re lig ious disag re eme nt and theism 215 Any rigorous model of equal weight must deal with several important problems.8 In future work I plan to develop a more rigorous model of equal weight, but, for present purposes, the straight averaging model will do. Straight averaging can be regarded as a first approximation of equal weight, and so can serve as a useful (albeit not completely accurate) model for illustrating some of the consequences of the EWV. Straight averaging seems to give approximately the right result (for what an EWV would intuitively say) for most kinds of distributions of degrees of confidence in p amongst a group of disagreeing peers. The disagreeing detectives case shows that sometimes in order to resolve disagreement amongst epistemic peers about p, the peers ought both to agree that p (or not-p). Given this, if the straight averaging view can be used to show how this could also happen for disagreements about whether God exists, then we will have good evidence that the EWV, however it is to be precisified, will allow for this possibility. Whether it actually happens, and which way the disagreement ought to be resolved, will of course depend upon the details of religious disagreement and a more precise model for equal weight. However, in this chapter, I aim simply to give good evidence that disagreement about whether God exists could turn out to be rationally resolved by agreement that God exists (or that God does not exist), rather than by suspending judgement. So, I am suggesting that SAEW be used as an approximate model for the notion of ‘weighting judgements equally’ that is used in EWV. The latter says, then, that SAEW should be used to resolve disagreements about more basic beliefs first, and then the less basic beliefs of the peers should be revised appropriately. Bayes’ theorem can be used to make these revisions. If there are any remaining disagreements at the level of less basic beliefs, then SAEW should be used to resolve those, and then any changes in even less basic beliefs should be made accordingly. The process is continued until all disagreements are resolved.

IV. Religious Disagreement Let us now turn to the question of whether the general phenomenon—which I have called ‘disagreement rationally requiring assent’ (DRRA)—observed in the disagreeing detectives case might also occur for religious disagreement. As noted above, the phenomenon occurs when: (1) epistemic peers disagree about p; (2) there is agreement amongst the epistemic peers about the force of some body of evidence e (or, agreement that e supports p to some degree, after having weighted disagreements about the evidential force of e equally), but disagreement about the force of f sufficient to justify suspension of judgement about the evidential force of f ; and (3) once the peers have suspended judgement about the force of f , e is then strong enough to justify belief in p for each of the peers. Now are there any bodies of evidence e and f concerning God’s

8 See Appendix B.

216 probabi lity in the phi lo sophy of re lig ion existence that might satisfy these conditions? Two good candidates suggest themselves for e: testimony to miracles, and the existence and amount of horrendous evils in the world.9 I will focus mostly in what follows on considering how miracles might produce DRRA. However, I will also briefly discuss how horrendous evils could produce DRRA. Miracles Let us divide evidence concerning God’s existence into two classes: the Background Evidence (BE) and the Miraculous Evidence (ME). Let the Background Evidence include everything that is relevant aside from the miraculous evidence; this will include cosmological arguments, ontological arguments, design arguments, religious experience, various problems of evil, and the problem of divine hiddenness. For the moment, we will leave the class of Miraculous Evidence somewhat vague: it includes whatever evidence there is concerning the occurrence of a particular miracle or set of miracles. Now, according to Bayes’ Theorem: Pr(T/BE & ME) Pr(T/BE) Pr(ME/T & BE) = × , Pr(∼T/BE & ME) Pr(∼T/BE) Pr(ME/∼T & BE)

(1)

where T = God exists. The left-hand side ratio gives us a measure of the degree of belief one ought to have in T on the basis of BE and ME. To work out its value, we need to obtain values for the two ratios on the right-hand side. People obviously disagree about the values of these ratios, so we can use SAEW to derive values for them. While it would require detailed sociological study to work out accurate values for these ratios, here it will suffice to give a very rough calculation of Pr(T/BE)/ Pr(∼T/BE). This will be enough to support my claim that disagreement about whether God exists could turn out to be rationally resolved by agreement that God exists (or that God does not exist), rather than by suspending judgement. There are six major (in the sense of number of adherents) positions on religion to consider: those of Christianity, Judaism, Islam, Buddhism, and Hinduism, respectively, and one which rejects all such views, which I shall simply call ‘non-religious’ (this includes the position of atheists and agnostics). Let’s assume that each of these positions has approximately the same number of experts at evaluating religious beliefs, and furthermore that the experts are epistemic peers.10 Three of these positions are monotheistic (Christianity, Judaism, and Islam), two are not monotheistic (Buddhism and non-religious). Hinduism is more complicated, but for simplicity sake, and to make the case a little harder for monotheism, I propose to count Hinduism as nonmonotheistic. Finally, assume that the monotheistic experts say that Pr(T/BE) ≈ 1 9 Religious experience is a further possibility. It has also been suggested to me in discussion that the fine-tuning design argument might be a good candidate for e. 10 There are two reasons why I’m only counting the experts. First, it seems that the experts’ views ought to count for more. Second, if we counted everyone, the monotheists’ degrees of belief would clearly swamp the non-monotheists (because there are more of them). But, for dialectical purposes, I want to make it harder for my argument to go through.

re lig ious disag re eme nt and theism 217 and the non-monotheistic experts say that Pr(T/BE) ≈ 0. Each of these assumptions is obviously an oversimplification, this last one I think more so than the others because monotheists and non-monotheists give a wide range of opinions about Pr(T/BE), but, again, I just want to get a rough value here. Given all of these assumptions, SAEW entails that Pr(T/BE) ≈ 0.5, and thus Pr(T/BE)/ Pr(∼T/BE) ≈ 1. Inserting this value into equation (1), it follows that Pr(T/BE) will be greater than 0.5 as long as there is some miraculous evidence ME for which the average view of the experts in the six different groups would be that Pr(ME/T&BE)/ Pr(ME/ ∼T&BE) > 1.11 Is there any such evidence? I’m not sure. I think there is good reason for thinking that taking ME = there are many people who testify to witnessing miracles of various kinds won’t work because, arguably, this is just as much to be expected on ∼T&BE as it is on T&BE. After all, even if God doesn’t exist, it wouldn’t be surprising for many people to witness odd events they can’t explain and attribute them to God, and of course there will be many other sorts of bad motivations (to get attention, claim divine approval, and so on) for making such claims that many people are bound to fall prey to at one time or another. So, it seems clear that only some more specific miraculous evidence would have a chance at making Pr(ME/T&BE)/ Pr(ME/∼T&BE) > 1 on average for the experts. There may well be such evidence. The evidence most discussed in philosophical and theological literature concerns the resurrection of Jesus.12 For all we know, it could turn out that the average value for experts who have considered this evidence is that Pr(ME/T&BE)/ Pr(ME/∼T&BE) > 1.13 There are several different ways that the average value of Pr(ME/T&BE)/ Pr(ME/∼T&BE) for the experts could turn out to be greater than 1. Here are two:

11 There is an interesting question about which average we are looking for: Praverage (ME/T&BE)/ Praverage (ME/∼T&BE) or (Pr(ME/T&BE)/ Pr(ME/∼T&BE))average . The former takes the ratio of the average values of the two probabilities, whereas the latter takes the average of the ratio of probabilities for all the experts. EWV entails that we should take the averages of whichever values are more epistemically basic for the individuals. So, if people’s beliefs about Pr(ME/T&BE) and Pr(ME/∼T&BE) are more basic, then what we want is the former average, whereas if people’s beliefs about the ratio Pr(ME/T&BE)/ Pr(ME/∼T&BE) are more basic, then what we want is the latter average. What if some people go one way and some the other? Then we should take the average value of those two different averages. I suspect that people have more of a knack for judging how much more or less likely one thing is than another, especially when dealing with the probabilities of events for which they do not have grounds for giving a precise value (dice and cards and the like are objects for which we can give pretty precise probabilities for certain events), and so I suspect we should be looking for the latter of the above averages. Consequently, my arguments throughout will be concerning the latter of the above averages. However, the results I argue for will also hold if we should be looking for the former of the above averages instead. I will illustrate this with a toy example in note 14 below. Thanks to a reviewer for comments that led to this note. 12 For a sampling of the literature on the resurrection of Jesus, see W. L. Craig (1989), J. D. Crosson (1995), J. D. Crosson and N. T. Wright (2006), S. Davis (2004) and M. Martin (2004) (and other papers in their extended dialogue in print), J. Earman (2000) (especially the collection of historical documents included in his book), M. Martin (1991), T. McGrew and L. McGrew (2009), R. Swinburne (2003), N. T. Wright (2003), and references therein. 13 Nobody has done a study on this. Indeed it would be a hard study to carry out because of the difficulty identifying the experts and ensuring that they were peers on the topic.

218 p robab i l ity i n th e ph i lo sophy of re l i g i on (1) Non-monotheists tend to give a value somewhat less than 1, but monotheists assign a value quite a bit higher than 1; (2) Many non-monotheists and most monotheists assign a value higher than 1 (the non-monotheists are non-monotheists because they think there is other evidence against theism and/or they assign a low prior to God’s existence). I’m not aware of any good reasons for thinking that these two possibilities are not actual. If either were actual, then (depending on the precise values) it could well turn out that the DRRA phenomenon occurs for God’s existence and that the rational resolution of disagreement amongst epistemic peers requires belief in theism. Indeed, for all we know, the phenomenon could occur even if Pr(T/BE)/ Pr(∼ T/BE) is somewhat less than 1 (this is why a rough calculation of this ratio was enough to make my point).14 Here’s a simpler way of describing how evidence from miracles could produce the DRRA phenomenon. Assuming the EWV, disagreement amongst peers about whether the background evidence supports God’s existence will end up raising (via SAEW) atheists’ and agnostics’ degrees of belief that the background evidence supports God’s existence. Once they have such a higher degree of belief, it becomes easier for evidence that a miracle occurred to have sufficient force to justify belief that God exists (especially if their assessment of that evidence is again raised by their monotheistic epistemic peers). Of course, it’s important to note that the monotheists don’t get off scot-free; their degree of belief that God exists rationally ought to go down. But, if it turns out that Pr(T/BE&ME)/ Pr(∼ T/BE&ME) remains greater than 1, then belief that God exists would still be justified for them.15 The argument I have given illustrates an interesting role that miracles can play in justifying belief in God, one that hasn’t been noticed in the literature on miracles. For 14 Suppose, contra to my suggestion in note 11, that the value we want is Praverage (ME/T&BE)/ Praverage (ME/∼T&BE). It is still possible that individual values for theists, atheists, and agnostic experts of Pr(ME/T&BE) and Pr(ME/∼T&BE) render this value > 1. Suppose there were four experts, two theist and two non-theist, whose personal values for these probabilities are as in the following table:

Theist 1 Theist 2 Non-theist 1 Non-theist 2

Pr(ME/T&BE)

Pr(ME/∼T&BE)

.9 .7 .1 .05

.05 .1 .2 .2

Notice that for each theist, Pr(ME/T&BE)/ Pr(ME/∼T&BE) > 1, and for each non-theist, Pr(ME/T&BE)/ Pr(ME/∼T&BE) < 1. However, Praverage (ME/T&BE) = .48 and Praverage (ME/ ∼T&BE) = .14, and so Praverage (ME/T&BE)/ Praverage (ME/∼T&BE) > 1. This toy example thus illustrates that what I am arguing to be possible genuinely is possible even if the value we need is Praverage (ME/T&BE)/ Praverage (ME/∼T&BE) instead of (Pr(ME/T&BE)/ Pr(ME/∼T&BE))average . 15 There are difficult issues about what minimum confidence level is required for all-out belief, but it could turn out that the value of the ratio exceeds whatever minimum value is required. Setting aside all-out belief, the DRRA phenomenon shows that P(T/BE&ME) could be fairly high, certainly greater than 0.5. Assuming that this probability measures degree of justification, a high degree of belief can be justified in T (even if all-out belief isn’t justified).

re lig ious disag re eme nt and theism 219 instance, in his recent book A Defense of Hume on Miracles, Robert Fogelin defends David Hume’s approach to arguments for God’s existence based on miracles. In a note Fogelin adds an important qualification to his Humean critique of the argument from miracles: If we have independent reason to think that a deity exists who sometimes sees fit to produce miracles for special purposes, then invoking the action of such a being might provide the best explanation for miraculous occurrences. This move would not be permitted, though, if the supposed occurrence of a miracle were being cited as proof of God’s existence. (Fogelin 2003: 90 n. 5)

My argument that the DRRA phenomenon might occur for evidence of miracles for God’s existence shows that Fogelin’s qualification is too strong. Evidence for the occurrence of miracles can justify belief in God’s existence even if one suspends judgement about God’s existence on the basis of background evidence. Notice that this lesson applies regardless of whether we hold the EWV. If you think, contra the EWV, that you can stick to your guns on your degrees of belief in the face of disagreement, it could still turn out that, if you suspend judgement in God’s existence on the basis of background evidence, evidence from miracles could be strong enough to rationally justify belief in God’s existence for you. Horrendous evils Probably the most powerful argument against God’s existence stems from the fact that there are horrendous evils for which we can see no justifying reason to explain why God would allow them to occur. Many monotheists feel the force of this argument, and indeed some theistic philosophers grant that horrendous evils provide some evidence against God’s existence, although they think it is outweighed by other evidence. The eminent theistic philosopher of religion Richard Swinburne takes such a view in the second edition of The Existence of God.16 For all we know, it could turn out that on average the experts take the evidence from horrendous evils to be evidence against God’s existence. Let’s divide the evidence into two classes: the evidence of horrendous evils (EV) and all other evidence concerning God’s existence, including miracles (BE). Then we have: Pr(T/BE) Pr(EV /T&BE) Pr(T/EV &BE) = × . Pr(∼T/EV &BE) Pr(∼T/BE) Pr(EV /∼T&BE)

(2)

Now that BE does not include evidence from horrendous evils and does contain evidence from miracles, BE contains almost entirely positive theistic arguments, along with the problem of divine hiddenness. I suspect, then, that the average value amongst experts for Pr(T/BE)/ Pr(∼T/BE) will be greater than 1, but it 16 To be more precise, Swinburne says that there is a good C-inductive argument against ‘bare theism’, which is simply the claim that God exists. But, he thinks that there is not a good C-inductive argument against Christian theism. See Swinburne 2004: 266–7.

220 p robab i l ity i n th e ph i lo sophy of re l i g i on is not clear how much greater than 1 it will be. The following both seem like live possibilities, in the sense that they are relevant and we don’t have good evidence for ruling either of them out: (1) Pr(T/BE)/ Pr(∼T/BE) is slightly greater than 1, but Pr(EV/T&BE)/ Pr(EV/∼T&BE) is considerably less than 1 (on average, for the experts); (2) Pr(T/BE)/ Pr(∼T/BE) is quite a bit greater than 1, but Pr(EV/T&BE)/ Pr(EV/∼T&BE) is enough less than 1 (on average, for the experts) to yield Pr(T/EV&BE)/ Pr(∼T/EV&BE) ≤ 1. In either case, all the experts rationally ought to agree that God does not exist (or probably does not exist), as long as Pr(T/EV&BE)/ Pr(∼T/EV&BE) is low enough. So, for all we know, it could turn out that the DRRA phenomenon occurs for evidence for horrendous evils against God’s existence.

V. Two Objections First, one might object that there is something odd about averaging peers’ values for Pr(p), especially if there is widespread disagreement (as opposed, say, to disagreement that has a Gaussian distribution). Perhaps, one might urge, in the face of such disagreement, one ought to suspend judgement about the value of Pr(p); that is, one shouldn’t assign a value to it because it is inscrutable. But, then, we won’t be able to assign values for at least one of the ratios on the right-hand side of each of equations (1) and (2), and thus, it would seem, we would also have to suspend judgement on Pr(T/BE&ME) and Pr(T/EV&BE). Thus, the DRRA phenomenon does not occur for the claim that God exists. However, despite the fact that it might sometimes be rational to suspend judgement about Pr(p), suspending judgement about some Pr(p) shouldn’t automatically rule out the DRRA phenomenon. After all, in the disagreeing detectives case, the detectives disagreed wildly about the reliability of the witness. Perhaps they should not have assigned a probability for her reliability. Still, it seems that they should agree that Kelly is guilty. So, perhaps this objection shows that there are limitations to modelling disagreement using a Bayesian framework, but it does not show that the DRRA phenomenon is not present in the cases I have discussed. After all, we can present the cases of miracles and horrendous evils in a more intuitive way, parallel to the disagreeing detectives case. Suppose you should suspend judgement on God’s existence on the basis of e, but you also have evidence f that—as you and your peer rationally agree—favours God’s existence. What should you both believe? Intuitively, parallel to the disagreeing detectives case, you should both believe that God exists. Second, one might argue as follows: ‘Look, you’ve granted that we don’t know to the needed degree of accuracy what the average values of the various ratios in equations (1) and (2) will be for the experts, because we simply haven’t done the relevant sociological research. Shouldn’t we then suspend judgement on what those average values are? And then, once we have suspended judgement on the values of the right-hand side ratios

re lig ious disag re eme nt and theism 221 in equations (1) and (2), we will also be rationally required to suspend judgement about Pr(T/BE&ME)/Pr(∼T/BE&ME) and Pr(T/EV&BE)/Pr(∼T/EV&BE), and thus suspend judgement about God’s existence’. My first reply is to point out once again that I have not attempted to argue for specific values of the average Pr(T/BE&ME)/Pr(∼T/BE&ME) and Pr(T/EV&BE)/Pr(∼T/EV&BE) ratios for the experts. I’ve only argued that it is within the realm of nearby epistemic possibility that those ratios have average values for the experts that would (if we were to become aware of it) rationally require the experts and the rest of us to believe that God exists (if the values turned out one way) or that God does not exist (if the values turned out another way), assuming the EWV. Here’s another way to put my point: just by learning the probability distribution for the experts, we might end up being rationally required either to believe that God exists or that he doesn’t on the EWV. My argument for this conclusion is not affected by the objection. My second reply is to say that I’m not persuaded that the EWV would entail that we should suspend judgement on God’s existence until we have actual evidence about the relevant probability distributions for the experts. Moreover, there are tricky issues about whether there really are epistemic peers on the issue of God’s existence. The existence of such peers becomes doubtful once we see that the occurrence of the DRRA phenomenon regarding God’s existence will likely depend on evidence for specific miracles. There are many different reports to witnessing miracles, nobody is aware of all of them, and many people are aware of different ones. Furthermore, fully assessing even the most discussed alleged miraculous occurrence—Jesus’ resurrection—would require, among other things, knowledge of the Jewish background, knowledge of events of the time, textual analysis of the relevant documents, knowledge of subsequent history, as well as sophisticated philosophical conceptions of evidence. Very few people are in possession of all of this knowledge. Most people possess at best different parts of this body of knowledge. If two or more people are not epistemic peers on this issue, then the EWV simply doesn’t apply to them, until they become peers.

VI. Conclusion Here I have presented a new argument to the conclusion that the EWV does not inevitably lead to the problem of spinelessness. Under certain circumstances, the rational way for epistemic peers to resolve disagreement about p is for both to believe that p. This will be the case when there is agreement amongst epistemic peers about the force of some body of evidence e but disagreement about the force of f , in cases where, if the peers suspend judgement about the force of f , then e justifies belief in p for each of them. I have also argued that evidence for miracles, or evidence provided by the existence of horrendous evils, can play the role of ‘e’ and therefore rationally require epistemic peers to believe that God does, or does not, exist. These arguments

222 p robab i l ity i n th e ph i lo sophy of re l i g i on invite a follow-up study which would enable us to calculate the experts’ degrees of belief in the relevant ratios, average them, and then see whether or not our epistemic peers are rationally obliged to believe in the existence of God.17 Does religious disagreement actually aid the case for theism then? Maybe; maybe not. It depends on what that follow-up study reveals. It also depends on whether the EWV can withstand further criticism. But, oddly enough, theistic and non-theistic equal weighters may well be able to use disagreement to their epistemic advantage.

Appendix A While intuitions differ on whether or not Gary and Isaac should suspend judgement, in my informal polling more people agree with me than disagree. While not guaranteeing that my view is right, this minimally shows that we ought to take it seriously; and thus that it is worthwhile to investigate what would follow from it for the EWV. One possible explanation for differing intuitions about the detectives case is that it concerns a judgement of guilt, and we are used to holding such judgements to a higher evidential standard than we often hold other beliefs. Suspending judgement about the reliability of the witness still casts enough doubt (because the witness could be right) that we shouldn’t believe that Kelly is guilty using the higher standards. However, using normal standards, we should judge that Kelly is guilty (although perhaps we wouldn’t be confident enough of this to convict Kelly in a court of law). To avoid potential noise in our intuitions due to differing standards, consider a different case that doesn’t involve judgements of guilt: Steve’s daughter Hannah didn’t come home after school today, and Steve and his wife Marge wonder where she is. There’s a note from school on the calendar indicating that there is a play practice after school today (and Hannah is in the play), and Hannah is conscientious about going to play practice. However, Steve seems to clearly remember Hannah saying this morning that she was going to a friend’s house after school, whereas Marge does not remember Hannah saying this. Suppose they are epistemic peers, so their respective memories about such claims are equally reliable (I know that this is probably hard to imagine for those academic philosophers who are husbands reading this chapter; our wives remember things much better than us, but suppose it for the sake of argument). What should they believe about Hannah’s location? They should suspend judgement about whether Hannah said she was going to her friend’s house, but the note is still good evidence that the practice is going on. Given that Hannah is conscientious in going to play practice, the note is good evidence that she is at play practice. So, they should believe that she is at play practice. They should be less certain of this than they would otherwise be if they hadn’t had the conflicting memories, but it seems that overall they should believe that Hannah is at play practice. Ultimately, even if my judgements about these cases aren’t fully persuasive, all I need for my argument in this chapter is EWV, as I elucidate it later, which these cases support. Later I 17 Many might find it worrisome that it now appears that justification for religious beliefs ultimately

depends upon sociological facts about other people’s degrees of belief. Indeed, I suspect that this worry lies behind the reluctance of many philosophers to embrace the EWV. Perhaps this is a genuine problem for the EWV. Here I have suspended judgement on this issue and simply assumed the EWV for the sake of argument to see if equal weighters have to fear spinelessness generally, and spinelessness about religion specifically. I’ve argued that they do not.

re lig ious disag re eme nt and theism 223 give some other more theoretical reasons for accepting EWV. Those reasons, together with the disagreeing detectives case and the Hannah case, strongly support EWV.

Appendix B As intuitive as the SAEW model seems, it faces a handful of problems. David Jehle and Branden Fitelson have recently argued that this model (which they call the ’straight averaging’ model) conflicts with some other intuitive principles from the literature on probability aggregation (see Jehle and Fitelson 2009). In effect, they argue that it is difficult to develop a model of equal weight that at once: (1) regards the Pr(p) for each person after weighting each other’s views equally as probability functions that respect conditionalization as a constraint on the relationship between the Pr(p) for each person after and before finding out about the disagreement; (2) that treats the Pr(p) for each person after weighting each other’s views equally as a function of the Pr(p) for each person prior to weighting each other’s views equally; while (3) preventing the changes in Pr1R (p) and Pr1S (p) from mandating changes in other probability assignments of peer-propositions that were initially agreed-upon. They present a series of modified straight averaging and approximate straight averaging models that are designed to better fit the principles, but none fit all of the principles. They suggest that principle (2), above, is the most reasonable one to reject, and they develop a model of equal weight that rejects it while accepting the other principles. Rather than offering a conclusive defence of that model, they take their discussion to simply raise issues about how to formulate the equal weight view and to offer a challenge to the defender of the view to be clearer about what the constraints are on equal weighting. Their argument, which merits much further discussion, presents one kind of challenge to modelling equal weight. There are yet other problems. Suppose we have a large number of epistemic peers who disagree about p and therefore have a range of different levels of confidence in p. Suppose we look at a graph of the distribution of the different levels of confidence in p in this group. Straight averaging says that everybody in the following cases should accept Pr(p) = 0.5 after becoming aware of their disagreeing epistemic peers: a straight line distribution, a bimodal distribution with each peak equidistant from 0.5, and a parabolic distribution with the peak at 0.5. Using straight averaging seems fairly intuitive for the first and third cases, but quite unintuitive for the second case. Each person in the bimodal case seems to have good evidence that Pr(p)  = 0.5 since the epistemic peers cluster around two probability values. In such a case, it seems more plausible to say that Pr(p) = x or y, where x and y are the probability values at which there are peaks. (That is, one ought to believe that the Pr(p) = x or Pr(p) = y, but suspend judgement on the actual value of Pr(p).)

References Christensen, D. (2007) ‘Epistemology of Disagreement: The Good News’, Philosophical Review, 116, 2: 187–217. Craig, W. L. (1989) Assessing the New Testament Evidence for the Historicity of the Resurrection of Jesus. Lewiston, NY: Edwin Mellen Press. Crosson, J. D. (1995) Who Killed Jesus? San Francisco: Harper.

224 p robab i l ity i n th e ph i lo sophy of re l i g i on Crosson, J. D. and N. T. Wright (2006) The Resurrection of Jesus. Minneapolis: Fortress Press. Davis, S. (2004) ‘It is Rational to Believe in the Resurrection’. In M. Peterson and R. van Arragon (eds), Contemporary Debates in Philosophy of Religion. Malden, MA: Blackwell, 164–73. Earman, J. (2000) Hume’s Abject Failure. Oxford: Oxford University Press. Elga, A. (2007) ‘Reflection and Disagreement’, Noûs, 41, 3: 478–502. Feldman, R. (2006) ‘Epistemological Puzzles about Disagreement’. In S. Hetherington (ed.), Epistemology Futures. Oxford: Oxford University Press, 216–236. (2007) ‘Reasonable Religious Disagreements’. In L. Anthony, (ed.), Philosophers Without Gods. Oxford: Oxford University Press, 194–214. Fogelin, R. (2003) A Defense of Hume on Miracles. Princeton: Princeton University Press. Frances, B. (2010) ‘The Reflective Epistemic Renegade’, Philosophy and Phenomenological Research, 81: 419–63. (forthcoming) ‘Discovering Disagreeing Epistemic Peers and Superiors’, International Journal of Philosophical Studies. Gärdenfors, P. (1986) ‘Belief Revisions and the Ramsey Test for Conditionals’, Philosophical Review, 95, 1: 81–93. Jehle, D. and B. Fitelson (2009) ‘What is the “Equal Weight View”?’, Episteme, 6, 3: 280–93. Kelly, T. (2006) ‘The Epistemic Significance of Disagreement’, Oxford Studies in Epistemology, 1: 167–96. Koons, R. (2009) ‘Defeasible Reasoning’. In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter Edition): http://plato.stanford.edu/archives/win2009/ entries/reasoning-defeasible Martin, M. (1991) The Case Against Christianity. Philadelphia: Temple University Press. (2004) ‘It is not Rational to Believe in the Resurrection’. In M. Peterson and R. van Arragon (eds), Contemporary Debates in Philosophy of Religion. Malden, MA: Blackwell, 174–84. McGrew, T. and L. McGrew (2009) ‘The Argument from Miracles: A Cumulative Case for the Resurrection of Jesus of Nazareth’. In W. L. Craig and J. P. Moreland (eds), Blackwell Companion to Natural Theology. Malden, MA: Blackwell, 593–662. Oppy, G. (2006) Arguing About Gods. Cambridge: Cambridge University Press. Swinburne, R. (2003) The Resurrection of God Incarnate. Oxford: Oxford University Press. (2004) The Existence of God. 2nd edition. Oxford: Oxford University Press. van Inwagen, P. (1996) ‘It is Wrong, Everywhere, Always, and for Anyone, to Believe Anything upon Insufficient Evidence’. In J. Jordan and D. Howard-Snyder (eds), Faith, Freedom, and Rationality. London: Rowman & Littlefield, 137–53. White, R. (2005) ‘Epistemic Permissiveness’, Philosophical Perspectives, 19: 445–59. Wright, N. T. (2003) The Resurrection of the Son of God. Minneapolis: Fortress Press.

12 Can it be Rational to Have Faith? Lara Buchak

I. Introduction My concern here is the relationship between faith and practical rationality. I seek to develop a unified account of statements of faith concerning mundane matters and those concerning religious faith. To do so, I consider the sense in which faith requires going beyond the evidence, and argue that faith requires terminating the search for further evidence. Having established this, I turn to the question of whether it can still be rational to have faith; arguing that, contrary to common assumptions, there need be no conflict between faith and rationality. We shall see that whether faith can be practically rational depends both on whether there are extrinsic costs associated with postponing the decision involving faith and the extent to which potential counter evidence would be conclusive.

II. Preliminaries I begin with the idea that faith statements in religious contexts and in more mundane contexts express the same attitude and so share some typical characteristics. By ‘faith statements’ I simply mean statements involving the term ‘faith’. The following are representative examples: I have faith in your abilities. I have faith in you. She acted on faith. It was an act of faith. I have faith in God’s goodness.

He has faith that his spouse won’t cheat on him. He has faith that you won’t reveal his secret. She has faith that her car will start. I have faith that God exists. I have faith in God.

These statements share three important features: they involve a relationship between the agent and a particular proposition, between the agent and a particular (actual or potential) action, and between the proposition and the evidence the agent currently possesses. The first thing to notice is that faith statements typically involve a proposition to which the actor involved acquiesces. This is obvious in the case of ‘faith that’

226 probabi lity in the phi lo sophy of re lig ion statements: when a person has faith that p, he acquiesces to p.1 It is also clear in the case of those statements that can be easily translated into ‘faith that’ statements: for example, the statement I have faith in your abilities is equivalent to I have faith that you will be able to do such-and-such. It is less obvious in the case of those statements claiming that an individual has faith in a person; however, upon further inspection, having faith in a person does typically require acquiescing to particular propositions about that person. For example, having faith in a person might involve acquiescing to the claim that the person will do the right thing or will succeed at a task, and having faith in God might involve acquiescing to (at least) the claims that God exists and that God is good. By the same token, performing an act of faith or acting on faith seems to involve acquiescing to a proposition, and which proposition one acquiesces to will be set by the context. For example, if setting down one’s own weapons is an act of faith, then this is because setting down one’s own weapons involves acquiescing to the claim that the other person will then set down his. The second thing to notice about faith statements is that the truth or falsity of the proposition(s) involved is ordinarily a matter of importance to the actor. For example, it does not seem apt to state that I have faith that the Nile is the longest river in Egypt, because I do not care whether or not this is true. We do not attribute faith to a person unless the truth or falsity of the proposition involved makes a difference to that person. I might consider whether to have faith that my spouse won’t cheat on me or that my car will start precisely because it makes a difference to me whether or not these things are true. Along the same lines, having faith typically involves an action: a person’s having faith in something should make a difference to her behaviour. However, this needn’t be an actual action. It would be enough for faith that if a person were put in a particular situation, she would then manifest the relevant behaviour (assuming that there are no forces that would stop her). Faith is thus linked to a disposition to act. This brings us to the next point about the relationship between faith and behaviour: it seems that one can have faith in a particular proposition relative to one action but not to another. For example, I might have faith that my car will start when I only need to drive to work but lack that faith when I am relied upon to carry a life-saving organ to the hospital (as evidenced by the fact that I may double-check my engine or arrange for a backup mode of transportation in the latter case but not the former). A person might have faith in God when it comes to giving weekly donations to the poor but lack faith in God when it comes to allowing himself to be martyred.

1 I speak of acquiescing to a proposition rather than believing it because I am not sure that if I have faith

in something, I thereby believe it. While it sounds infelicitous to say ‘I believe that ∼X but I have faith that X’, there may not be anything wrong with saying ‘I don’t know whether X—I have no idea whether I believe that X or not—but I have faith that X’. So as not to prejudge that issue, I make a weaker claim: that having faith involves taking the proposition to be true, that is, ‘going along with it’, but not necessarily adopting an attitude we might describe as belief.

can it be rational to have faith ? 227 There are two ways in which we might interpret the fact that one might have faith when it comes to the performance of some actions but not others: we might say that faith is context-dependent, or we might say that faith comes in degrees. There is something to be said in favour of each of these approaches. However, whether one has faith in X expressed by a particular act A will be determinate on either approach, and since this will be our basic unit of analysis in this paper, we needn’t choose between them. The next thing to bring into the picture is the relationship between the agent who has faith in X (expressed by some act A) and the evidence he has for X. We make assertions of faith only when the status of the proposition involved is uncertain or when the evidence we have is inconclusive. For example, when a friend is worried about the outcome of an exam, we might reassure her by saying ‘I have faith that you passed’; however, once she shows us that she got an A we would no longer say this. Clearly, this is not because we are less willing to acquiesce to the claim that she passed, but because we now know for certain that she did. For similar reasons, it seems odd to claim to have faith in logical truths. These considerations suggest that a person cannot have faith in propositions of which he is antecedently certain or for which he has conclusive evidence.2 Are there further restrictions on which propositions a person can have faith in? I don’t believe so. Indeed, a person may have no evidence at all for the proposition he has faith in, or even may have evidence that tells against the proposition. For example, we could imagine someone saying ‘Although she’s spilled all the secrets I’ve told her so far, I have faith that this time will be different’, or ‘I don’t think there’s any evidence that God exists, but I have faith that he does’. Therefore, that a person has faith that X implies nothing about his evidence for X, aside from its inconclusiveness. Statements in which the actor has faith despite no or contrary evidence do seem correctly described as cases of faith, even though they are not cases in which we are inclined to think that the actor is wise to have faith; rather, we think his faith is misplaced. We will later see that we can do justice to the distinction between well-placed faith and misplaced faith. My final preliminary observation is that having faith seems to involve going beyond the evidence in some way. The bulk of my argument will be devoted to spelling out

2 Although the following worry arises from the possibility of overdetermination. I might have faith in my friend, and therefore have faith that my friend hasn’t transformed me into a brain in a vat for his own merriment (this example is due to an anonymous referee), and yet I might be antecedently certain that I’m not a brain in a vat (on the basis of philosophical arguments, perhaps). Or I might have complete faith in a friend’s testimony, and thus have faith in anything he says; however, it might be that he sometimes says things of which I am already certain. What should we say in these cases? One possible response is to say that a friend’s testimony simply can’t produce faith in propositions of which I am already certain. After all, we may think it sounds strange to say I am independently convinced that I can’t be a brain in a vat, and I also have faith that you haven’t envatted me. Another possibility is to claim that these statements, to the extent that we can imagine circumstances in which they could be uttered felicitously, are really modal in character: the actor is claiming that if she wasn’t independently convinced that she wasn’t a brain in a vat, she would have faith that you haven’t envatted her.

228 p robab i l ity i n th e ph i lo sophy of re l i g i on in what way one must go beyond the evidence in order to count as having faith. I postpone discussion of this to the next section. We can now begin to give a formal analysis of faith. As we’ve seen, the term ‘faith’ appears in many different grammatical constructions: you might have faith in a person, you might have faith in a proposition, you might perform an act of faith, or you might act on faith. We require an account that makes sense of all of these uses of the term. I’ve already pointed out that faith typically involves a proposition as well as an action to which the truth or falsity of the proposition makes a difference. I propose, then, to make faith that X, expressed by A the basic unit of analysis, where X is a proposition and A is an act, and define the other constructions in terms of this one. It is important that our analysis expresses the relationship between the proposition and the act. I have explained that a person can have faith that X only if he cares whether X is true or false, and presumably this is because the person would like to perform the act if X is true but would like to do some other act if X is false. So, as a first pass, we might say: A person has faith that X, expressed by A, only if that person performs act A when there is some alternative act B such that he strictly prefers A&X to B&X and he strictly prefers B&∼X to A&∼X.

Thus, we might say that the agent bases A on the truth of X, since there is some alternative action that the agent would perform if he knew that X were false. This is not yet the whole story, but it does allow us to go a step further and identify what it is to have faith in a person, and to perform an act of faith: A person P has faith in another person Q if and only if there is some act A and some proposition(s) X that express(es) a positive judgement about Q such that P has faith in X, expressed by A.3

So, Bob might have faith in Mary because he has faith that Mary won’t reveal a secret he tells her, expressed by the act of telling her his secret. Paul might have faith in God because he has faith that God exists and that God is good, expressed by the act of praying. Again, faith only requires a disposition to choose particular acts, and these acts need not be actually available. We can now take the next step and identify what it is to perform an act of faith, or to act on faith: A person performs an act of faith (or acts on faith) if and only if he performs some act A such that there is a proposition X in which he has faith, expressed by A.

3 Notice that the judgement must be positive from the point of view of the agent, in the sense that the agent has a preference for A&X, otherwise the account would be subject to the following counterexample: we think that if a person prefers that his friend refrain from smoking, even though he thinks his friend is inclined to smoke, he can’t appropriately be said to have faith that she will smoke.

can it be rational to have faith ? 229 With these preliminaries in place, the rest of this paper will elaborate what else faith that X, expressed by A, requires and under what circumstances it is rational to have such faith.

III. Going Beyond the Evidence: Three Views Before outlining my own view I consider three initially promising ways to make sense of the requirement that faith goes beyond the evidence. I conclude that each of these attempts fails to reveal a genuine requirement of faith.4 The first analysis claims that faith in X requires believing X to a higher degree than one thinks the evidence warrants.5 More precisely, for an agent to have faith in X, he must think that the evidence warrants believing X to some degree, say, r, but nonetheless believe X to degree q, where q > r. As a special case of this, one might think that faith requires believing X to degree 1, even though one thinks that the evidence warrants a definite credence less than this. On this analysis, having faith involves being entrenched in a kind of partial belief version of Moore’s paradox: one thinks something like X is likely to degree r, but I believe X to degree q. Ignoring the issue of whether this could ever be rational—since we don’t want to prejudge the main issue by assuming there must be cases of rational faith—there are two problems with taking this to be a requirement of faith. First, it seems hard to imagine someone actually having faith in this sense and, especially, recognizing that he has faith in this sense. Yet having faith seems to be a common occurrence, one that does not involve psychological tricks or self-deception. Second, because it is unclear that one can reliably or stably have faith in this sense, or even take steps to set oneself up to have faith in this sense, it does not seem to be the kind of thing that ethics would require. And yet, religious ethics and the ethics of friendship do seem to require faith in certain cases. The second analysis is more initially plausible. According to this, for the person who thinks that the evidence warrants believing X to degree r, faith requires acting as if he has degree of belief q—that is, performing the actions that he would perform if he had degree of belief q—where q > r. Thus, one can maintain one’s degree of belief in r—and so avoid epistemic inconsistency—while still behaving, as regards the likelihood of X, in a way that goes beyond the evidence. Indeed, the paradox sounds less paradoxical when degree of belief is cashed out in terms of betting behaviour. One thinks, the evidence warrants my adopting a betting quotient of r, but I adopt a betting quotient of q. Again, as a special case of this, we might think that faith requires adopting a betting quotient of 1, that is, using p(X) = 1 when making decisions. This would 4 I presuppose a standard ‘partial belief ’ framework, where beliefs come in degrees (sometimes called ‘credences’) between 0 and 1. 5 In some of the analyses under review here the act does not figure into the proposal. Therefore, for readability, I will say ‘faith in X’ when I really mean ‘faith that X expressed by A, for some particular A’.

230 probabi lity in the phi lo sophy of re lig ion involve not considering or caring about states of the world in which ∼X holds when making decisions. I admit that this analysis has some plausibility. However, I again think that there are problems concerning both the phenomenology and ethics of faith. The phenomenological worry is that, on this analysis, faith requires simultaneously keeping track of two things: one’s actual credences, and the ‘faith-adjusted’ credences that one employs in decision making. However, the phenomenology of faith doesn’t seem to involve a lot of mental accounting. Yet perhaps this is not a serious problem, because the defender of this view could argue that since faith is relative to particular acts, one only needs to consider one’s faith-adjusted credences when making the relevant decision. There are two more serious problems arising from the fact that although religious ethics and the ethics of friendship endorse faith in many situations, they wouldn’t endorse certain demands that this analysis suggests. First, consider what this analysis recommends that a faithful person do when asked whether he believes that X. Since this is an action, presumably the faithful person ought, if he ought to have faith, to figure out what to do using his faith-adjusted credences, not his actual ones. So he ought to claim to believe X more strongly than he does; that is, he ought to lie. But those that endorse faith often strongly denounce lying. The second problem is brought to the fore when we consider the special case view that faith requires acting as if p(X) = 1. On this view, the faithful person ought to take any bet that is favourable on the condition that X obtains, regardless of the stakes. So, if asked to bet $1m on a gamble that pays 1 penny if X obtains, the person with faith in X ought to say yes: after all, he can disregard the possibility of ∼X for the purposes of decision making. I’m extremely doubtful that religious ethics would endorse the claim that the truly faithful ought to risk $1m for a mere penny if God exists, especially since they recognize that the evidence isn’t conclusive.6 So we can dismiss the first two analyses which held that in order for a person to count as having faith in X, he must treat his credence in X as higher than it in fact is, either by actually raising it or by acting as if it were higher. Perhaps these analyses have gone astray because they take for granted an inadequate account of when faith enters into one’s belief formation process. They both assume that one examines all of the evidence dispassionately, forms a belief, and then decides whether to adjust this belief in light of faith. But perhaps the relationship between faith and belief formation is more complex than this assumption recognizes. Instead, faith might require taking evidence into account in a particular way—a way that favours X or gives the truth of X the benefit of the doubt, so to speak. Following this line of thought, a third

6 Perhaps the defender of the second analysis could claim that his view doesn’t entail an affirmative answer because betting itself, when the payoffs are so frivolous, has inherent disutility. But it is not clear that he can respond in this way to the case in which the ‘payoffs’ are goods of real value, for example, in the gamble that results in a million lives lost if God doesn’t exist and one life improved mildly if God does exist.

can it be rational to have faith? 231 analysis of faith holds that faith requires setting one’s prior probability to p(X) = 1 before examining the evidence. On this view, one interprets evidence, not with an eye towards finding out whether or not X holds, but in light of the assumption that X does hold. On this view, we might say that faith goes before the evidence, not beyond it. Note that this third analysis is different from the ‘special cases’ of the first two analyses. On those analyses, the faithful agent sets p(X) = 1 even though he believes that the evidence warrants something less. On the present analysis the agent doesn’t have an opinion about what the evidence warrants that is separate from the question of whether he has faith. So let’s say that I have faith that my friend won’t reveal a secret I told him, and I overhear a third party complaining that my friend is a gossip. On the first analysis, I consider this to be evidence against the claim that my friend will keep my secret, but I nonetheless ignore it and continue to have a high degree of belief in the claim. On the second analysis, I consider this to be evidence against the claim, and I lower my degree of belief in the claim, but I nonetheless continue to act as if I have a high degree of belief (I carry on as if no one knows my secret, and I continue to confide in this friend). On the analysis we are now considering, I don’t consider this to be evidence against the claim, precisely because I have faith in the claim. Indeed, there will be no possible evidence that tells against X. This third analysis has a number of advantages. For one, it sheds light on the fact that there seems to be no good answer to the question of how a rational person ought to set his priors. On this view, the reason that there is no good answer is that epistemic rationality stops just short of this question and faith takes over: one can’t avoid having faith in something, because one can’t avoid setting one’s priors.7 This supports William James’ claim that one’s non-rational or ‘passional’ nature must play a role in what to believe when reason alone doesn’t dictate an answer, and that the passional nature generally comes into play in figuring out how to interpret evidence (see James 1896). It also supports an argument of Søren Kierkegaard’s pseudonymous Johannes Climacus that reason alone cannot produce faith; instead, faith requires an act of will (see Kierkegaard 1846). Roughly, Climacus argues that one can never get to religious faith by engaging in objective inquiry because religious faith requires total commitment to particular historical claims. Objective inquiry can never yield certainty in these matters: it always leaves room for doubt. On Robert Adam’s interpretation of Kierkegaard, total commitment to a belief requires a commitment not to revise it in the future (see Adams 1976). Thus, it requires setting p(X) = 1 and interpreting any new evidence in light of this. However, despite its attractiveness, this view is incorrect because it is vulnerable to similar phenomenological and ethical objections to the ones discussed above. Adams himself raises the ethical objection: ‘It has commonly been thought to be an important 7 Technically, one could avoid having complete faith in anything, since one could avoid setting p(X) = 1 for any X. However, if we think that degrees of faith correspond to setting lower priors, then one would have some degree of faith in many things.

232 probabi lity in the phi lo sophy of re lig ion part of religious ethics that one ought to be humble, teachable, open to correction, new inspiration, and growth of insight, even (and perhaps especially) in important religious beliefs’ (Adams 1976: 233). We might add that the ethics involved in friendship similarly does not seem to require that we remain determined not to abandon our belief in a friend’s trustworthiness come what may. The phenomenological objection can be bought out by considering that anyone who is acting on faith typically feels like she is taking a risk of some sort. The act A that you are performing on faith (that X) is supposed to be better than some alternative if X holds and worse than that alternative if X does not hold. But if one is certain that X is true, then doing A is not a risk at all! On the contrary, A is simply an act that, from your point of view, will undoubtedly turn out well. It is like the act of taking a bet on which you win $100 if water is H2 O and lose $100 if it is not. One might reply that from an objective standpoint, doing A is a risk—because setting one’s priors is risky in some sense. But even if that is the case, the view still fails to explain the phenomenology of acts of faith, since they feel risky even from an internal perspective. What is distinctive about taking a leap of faith, so to speak, is that you are fully aware that it might turn out badly—even if you think it unlikely that it will. An additional objection to this third analysis is that it cannot distinguish between cases of well-placed faith and cases of misplaced faith. Recall the above example of the person who knows that her friend has spilled all of her secrets so far but who has faith that he will not spill future secrets. We likely regard this as a case of misplaced faith. At any rate, when we compare this person to the person whose friend has never spilled a secret and who has faith that he will not spill future secrets, we think that this second person’s faith has a lot more in its favour. But we cannot make sense of this on the present view, since rationality has no conclusions about which priors are laudable and since faith enters the picture before any evidence is interpreted. So, although this third analysis was initially promising, it does not ultimately succeed. On my view, whether someone has faith is not determinable from his priors: a person who starts out sceptical, but who then amasses evidence in favour of X, could indeed end up choosing to have faith in X (consider the conversion of St Paul). Furthermore, a person who begins by assuming that X must be true doesn’t thereby count as having faith in X: credulity and faith come apart. So do credence and faith, as we will see in the next section.

IV. Faith and Examining Further Evidence There is something to Kierkegaard’s idea that one can never arrive at faith by engaging in empirical inquiry—that faith instead requires an act of will. However, this is not because faith requires a kind of certainty that empirical inquiry cannot provide, nor because faith must precede inquiry. Instead, it is because engaging in an inquiry itself constitutes a lack of faith. That is, faith requires not engaging in an inquiry whose only

can it b e rati onal to have faith ? 233 purpose is to figure out the truth of the proposition one purportedly has faith in. So the sense in which faith in X requires some response to the evidence aside from that normally warranted by epistemic norms is that it requires a decision to stop searching for additional evidence and to perform the act one would perform on the supposition that X. Consider an example. If a man has faith that his spouse isn’t cheating, this seems to rule out his hiring a private investigator, opening her mail, or even striking up a conversation with her boss to check that she really was working late last night—that is, it rules out conducting an inquiry to verify that his spouse isn’t cheating. If he does any of these things, then she can rightfully complain that he didn’t have faith in her, even if she realizes that, given his evidence, he should not assign degree of belief 1 to her constancy. Similarly, if I have faith that my friend will keep a secret, this seems to rule out asking a third party whether he thinks that friend is trustworthy. To use a religious example, when so-called ‘doubting’ Thomas asks to put his hand in Jesus’ side to verify that he has been resurrected in the flesh, this is supposed to indicate that he lacks faith. We can say something even stronger: faith seems to require not looking for further evidence even if one knows that the evidence is readily available. For example, consider a case in which a man simply stumbles across an envelope which he knows contains evidence that will either vindicate his wife’s constancy or suggest that she has been cheating. He seems to display a lack of faith in her constancy if he opens it and to display faith in her constancy if he does not. And this seems true even if the evidence has been acquired in a scrupulous way: we might imagine the wife herself presents the envelope to the man, as a test of his faith.8 So we now have the following first pass at a full analysis of faith: A person has faith that X, expressed by A, if and only if that person performs act A when there is some alternative act B such that he strictly prefers A&X to B&X and he strictly prefers B&∼X to A&∼X, and the person refrains from gathering further evidence to determine the truth or falsity of X, or would refrain, if further evidence were available.

To make this more precise, we might state this final condition in preference terms: . . . and the person prefers to decline evidence rather than to view it.

This formulation has an unfortunate upshot, though: it implies that anyone who has faith that X, expressed by some act A, must decline evidence in the matter of X even if they want the evidence for purposes other than deciding between A and B. For example, consider the Christian apologist who has faith that Jesus was resurrected (expressed by, say, the action of going to church every week) but who combs through the historical evidence surrounding Jesus’ resurrection in the hopes of finding evidence 8 Indeed, my account can easily explain why presenting him with the envelope could be a test of faith: it is a test to see whether he will choose to acquire further evidence.

234 p robab i l ity i n th e ph i lo sophy of re l i g i on to convince someone who does not believe. Or consider the person who intends to open the private investigator’s envelope publicly, precisely to show that he has faith in his spouse’s constancy.9 On the current analysis, neither of these acts can be acts of faith: indeed, performing them entails that the agent does not have faith in the proposition in question. The reason that we would say that the apologist has faith in the resurrection even though he continues to look for evidence is that he doesn’t consider his decision to attend church dependent on the outcome of his investigation. Indeed, if he had no desire to convince other people, he would not look for evidence. Similarly, the reason we know that the husband has faith in his spouse, expressed by, say, the act of remaining constant himself, is that his constancy doesn’t depend on the contents of the envelope, even though it does depend on his (current) beliefs about whether his spouse is cheating. So what these examples show is that the claim that the faithful person does not look for evidence at all is too strong. Instead, the faithful person does not look for evidence for the purposes of deciding whether to do A. Thus, if he does look for evidence, he considers this search irrelevant to his decision to do A. A precise way to spell out that the act doesn’t depend on the evidence is that the faithful agent is willing to commit to A before viewing any additional evidence in the matter of X; indeed, he wants to commit to A. In preference terms, he prefers to commit to A before viewing any additional evidence rather than to first view additional evidence and then decide whether to do A. This covers both the case of the person who looks for no additional evidence and the person who does look for evidence, but not in order to decide whether to do A. We can now formulate my final analysis: A person has faith that X, expressed by A, if and only if that person performs act A when there is some alternative act B such that he strictly prefers A&X to B&X and he strictly prefers B&∼X to A&∼X, and the person prefers {to commit to A before he examines additional evidence} rather than {to postpone his decision about A until he examines additional evidence}.

I purposely leave it ambiguous whether this second preference should be strict or weak, as I think a case can be made for either view, and I return to this issue in Section VII. As mentioned above, my analysis vindicates part of Kierkegaard’s insight that faith does require total commitment, and that looking for evidence reveals that one is not totally committed. But what one must commit to is an act, not a belief: specifically, one must commit to performing an act regardless of what the evidence reveals. My analysis also vindicates the idea that faith requires an act of will—on my account one consciously chooses not to look for more evidence (even though doing so might be tempting!)—which is difficult to explain if faith requires a certain degree of belief and belief is not directly under one’s volitional control. One upshot of my analysis is that it is possible for two people to have the same evidence, the same credence function, and the same utility function, and perform 9 Thanks to Sherrilyn Roush for this example.

can it be rational to have faith? 235 the same act, and yet one of these acts displays faith while the other doesn’t. So, for example, assume Ann and Erin have the same evidence about Dan’s secret-keeping ability; that both have p(Dan will keep a secret) = 0.9; that both have the same utility function (that is, the stakes are the same for both of them). Now assume that each has a choice whether to ask a third party what he thinks about Dan’s secret-keeping ability before deciding whether to tell Dan her secret. Ann decides to simply tell her secret; Erin decides to ask the third party, and then ends up telling her secret to Dan on the advice of this third party. Here, Ann displays faith that Dan will keep a secret (expressed by the act of revealing her own secret), whereas Erin does not display faith, even though she also performs this act. So the same act in the same circumstances can be done with or without faith. The argument so far has told us nothing about the circumstances, if indeed there are any, in which faith can be rational. I now turn to this question. First I briefly explain the distinction between epistemic rationality and practical rationality, beginning with the former.

V. Epistemic and Practical Rationality I will assume a broadly evidentialist conception of epistemic rationality: one should proportion one’s beliefs to one’s evidence. One should not, for example, simply believe what one likes or believe what would make one happy. More generally, one should not take non-truth-conducive reasons as reasons for belief. I will also assume a subjective Bayesian account of partial belief: degrees of belief obey the probability calculus; one updates one’s beliefs by conditionalizing on new evidence; and two people can (rationally) have different degrees of belief in a proposition if and only if they have different evidence that bears on that proposition, or they believe the same evidence bears on that proposition differently, or they have different priors.10 An important feature of this account for our purposes is that a rational person can only change his credences in response to evidence—and, in fact, must update them in response to new evidence, at least in matters he cares enough about to form beliefs. To count as epistemically rational, you must proportion your beliefs to the evidence and your beliefs must be coherent (in other words, your credences must obey the probability calculus). Practical rationality, on the other hand, concerns selecting the means to achieve one’s ends. Practical rationality is the kind of rationality that decision theory is meant to formalize: the values of the ends are made precise by means of a utility function, 10 Depending on how one defines evidentialism, it might be strictly speaking incompatible with subjec-

tive Bayesianism, since the latter allows any set of prior degrees of belief to count as rational, so that two people could share a body of evidence and still (rationally) have different degrees of belief. On the other hand, a natural version of evidentialism implies that there is only one rational way to respond to a given body of evidence. We could alleviate this problem by pointing out that rationality is a feature of belief change, or that evidentialism and subjective Bayesianism agree about what to do once the priors have been set.

236 probabi lity in the phi lo sophy of re lig ion and the likelihoods of any particular acts achieving those ends are made precise by means of a credence function.11 On the standard view, subjective expected utility (EU) theory, rational agents maximize expected utility: they prefer the act with the highest mathematical expectation of utility, relative to their utility and credence functions. So if we think of an act as a gamble that yields a particular outcome in a particular state of the world—for example, g = {O1 , E1 ; O2 , E2 ; . . . ; On , En } is the act that yields Oi if Ei is true, for each i—then the value of this act is: EU(g) =

n 

p(Ei )u(Oi )

i=1

In my view, EU theory is too restrictive. However, since it is the widely accepted view, and since I agree that expected utility maximizers are practically rational, I will postpone discussion of my alternative view to Section IX. Before turning to the question of whether faith can be rational, it is worth clearing up a worry: that the definitions of rationality that I’ve adopted might not be strong enough. On the present definition of epistemic rationality, one may adopt any prior degrees of belief, including any conditional priors: priors about the relationship between particular hypotheses and particular pieces of evidence. For example, a person may rationally believe he has been abducted by aliens, as long as he also believes that the evidence he has supports this to the degree that he believes it. A similar point holds about preferences in the case of practical rationality. Both epistemic rationality and practical rationality, as I define them here, are notions of consistency: the only restriction on degrees of belief is that they are consistent with one another, and the only restriction on preferences is that they are consistent with one another, given one’s degrees of belief in each possible state of the world. However, there is another notion of rationality, which rules out believing one has been abducted by aliens and rules out certain preferences, which we might call reasonableness. I cannot fully respond to this worry in depth here. But it is important that the consistency notion of rationality and the reasonableness notion come apart quite readily. Consistency restrictions are structural: they rule out particular patterns of belief or desire, regardless of the content of these attitudes. On the other hand, reasonableness restrictions are substantial: they rule out particular beliefs or desires, regardless of which other beliefs or desires one has. Therefore, we can talk about what they require separately, and this project is about the requirements of rationality in the consistency sense. Or, since this sense of rationality exhausts the subjective sense of rationality, this project addresses the question of whether it can ever be rational from an agent’s own point of view to have faith. If we want to answer the further question of whether it is reasonable to have faith—that is, of whether a person has objective reasons to have faith—we can address this separately. Indeed, nothing in my argument relies on the 11 On the debate between realism and functionalism about the utility function, see J. Dreier (1996).

can it be rational to have faith? 237 content of the propositions believed or desired. Hence the question of whether faith is reasonable can be answered by asking whether there are any contents for which it is reasonable to have the patterns of belief and desire presented in my examples.

VI. Practical Rationality and Evidence-gathering It should be clear that on the first two analyses of faith, considered in Section III above, faith is irrational. On the first analysis, which requires the agent’s credences to be higher than those the evidence suggests, faith is epistemically irrational, and on the second analysis, which requires the agent to act as if his beliefs are different than they are (that is, to take something other than his credences as ‘what he believes’ for the purposes of decision making), faith is practically irrational, however we spell out practical rationality. On the third analysis, faith is always rational, provided one has consistent credences and preferences (though one can’t, for example, have faith in two contradictory propositions). On my analysis faith can also be epistemically rational: that one has faith in X implies nothing about one’s degrees of belief or the consistency thereof.12 Therefore, one can clearly meet the requirements of epistemic rationality, as I’ve stated them, while having faith—whether one has faith is completely separate from whether one is epistemically rational.13 But can faith also be practically rational? Faith requires two preferences: a strict preference for A over the alternatives and a preference not to look for further evidence for the purposes of this decision (or to commit to A rather than seeing further evidence before deciding). Therefore, faith can only be practically rational if both of these preferences are practically rational. Assessing the rationality of the first preference is fairly straightforward: strictly preferring A to the alternatives is rational just in case A has a higher expected utility than the alternatives, given the agent’s credences. So the first (fairly obvious) restriction on when faith is rational is this: one’s credence in X must be sufficiently high as to make A the practically rational act. What about the second preference? Can it be rational to prefer committing to A rather than seeing more evidence before deciding? To address this question, it will help to have a canonical example of the kinds of situations in which the question of faith arises. Again, these situations involve an agent performing an act A in a situation in which the status of some proposition X is in question, and in which he prefers A&X to B&X and prefers B&∼X to A&∼X for some alternative act B. Let us simplify by assuming that there are only two alternatives, and that the alternative act is such that

12 Technically, it implies that p(X)  = 1, but if p(X) = 1 then X is not an appropriate object of faith, so this is not a restriction on one’s credences, but on what can count as an object of faith. 13 Even if epistemic rationality requires that one look for further evidence, it will not conflict with faith on my definition, since having faith doesn’t forbid one from looking for evidence; it only dictates that one must prefer committing to an act before seeing more evidence to postponing the decision.

238 probabi lity in the phi lo sophy of re lig ion its value does not depend on X, that is, the agent is indifferent between B&X and B&∼X.14 To put some concrete utility values in place: u(A&X) = 10 u(B&X) = 1 u(A&∼X) = 0 u(B&∼X) = 1 To generalize, let us assume there is a high value, a middle value, and a low value such that u(A&X) = H, u(B&X) = u(B&∼X) = M, u(A&∼X) = L, and H > M > L. Here are some examples of decisions that might include the relevant values. Consider an agent who is deciding whether to become a monk and does not have conclusive evidence that God exists. If God exists, then becoming a monk is very good—the agent will experience all the goods of the religious life. But if God does not exist, then becoming a monk will result in the agent living a life that is ultimately wasted. On the other hand, failing to become a monk is fine, but not great, either way, if we assume that from a religious point of view becoming a monk is supererogatory: the agent lives roughly the same life as a non-monk whether or not God exists. To take another example, consider an agent who is deciding whether to use his van to transport 10 critically injured patients to the hospital rather than using his car to transport 1 and who does not have conclusive evidence that his van works (but he has, say, near-conclusive evidence that his car works, or a backup plan in case it doesn’t). Or consider an agent who is deciding whether to reveal a secret to someone else, and who does not have conclusive evidence that the friend will keep it; or an agent who is deciding whether to marry a particular person, and does not have conclusive evidence that this person will make a good spouse. In each case, performing the act could turn out very well or poorly, whereas not performing the act is the same either way.15 We could think of the act as an opportunity for something great—but a risky opportunity—that one might take or pass up.

14 What if X is something that the agent strongly prefers to be true, so that A&X and B&X both have a high utility? One might think, for example, that faith in God is typically like this, since the agent often prefers worlds in which God exists to worlds in which God does not exist, regardless of the agent’s choices. In fact, we could build this into the example without changing any of the results. In our discussion, there are only two options and two possible outcomes for each option, and we are only concerned with which of the two options has a higher EU and by how much, that is, with the difference between EU(A) and EU(B). We know that EU(A) – EU(B) = p(X)(u(A&X) – u(B&X)) + p(∼X)(u(A&∼X) – u(B&∼X)). Therefore, the only thing that matters is the difference between A&X and B&X on the one hand, and between A&∼X and B&∼X on the other hand. So if we uniformly increase the value of the outcomes in which X holds, say by replacing the above values with u(A&X) = 110, u(A&∼X) = 0, u(B&X) = 101, u(B&∼X) = 1, to symbolize that the truth of X has inherent value to the agent, this equation will have the same value, and consequently, all of the results of sections VI–VIII will still hold. 15 One might argue that the act couldn’t be the same either way, since, e.g. passing up the possibility of a faithful spouse (or an opportunity to save 10 lives) is worse than passing up the possibility of an unfaithful spouse (or an opportunity not to save any lives). I think the question of how decision theory should handle these nuances is an interesting one, but for now, I will just assume that facts about what might have happened do not make a difference to the agent’s utility function—or at least that they make a negligible difference.

can it b e rati onal to have faith ? 239 Let us assume that A has a higher EU than B, and so A is practically rational given the agent’s current information. Now we want to know whether practical rationality requires that the agent gather more information before she makes her decision. There is a theorem that bears directly on this question: I. J. Good (1967) showed that gathering additional evidence (in Good’s terminology, making a new observation) and then using it to update one’s beliefs before making a decision always has a higher expected utility than making a decision without doing so, provided the following two conditions are met: 1) It is not the case that the agent will perform the same act regardless of the evidence she receives. 2) Gathering additional evidence is cost-free. If condition (2) holds but, contrary to (1), the agent will perform the same act regardless of the evidence, then gathering the evidence will have the same expected utility as not doing so. It is helpful to say something about Good’s setup and about how he calculates the expected utility of gathering and using additional evidence and that of not doing so. The expected utility of not gathering additional evidence is simply the EU of the act with the highest expected utility: in our case, EU(A). (So, for example, if p(X) = 0.59, then EU(don’t gather) = 5.9.) The expected utility of gathering the evidence is more complicated. We consider, for each piece of evidence the agent might receive, which act will have the highest EU relative to the agent’s new credence function after receiving that piece of evidence. The EU of this act is the agent’s utility upon receiving that evidence, since we assume that the agent will pick the act that maximizes expected utility. (So, for example, the EU upon receiving a bit of evidence E that will result in p(X) = 0.95 will be EUnew (A | E) = 9.5, and the EU upon receiving a bit of evidence ∼ E that will result in p(X) = 0.05 will be EUnew (B | ∼E) = 1.) We then need only to weight each of these values—the value of each piece of evidence, so to speak—by the probability of receiving that piece of evidence, to determine the expected utility of gathering additional evidence. (So, for example, if p(E) = 0.6 and p(∼E) = 0.4, then EU(gather) = 6.1.) I note that, throughout, Good assumes that gathering evidence itself has no utility costs. Formally, where Oi (Aj ) is the outcome of action Aj in state Si and p(Si | Ek ) is the agent’s conditional credence for Si given Ek :  EU(don’t gather) = maxj p(Si )u(Oi (Aj )) i   (p(Ek ) maxj p(Si |Ek )u(Oi (Aj ))) EU(gather) = k

i

In our scenario, with two possible actions whose utility values depend only on the status of X, we have:

240 probabi lity in the phi lo sophy of re lig ion EU(don’t gather) = p(X)H + p(¬X)L  EU(gather) = p(Ek ) max{M, (p(X|Ek )H + p(¬X|Ek )L)} k

Good proves that unless the agent will do the same act regardless of the result of the experiment—that is, unless the same act maximizes expected utility for each possible piece of evidence Ek —then EU(gather) is always higher than EU(don’t gather). If the agent will do the same act regardless of the result, then these values are the same. In other words, it is always rationally permissible to make a new observation and use it, and it is rationally required that one do so if some piece of evidence that might result from doing so will lead one to do B instead of A.

VII. Commitment and Interpersonal Cost We can now consider under which circumstances, if any, one is rationally permitted, or rationally required, to have faith. In this section and the next I spell out when it is rational to refrain from examining additional evidence. I assume throughout that the other conditions for the rationality of faith have been met: that the agent’s credences are coherent and are such that doing A rather than B maximizes expected utility. I also assume, as Good does, that for each ‘experiment’ (or bit of searching) the agent is considering performing, the agent can assign credences to the possible results Ei and conditional credences p(X | Ei ) to the hypothesis given each possible result; and that if the agent performs the experiment, he will update his credence in X on the result that he obtains and choose the action that maximizes expected utility on his updated credence. As we saw in the previous section, one case in which it would be rationally permissible not to examine additional evidence is the case in which the agent will do the same thing no matter which result obtains. According to expected utility theory, doing so will be rational just in case the conditional probabilities p(X | Ei ) are all such that A maximizes expected utility after updating on each result. In our example, this will be the case in which p(X | Ei ) > 0.1 for all Ei , or, more generally, the case in which p(X | Ei ) > (M – L)/(H – L) for all Ei . In less technical terms, this will hold when none of the observations the agent is considering would tell against X conclusively enough to dissuade the agent from A. However, in these cases the agent will never strictly prefer not examining the evidence; instead, he will be indifferent to whether he examines the evidence or not (provided, again, that doing so is cost-free). So if we think that faith requires a strict preference against examining additional evidence—and as I mentioned in Section IV, I am not sure which stance to take—then pointing to situations in which the agent will do the same regardless of the evidence will not help us. The other condition under which it would be rational to eschew additional evidence is the situation in which the evidence is costly. Since one of the conditions of Good’s Theorem is that costs are negligible, he doesn’t explain how to proceed when there are

can it be rational to have faith? 241 costs. But there is a fairly obvious way to determine whether the benefit of performing the experiment outweighs the cost, in expected utility terms. Specifically, we can measure the benefits of the experiment as the difference between EU(gather) and EU(don’t gather), where Sk is a relation defined to hold of all evidential results Ek such that the agent will want to choose A if he sees that result:  EU(gather) – EU(don’t gather) = p(Ek )(M − (p(X|Ek )H + p(¬X|Ek )L)) k ¬Sk

The multiplicand in each term of the summation is the EU of performing B minus the EU of performing A, relative to the new evidence Ek . If the experiment has some monetary or cognitive cost that is measurable in utility and that does not depend on which action the agent chooses, the result of the experiment, or the state of the world, then it is a simple matter to determine whether the benefit outweighs the cost.16 However, the costs of gathering evidence needn’t be monetary or cognitive. Indeed, there are two types of (non-monetary) costs that seem particularly relevant to situations involving faith: interpersonal costs and the costs of postponing a decision. We need to examine each of these, to see whether they could make faith rational. The first kind of cost comes up primarily in contexts in which one has faith in another person. In these cases, lacking faith might in itself cause harm to the relationship. For example, one’s spouse or friend might be upset if one doesn’t have faith in her, or one might miss out on certain goods that mutual faith is a prerequisite for (a feeling of connection or security, perhaps). In the religious case, it might be that one’s relationship with God will be lacking if one does not have faith and will be lacking for that fact. We might think of these costs as intrinsic costs to lacking faith. Before concluding that these costs can play a role in rationalizing faith, though, we should consider why it should be upsetting to someone that another person lacks faith in them. I suspect that the most common reason is that a lack of faith indicates that the agent is not as he ought to be with respect to his beliefs and desires regarding the other person. For example, a husband assigns low credence to the possibility that his wife is faithful even though she’s given him evidence through her actions and character that should be sufficient for a high credence, or he assigns a low utility to continuing the marriage with her. But these aren’t complaints against the husband’s lack of faith per se. In one case, it is a complaint that he is being irrational in a particularly hurtful way. In the other, it is a complaint that he has the wrong values given their relationship. So there is no ‘additional’ cost to lacking faith, beyond the costs of these actions. I tentatively conclude that the intrinsic cost of lacking faith might be a way in which evidence could be costly for a rational agent, but that this position would need to be further supported. 16 Since we can only calculate the utility of fully specified outcomes, the cost will technically be a utility difference c such that u(A&X&don’t gather) – u(A&X&gather) = u(B&X&don’t gather) – u(B&X&gather) = u(A&∼X&don’t gather) – u(A&∼X&gather) = u(B&∼X&don’t gather) – u(B&∼X&gather) = c.

242 probabi lity in the phi lo sophy of re lig ion

VIII. The Cost of Postponement The second kind of cost that might be associated with gathering evidence is the cost of postponing the decision. In the most extreme case, looking for further evidence amounts to losing the option of doing A. For example, it might be that one’s friend is only available to hear one’s secret today, so if one does not reveal it, one will lose that option forever. Or it might be that one’s potential spouse has given one an ultimatum. Or it might be that one needs to choose a vehicle to drive critically injured patients to the hospital, and any delay will result in their certain death. In these cases, the overall value of gathering further evidence will be negative: it will be the difference between the expected value (on one’s current credences) of doing A and that of doing B, since deciding to gather further evidence is equivalent to deciding to do B. In a less extreme version, A might be an action that provides more good to the decision-maker the earlier he chooses it (in the event that X holds), so the utility of choosing A tomorrow might be slightly lower than the utility of choosing A today. If we imagine that the agent always prefers a day of marriage with a faithful spouse to a day of bachelorhood, or that in the event that God does exist, the agent prefers a day spent as a monk to a day spent as an ordinary citizen, then each day of postponing the decision is costly. For decisions where postponement is costly but does not prevent the agent from eventually choosing A, under what conditions does this cost outweigh the benefit of gathering additional evidence? To answer this question, let us assume that the only cost associated with postponing the decision occurs in the event that one eventually does A and that X obtains. Then let us fix the cost of postponing the decision while one does a particular experiment as c: specifically, c is the difference between u(A&X&don’t gather) and u(A&X&gather), that is, the difference between doing A when X obtains without gathering more evidence (or while committing to A regardless of the evidence) and doing A when X obtains after gathering more evidence. Then the benefit of performing the experiment—according to the agent’s current credences—is: EU(gather with cost) – EU(don’t gather) =  k ¬Ck

p(Ek )(M − (p(X|Ek )H + p(¬X|Ek )L)) − c



p(Ek )p(X|Ek )

k Ck

Here, Ck is defined to hold when, if the agent were to receive evidential result Ek , he would still want to choose A, even in light of the additional cost of doing A if X. Note that the first term will always be positive, so the magnitude of the second term must be sufficiently high to offset it in order for the value of performing the experiment to be negative. We can say something about when the value of performing the experiment is apt to be negative. It will be lower when c is higher; that is, when the costs of doing the experiment in the circumstances in which it is costly are higher. It will also be

can it be rational to have faith? 243 lower when M is lower, and H and L are higher; that is, when there is less of a risk  p(Ek )p(X|Ek ) is high. This involved in doing A. Next, it will be lower when k ¬Ck

corresponds to the possibility of the agent doing B as a result of the experiment, even though X in fact holds: the agent might, for example, get ‘misleading’ evidence that leads to the rational performance of an action that in fact has a lower payoff than its  p(Ek )p(X|Ek ) is high. This corresponds alternative. Finally, it will be lower when k Ck

to the probability of the agent doing A when X holds, and thus having to pay a cost he otherwise would not have had to pay. In conjunction with the previous fact, the experiment will have a lower value when p(X) is antecedently high. Thus, holding fixed H, M, and L, in situations in which gathering the evidence proves costly in the event that X obtains and the agent does A, refraining from gathering further evidence is more likely to be rationally required (1) when this cost is high; (2) when the experiment is likely to result in misleading evidence against X, that is, evidence that makes one ‘miss out’ on the possibility of doing A when X in fact holds; or (3) when one already has a high credence in X. The fact that costs associated with postponing a decision can make faith rational vindicates an observation made by William James (1896), though he did not express it in these terms. James argued that when a decision about what to believe is momentous—in that it involves a once-in-a-lifetime opportunity, for example—then it must be made by the will, and that postponing the decision is a decision in itself. He used this observation to argue that it is rationally permissible to choose to believe in God even when one does not have conclusive evidence for God’s existence. I don’t think that it is rationally permissible to believe that God exists when one does not have conclusive evidence, if this means setting one’s credences differently from what one has evidence for (though I’m not saying that this is what James is suggesting). However, I do think that it is sometimes rationally permissible (and indeed, sometimes rationally required!) to have faith in God—as evidenced by doing some particular religious act without looking for further evidence—in circumstances in which postponing the decision to act is costly. The upshot of this discussion is that, if we accept EU theory as the correct theory of practical rationality, then faith can be rational—depending, of course, on one’s credences and the situation in which one finds oneself. We have seen three important results in this regard. First, if we think that faith requires only a weak preference for not gathering additional evidence—that is, if you count as having faith when you are indifferent between making the decision on current evidence and postponing the decision—then faith is rationally permissible, but not rationally required, in cases in which no piece of evidence that one could potentially gather would alter the agent’s decision about what to do. This will hold when no piece of evidence will tell conclusively enough against X such that doing A will no longer maximize expected utility. However, if we think that faith requires a strict preference for no

244 p robab i l ity i n th e ph i lo sophy of re l i g i on additional evidence—that is, you must strictly prefer making the decision on current evidence—then faith will not be rationally permissible in these circumstances.17 Second, faith (under both the strict and weak reading of preference) will be rationally required in circumstances in which there is an interpersonal cost to looking for more evidence; that is, in which lacking faith is intrinsically worse than having faith. However, it is unclear whether such circumstances obtain. In my opinion, the right explanation for the fact that there are relational goods one can’t get unless one has faith isn’t that faith is in itself valuable, but rather that there are some goods that one can’t get if one is more suspicious of another person than the evidence warrants, or if one hesitates to act on a matter involving the relationship. Third, and most crucially, faith (under both readings) is rationally required in circumstances in which the costs of delaying the decision are high enough to outweigh the benefit of additional evidence. Holding fixed the costs of delay, whether these costs outweigh the benefits, depends both on one’s credence in the proposition one has faith in and on one’s beliefs about the potential evidence one might encounter.

IX. Risk Aversion and the Possibility of Misleading Evidence There are two reasons to think that our results so far are incomplete. First, one might think that faith requires more than a preference for not gathering additional evidence—or for committing to an action before the evidence comes in. It requires a preference for not gathering additional evidence even when this evidence is cost-free. For example, we may think that the person who examines the private investigator’s envelope even when there are no ‘postponement costs’ lacks faith. Second, one might think that EU maximization is too strong a criterion of rationality, and that one can be practically rational without being an expected utility maximizer. While I am sympathetic to the general aim of decision theory, and hold that EU theory is largely correct in its analysis of practical rationality, I nonetheless think that expected utility theory employs too narrow a criterion of rationality and should therefore be rejected. There are two motivations for this. The first is that EU theory incorrectly analyzes risk-averse behaviour for rational agents. An agent is risk-averse if he prefers a sure-thing sum of money to a gamble that yields that sum on average: for example, if he prefers $50 to a coin-flip between $0 and $100.18 On EU theory, the only way an agent can rationally have this preference is if he values monetary

17 Perhaps we could argue that faith is rational in these circumstances by stipulating that every experiment has some cost. However, when we consider that faith requires not just a (strict) preference for avoiding evidence in the matter of X when deciding whether to do A or B, but more precisely a (strict) preference for committing to a decision before seeing the evidence, we realize that we would have to stipulate that not committing before performing the experiment always has a cost, and this is less plausible. 18 For a more general definition of risk aversion, see M. Rothschild and J. Stiglitz (1970).

can it be rational to have faith? 245 outcomes in a particular way: in our example, if the utility difference between $50 and $0 is greater than the utility difference between $100 and $50. Intuitively, though, there are two different reasons for having this preference. On the one hand, one might genuinely value increments of money in this way (one’s values for money might ‘diminish marginally’). On the other hand, one might think that an increment of $50 adds the same value regardless of whether one already has $50, but care about other properties of the gamble besides its average utility value: for example, the minimum value it might yield, the maximum, or the spread of possible outcomes. On EU theory, anyone who is risk-averse for this latter reason is thereby irrational. However, as I argue elsewhere, I think that both kinds of considerations could figure into a rational agent’s preferences.19 The second motivation for rejecting EU theory is that it misses an important subjective component of practical rationality. Once we know the values an agent attaches to the relevant ends and the credences he assigns to the relevant states, we are still missing an important factor in determining what he ought to do—namely, how he weighs the prospect of a high probability of realizing some ends he values against a small probability of realizing some ends he values much more. Above, I explained that decision theory formalizes means-ends reasoning: utility corresponds to how much an agent values particular ends, while credence corresponds to the likelihood with which the agent thinks some particular means will realize one of these ends. We might think of the additional factor, which is missing from EU theory, as the factor corresponding to whichever means the agent considers most effective, not in realizing some particular end or another but in realizing his ends as a whole, given that there are many particular ends at stake. In other words, we might think of this factor as corresponding to how the agent structures the realization of his goals. This involves deciding whether to prioritize definitely ending up with something of some value or instead to prioritize possibly ending up with something of extraordinarily high value, and by how much. That is, it is up to the agent how the minimum value, the maximum value, the spread, and other ‘global properties’ factor into his decision. What is at issue here is the following: when an agent is making a single decision, ought he to care only about how a decision would turn out on average if it were to be repeated, or can he place some weight on, for example, how bad the worst-case scenario is? In an earlier paper, I demonstrated that agents who conform to an alternative theory rather than EU theory—agents who are risk-averse in the second sense described above—will sometimes rationally reject cost-free evidence (see Buchak 2010). In particular, in scenarios like those outlined in Section VI above, it will be rationally required for these agents to commit to an action A before looking at additional evidence rather than to look at additional evidence and then decide. This will be the case in situations with the following properties: (1) the agent already has

19 See L. Buchak, ‘Risk and Rationality’ (MS).

246 probabi lity in the phi lo sophy of re lig ion a high degree of belief in X (that is, p(X) is antecedently high); and (2) evidence that lowers the agent’s credence in X sufficiently to make her do B does not tell conclusively against X (for example, for an experiment with two possible outcomes, evidence E in favour of X and evidence ∼E against X, p(X | ∼E) is low but still significant). In short, agents who care about global properties are concerned with a particular risk involved in looking for additional evidence: the risk of coming across evidence that makes it rational to do B even though X is true. In other words, they are concerned, and rationally so, about the risk that additional evidence will be misleading. I already mentioned that the possibility of misleading evidence will make it rational for the EU maximizer to reject costly evidence. However, on my view, the risk of misleading evidence makes faith rational even in cases in which there is no cost to looking for evidence. If one accepts my view, then faith is rational in more cases than it is on standard decision theory.

X. Conclusion We have seen that whether faith that X, expressed by A, is rational depends on two important factors: (1) whether one has a high enough (rational) degree of belief in X, and (2) the character of the available evidence. Specifically, faith in X is rational only if the available evidence is such that no possible piece of evidence tells conclusively enough against X. There are two interesting practical upshots of this conclusion. First, notice that in a standard class of cases, when one has a high degree of belief in a proposition, the odds of any particular experiment being such that it could drastically lower one’s degree of belief decreases the larger the collection of evidence the agent already has.20 So, in a rough-and-ready way, we might say that faith in X (expressed by some particular act A) is practically rational to the extent that the agent’s belief in X is already based on a large amount of evidence. Second, whether faith is rational depends on the kind of world we are in. Faith will be rational to the extent that evidence isn’t ordinarily very conclusive, or to the extent that our decisions usually do have postponement costs. We won’t be able to vindicate the claim that faith is rational regardless of the circumstances. But we can explain why having faith is rational in certain worlds, perhaps worlds like ours. Individuals who lack faith because they insist on gathering all of the available evidence before making a decision stand to miss out on opportunities that could greatly benefit them.21 20 As James Joyce avers, it is ‘usually the case that the greater volume of data a person has for a hypothesis the more resilient her credence tends to be across a wide range of additional data’ (2005: 161). 21 I am indebted to Matthew Benton, Jake Chandler, Joshua Hershey, John Pittard, Jada Twedt Strabbing, and two anonymous reviewers, for extensive written comments on earlier drafts of this paper. This paper has also benefitted from discussions with David Baker, Branden Fitelson, Marco Lopez, Tristram McPherson, Lydia McGrew, and Sherrilyn Roush. Finally, I am grateful for helpful comments from participants at the Conference on Formal Methods in Philosophy of Religion at the Katholieke Universiteit in Leuven,

can it be rational to have faith? 247

References Adams, R. (1976) ‘Kierkegaard’s Arguments against Objective Reasoning in Religion’, The Monist, 60, 2: 228–43. Buchak, L. (2010) ‘Instrumental Rationality, Epistemic Rationality, and Evidence-gathering’, Philosophical Perspectives, 24, 1: 85–120. ‘Risk and Rationality’. Unpublished MS. Dreier, J. (1996) ‘Rational Preference: Decision Theory as a Theory of Practical Rationality’, Theory and Decision, 40: 249–76. Good, I. J. (1967) ‘On the Principle of Total Evidence’, British Journal for the Philosophy of Science, 17, 4: 319–21. James, W. (1896) The Will to Believe: And Other Essays in Popular Philosophy. London: Longmans, Green & Co. Joyce, J. (2005) ‘How Probabilities Reflect Evidence’, Philosophical Perspectives, 19: 153–78. Kierkegaard, S. (1941) Concluding Unscientific Postscript. Trans. D. F. Swenson. Princeton, NJ: Princeton University Press. Originally published in 1846. Rothschild, M. and J. Stiglitz (1970) ‘Increasing Risk: I. A Definition’, Journal of Economic Theory, 2, 3: 225–43.

Belgium; the Philosophy of Religion Society at San Francisco State University; and the Cal Undergraduate Philosophy Forum, Branden Fitelson’s Epistemology seminar, and PhilFemme, all at UC Berkeley.

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Index of Names Adams, R. 128, 145, 231–2 Adams, W. 58 Adler, J. 14 Aguirre, A. 113 Alston, W. P. 4, 8, 65, 73, 128 Aquinas, T. 8

Eagle, A. 1 Earman, J. 1, 6, 29, 48, 55–6, 64, 217 Easwaran, K. 185 Edwards, W. 184 Eells, E. 85 Elga, A. 209–11

Babbage, C. 32, 41, 47–8, 54–5 Bartha, P. 188–91, 193–4 Bartholomew, D. J. 3 Behe, M. 9 Bentham, J. 50 Bergmann, M. 130 Bostrom, N. 9 Bosworth, N. 63 Bovens, L. 66 Buchak, L. 245 Buchler, J. 33–4, 44 Butler, J. 27

Feldman, R. 17, 209–10 Ferreira, J. L. 66 Fitelson, B. 1, 9, 11–12, 84–5, 137–8, 146, 161, 210, 223, 246 Fogelin, R. 219 Frances, B. 211 Fraser, A. C. 50

Campbell, G. 7, 49–52, 60, 62 Carnap, R. 1, 12, 129, 147–9, 152, 154, 161–3 Carr, B. 94, 113 Chandler, J. 11, 137 Chesterton, G. K. 62 Chihara, C. S. 38 Christensen, D. 209–10 Cicero, M. T. 8 Collins, R. 96–7, 103, 112, 114, 120, 122 Conway Morris, S. 92 Conway, J. 178–9 Corner, D. 6, 171 Cowell, R. G. 81 Craig, W. L. 217 Crosson, J. D. 217 Davies, P. 112 Davis, S. 217 Dawkins, R. 92–3 De Condorcet, N. 7, 46–8, 52–6, 59–60 De Retz, J. 57 Dembski, W. 9 Diderot, D. 15 Dougherty, T. 121 Draper, P. 11 Dreier, J. 236 Duff, A. 178, 187, 189, 192

Gale, R. M. 202 Gärdenfors, P. 212 Gardner, M. 8, 67 Geivett, R. D. 4 Gillies, D. 1, 6, 33, 35 Good, I. J. 18–19, 239–40 Goodman, N. 148 Gould, S. J. 92 Gregory, O. 194 Hacking, I. 4, 11, 13–15, 132, 167–9, 172–7, 179, 183, 188 Hájek, A. 1, 14, 168, 182, 185, 187, 189–90, 192 Hall, N. 8, 108 Harrison, V. S. 3, 19 Hasker, W. 145 Hawking, S. 112 Hick, J. 145 Himma, K. E. 9 Holder, R. D. 6, 54–5, 59 Howard-Snyder, D. 11 Howson, C. 10 Huber, F. 1 Hume, D. 2, 4–7, 11, 28–32, 35–7, 41, 47, 49–52, 54–8, 60, 64, 80, 219 Humphreys, P. 1 Huxley, T. H. 117 Hynek, J. 40 James, W. 13, 65, 73, 188, 231, 243 Jantzen, B. 28 Jeffrey, R. C. 16, 178, 181 Jeffreys, H. 10 Jehle, D. 210, 223

250 inde x of nam e s Jenkin, R. 61 Jordan, J. 13–14, 193 Joyce, J. 13, 138, 246 Kelly, T. 14, 210–14, 220, 222 Keynes, J. M. 53 Koons, R. 212 Koperski, J. 94 Kreps, D. 13 Kruskal, W. 36 Langley, S. P. Lauritzen, S. L. Leeds, S. 66, 73 Legg, C. 34–6, 44 Leslie, J. 9, 110 Levine, M. 6 Lewis, D. 115, 184 Lindman, H. 184 Lipton, P. 79 Machina, M. 13 Mackie, J. L. 11 Maher, P. 161 Malcolm, N. 17 Manson, N. A. 9 Martin, M. 180, 182, 217 Maynard Smith, J. 16 McCloskey, H. J. 11 McGrew, L. 59, 79, 91, 94, 98, 217, 246 McGrew, T. 6, 59, 94, 217, 246 McLennen, E. F. 167, 170–2, 178 Mellor, D. H. 1 Merrill, K. R. 35–6 Millican, P. 6 Mitchell, B. 3 Mlodinow, L. 112 Monton, B. 97 Mougin, G. 193–5 Nover, H. 182 Oppy, G. 3, 181, 211 Orr, H. 9 Otte, R. 38, 128–9, 134, 141 Paley, W. 8–9, 79 Parfit, D. 110 Parkinson, R. 59 Pascal, B. 4, 12–16, 71, 165, 167–8, 170–83, 187–91, 194–95, 197–8, 201–2, 205 Pearl, J. 8–9, 67, 79, 81 Peirce, C. S. 6–7, 27–44 Pennock, R. T. 9 Petersen, M. 13 Petrie, W. 42–3

Phillips, D. Z. 17 Plantinga, A. 4, 12, 128, 135, 145–6 Poston, T. 121 Price, G. R. 16, 210 Pruss, A. 146 Ratzsch, D. 9 Rees, M. 113 Reid, T. 7, 48–9, 62 Reisner, A. 14 Rényi, A. 184 Resnik, M. D. 13 Rice, H. 110 Robinson, A. 178 Rothschild, M. 244 Rowe, W. L. 4, 11, 128–30, 134 Royall, R. 132 Ryan, J. 13 Saka, P. 14 Salmon, W. C. 129 Sarkar, S. 9 Saunders, S. 115 Savage, L. J. 16, 184 Schellenberg, J. L. 19 Schlesinger, G. 15, 181–2, 190–2 Seidenfeld, T. 19 Shafer, G. 8, 66 Shah, N. 14 Shao, J. 39 Shimony, A. 184 Skyrms, B. 16, 197 Smolin, L. 111–12 Sobel, J. H. 6, 8, 14, 181–2, 192–3 Sober, E. 9, 79–80, 91, 93, 98, 130, 132, 193–5, 211 Sorensen, R. 182 Starkie, T. 49, 59 Stephens, C. 9 Stiglitz, J. 244 Stoeger, W. R. 112 Strong, A. H. 49 Susskind, L. 112 Sweetman, B. 4 Swinburne, R. 3–4, 10, 91, 98, 103–4, 106, 108–10, 112, 116, 121–2, 145, 193, 217, 219 Taliaferro, C. 2, 19 Tegmark, M. 111, 114–15 Tertullian 16–17 Tooley, M. 11–12, 146 Trakakis, N. 3, 11 Tremlett, G. 72 Tu, D. 39

i nde x of nam e s 251 van Fraassen, B. C. 136 van Inwagen, P. 122, 130, 210 Venn, J. 7, 52–3, 56–7, 60 Voltaire 60 Walley, P. 136 Watts, F. 3 Weibull, J. 197 Weisberg, J. 80

White, R. 209 Wiener, P. P. 28, 37 Williamson, T. 172 Wittgenstein, L. 17 Wolterstorff, N. 4 Wright, N. T. 217 Wykstra, S. J. 130 Yandell, K. 19

Index of Subjects abduction 7, 36–7, 43 basic beliefs 212–13 Bayes factor 7, 54–7, 59–62 Bayes’ theorem 28, 30, 46–7, 54, 79, 99–100, 106, 118, 121–2, 214–16 Bayesian network 81–2, 87–8, 91, 93, 95 biological evolution 9, 16, 79–80, 89–90, 92–4, 98, 103, 108, 110, 116, 121, 135, 197, 202 Condorcet formula 7, 46–7, 52–6, 59 confirmation theory 1, 37–8, 81–2, 137 contrastive confirmation 11, 131, 133, 136–7, 140–1 decision theory 4–5, 13n28, 15–16, 167, 174, 179, 181–2, 186, 190–5, 235, 238, 244–6 degree of confirmation 84–7, 137 deliberational dynamics 15–16, 196–201, 202–3 disagreement, revealed and the equal weight view 210, 214–15, 222–3 religious 16–19, 211–23 without suspension of judgement 211–13 dominance 3, 13–14, 167–70, 172–4, 181–2 super- 167, 170–3, 175 superduper- 14, 174 Weak 13–14, 169–70 epistemic distance 130 evolutionary game theory 197, 201 expected utility indeterminate 180–3 infinite 14, 176–7, 180, 187, 190, 195, 201 relative 16, 193, 198 explaining away 9–10, 9n16, 79–90, 92, 94–8 faith nature of 16–19, 225–35 rationality of 236–6 favouring, evidential See contrastive confirmation fideism 17 fine-tuning 9–10, 80, 90, 94–8, 103–4, 108–11, 116, 118, 120–2 frequency-dependent selection 16 Hawk-Dove game 16

independence, probabilistic 6, 28, 31, 32, 35n10, 36n11, 39n15, 80, 85, 89, 100–1, 170–1, 173 inductive logic and the objectivity of prior probabilities 146–7 and family-relative structure-descriptions 163 and state-descriptions 148–9, 162–3 and structure-descriptions 147–50 Carnap’s λ-continuum 162 inquiry, and misleading evidence 244–6 decision-postponement cost of 242–4, 246 interpersonal cost of 241, 244 intelligent design movement 9, 94 law of large numbers 172, 177 Law of Likelihood 11–12, 132, 134, 137 law of nature 29, 43, 50, 90, 95–6, 103–7, 109–13, 115–19, 121 likelihoodism 131–4, 136–8, 140–1 method of balancing likelihoods 6, 31–2, 34, 43–4 miracle 27–9, 34, 36, 44, 47–58, 60–2, 216–21 Monty Hall puzzle 8 multiverse hypothesis 9, 10, 80, 94–8, 104, 111–20 varieties of 111–16 natural selection 9, 10, 16n32, 111, 117, 135. See also biological evolution natural theology 8, 27, 103, 107–9, 122 observational selection effects 80 one-universe hypothesis, simplicity of 10, 110–11 Pascal’s Wager 4, 12–16, 165, 167–8, 178, 181–2, 185, 187–9, 190n6, 197, 201 and mixed strategies 14, 182, 184, 189n4, 192, 195 argument from dominance 13–14, 167–8 argument from dominant expectation 13–14, 174–7 argument from expectation 13–14

i nde x of sub j e c t s 253 Many Gods objection 15, 185, 187–90, 194, 200–5 Many-Wagers Model 188n2, 198, 196–205 posterior odds 31, 40–1, 66, 70 pragmatic reasons for belief 13, 14, 188n2 principle of credulity 48–9, 52 principle of preservation 18n37, 212n4 principle of regularity 184–5 principle of veracity 48, 51 probability and propensities 33, 35, 48, 51, 110, 116 and relative frequencies 6, 33, 35, 42, 52 epistemic 121–2 logical 121–2, 148, 162. See also inductive logic nature of 1n1, 32–6, 47, 52, 80, 91, 120–1 subjective 33, 129, 147, 188, 194–7, 204, 235 problem of evil 2, 4, 10–12, 127, 142, 144 evidential 4, 12, 34, 128–9, 134, 136–8, 140, 144, 146 logical 4, 127, 144 problem of new hypotheses 37–9 properties morally significant 150, 152–6, 159 rightmaking 144, 150, 152, 155–6 wrongmaking 144–6, 150–5, 159–61, 163–4 protocol 8, 66–70 rationality epistemic 231, 235–7 practical 225, 235–7, 239, 243–5 reference class problem 34, 35n9, 52–3 reformed epistemology 4

relational confirmation See contrastive confirmation replicator dynamics 189, 197–8 risk aversion 244–6 sampling bias in witness reports 6, 34–5, 39–42 Schlesinger’s Principle 15, 190–2 Scottish Common Sense School 48 simplicity, and prior probability 6, 10, 12, 28, 31–2, 38, 41, 48, 54, 60n7, 80, 84–6, 91n8, 97, 104–15, 121, 170, 231 nature of 10, 90, 95, 105–6, 110, 113, 116, 122, 216 of the theistic hypothesis 108–10 sceptical theism 11–12, 130, 142 spinelessness, problem of 210–11, 221, 222n17 testimony, non-binary 54–5, 57–9 theodicy 109, 117–18, 130–1, 134–5, 145 Three Prisoners puzzle 8, 66–7 utility expected 236, 238–46 infinite 14–15, 174, 178, 180, 182–3, 187–8, 189n3, 191, 205 relative 15–16, 191–200 value of cost-free information 18–19, 239–40, 245–6 witness reliability 34–5, 50, 52–3, 55–6, 61, 212, 169–70 and the principle of credulity 48–50, 52 domain-specificity of 50–2