The Force of Argument : Essays in Honor of Timothy Smiley [1 ed.] 9780203859810, 9780415801201

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The Force of Argument

Routledge Studies in Contemporary Philosophy 1. Email and Ethics Style and Ethical Relations in Computer-Mediated Communication Emma Rooksby

11. Real Essentialism David S. Oderberg

2. Causation and Laws of Nature Max Kistler

12. Practical Identity and Narrative Agency Edited by Catriona Mackenzie and Kim Atkins

3. Internalism and Epistemology The Architecture of Reason Timothy McGrew and Lydia McGrew

13. Metaphysics and the Representational Fallacy Heather Dyke

4. Einstein, Relativity and Absolute Simultaneity Edited by William Lane Craig and Quentin Smith

14. Narrative Identity and Moral Identity A Practical Perspective Kim Atkins

5. Epistemology Modalized Kelly Becker

15. Intergenerational Justice Rights and Responsibilities in an Intergenerational Polity Janna Thompson

6. Truth and Speech Acts Studies in the Philosophy of Language Dirk Greimann & Geo Siegwart 7. A Sense of the World Essays on Fiction, Narrative, and Knowledge Edited by John Gibson, Wolfgang Huemer, and Luca Pocci 8. A Pragmatist Philosophy of Democracy Robert B. Talisse 9. Aesthetics and Material Beauty Aesthetics Naturalized Jennifer A. McMahon 10. Aesthetic Experience Edited by Richard Shusterman and Adele Tomlin

16. Hillel Steiner and the Anatomy of Justice Themes and Challenges Edited by Stephen de Wijze, Matthew H. Kramer, and Ian Carter 17. Philosophy of Personal Identity and Multiple Personality Logi Gunnarsson 18. The Force of Argument Essays in Honor of Timothy Smiley Edited by Jonathan Lear and Alex Oliver

The Force of Argument Essays in Honor of Timothy Smiley

Edited by Jonathan Lear and Alex Oliver

New York

London

First published 2010 by Routledge 270 Madison Avenue, New York, NY 10016 Simultaneously published in the UK by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 2010 Taylor & Francis All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data The force of argument : essays in honor of Timothy Smiley / edited by Jonathan Lear and Alex Oliver. p. cm. — (Routledge studies in contemporary philosophy ; 18) Includes bibliographical references and index. 1. Logic. 2. Philosophy, Modern--21st century. I. Smiley, T. J. (Timothy John) II. Lear, Jonathan. III. Oliver, Alex. BC38.F67 2010 160—dc22 ISBN 0-203-85981-2 Master e-book ISBN

ISBN10: 0-415-80120-6 (hbk) ISBN10: 0-203-85981-2 (ebk) ISBN13: 978-0-415-80120-1 (hbk) ISBN13: 978-0-203-85981-0 (ebk)

Contents

Preface Acknowledgments 1

Philosophy In and Out of the Armchair

ix xiii 1

KWAME ANTHONY APPIAH

2

Restricted Quantifiers and Logical Theory

19

THOMAS BALDWIN

3

Logical Form

48

JAMES CARGILE

4

The Socratic Elenchus: No Problem

68

JAMES DOYLE

5

What Makes Mathematics Mathematics?

82

IAN HACKING

6

Smiley’s Distinction Between Rules of Inference and Rules of Proof

107

LLOYD HUMBERSTONE

7

Relative Validity and Vagueness

127

ROSANNA KEEFE

8

The Force of Irony JONATHAN LEAR

144

viii Contents 9

The Matter of Form: Logic’s Beginnings

165

ALEX OLIVER

10 Abstractionist Class Theory: Is There Any Such Thing?

186

MICHAEL POTTER

11 A Case of Mistaken Identity?

205

GRAHAM PRIEST

12 Inferential Semantics for First-Order Logic: Motivating Rules of Inference from Rules of Evaluation

223

NEIL TENNANT

Bibliography of Works by Timothy Smiley List of Contributors Index

259 261 263

Preface

Throughout the second half of the twentieth century Timothy Smiley gave lectures on logic at Cambridge University. It is not an exaggeration to say that the lectures are a life-time memory for many who heard them. A remarkably large number of those students decided to pursue philosophy as a profession and have gone on to distinguished careers of their own: when asked, they cite Smiley’s lectures as an inspiration. Smiley often spoke on topics that, from the outside, might seem arcane: as a notable example, the theory of defi nite descriptions. How could students’ life-decisions be influenced by an inquiry into the workings of the word ‘the’? Bertrand Russell once said that this word is so important that ‘like Browning’s grammarian with the enclitic δε, I would give the doctrine of this word if I were “dead from the waist down” and not merely in prison’ (Russell 1919: 167). But in Smiley’s class, the energy that lit up the room came from a different source. Students could see unfolding before their eyes how logic works. This was not the study of philosophy, it was a philosophical engagement with the myriad manifestations of logical consequence and deductive inference. And Smiley was always asking questions one could not fi nd in a logic text—or worse, questions one could find, only to be accompanied by a pat answer. For example, why does a deductive argument need to be logically valid? Instead of the pieties about necessary truth-preservation, Smiley would point out that with many of the deductions actually made in life, the reasoner does not need or care to consider all logical possibilities; his reasoning would be better off if it concentrated on the real possibilities that might arise. In effect, Smiley invited his students to join him in thinking a problem through; and it is this activity of thinking—with all the dazzling brilliance, cleverness, depth, and utter honesty that Smiley brought to it—that students did not want to give up. And there was mixed in for leavening the impish delight of seeing how Russell or Frege or Łukasiewicz or . . . got it wrong. ‘I suppose I do what I do from a mixture of curiosity and mischief’, Smiley has said of his work, ‘a reluctance to let sleeping dogs lie and a desire for clear-cut, unexpected results’ (quoted in Pyke 1993). Well, by now Smiley has piled up an astonishing array of such results, and there are yelping dogs

x

Preface

everywhere. In an enduring engagement with Aristotle he has transformed the way we must consider the syllogism from what had been the received wisdom of the earlier part of the twentieth century. More important, he has established Aristotle not simply as a towering figure in the history of logic, but as a logician of the fi rst rank who still needs to be reckoned with. In lectures arising from his doctoral work, he was the fi rst to announce completeness proofs for systems of propositional modal logic, interpreted in terms of possible worlds (see Copeland 2002). From his lectures on the theory of descriptions—the news of which was passed around by word of mouth for years—there emerged his British Academy lecture ‘The Theory of Descriptions’ which readers regularly use the words ‘gem’ and ‘masterpiece’ to describe. In this paper, Smiley takes on Russell’s and Frege’s treatment of empty terms. A lot hangs on this. As is well known, the classical predicate calculus omits such terms and, as a result, cannot represent defi nite descriptions or treat (partial or total) functions directly. Instead it relies on Russell’s unwieldy reductive method of replacing function signs by descriptions and then eliminating descriptions. By contrast, Smiley proposed a logical theory which takes names and descriptions, including empty ones, as genuine terms, and accommodates functions, including partial ones, in a natural way, while preserving bivalence. (The technical details were already settled in his ‘Sense without denotation’, a pioneering article in what has become known as Free Logic). From the point of view of elegance, simplicity, and naturalness, Smiley’s theory represents a transformation in the formalization of functions. And he showed beyond question that the rationalizations Russell and Frege each used for their treatments of empty terms do not stand up to scrutiny. But Smiley has done more than achieve significant logical results: he has investigated philosophical method. No one has done more to bring to the uncomfortable attention of philosophers and logicians our tendency to elevate habit into necessity. Over and over again, the predicate calculus is treated as though it simply were logic. The best reason for this is no better than: this is what philosophers and logicians have been brought up on; it is what we are familiar with. As a result, not only are signifi cant logical problems ignored, the evident limitations of predicate calculus are covered over with a tissue of bogus methodology, self-serving appeals to hidden logical reality, and false history. In a similar vein, in ‘What are sets and what are they for?’ Smiley and Alex Oliver provide a critique of the widespread belief that set theory can provide a foundation for mathematics. Smiley has been exemplary in opening up fields of logical inquiry that do not fit the standard template of predicate calculus. In MultipleConclusion Logic, written with David Shoesmith, Smiley points out that ordinary arguments are ‘lop-sided’: they can have any number of premises, but only one conclusion. Multiple-conclusion logic allows there to be any number of conclusions as well; and an inference is valid if it is

Preface xi impossible for all the premises to be true and all the conclusions false. Thus a multiple-conclusion proof will set out the field in which truth must lie. Shoesmith and Smiley demonstrate many elegant results about multiple-conclusion logic—for example, that a categorical axiomatization of propositional calculus can be obtained using multiple-conclusion rules—but overall the reader can see how the symmetry of multiple-conclusion logic yields unconditional results which need qualifi cation in the asymmetrical setting of single-conclusion logic. Thus the inquiry also sheds significant light back onto ordinary logic. In series of papers with Alex Oliver, Smiley has studied the logic of plurals—‘the real numbers’, ‘the editors of this collection’. Oliver and Smiley show that no attempt to reduce such plural terms to a singular term standing for a set or group can succeed; and they set out to explore logic with genuinely plural terms. In related work, they have taken on the logic of lists—as in ‘Whitehead and Russell wrote Principia’—and have shown that it requires either predicates that can take variably many arguments or a plural term-forming functor that can operate on variably many arguments. Either way, one needs a ‘multigrade’ apparatus that is excluded from the predicate calculus. Against the Fregean tide that has seen logic as concerned with a special kind of general truth—logical truth—Smiley has emphasized that the relations of logical consequence and deducibility are the primary objects of logical inquiry. And in ‘Rejection’ he questions the Fregean idea that rejection of a proposition can be equated with the assertion of its negation. Rejection, Smiley argues, is a distinct activity from assertion and, indeed, is needed to explain the meaning of not. If one were to take rejection seriously as a counterpart activity to assertion, one would need to re-think the Fregean arguments that dominated twentieth-century logic. Smiley has also made ground-breaking contributions to many-valued logic, the theory of conditionals, to our conceptions of entailment, logical consequence and logical form, and our understanding of relative necessity and validity. But our purpose here is not to summarize Timothy Smiley’s work but to thank him (for such a summary, see Oliver 2005; a bibliography of Smiley’s work is included in this volume). Throughout his career, Smiley has displayed unplumbable generosity towards students and colleagues. Whether he is pointing a student towards a fruitful area of research, or helping him see that his current conceptualization is not yet quite right; whether he is spurring on a colleague with a unique blend of serious challenge, honest encouragement, and teasing; whether he is meticulously correcting draft after draft of a student’s paper (literally dotting the i’s and crossing the t’s), Smiley has done the best he can to bring out the best in all around him. For five decades he and his wife Benita have opened their home to students who in a weary moment, or a moment of insecurity, could use a home-cooked meal and a laugh. A small sample of those

xii

Preface

who are grateful have put together this volume on the many ways that good arguments actually work. Jonathan Lear Alex Oliver

July 2009

REFERENCES Copeland, B.J. (2002) ‘The genesis of possible world semantics’, Journal of Philosophical Logic 31: 99–137. Oliver, A. (2005) ‘Smiley, Timothy’ in S. Brown (ed.) The Dictionary of 20 th Century British Philosophers, Bristol: Thoemmes, pp. 970–5. Pyke, S. (1993) Philosophers, Manchester, England: Cornerhouse. Russell, B. (1919) Introduction to Mathematical Philosophy, London: Allen and Unwin.

Acknowledgments

The editors would like to thank Rob Trueman and Jonny Thakkar for invaluable assistance, and also Mrs. A.N. Taylor for her kind permission to reprint her photograph.

1

Philosophy In and Out of the Armchair Kwame Anthony Appiah

1

CONCEPTUAL ANALYSIS

One thing philosophers claim to do is: analyze concepts. We do this in language, so our analyses appear as sentences that we claim are conceptual truths. We utter sentences containing a word that expresses a certain concept: ‘green’, say, which expresses the concept green. But a conceptual truth is not just a true sentence that uses a concept: ‘The concept green applies to my shirt’ is a truth but not a conceptual truth. Rather, a conceptual truth is a truth that anyone who has the necessary concepts is in a position to know: e.g. ‘Green is a color’. Nothing more than knowledge of concepts is required to know that this is true. And, surely, knowledge of the concepts expressible in our language is just what all of us have with us wherever we go. As a result, as Tim Williamson once put it, ‘If anything can be pursued in an armchair, philosophy can’ (Williamson 2005: 1). Now, if knowledge of concepts led immediately to knowledge of all conceptual truths, conceptual analysis would be trivially easy. But there are conceptual truths that people understand, but do not know to be true: e.g. ‘You cannot trisect an arbitrary angle with straight edge and compass’. How can this be, if possession of concepts is all that is required? This question raises a version of the paradox of analysis, which asks how an analysis can both be informative and correct. If it is informative, after all, then its truth doesn’t follow from conceptual knowledge alone. Perhaps it is obvious to you what has gone wrong here. If it is, you can stop now. I aim to assemble the materials for an understanding of why—on almost any plausible semantic view—conceptual truths can indeed be hard to fi nd, even though there is a sense in which the materials for fi nding them are, indeed, often available to anyone who understands the sentences we use to express them. I aim, too, to show that there are many conceptual truths that are most easily found by doing experiments, even if we could, in principle, have hit upon them in other ways. And I also want to show that there are conceptual truths that are inaccessible from an armchair (unless the armchair comes with a good supply of empirical information). So my fi rst reason for being interested in these issues is that we need to get clear

2

Kwame Anthony Appiah

about them if we’re to decide what sorts of question we should, in fact, aim to work out in the armchair. But another thing philosophers do is: try to keep clear about what we are doing. And any philosophically satisfying account of conceptual analysis will have, as a result, to be consistent with a philosophically satisfying account of concepts. In the last century or so, this has usually been taken to involve explaining linguistic meaning: in large measure because older approaches, which treated concepts as mental seemed not to have advanced very much, in part out of a general tendency (of which behaviorism in philosophical psychology is one symptom) to favor analyses that focus on things—like utterances and inscriptions—that, unlike ideas and thoughts, inhabit the public world.1 And so we must start with accounts of assertoric meaning, even if, as I shall suggest, this is not where we should end up.

2

FOUR THEORIES OF MEANING

In the analytic tradition there are four deep traditions of thought about assertoric meaning. In one—broadly pragmatist—tradition, the content of an assertion is identified with what assenting to it would lead us to do. Frank Ramsey, a student of pragmatism, put the idea this way in ‘Facts and Propositions’: The essence of pragmatism I take to be this, the meaning of a sentence is to be defi ned by reference to the actions to which asserting it would lead . . .2 (Ramsey 1927: 51) In a second—verificationist—tradition, the content of an assertion is identified with the experiences that would (and would not) warrant its utterance, its verification conditions (along with its falsification conditions, although from now on I’ll not refer explicitly to them).3 And in the third— realist—tradition, the content of an assertion is identified with its truth– conditions, with the states of the world that would make it true.4 These three lines of thought can be woven together. We could say, with the realist, that knowledge of meaning is knowledge of truth conditions, and still think that such knowledge would issue in knowledge of the verifi cation conditions. Someone who knows when a sentence would be true might indeed be expected to recognize evidence for and against its truth. We could add, following another of Ramsey’s ideas, that someone who knows what a sentence means and believes it to be true will be led to act in those ways that would be utility-maximizing if the sentence were true. Still, if the realist is right, the verificationist is probably wrong, because there will be more to meaning than verification conditions, unless it turns out that two sentences supported by exactly the same evidence must have

Philosophy In and Out of the Armchair

3

the same truth conditions, in which case realism and verificationism will be equivalent. The pragmatist theory might be thought to be the combination of realism about the content of sentences with the view that assertions express beliefs and beliefs are states that lead us to act in ways that would be rational if the sentences expressing them were true. The fourth tradition, which is Wittgensteinian, identifies meaning with use (Wittgenstein 1967). To understand a sentence, on this view, is to know how to use it properly. The use-theory is well placed to treat language that is not assertoric, since imperatives (including questions) and optatives and exclamations can be used appropriately or inappropriately as much as assertions. But for assertions the use-theory could turn out to be consistent with all the others, since to know how to use an assertoric sentence it might be both necessary and sufficient to know its truth-conditions, its verification conditions, or which acts the belief it expressed made apt. Each of these traditions would need more careful articulation if we were to assess its plausibility. But there is something they largely share, an assumption made explicit in this passage from Michael Dummett: What a theory of meaning has to give an account of is what it is that someone knows when he knows a language; that is, when he knows the meaning of the expressions and sentences of the language. (Dummett 1975: 99) Even Wittgensteinians could agree with this. They could hold that knowledge of meaning is knowing how to use a sentence, so meaning is what a competent speaker knows. But they could also claim that it is irreducibly a form of know-how, and so not analyzable in terms of knowing any theory, since (they might claim) know-how need not be underwritten by propositional knowledge. But the other views develop most naturally as theories in which knowledge of meaning is propositional: knowing the truth conditions of a sentence is knowing that it would be true under such-and-such circumstances; knowing the verification conditions is knowing that it would be reasonable to assert it when such-and-such evidence was available; knowing the pragmatic meaning is knowing that it would be appropriate to act in such-and-such ways if one believed it true. 5 So it is a deeply embedded assumption of much of the semantic theorizing of the last century or so that the capacities that underlie the competence of speakers of a language are underwritten by propositional knowledge about the sentences they utter. Granted that this is so, it is likely that you can generate some version of the paradox of analysis for most of these views. Consider, by way of example, some sentence that is offered in the course of a philosophical analysis as ‘true in virtue of its meaning’. Let it be, say, K:

You can only know that something is so if it is, in fact, so.

4

Kwame Anthony Appiah

If this sentence is true in virtue of its meaning then, the proponent of the paradox of analysis says, anyone who knows what it means—anyone, then, who understands it—will know that it is true. So such an analysis, if correct, cannot be informative. This line of argument has always struck me as oddly unconvincing. On an account of meaning in which grasp of meaning is propositional knowledge, there will be some proposition, M(K), which states the meaning of K: Understanding K is knowing that M(K). To say that it follows from the meaning of K that K is true is presumably to say that E:

That M(K) entails that K is true.

Now, as we saw, on a propositional theory anyone who understands K will know that M(K). But we can deny that it follows from the truth of E that someone who knows that M(K) also knows that K is true, provided we accept N:

You need not know all the propositions entailed by what you know.

And, while there have no doubt been people who were tempted to deny N, it is, absent a counterargument, something that it is very natural to accept. There are puzzles that remain about how we can be said to know something without knowing something that follows pretty directly from it, and I shall return to those puzzles later. For the moment, though, I want to go back and trace a different trajectory in the history of modern philosophical semantics, which leads not to the problem of the paradox of analysis but to a rejection of the very idea of meaning.

3

QUINEAN EXCEPTIONALISM

Willard Van Orman Quine is not easy to fit into one of the four traditions I have identified, but that, I think, is because he did not have, in the sense I have sketched, a theory of meaning at all. What he had instead was an account of how language worked that was meant to allow us to do without the idea of meaning. In ‘Two Dogmas of Empiricism’ he proposed we should give up both the analytic-synthetic distinction and the doctrine he called ‘reductionism’, which he glossed as ‘the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience’ (Quine 1951: 20). Reductionism of that sort has gone the way of the dodo. But, whatever their official positions when faced directly with the

Philosophy In and Out of the Armchair

5

question whether Quine was right about analyticity, many philosophers still practice conceptual analysis in a way that seems to suppose the very distinction between a kind of truth that obtains, as Quine put it, ‘independently of fact’, and a kind of truth that depends on the way the world actually is. I am not alone in fi nding Quine’s arguments against analyticity unconvincing (see e.g. Grice and Strawson 1956; Putnam 1962). Sometimes in ‘Two Dogmas’ he argues against particular proposals of Carnap’s: but you do little permanent damage to the very idea of analyticity by rejecting one philosopher’s account of it. Quine’s central thrust, however, amounts to a challenge to provide an explanation of analyticity in terms of notions that he thinks unobjectionable. And since he objects to the notions of meaning and of synonymy and of cognitive significance, this is, unsurprisingly, hard to do. Of course, you cannot give an analysis of analyticity in terms that do not presuppose the availability of an idea of meaning. What is more interesting in Quine’s essay, then—along with the critique of reductionism, which we have fully absorbed—is his alternative picture of the content of assertions and thoughts. It appears first in his explicit rejection of the verificationist account. The basic difficulty, Quine says, is that verificationism supposes that ‘that each statement, taken in isolation from its fellows, can admit of confirmation or infirmation at all’ (Quine 1951: 38). The alternative to reductionism, then, is a semantic and epistemic holism: in which, in his famous formula, ‘our statements about the external world face the tribunal of sense experience not individually but only as a corporate body’. Now in rejecting this verificationist tradition, Quine claimed to be ‘shift[ing] toward pragmatism’. But this is not pragmatism in the sense that I have been discussing. What he means by pragmatism is that considerations of ‘conservatism’, ‘simplicity’, ‘convenience’ and the like play a role in the way we adjust our beliefs to experience; and, though he does not say so, I take it that what is ‘pragmatic’ about this is that such considerations relate to the practical utility of our theories and not—or at any rate not obviously—to their truth.6 So, as I say, it is hard to fit Quine into my categories. In ‘Two Dogmas’, Quine urges us to do without analyticity because he wants us to do without the idea of meaning (and, equivalently for him, without the idea of any particular belief’s having a specifiable content). You can think of a person’s network of beliefs as like a vast system of sentences that she holds true. Experience leads her to add some new sentences and drop old ones; sometimes this occurs near the periphery of the network only, when we respond to ordinary sensations with perceptual judgments. But these adjustments at the periphery may lead to changes deeper in the network. And, Quine claimed, no sentence (no belief) is in principle unrevisable; there are sequences of experience, he famously went on to argue, that will make it pragmatically useful to abandon even (what were formerly regarded as) truths of logic. Part of what is odd about this picture is that it leaves out something that was central to the Wittgensteinian picture: Quine, like Carroll’s Humpty

6

Kwame Anthony Appiah

Dumpty, ignores the ways in which language is a public, i.e. shared, thing.7 ‘“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less”’ (Carroll 1872: 188). And Quine appears to be saying, similarly, that each of us can choose whether or not to abandon a sentence as it suits us. This surely ignores the fact that we use language in ways that are meant to be responsive to norms that are collectively sustained. It may, in some sense, be up to me what I believe, because my beliefs are not necessarily responsive to publicly created norms— though, since many of my beliefs are irreducibly couched in terms borrowed from public language, even this may not be true in general. But when I make a claim in language it can be criticized by others as failing to meet linguistic norms, not all of which are syntactic or phonological. If I say, G:

Green is not a color,

I may be saying something true. ‘Green’ might be someone’s name. But if I use the word ‘green’ with its ordinary primary sense, G is false for reasons that anyone who grasps the concept green must know. When utterances of G express a truth, the word ‘green’ does not express the concept green. Quine’s claim is that none of our public norms requires us to hold the relations between terms constant; it is hard to see why we should believe him on the basis of the arguments in ‘Two Dogmas’. Indeed, the very image of a web of belief supposes that there are nodes and connections, and the natural interpretation of the metaphor is that the nodes are beliefs and the connections are relations of (epistemic) support. The belief expressed by the sentence ‘The grass is green’ supports the one expressed by ‘The grass is colored’. If that is true, language has a semantic structure, and there is space for the idea of meaning; for relations between these sentences will be fi xed ‘independently of fact’ and sentences expressing these relations will be conceptual truths. This is the essence of Donald Davidson’s later objection that, in holding on to the idea of a conceptual scheme, Quine had not escaped the last of empiricism’s dogmas: the idea that one can distinguish, even if only holistically, between scheme and content, between the contribution of language and the contribution of the world (Davidson 1974). So, Davidson too has no reason to think we can engage in conceptual analysis: there is a sense, indeed, in which he doesn’t believe there are concepts at all.

4

DAVIDSONIAN EXCEPTIONALISM

On three of the four views that I sketched at the start, conceptual analysis is possible because there are propositions that underlie competence.8 There is the puzzle about how, knowing these propositions, we can fi nd the enunciation of some of their entailments illuminating: we owe a response to the paradox of analysis. But on the Quinean and Davidsonian views there are,

Philosophy In and Out of the Armchair

7

in effect, no such propositions. Conceptual analysis is impossible, because there are no meanings. That Davidson has no place for conceptual analysis grounded in the competence of native speakers is obscured by the fact that he actually proposed something that he called a theory of meaning. But this was an alternative to what earlier philosophers had been seeking, something to replace it. It was not the old grail, but a new quest. It involved mobilizing the structure of a Tarskian theory of truth, so that it appeared, at first, to issue in realist truth-conditions. For Davidson, as for Tarski, the ‘iff’ in T-sentences like the notorious ‘Snow is white’ is true iff snow is white was a material biconditional. But this was somewhat misleading because, unlike Tarski, he did not require that the sentence to the right of the ‘iff’ should be a translation in the metalanguage of the sentence mentioned on the left. Davidson’s view was, rather, that we should accept sentences of this form if they are derivable from a T-theory for the whole language that gives the best interpretation of the verbal behavior of its speakers. And that best interpretation will be one that fits with the best interpretation of their behavior overall.9 The basic idea is simple. Ordinarily, when someone utters an (assertoric) sentence S, this is evidence that they hold that sentence to be true. So the best T-theory will be one that entails for each sentence a condition that we have good reason to think speakers of the language hold-true when they utter that sentence. In trying to make the best sense of the behavior of the speakers of a language, we must treat them as roughly rational: only if they are rational can we see their acts as the product of intentions and beliefs and so ascribe such states to them. We must suppose both that they are practically reasonable, mobilizing means and ends in a rational way, and that they are epistemically responsible, believing what they have good evidence to believe. Davidson thinks that these requirements—the ‘principle of charity’—will mean that we cannot make sense of a person unless we suppose that most of their beliefs are correct; and that we must begin with the assumption, as well, that normally, they say only what they believe.10 Because the evidence that we mobilize in constructing the T-theory for a language is the totality of the behavior of its speakers, the theory’s conditions of adequacy are holistic in two dimensions: we must look at all the behavior of all the speakers. So even if I am aiming to construct such a theory for my own language, I cannot rely, as a result, merely on consulting my own speech dispositions. As I said, then, a Davidsonian theory of meaning cannot underwrite a practice of armchair conceptual analysis.

5

THE QUINE-DAVIDSON ALTERNATIVE

I am not endorsing this picture. I shall, in fact, offer an alternative one later. But for the purposes of defending a practice of conceptual analysis, it is

8

Kwame Anthony Appiah

useful to point out that there is still on the Quine-Davidson view a proper place for something that will look very much like conceptual analysis. (This is a good thing since, after all, much of what they both did looked awfully like a kind of conceptual analysis.) On Davidson’s view, I take it, the way to defend a proposal like K: You can only know that something is so if it is, in fact, so, is to see it as an expression of something that follows from the best interpretative theory of English speakers (including its T-theory for English) which is, roughly, that K′:

when speakers utter a sentence of the form ‘A knows that S’ they will (standardly) hold S to be true.

Of course, not having a distinction between conceptual and non-conceptual truths, he need only think of this as one of the truths we have good reason to believe about English as she is currently spoke (cf. Carolino and Da Fonseca 2004). And the evidence that K′ is true must be found by enquiring whether it fits well holistically with all the available evidence of speakers’ linguistic and non-linguistic behavior. This is an inquiry that looks like it might require us to do a good deal more than sit in the armchair. And that might seem like the basis for an objection to the view. After all, what evidence did Davidson have about the general behavior and linguistic dispositions of the millions of speakers of (his dialect of) English? But, for Davidson, recall, it was one of the dogmas of empiricism that we could distinguish between conceptual and empirical truth. This meant that he didn’t really have the resource for an epistemological distinction between empirical and non-empirical, a priori and a posteriori. So it would be question-begging to put the point by saying that Davidson spent a lot of time in his armchair engaging in an empirical inquiry. But there remains something puzzling on this view about what it is to be a competent speaker of a language. Isn’t Davidson, in fact, in a special position in relation to his own language, one that entitles him to take his own dispositions as evidence about more than himself? Why else, after all, should we take claims about English to be answered by looking at some particular group of speakers? The natural answer is, surely, that it is their language. If a word is a word in American English, proper usage gets decided by Americans. But it cannot be true, on pain of circularity, that each American speaker has to consult the usage of all the members of that community in order to know how to use her language. Instead we must see them all as participants in a convention, of the sort David Lewis analyzed, in which they each aim to use the words in conformity with a mutual understanding. If that is what they are doing, they must be able to associate each sentence with

Philosophy In and Out of the Armchair

9

a content of some sort; and that content must depend on the words and their arrangement in the sentence. Davidson doesn’t tell us how we actually do this. The T-theory for Davidson is a theorist’s construct, not something whose grasp explains the speaker’s competence. We must try to invent a story for him. The story must be consistent with the holism and the revisability of each assignment of truth-value in the light of new evidence. (If there were sentences that no experience could lead us [legitimately] to abandon, there would be conceptual truths.) It seems to me, however, that the demand for holism and the demand for revisability are at odds. Suppose we think of an overall state of belief as an assignment to a body of sentences of one of at least three belief-values: held-true, held-false, and unassigned.11 If there were literally no conceptual structure, as Davidson held—which seems to be what is required for the revisability claim to be right—then there would be no constraints on these assignments (beyond, perhaps, the demand that at most one of these values gets assigned to each belief). Otherwise, if, for example, it were impossible for there to be assignments in which something of the form X knows that it’s snowing was held-true and It is snowing was not held-true, then K would be a conceptual truth, of the sort that Davidson appears to think impossible.12 But revisability of this sort undermines the holism: how can we rationally adjust overall belief-values in the light of a new assignment of values based on, say, a series of perceptual experiences, if assignments to one belief do not in any way constrain assignments to another? We can make the same point another way: what does it mean to suppose, with the principle of charity, that people are reasonable, unless there are constraints on inference imposed by reason? Notice that nothing I have said undermines Davidson’s view that claims about meaning, even in our own language, are responsible to the totality of evidence about speakers of the language. I have just argued that there must be some conceptual structure, if our beliefs are to constitute the sort of holistic system Quine, following Duhem, and Davidson, following Quine, insisted they do. The holism consists in the fact that we are rationally required to adjust our holdings-true and -false of some beliefs in the light of our holdings-true and -false of others. But it does not follow that these rational constraints must be accessible to believers by introspection or intuition of some sort. It is one thing to be a creature that follows the demands of reason, another to be able to articulate reason’s demands.

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So it seems to me we can go a long way in the Quine-Davidson direction and still keep a place for conceptual truth; though we must then agree with them that access to conceptual truth need not be available from the armchair. Why two such successful armchair workers were able to proceed without much attention to the wider world is a question to which I will return.

6

THE NATURE OF MENTAL REPRESENTATIONS: A SKETCH

I have proceeded so far without a positive story of my own. Let me sketch one now. It begins with mental representation rather than linguistic representation. But this account imagines many mental representations as being, in certain important respects, like sentences. In particular, while I believe some mental representations are more like pictures than sentences, I want to focus on those vast majority of the beliefs we are likely to express in language. These, I claim, share with sentences the crucial feature that they have a form, something analogous to a syntax. Beliefs are also like sentences in that they are representations; they consist of elements—including some that are subject-like and some that are predicate-like—that represent objects, properties, and relations in the world. What makes it true of an element that it represents a certain object, property or relation is that there is connection of the right causal sort between that object and the representational element.13 So the account is, in this way, broadly functionalist: it explains what mental representations are by explaining their role in the functioning of the mind. The details do not matter for the arguments I want to make here. The key features of the resulting model are these. First, representations are individuated by three features. One is which class of propositional attitudes they belong to: are they beliefs, desires, intentions, perceptions, imaginings, and so on? Second they represent the world a certain way—they have, I shall say, a certain r-content. Beliefs represent how the person takes the world to be; desires how they would like it to be, and so on. The belief that the world is such-and-such a way and the desire that it should be that way have the same r-content. And third, they have a form. As a result, two distinct mental representations can represent the world as being the same way. Here natural language provides obvious analogies: ‘John loves Mary’ and ‘Mary is loved by John’ represent the same state of the world, but in representations of different form. But the thought that a representation expresses—its content in the sense that the content of a belief is the content of a sentence that expresses it—is fi xed by both its form and its r-content. R-content is only one dimension of content. The second crucial point is that the processes that generate representations from representations—processes that we can call computations— are formal. That is, when beliefs give rise to other beliefs (or beliefs and desires together give rise to intentions) the form of the new states is causally

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11

determined by the forms of the old ones. In consequence, the laws governing the production of mental representations from other representations make no reference to their contents. The third point is that the correct account of r-content is externalist. That is, what a representation represents depends, at least sometimes, on facts outside the mind of the person whose representation it is. R-content, as Putnam might have put it, ain’t in the head. So, for example, that a certain representational element represents the color green, is a fact about the connections between that element and green things in the world; and it does not depend on those connections being represented by the agent. This claim may be controversial, but Putnam’s work should remind us that there are relatively uncontroversial cases, where there is, as he put it, linguistic division of labor. What makes it true, as he pointed out, that I refer to different trees with the words ‘elm’ and ‘beech’ isn’t a consequence of there being, represented in my own head, facts that distinguish one kind of tree from another. And the same goes for my beliefs about elms and beeches. The representational element in my head whose presence makes a belief a belief about elms has the reference it has in part because of facts that connect it to uses of the word ‘elm’ and the connection of that word to elm trees. As a result, no amount of introspection is going to produce an account of what distinguishes elms from beeches. These concepts cannot be analyzed from the armchair. Given this sort of account of mental representations, we can explain linguistic representations by associating assertoric sentences with beliefs. We can grant that what beliefs a person may have can depend—as in the case of beliefs about elms—on whether certain linguistic items are in their repertory. For the causal story that explains what the contents of belief are may include facts about the uses of words by others. (This is obvious enough when it comes to beliefs that N has a certain property, where ‘N’ is a proper name.) But understanding a sentence on this view is fundamentally a matter of its eliciting (in normal circumstances) a mental representation that has both the right form and the right r-content. We now have all the resources we need to answer the question about conceptual truth I have been deferring: the answer to one form of the paradox of analysis. Because with these materials in hand we can see why you can have the resources in your conceptual repertoire to grasp a truth and still not have grasped it. Suppose, to revert to my earlier example, you know that M(K). Even if there are computations that will get you from M(K) to ‘K is true’, and even if you have the capacity to perform those computations, it does not follow that you have done so. To get from M(K) to ‘K is true’ you will have to think things through (as we might more naturally put it) and you will have to think them through correctly. But you can count as having the capacity for a certain computation even if you don’t always exercise it correctly. If, generally speaking, I can infer Q from P-or-Q and not-P and infer P from P-or-Q and not-Q, and infer P-or-Q from P and also from Q

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(and I am not disposed to make inferences from or to disjunctions that are fallacious) I have the computational resources that display a command of inclusive disjunction. It doesn’t follow that I don’t have grasp of disjunction if, on some occasion, I mistakenly ‘infer’ not-Q from P-or-Q and P. All it shows is that sometimes I can fail to do something I am generally able to do: in this case, make valid inferences involving disjunction. That is why N:

You need not know all the propositions entailed by what you know,

is true; and if N is true, the version of the paradox of analysis with which I began does not arise. Here, I think, is the truth in Davidson’s claim that we can only understand people by taking them to be more-or-less rational. We cannot suppose someone to have a disjunctive thought unless we suppose that they can normally make the right disjunctive inferences. But, on the other hand, we do not lose our capacity to make sense of someone in general if, on a particular occasion, they fail to reason as they should. Of course, supposing someone to have a capacity for an inference of a certain kind when there are no circumstances in which they would exercise that capacity, would make it possible to ascribe any capacity at all to anything. The right way to draw the boundary between this too generous policy of ascription and the too stingy one of denying a capacity to anyone who ever makes a mistake is this: you can treat something as an aberration provided there is some explanation of why it occurred. As I wrote in Assertion and Conditionals, For computational errors have to be deviations from the normal functioning of the physical system that embodies the functional states. It follows that in calling something a computational error, we are committed to their being some explanation of it. (Appiah 1986: 73–4) That there is an explanation doesn’t mean that we have it: so we can exercise charity in Davidson’s sense—supposing someone has a capacity they have recently failed to display—without having to know what accounts for the failure.

7

NECESSITY AND ANALYTICITY

Since Kripke’s Naming and Necessity philosophers of language generally have been aware of the possibility that there are necessary truths that are not conceptual in the sense that I identified at the start: knowable by those who possess the relevant concepts (Kripke 1972). The simplest kind of example is the sentence Frege used to introduce the distinction between sense and reference:

Philosophy In and Out of the Armchair M:

13

The Morning Star is the Evening Star

(taking ‘The Morning Star’ and ‘The Evening Star’ to be proper names). Once we have fi xed the reference of each of these names as the planet Venus, there is no possible world at which M is not true, Kripke argued, because there is no world in which Venus is not Venus.14 But we can see that M is a truth discoverable only a posteriori. Nevertheless, the reason that it is true is, in an obvious sense, a consequence of the semantic rules of English. For those rules, however the right account of the semantics of proper names works, make the two names name the same object, even though that fact is not one that every competent speaker need know. So you could say that this is a conceptual truth, if you liked, but not the sort of conceptual truth that you should expect to discover in the armchair. The possibility exists because the correct account of concepts must be, as I said, externalist. There can be facts about how our language works that are not represented in our minds. Beliefs about names are the extreme example of this, but there are, as Kripke also pointed out, other interesting possibilities. One is that if many substance-concepts like water and H 2O are natural kind concepts, then there will be other conceptual truths that are plainly discoverable only a posteriori, Water is H 2O among them. (This point is obviously related to Putnam’s insight about elm and beech.)

8

ESCAPING THE ARMCHAIR

We have here two models of how a kind of conceptual truth can be hard to access in the armchair. In the first, a truth may be hard to find because it requires many computational processes to uncover, and carrying them out correctly—engaging in the kind of armchair reflection that we associate with the practice of philosophy—is hard. The sorts of inferential processes that lead from one thought to another are often quite hard to do ‘in one’s head’. And, in fact, if we think about it, real philosophical work in an armchair often involves the use of inscriptions (pencil-and-paper, once, now as often on a computer screen). The capacity for mental logic, like the capacity for mental arithmetic, is one that most people have to a very limited degree. In the second model, that of the necessity of identity, we see that conceptual truths may arise from the way language users are embedded in the world. And those ways are not transparent to language users themselves, because the facts of their embedding are not all mentally represented. There is one more important reason for thinking that there may be conceptual truths that are, if not inaccessible, then at least very hard to access

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in the armchair. Even if it makes sense to suppose that our understanding of sentences is the result of a form of propositional knowledge, that knowledge is clearly implicit. It may be that a form of propositional knowledge underlies my capacity to use the word ‘know’. But my uses of it on particular occasions do not involve the articulation of that knowledge: they involve acting on it. As a result, it may be true e.g. that 1 I won’t say someone knows something that I take to be false, and 2 This fact follows from propositions I believe that underlie my competence but, nevertheless, 3 I do not have conscious knowledge of 1. If I reflect on stories like the original Gettier example, my competence will display itself in my correctly seeing that I should not say that Smith knows that the man who will get the job has ten coins in his pocket. (Gettier 1963). But, in the absence both of a conscious articulation of the knowledge that underlies my competence and of my making the necessary further inferences, there is no reason to suppose that I will know the conceptual truth I dubbed K. The way I would find it out would be by considering examples and asking what people would say about them, hypothesizing that K is true, and looking for counterexamples. I would do what Socrates did in the Theaetetus when he discovered the truth of K. (Socrates’ method would be justifiable on a Wittgensteinian account of conceptual knowledge as irreducible know-how, too: you can look for patterns in your own linguistic dispositions by eliciting them, whether or not they are grounded in propositional knowledge.) So there may be truths about conceptual relations, rooted in the unconscious knowledge that guides our linguistic understanding, that are not consciously accessible to a person who has those concepts. But, on the other hand, for our language to work it has to be true that usually when I think a certain state of affairs is aptly represented by a sentence, that is the judgment that would be made by other speakers of my language. Provided it is also true that what I think I would say about an imagined situation I would in fact say, then thinking about imaginary situation will allow me to predict not just what I would say but what most speakers would. As a result, there is less of a puzzle about why this sort of testing of hypotheses against examples can go on in an armchair. Because we share our competence, each of us has good reason to think that we can speak for others.

9

THE STATUS OF CONCEPTUAL CLAIMS

Nevertheless, the claim that a sentence expresses a conceptual truth is itself an empirical claim. It presupposes that we have got the sentence’s meaning

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15

right, and we have only got it right if we know what it means.15 And that I know what it means is an empirical fact, if it is a fact. It is a fact about the conventions in which we participate, of course, so while I may be wrong about a sentence’s meaning, we cannot usually all be wrong about it. Still, so it seems to me, it is worth checking whether the dialect we philosophers speak to one another is, in fact, the same language spoken in the wider world. And, as a result, I think that the form of experimental philosophy that consists in sampling the actual utterances of speakers to check on our semantic intuitions is eminently worthwhile. Joshua Knobe has used such methods to explore our concept of intention and has argued that people generally have a concept of intention on which it makes sense to suppose that someone intends only those side effects that are morally wrong (Knobe 2004; for more on this approach see Appiah 2008). There are, as he agrees, many ways of explaining the way actual people respond to the scenarios he offered them, but this is one possible explanation. So conceptual discovery can be guided by empirical work on the way people actually use words. And some conceptual discoveries—on the model of the necessity of identity—can only be made by looking at the world. Still, there is work to be done in the armchair, even if the laptop and pencil and notebook will be close at hand. But discovering how our concepts work—descriptive analysis—is only part of the story. For there has always been place for revisionary analysis, too. To see why, just recall the pragmatism that Quine endorsed: the picture of language as a tool that needs to be shaped to be useful, the picture that seems to be there, too, in the Wittgensteinian tradition. Sometimes, in philosophy as in physics or in biology, there is an argument for introducing a new concept, a new tool. Space-time, gene, semantic externalism: each of these was introduced by way of arguments for noticing a feature of our world worth noticing. None of these words acquired its meaning by someone’s offering a simple defi nition. Rather they were embedded in many sentences: many truths together fi xed their sense. Frank Ramsey, with whom I began, invented a strategy for understanding what was going on in these cases. Write the conjunction of these many sentences, R. Take out the new word from that conjunction, then replace each of the blanks with a variable, x, to produce Rx. Then our understanding of that new word is captured by saying, Space-time (or ‘gene’ or ‘semantic externalism’) is the x such that Rx. Provided R is consistent, this specifies a concept. When we introduce a word it can be unclear which of the sentences belong in that conjunction and so there can be discussion about which concept we have introduced. And the right way to answer that question is pragmatically: showing what can be done with one concept rather than another, arguing which concept is the most useful, and, sometimes, arguing that we need two or more terms, because several concepts have their place. Many words are like this:

16 Kwame Anthony Appiah there are many candidates for the concept ‘X’ expresses; there are, as we might say, many different conceptions of X. Arguments of this sort can often be carried out in the armchair, not because they don’t depend on empirical fact, but because the relevant empirical facts are widely known. I have applied this strategy to the concepts of race and social identity in recent years, proposing conceptions of each of them that I think are more useful than some of those offered by others (see e.g. Appiah 2007). My thoughts in the armchair are enriched by sociology, literature, history, and anthropology. I can stay seated because others have ranged more widely. This is conceptual analysis, too, even though it aims to revise not reflect our existing ways of thought.

10

ENVOI

My remarks about conceptual analysis presuppose an account of concepts that connects them with ideas, with mental representations. I have explained features of conceptual analysis by reference to facts such as these: the content of mental representations involves their form; the correct account of their r-content is externalist; conceptual knowledge involves unconscious ideas; and the introduction of new terms connects them only loosely with distinct ideas. For the purposes of understanding what we are doing in and out of the armchair, it is not enough to understand the relations between words and the world; you must take account of the relations of both to our ideas as well.16

NOTES 1. Hacking (1975) gives an illuminating account of the relevant history. 2. What Ramsey means, surely, is the ‘actions that someone who believes what the sentence asserts would be led to’; so ‘assenting’ would have been better than ‘asserting’. 3. The passage from Ramsey that I just cut short continues, ‘. . . or, more vaguely still, by its possible causes and effects’. So if you think of experiences as the causes of judgments, then the pragmatist theory can be seen as having all the resources of the verificationist and then some. 4. Since, on most views, there are only two truth values, true and false, fi xing the truth-conditions fi xes the falsity-conditions as well, so we don’t need both; whereas the verification conditions of a sentence don’t fi x its meaning on their own, since two sentences could have the same verification conditions but distinct falsification conditions: e.g. ‘It’s green’ and ‘I have good reason to believe its green’. These questions are explored in Appiah (1986). 5. You will notice that these analyses tend to follow an insight that Dummett has insisted is due to Frege, namely that in explaining meaning you should begin with the meaning not of words but of sentences (see Dummett 1993: 5). 6. I say ‘not obviously’ because, in James’s pragmatism, considerations of this sort play into the defi nition of truth. ‘“The true”, to put it very briefly, is

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7. 8. 9.

10.

11. 12.

13. 14. 15. 16.

17

only the expedient in the way of our thinking, just as “the right” is only the expedient in the way of our behaving. Expedient in almost any fashion; and expedient in the long run and on the whole, of course’ (James 1907: 106). Quine was always clear, of course, that language was public in one sense, because he was a behaviorist. But that is consistent with not remembering that language is responsive to norms we share. Only the Wittgensteinian option that characterizes competence as irreducibly practical knowledge is non-propositional. I shall say something about how conceptual analysis might be understood on this view later. I ignore the—very interesting—question what we should do if there are several theories that fit the overall behavior equally well, but better than any other candidates. (Thanks to Alex Oliver for pointing this out.) This is actually rather likely, given the general underdetermination of empirical theory by evidence. Davidson makes clear that it is the intelligibility rather than the truth of other people’s beliefs that is fundamental in the preface to Essays in Truth and Interpretation: ‘The aim of interpretation is not agreement but understanding. My point has always been that understanding can be secured only by interpreting in a way that makes for the right sort of agreement. The “right sort”, however, is no easier to specify than to say what constitutes a good reason for holding a particular belief’ (Davidson 2001: xvii). In a richer picture, we might think of the assignments as more like probabilities, though probabilities have too rich a structure to be plausibly psychological realized. The thought that any assignment of values to each S is possible, doesn’t, of course, entail that all overall assignments of values are possible (any more than it follows because each of us can be Prime Minister, we all can be). My point is that making the assignment of truth-value to S depend on the assignment of truth-value to another S looks like making a conceptual connection; a connection ‘independent of fact’. The details here are to be found in my book (1985). This book was based on my doctoral dissertation, which was examined by Timothy Smiley and Dorothy Edgington. This claim is controversial and there are anti-essentialists who deny this version of the necessity of identity. I shall assume, without arguing, that they are wrong. On the other hand, if the thought that a sentence expresses is a conceptual truth, its truth is knowable by the thinker a priori. I am very grateful to the editors of this volume for helpful comments on an earlier draft. I wish I could have responded to them more fully.

REFERENCES Appiah, A. (1985) Assertion and Conditionals, Cambridge: Cambridge University Press. . (1986) For Truth in Semantics, Oxford: Blackwell. Appiah, K.A. (2007) The Ethics of Identity, Princeton: Princeton University Press. . (2008) Experiments in Ethics, Cambridge, MA: Harvard University Press. Carolino, P. and Da Fonseca, J. (2004) English as She is Spoke: Being a Comprehensive Phrasebook of the English Language, Written by Men to Whom English Was Entirely Unknown, P. Collins (ed.), New York: McSweeney’s.

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Carroll, L. (1872) Through the Looking Glass, and What Alice Found There, page references are to Alice’s Adventures in Wonderland & Through the Looking Glass, New York: Signet Classics, 2000. Davidson, D. (1974) ‘On the very idea of a conceptual scheme’, Proceedings and Addresses of the American Philosophical Association 47: 5–20. . (2001) Essays in Truth and Interpretation, 2nd edn, Oxford: Oxford University Press. Dummett, M. (1975) ‘What is a theory of meaning?’ in S. Guttenplan (ed.) Mind and Language, Oxford: Clarendon Press, pp. 98–138. . (1993) Frege: Philosophy of Language, 2nd edn, Cambridge, MA: Harvard University Press. Gettier, E. (1963) ‘Is justified true belief knowledge?’, Analysis 23: 121–3. Grice, H. and Strawson, P. (1956) ‘In defense of a dogma’, The Philosophical Review 65: 141–58. Hacking, I. (1975) Why Does Language Matter to Philosophy?, Cambridge: Cambridge University Press. James, W. (1907) Pragmatism: A New Name for Some Old Ways of Thinking, page references are to the 1975 edn, Cambridge, MA: Harvard University Press. Knobe, J. (2004) ‘Intention, intentional action and moral considerations’, Analysis 64: 183–7. Kripke, S. (1972) ‘Naming and necessity’ in Semantics of Natural Language, D. Davidson and G. Harman (eds), Dordrecht: Reidel, pp. 253–355, 763–9. Putnam, H. (1962) ‘It ain’t necessarily so’, Journal of Philosophy 59: 658–71. Quine, W.V.O. (1951) ‘Two dogmas of empiricism’, The Philosophical Review 60: 20–43. Ramsey, F.P. (1927) ‘Facts and propositions’, page references are to the reprint in F.P. Ramsey: Philosophical Papers, D.H. Mellor (ed.), Cambridge: Cambridge University Press, 1990, pp. 34–51. Williamson, T. (2005) ‘Armchair philosophy, metaphysical modality and counterfactual thinking’, Proceedings of the Aristotelian Society 105: 1–23. Wittgenstein, L. (1967) Philosophical Investigations, G.E.M. Anscombe (trans.), 3rd edn, Oxford: Blackwell.

2

Restricted Quantifiers and Logical Theory Thomas Baldwin

For many years Tim Smiley gave a wonderful series of lectures on Russell’s Theory of Descriptions (the existential quantifier here has narrow scope: the lectures were different each year). He published his conclusions about Russell’s theory in his brilliant British Academy lecture on ‘The Theory of Descriptions’ (1981). That lecture includes a brief discussion of restricted quantifiers; but when I attended the lectures (back in 1967!), this topic had a larger place and started me thinking about the issues it raises.

1

DAVIDSON AND LOGICAL FORM

Some years ago Donald Davidson wrote: Someone interested in logical form . . . would maintain that ‘All whales are mammals’ is a universally quantified conditional whose antecedent is an open sentence true of whales and whose consequent is an open sentence true of mammals. The contrast with surface grammar is striking. The subject-predicate analysis goes by the board, ‘all whales’ is no longer treated as a unit, and the role of ‘whales’ and ‘mammals’ is seen, or claimed, to be predicative. What can justify this astonishing theory? Part of the answer—the part with which we are most familiar—is that inference is simplified and mechanized when we rewrite sentences in some standardized notation. If we want to deduce ‘Moby Dick is a mammal’ from ‘All whales are mammals’ and ‘Moby Dick is a whale’, we need to connect the predicate ‘is a whale’ in some systematic way with a suitable feature of ‘All whales are mammals’. The theory of the logical form of this sentence tells us how. (Davidson 1980: 138) Davidson does not appear to have realized that there is an alternative theory of the ‘logical form’ of sentences such as ‘All whales are mammals’ which elucidates inferences such as Davidson’s simple ‘Moby Dick’

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inference without treating these sentences as universally quantified conditionals. The key to this alternative theory is the use of restricted quantifiers and my main aim in this paper is to discuss the merits and limitations of this theory. Before doing this, however, it is worth briefly reviewing the ‘astonishing’ theory Davidson’s remarks assume, familiar though it is. Davidson’s suggestion is that the inference All whales are mammals Moby Dick is a whale So Moby Dick is a mammal is ‘simplified and mechanized’ by treating it as having the following logical form: (∀x) (If x is a whale, then x is a mammal) So If Moby Dick is a whale, then Moby Dick is a mammal Moby Dick is a whale So Moby Dick is a mammal But does this account really ‘simplify’ the inference? Why should this simple inference be regarded as dependent on the logic of conditionals, one of the most contested areas of logical theory? This concern is further extended by considering Aristotle’s famous master syllogism—Barbara: All As are Bs All Bs are Cs So All As are Cs When this pattern of inference is handled in Davidson’s recommended fashion and set out in natural deduction style with the usual rules of inference, it comes out as: Premise Premise

P1 P2

Assume

A1

(∀x) (If Ax then Bx) (∀x) (If Bx then Cx) If Ay then By If By then Cy Ay By Cy If Ay then Cy (∀x) (If Ax then Cx)

P1 ∀-E P2 ∀-E A1, P1 If-E A1, P1, P2 If-E P1, P2 If-I P1, P2 ∀-I

The dependence here on the logic of conditionals is certainly ‘astonishing’ in the light of the intuitive simplicity of the original inference.

Restricted Quantifi ers and Logical Theory 21 Davidson’s attitude to this astonishment, however, was that it is not a reason for suspicion: for one should not expect the logical form of an inference to be manifest in the surface structure of natural language: Logical form was invented to contrast with something else that is held to be apparent but mere: the form we are led to assign to sentences by superficial analogy or traditional grammar. What meets the eye or ear in language has the charm, complexity, convenience, and the deceit of other conventions of the market place, but underlying it is the solid currency of a plainer, duller structure, without wit but also without pretence. This true coin, the deep structure, need never feature directly in the transactions of real life. (Davidson 1980: 137) This line of thought expresses a Platonic conception of form: logical forms belong to a deep structure that is typically hidden by the appearances of ‘mere’ surface structure. The ordinary language in which we conduct ‘the transactions of real life’ is a Platonic cave of misleading appearances; only enlightened philosophers who can use the disciplines of logic and semantics to see through these appearances are able to appreciate the ‘true coin’, the logical form of the arguments advanced in these discussions. Aristotle criticized Plato for conceiving forms as a domain of transcendent objects, which, far from elucidating the structure of the physical world we experience, undermine it by treating it as a mere appearance, and he advanced a different conception of forms as themselves constitutive of the structure of the physical world. Aristotle’s syllogistic theory manifests a similar approach to logic by assigning logical significance to the surface structure of ordinary language and thereby offers an approach to logic which offers an alternative to Davidson’s Platonism. For although some degree of simplification and schematic regimentation has to be employed in order to frame generalizations and construct formal proofs within a systematic logical theory, ordinary arguments such as Davidson’s Moby Dick inference are undermined if they are taken to depend on a deep structure which uses resources such as the theory of conditionals that far exceed the manifest form of the language employed. So from this Aristotelian perspective, the astonishment noted by Davidson provides a reason for suspecting that the logical theory he is assuming is not appropriate for the task of identifying the logical form of the arguments in question and for looking for an alternative theory.

2

RESTRICTED QUANTIFIERS

The alternative theory I want to consider here is that which replaces the unrestricted quantifiers of standard logical theory with restricted

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Thomas Baldwin

quantifiers and suitable rules of inference for them. Just what should be counted as a restricted quantifier will require considerable discussion, but for the moment it will suffice to approach the matter in an informal manner. Natural language quantifiers such as ‘All whales’ and ‘Some dogs’ are formed by attaching a determiner (‘All’, ‘Some’) to a common noun which specifies a kind of thing. In standard logical theory the role of these quantifiers is conceived as comprising three points: (i) the determiner is interpreted as a variable binding quantifier (‘For all x . . .’ etc); (ii) the role of the common noun is interpreted by giving it a predicative role in an open sentence with a variable which connects it with the quantifier from it is derived (‘x is a whale’); (iii) this open sentence is exported into the regimented version of the verb phrase to which the natural language quantifier was attached, where the open sentence occurs either as the antecedent of a conditional (when it comes from universal quantifiers such as ‘All whales’ and ‘No whales’) or as the fi rst conjunct of a conjunction (when it comes from particular quantifiers such as ‘Some dogs’). Thus, as we saw, ‘All whales are mammals’ is given the form: (∀x) (If x is a whale, then x is a mammal) and ‘Some dogs bark’ is taken to have the form (∃x) (x is a dog & x barks) Both the second and the third elements of this account can be questioned. To question the second step is to suggest that it is worth giving some logical significance to the distinction between common nouns (‘whale’) and predicates (‘ . . . is a whale’), and following up this suggestion takes one into the logic of sortal quantification in which common nouns are taken to identify sorts of thing (whales, dogs etc) of which quantified propositions are true or false. This logic rapidly becomes complicated once one considers familiar inferences such as contraposition, which affi rm the equivalence of ‘All whales are warm-blooded’ and ‘All non-warm-blooded objects are non-whales’, since not only do the terms here (‘whale’, ‘warm-blooded’) switch roles but one also has to accommodate the role of negation in specifying a sort of thing (‘non-warm-blooded objects’). These issues are well worth investigation, but I shall not to pursue them here.1 Instead I shall concentrate on the implications of rejecting the third element of the standard account in order to assess the merits of a theory which attaches logical significance to restricted quantifiers. On this approach the open sentence ‘x is a whale’ remains ‘within’ the quantifier, restricting its application, instead of being exported as the antecedent of a conditional to which an unrestricted quantifier is attached. Thus the logical form of ‘All whales are mammals’ is taken to be manifest when it is expressed as:

Restricted Quantifi ers and Logical Theory

23

(∀x: x is a whale) (x is a mammal) In the light of the preceding discussion, the important point here is the absence of any conditional; equally, the challenge is to provide a logical theory which elucidates the inferences in which this sentence occurs without tacitly drawing on the logic of conditionals. As James McCawley showed some years ago, it is not difficult to set out rules of inference for restricted quantifiers (McCawley 1981; ch. 4). R∀-E

(∀x: Fx) (Gx), Fy

R∀-I

If Γ, Fy R Gy then Γ R (∀x: Fx) (Gx) where ‘y’ does not occur in ‘Γ’ and ‘Fy’ and ‘Gy’ are free for ‘x’

R∃-E

If Γ, Fy, Gy R C then Γ, (∃x: Fx) (Gx) where ‘y’ does not occur in ‘Γ’ or ‘C’

R∃-I

Fy, Gy

R

R

Gy

R

C

(∃x: Fx) (Gx)

Together with the usual sentential logic, these rules defi ne ‘ R’. They can be compared with the familiar rules for unrestricted quantifiers, which defi ne ‘ U ’. U∀-E

(∀x) (Gx)

U∀-I

If Γ U Gy then Γ U (∀x) (Gx) where ‘y’ does not occur in ‘Γ’ and ‘Gy’ is free for ‘x’

U∃-E

If Γ, Gy U C then Γ, (∃x) (Gx) U C where ‘y’ does not occur in ‘Γ’ or ‘C’

U∃-I

Gy

U

U

Gy

(∃x) (Gx)

As will be obvious the U-rules are largely obtained from the R-rules by deleting occurrences of the restrictive condition both within the quantifier (‘Fx’, where ‘x’ occurs as a bound variable) and also as an extra premise (‘Fy’, where ‘y’ occurs as a free variable). Equivalently, the R-rules are obtained from the U-rules by adding a suitable extra premise (‘Fy’) along with the restriction on the quantifier (‘Qx: Fx’). This comparison does not of course vindicate the R-rules; instead their rationale comes from the easy way in which they ‘simplify’ and ‘mechanize’ familiar patterns of inference. Thus Davidson’s Moby Dick inference is just an immediate instance of the fi rst rule for the elimination of a restricted universal quantifier (R∀-E). And the vindication of Aristotle’s Barbara is also straightforward:

24

Thomas Baldwin Premise Premise Assume

P1 P2 A1

(∀x: Ax) (Bx) (∀x: Bx) (Cx) Ay By Cy (∀x: Ax) (Cx)

A1, P1 R∀-E A1, P1, P2 R∀-E P1, P2 R∀-I

This is about as straightforward as one can envisage. On the one hand Barbara is not treated as a fundamental schema; equally, it is not held to need vindication from the logic of conditionals. Instead it is treated as an inference inherent in the logic of the universal quantifier—which is surely as it should be. Aristotle’s rule for the simple conversion of particular judgments, that ‘Some A are B’ implies ‘Some B are A’, receives a similarly satisfying treatment: Premise Assume Assume

P1 A1 A2

(∃x: Ax) (Bx) Ay By (∃x: Bx) (Ax) (∃x: Bx) (Ax)

A1, A2 R∃-I P1 R∃-E

Again, the rule is not treated as fundamental, but receives a straightforward vindication not dependent on the logic of conjunction, as is required when the inference is handled with unrestricted quantifiers. Obversion (‘No A are B’ implies ‘No B are A’) too is readily explicated (assuming the usual rules for negation): Premise Assume

P1 A1

Assume

A2

(∀x: Ax) (∼Bx) Ay ∼By P1, A1 R∀-E By ∼Ay P1, A2 R∼I (∀x: Bx) (∼Ax) P1 R∀-I

The ease with which these Aristotelian rules are explicated within R, this simple logic of restricted quantifiers, raises now a question about the existential import of the restricted universal quantifier: does ‘(∀x: Ax) (Bx)’ imply ‘(∃x: Ax) (Bx)’, as Aristotle maintained? After all, the non-Aristotelian, Boolean feature of U, the familiar logic of unrestricted quantifiers, is associated with the interpretation of natural language restrictive noun phrases in universal quantifiers as antecedents of material conditionals which are vacuously true overall where their antecedents are false. That is, it is argued, ‘All female presidents of the USA have been communists’ turns out to be true because once it is interpreted as: (∀x) (x was a female president of the USA ⊃ x was a communist)

Restricted Quantifi ers and Logical Theory

25

its truth follows directly from the fact that there have been no female presidents of the USA. Within R that line of argument is not directly available, since restrictions on the universal quantifier are not interpreted as the antecedents of conditionals. So, one may ask, what is the relationship here between ‘(∀x: Ax) (Bx)’ and ‘(∃x: Ax) (Bx)’, or rather, between ‘(∀x: Ax) (Bx)’ and the existence of some As, which can be expressed here as ‘(∃x: Vx) (Ax)’ where ‘Vy’ is taken within the logical theory to be a universal predicate (such as ‘y = y’) such that we have ‘ R Vy’. The answer to this question is that once one incorporates into R classical sentential logic, then ‘∼(∃x: Vx) (Ax)’ implies ‘(∀x: Ax) (Bx)’. For sentential logic gives one: P, ∼P

R

Q

Hence, in particular, (∃x: Vx) (Ax), ∼(∃x: Vx) (Ax)

R

By

But by R∃-I Vy, Ay

R

(∃x: Vx) (Ax)

Hence by Gentzen’s cut rule Vy, Ay, ∼(∃x: Vx) (Ax) And since ‘

R

R

By

Vy’

Ay, ∼(∃x: Vx) (Ax)

R

By

which, by R∀-I, yields ∼(∃x: Vx) (Ax)

R

(∀x: Ax) (Bx)

Thus it turns out that R is after all Boolean and not Aristotelian in its treatment of universal claims, even though the material conditional is not introduced into the interpretation of these claims. Of course one can question the starting point of this argument in sentential logic, ‘P, ∼P R Q’; but doing so is likely to take one into intensional interpretations of disjunction, and although these are interesting, they have no intrinsic connection with the interpretation of quantifi ers as restricted or unrestricted. So that line of thought is not relevant here, and the conclusion that R is Boolean in its treatment of the universal claims should be accepted.

26 3

Thomas Baldwin R AND U

This conclusion that R is Boolean suggests that, setting aside the fundamental difference concerning the logical syntax of quantifiers, on all major points of substance R and U are in agreement. Given the differences between R and U concerning quantifiers, there is, of course, no question of showing that all R-theorems are U-theorems and vice-versa. But one can nonetheless attempt to construct U-analogues of R-theorems by translating R-formulae into U-formulae in such a way that R-derivations are translated into U-derivations. The fi rst step here lies within R itself. It is easy to demonstrate the following equivalences (where ‘Vy’ is, as before, recognized within R as a universal predicate, so that R Vy): (∀x: Ax) (Bx) (∃x: Ax) (Bx)

R R

(∀x: Vx) (Ax ⊃ Bx) (∃x: Vx) (Ax & Bx)

One can now use these equivalences to motivate a systematic transformation within R of any R formula PR into PV (which is also an R formula) in accordance with the following rules: (i) where PR includes a formula (open or closed) of the form ‘(∀x: Ax) (Bx)’ (but where ‘Ax’ is not ‘Vx’), PV is the result of replacing that formula within PR with ‘(∀x: Vx) (Ax ⊃ Bx)’; (ii) where PR includes a formula (open or closed) of the form ‘(∃x: Ax) (Bx)’ (but where ‘Ax’ is not ‘Vx’), PV is the result of replacing that formula within PR with ‘(∃x: Vx) (Ax & Bx)’; (iii) otherwise PV is PR Given that R formulae are fi nite, this process of systematic transformation will terminate once all the quantifiers have been transformed into V-restricted quantifiers. The equivalences above ensure that PR PV. It R follows that all derivations within R can be transformed into derivations which involve only the PV formulae equivalent to the PR formulae used in the original derivation—i.e. where PR , QR , . . . R SR , one can construct a derivation PV, QV, . . . R SV. This latter derivation will typically involve extra steps which draw on the connectives (conditional or conjunction) introduced by the transformation of PR etc into PV etc. Thus, whereas (as I showed above) Barbara in PR format does not require use of the rules for conditionals, once this derivation is transformed into PV format, then the familiar extra steps which do make use of the rules for conditionals are required. But the validity of these new steps is guaranteed by the equivalence (∀x: Ax) (Bx) (∀x: Vx) (Ax ⊃ Bx) which of course itself draws R on these same rules of inference for conditionals. This result prepares the ground for the construction of U-analogues of R-derivations. This construction requires a further systematic, but very simple,

Restricted Quantifi ers and Logical Theory

27

transformation, namely the transformation of all the V-restricted quantifiers in the PV R-formulae into the unrestricted quantifiers of U-formulae by simply deleting the (vacuous) restriction ‘Vx’. Call the result of such a transformation a PW formula. Thus where PV includes an R-formula of the form ‘(∀x: Vx) (Ax ⊃ Bx)’ PW will include a corresponding U-formula of the form ‘(∀x) (Ax ⊃ Bx)’ etc; and given the fi niteness of the formulae involved, this process of transformation will terminate with a U-formula all of whose quantifiers are unrestricted. What then is the relationship between an R-derivation PV, QV, . . . R SV and a putative U-derivation PW, QW, . . . U SW? Since these R-derivations differ from U-derivations only because of the presence of restricted quantifiers with the vacuous restriction ‘: Vx’, the issue turns on whether the transformation of a sequence of R-formulae PV . . . which constitute a valid R-derivation into a sequence of U-formulae PW . . . is sufficient to ensure that this latter sequence is a valid U-derivation. A comparison between the R-rules (R∀-E, R∀-I, R∃-E, R∃-I) and the U-rules (U∀-E, U∀-I, U∃-E, U∃-I) will show that this condition is indeed satisfied subject to the assumption that Vy which legitimates within U the premises ‘Vy’ which occur within the U R-derivations. These premises do no work in the U-derivations, but they do no harm either; an alternative option would be to delete them all in the course of transforming an R-derivation into a corresponding U-derivation. The basic point here is just that at each stage in a valid R-derivation the legitimate application of an R-rule to an R-formula PV can be matched by the legitimate application of a U-rule to a U-formula PW. An analogous point applies in reverse, except that now it is necessary to introduce into the R-sequence obtained from a U-derivation by reversing the previous transformation suitable extra premises ‘Vy’ in order to be able to apply the R-rules to the restricted quantifiers. Since it is given that R Vy the addition of these premises is legitimate and suffices to transform the R-sequence into a valid R-derivation. Putting this conclusion together with the previous one, it follows that any valid R-derivation can be transformed into an equivalent U-derivation—i.e. where ΓR is a set of R-formulae and ΓW is the result of transforming each of them in accordance with these rules: ΓR

R

PR iff ΓW

U

PW

The comparable result for derivations involving arbitrary U-formulae is easier to establish. In this case all that is needed is to transform all the unrestricted quantifiers of an arbitrary U-formula PU into V-restricted quantifiers and thereby obtain an R-formula PX . Inspection of the U-rules and R-rules for the quantifiers will show that this transformation takes one from U-rules to special cases of the corresponding R-rule where ‘Vx’ occurs as the restriction on the quantifier; and because of the assumption within R that R Vy, it is always legitimate to add extra premises ‘Vy’ to the R-sequence in order to obtain legitimate applications of the R-rules

28

Thomas Baldwin

which correspond to the U-rule applied in the original derivation. Hence, with this transformation in place, any sequence of U-formulae PU . . . which constitutes a valid U-derivation is transformed into a comparable sequence of R-formulae PX . . . which constitutes a valid R-derivation. Equally, starting from a valid R-derivation of R-formulae PX . . . , and then reversing this transformation yields a U-sequence PU . . . which is a valid U-derivation. Hence: ΓU

U

PU iff ΓX

R

PX

These results show that as far as proof theory goes the two logical theories are broadly equivalent: any derivation in one theory can be transformed into an equivalent valid derivation in the other. This conclusion is reinforced by a further consideration. So far I have been comparing U and R, treating them as separate logical theories with their own distinctive formulae and rules of inference; but one can also merge them into a combined UR logical theory which admits quantifiers of both types, employing both sets of rules. Within UR it is then very easy indeed to prove the obvious equivalences: (∀x) (Ax ⊃ Bx) (∃x) (Ax & Bx)

UR UR

(∀x: Ax) (Bx) (∃x: Ax) (Bx)

This result confi rms that the transformations constructed above generate equivalent derivations, and shows that once both types of quantifiers are in play there is no substantial inferential gain from using one type rather than the other. What now about model theory? Here it is not the different rules of inference that matter but the rules specifying the satisfaction-conditions for formulae with quantifiers of the two kinds. The rules for the unrestricted quantifiers of U are familiar. In the standard Tarskian approach, variables are enumerated and assigned values relative to their place in a sequence of members of the domain; and once one factors in the relativity of truth and satisfaction to interpretations (‘I’) and other complexities required for handling formulae with multiple quantifiers, the following account is obtained: U∃&-sats

s satisfiesI ‘(∃xi) (Axi & Bxi)’ iff (∃s′) (s′ ≈i s & s′(i) ∈ A I & s′(i) ∈ BI)

where ‘A I ’ is used in the metalanguage to specify the subset of the domain which is assigned by interpretation I to the object-language condition ‘Ax’ etc. It is easy to see how this approach can be adapted to deal with restricted quantifiers and thereby deal with R-formulae such as ‘(∃xi: Axi) (Bxi)’:

Restricted Quantifi ers and Logical Theory R∃-sats

29

s satisfiesI ‘(∃xi: Axi) (Bxi)’ iff (∃s′: s′ ≈i s & s′(i) ∈ A I) (s′(i) ∈ BI)

But what still needs some discussion is the hypothesis that these satisfaction conditions are equivalent. We can assume that in both cases the domain and the interpretations of the object-language predicates are the same; the difference comes only from the rules such as those above which specify the satisfaction-conditions for formulae with quantifiers of the two kinds, restricted and unrestricted. Although these rules apply to distinct object-languages, the languages of U and R, they are themselves formulated in a metalanguage which uses quantifiers of both types. Hence the logic of this metalanguage is the logic UR sketched above, and it is then obvious from the UR equivalence set out above for the existential quantifier that the two conditions are equivalent. The intuitive grounding for this comes from the fact that where U∃&-sats makes satisfaction dependent on there being a member of the domain which belongs both to A I and to BI , R∃-sats makes satisfaction dependent on there being a member of the intersection of the same domain with A I which is also a member of BI . It is obvious that there is here one and the same set-theoretic condition described in slightly different ways. In the light of this equivalence and the comparable equivalence of the satisfaction-conditions of the two universal quantifiers— U∀⊃-sats

s satisfiesI ‘(∀xi) (Axi ⊃ Bxi)’ iff (∀s′) ((s′ ≈i s & s′(i) ∈ A I) ⊃ s′(i) ∈ BI)

R∀-sats

s satisfiesI ‘(∀xi: Axi) (Bxi)’ iff (∀s′: s′ ≈i s & s′(i) ∈ A I) (s′(i) ∈ B I)

the model-theoretic equivalence of U and R is easy to establish. In particular, model-theoretic analogues of the earlier proof-theoretic results are immediate. Where PR is an arbitrary R-formula, ΓR is a set of such formulae, PW is the transformation of PR in accordance with the fi rst set of rules set out above (so that ‘(∀x: Ax) (Bx)’ is transformed into ‘(∀x) (Ax ⊃ Bx)’ etc), and ΓW is a similar transformation of ΓR , it follows from the equivalence between these pairs of satisfaction-conditions that ΓR

R

PR iff ΓW

U

PW

For given the equivalences above, any countermodel to one side of this can be easily adapted to provided a countermodel to the other side. A bit more work is needed to construct R-deductions equivalent to arbitrary U-deductions. In this case we need to start from the familiar basic U-clauses: U∃-sats

s satisfiesI ‘(∃xi) (Axi)’ iff (∃s′) (s′ ≈i s & s′(i) ∈ A I)

30

Thomas Baldwin U∀-sats

s satisfiesI ‘(∀xi) (Axi)’ iff (∀s′) (s′ ≈i s ⊃ s′(i) ∈ A I)

and compare these to the following instances of the previous R-clauses, where the distinguished ‘universal’ predicate V is given a constant interpretation as the domain D: R∃V-sats

s satisfiesI ‘(∃xi: Vxi) (Axi)’ iff (∃s′: s′ ≈i s & s′(i) ∈ D) (s′(i) ∈ A I)

R∀V-sats

s satisfiesI ‘(∀xi: Vxi) (Bxi)’ iff (∀s′: s′ ≈i s & s′(i) ∈ D) (s′(i) ∈ A I)

As before, it is obvious that the two U / R pairs of similar conditions are equivalent since in the context of model theory everything belongs to the domain. Hence where PU is an arbitrary U-formula, ΓU is a set of such formulae, PX is the transformation of PU into an R-formula in accordance with the second rule set out above (so that ‘(∀x) (Ax)’ is transformed into ‘(∀x: Vx) (Ax)’ etc), and ΓX is a similar transformation of ΓU , it follows that ΓU

U

PU iff ΓX

R

PX

These results match those for proof theory. Not only does it confi rm those results, but it also makes it easy to export the familiar soundness and completeness results of U (classical fi rst-order logic) to R (fi rst-order logic for restricted quantifiers). For given Γ

U

P iff Γ

U

P

it follows in particular that ΓW

U

PW iff ΓW

U

PW

and one can then combine this with ΓR

R

PR iff ΓW

U

PW

ΓR

R

PR iff ΓW

U

PW

R

PR

and

to conclude ΓR

R

PR iff ΓR

Restricted Quantifi ers and Logical Theory 31 Overall, then, R and U are essentially equivalent logical theories, even though their theorems are different. While it is reassuring to confi rm that nothing of substance would be lost by switching from U to R, one may well now ask what would be gained? Clearly one does not gain the ability to prove any essentially new theorems within the familiar arena of classical logic, the logic of the quantifiers ‘All’, ‘Some’, and ‘No’. But it would be astonishing if there were essentially new theorems to be unearthed here; so this lack should not be a cause of disappointment. Furthermore this lack does not undermine the points made earlier in this paper, that R generates better proofs of some familiar theorems than U does because it gives logical significance to the intrinsic structure of natural language quantifiers instead of pulling them apart and relying on extrinsic logical connections such as conditionals to reconstitute the logical potential of the quantifiers. One can also ask whether there are aspects of model theory where the use of restricted quantifiers in the metalanguage provides an improved account of the structure of the theory. An obvious point arises from the way in which the clause ‘s′ ≈i s’ figures in the satisfaction conditions for quantifiers, since, however it is presented, it is clear that this clause has a purely restrictive function. This is not of a matter of any great importance, but there is a different point which applies to model theory generally and which is of considerable importance. Within model theory (e.g. in the context of discussions of compactness) it is important to distinguish logical consequence (Γ P) from logical truth ( Q). Yet this distinction relies tacitly on the use of a restricted quantifier in the characterization of logical consequence: ‘Γ P’ abbreviates the statement that all interpretations which make the members of Γ true also make P true. If the quantifier here is construed as unrestricted, then the statement is just that all interpretations are such that if they make the members of Γ true they also make P true. In effect, therefore, where ‘&(Γ)’ is the formula which is the conjunction of all the members of Γ, this affi rms the conditional logical truth, &(Γ) ⊃ P, and thereby eliminates the distinction between logical truth and logical consequence. Of course one can consider the relationship between Γ P and &(Γ) ⊃ P, and in many cases establish that they are equivalent. But, arguably, this does not hold universally, for example where Γ is infi nite; and even where it does hold, it is something to be established by a proof and not just imposed by a ruling that all quantifiers are unrestricted.

4

THE FOUNDING FATHERS

In the light of the intuitive merits of R over U it is on the face of it surprising that it has been standard practice to assume in logical theory that quantifiers are basically unrestricted (for confi rmation of this, look at almost any introduction to logic2). The reasons for this, I think, lie in the context in which quantification theory was fi rst developed and these reasons are also

32

Thomas Baldwin

worth exploring because they reveal some theoretical advantages which U possesses as compared with R which significantly qualify the account so far of the comparative merits of R and U. Frege of course comes fi rst here with his Begriffsschrift of 1879 where he introduces (§11) the ‘concavity’ with its distinctive variables (‘German’— i.e. gothic letters) to capture the logical role of generality in judgment (Frege 1879: 24). From the way Frege sets this up and combines it in §12 with the notation for the conditional to express universal judgments, it is clear that he is using the concavity as an unrestricted universal quantifier (Frege 1879: 28).3 In Begriffsschrift this new notation is not clearly conceptualized, but in Grundgesetze der Arithmetik §21 he explains it in terms of the distinction between fi rst-level and second-level concepts: the concavity plus bound variable expresses a second-level concept which combines with a fi rst-level concept to produce a complete thought (Frege 1893: 74). One might suppose that this approach can be readily extended to restricted quantifiers by thinking of them as second-level relations which apply to a pair of fi rst-level concepts. But this oversimplifies the situation: what one defi nes in this way is a ‘binary quantifier’ and I shall discuss in the next section the relationship between restricted quantifiers and binary quantifiers. Instead, restricted quantifiers such as ‘(Some x: . . .x. . .) (. . .x. . .)’ are best thought of as second-level functions which combine with a fi rst-level concept to produce a second-level concept of the form ‘(Some x: Fx) (. . .x. . .)’; and this in turn combines with a fi rst-level concept to produce a complete thought of the form ‘(Some x: Fx) (Gx)’. This mode of analysis is legitimate within Frege’s categories. Thus the question arises as to whether Frege ever justifies his practice of using unrestricted quantifiers as opposed to restricted ones. So far as I am aware, he never does so explicitly, but there is one place where he considers a related issue. In Grundgesetze §65 he argues that logic needs concepts with sharp boundaries and complete defi nitions, and in the course of discussing this last point he considers whether, when defi ning addition, instead of introducing the unrestricted law: If a is a number & b is a number then a + b = b + a one might restrict the domain to numbers and express the law as just a+b=b+a To this Frege objects that the unrestricted law is equivalent to If a + b ≠ b + a & a is a number, then b is not a number and comments ‘here it is impossible to maintain the restriction to the domain of numbers’ (Frege 1903: 170). Frege’s comment is quite right, but

Restricted Quantifi ers and Logical Theory

33

it does not undermine the kind of use of the explicitly restricted quantifiers I have been considering. For Frege’s unrestricted law is equivalent to the following claim made with a restricted quantifier: (∀a, b: a is a number & b is a number) (a + b = b + a) and this is equivalent to (∀a, b: a + b ≠ b + a) (∼(a is a number & b is a number)) This equivalence expresses the fact that just as restricting the domain of objects to pairs of numbers yields objects which satisfy the condition a + b = b + a, restricting the domain to pairs of objects for which the condition a + b = b + a does not obtain yields objects which are not numbers. Different restrictions are of course employed in these two cases, but that is exactly what the use of restricted quantifiers leads one to expect. Introducing a quantifier restricted to numbers does not bring with it the intention to continue to ‘restrict the domain to numbers’ when one contraposes in the way that Frege does. Frege’s argument here perhaps connects with his thesis that logic deals with absolutely general laws, and not, therefore, with laws restricted to numbers or any other domain. Yet, as the example here shows, it is not necessary to challenge Frege’s thesis concerning the generality of logic in order to legitimate the use of restricted quantifiers within logic. The restrictions within the quantifiers are introduced as a way of capturing the logical structure of quantifi ers, and the laws concerning their logical significance such as the rules of inference employed in the previous section have just the same generality as the rest of logic. So it would be a mistake to think that Frege’s thesis of the generality of logic requires that logic can avail itself only of unrestricted quantifiers. Yet even when all this is said, I think the proponent of R does have to make a significant concession here. Frege’s example involves contraposition and multiple quantifiers. As will be manifest, in order to capture this case with restricted quantifiers I have had to use the idiom ‘(∀a, b: . . .)’, that is to quantify over pairs of members of the domain. There is no intrinsic problem about this, but one might ask whether this extra degree of complexity is essential. After all there is no problem with the use of individual quantifiers in (∀a: a is a number)(∀b: b is a number) (a + b = b + a) As Frege observes, however, the trouble comes when one seeks to contrapose; one cannot say what is wanted as, say, (∀a: (∀b: a + b ≠ b + a)) (∼(a is a number & b is a number))

34

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since the occurrence of ‘b’ in ‘b is a number’ falls outside the scope of the quantifier ‘∀b’. So in this case the use of the pair quantifier is essential if the quantifier is restricted; whereas with the unrestricted quantifiers Frege commends there is no difficulty about using individual quantifiers: the inference is just (∀a)(∀b) (if a is a number & b is a number then a + b = b + a) so (∀a)(∀b) (if a + b ≠ b + a then ∼(a is a number & b is a number)). I should add that this situation is consistent with the results of the previous section, since in accordance with the rule for obtaining a valid R-inference from a U-inference this U-inference can be transformed into the R-inference (∀a: Va)(∀b:Vb) (if a is a number & b is a number then a + b = b + a) so (∀a: Va)(∀b: Vb) (if a + b ≠ b + a then ∼(a is a number & b is a number)). But here, of course, the restriction is vacuous and the R-inference just mimics the U-inference—whereas the protagonist of R had hoped that by means of significantly restricted quantifiers it would be possible to provide a more illuminating account of inferences involving quantifiers. I have argued that that condition is satisfied in the case of Barbara and other straightforward Aristotelian inferences; but what we see here is that once logical inference involves the logical manipulation of complex formulae with multiple quantifiers we get a more straightforward grasp of the inferential steps where all the quantifiers are unrestricted. This is not just a matter of our greater familiarity with unrestricted versus restricted quantifiers. The important point is, rather, that the scope restrictions inherent in the use of restricted quantifiers inhibit the kinds of transformation which are required following contraposition and similar inferential steps. For this reason, even the defender of R should acknowledge that once the lines of reasoning under consideration involve multiple quantifiers and relational predicates, it is likely that the steps in the reasoning will be more straightforward where unrestricted quantifiers are employed than where all the quantifiers are significantly restricted. One obvious context for such reasoning is mathematics, which is of course exactly the context for which Frege developed and presented his theory, as in Grundgesetze der Arithmetik. The situation here was in effect clarified and extended by the other pioneer of quantification theory, C.S. Peirce, who first developed a notation which is essentially the same as that used today, though Peirce himself gives credit to his colleague O.H. Mitchell for the basic idea. Peirce developed his theory in the context of his reflections on Boole’s algebra of logic, which lacks a satisfactory expression for generality, especially existential claims, and he expresses his key line of thought in the following passage in a manuscript ‘On the Algebra of Logic (Second Paper)’ of 1884:

Restricted Quantifi ers and Logical Theory 35 We now pass from the consideration of a single individual to that of the whole universe; and fi rst we have that rude distinction of ‘some’ and ‘all’ which may be said to discriminate logic from mathematics. After the whole Boolian school had for thirty years been puzzling over the problem of how to take account of this distinction in their notation, without any satisfactory result, Mr. Mitchell, by a wonderfully clear intuition, points out that what is needed is to enclose the whole Boolian expression in brackets, and to indicate to what proportion of the universe it refers by exterior signs. Denoting by A any expression such as we have hitherto considered, we may write ΠA to signify that A is true of every individual of the universe, and ΣA to mean that it is true of some individual of the universe. (Peirce 1884: 114) This passage does not reveal fully how sophisticated Peirce’s position was at this time, but in his next paper ‘On the Algebra of Logic: A Contribution to the Philosophy of Notation’ (which was published in 1885) he uses bound variables to express complex distinctions. Thus, to take one of his examples: he proposes that ΠiΣjlijbij means that everything is at once a lover and a benefactor of something; and ΠiΣjlijbji that everything is a lover of a benefactor of itself. (Peirce 1885: 180) As Peirce develops the position further, it becomes apparent that although the statements he expresses by means of this notation are indefi nitely complex, he sticks with the 1884 principle that one should aim ‘to enclose the whole Boolian expression in brackets, and to indicate to what proportion of the universe it refers by exterior signs’; that is, in his notation, one should place all the quantifiers together at the start of the proposition, using the variables to connect them to the appropriate predicates. Thus one of his examples in the 1885 paper has nine quantifiers bunched together at the start (Peirce 1885: 181). In effect Peirce is developing his notation in such a way that his favored way of expressing a quantified proposition, however complex, is in prenex normal form, and in the 1885 paper he argues that all logical formulae with these quantifiers can be expressed in this way (Peirce 1885: 182). The significance of this step is two-fold: fi rst, the adoption of prenex normal form greatly facilitates algebraic transformations and thus the construction of complex lines of argument; second, the quantifiers involved are all unrestricted, and essentially so, for the point of prenex normal form is precisely to separate the quantifiers from the other elements

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of logical structure, leaving a single complex formula with bound variables to be fed into indefi nitely complex algebraic transformations. And in the increasingly mathematical development of logical algebra into metalogic, fi rst by Skolem and then by Herbrand, Gödel, and others, these prenex normal forms played an important part in the construction of important theorems such as Gödel’s completeness theorem. This observation reinforces that suggested by the discussion of Frege: the protagonist of R has to accept that once logic gets mathematical, the quantifiers employed are typically unrestricted, since these facilitate the algebraic transformations that are central to the complex proofs characteristic of mathematical logic. This recognition is nonetheless consistent with the earlier point, that R provides a much more intuitive and natural way of expressing straightforward inferences which depend on the use of natural language quantifiers. Thus there is a rough and ready division of labor here: U is essential for mathematical logic whereas R captures the ‘logic of natural language’. This conclusion helps to explain the predominance of U in expositions of fi rst-order logical theory. For at the start of the twentieth century fi rst-order logic was introduced in the context of the logicist program of showing that mathematics is logic. But the central figure here was in fact Bertrand Russell. For although Frege and Peirce were the founders of the logical theory of quantifiers, in neither case was this aspect of their work much noted at the time of its publication. Instead it was primarily through Russell’s publications during the fi rst decade of the twentieth century, leading up to Principia Mathematica (1910–13), that fi rst-order logic achieved the status as the foundation for logical theory that it now enjoys.4 Having said this, however, it turns out that on the issue of restricted vs. unrestricted quantifiers itself, Russell’s position was complicated and ambivalent. Russell generally frames the issue here as one concerning restricted vs. unrestricted variables, but in his discussions of this issue a central question is the interpretation of ‘denoting phrases’ such as ‘All men’. His most extensive discussion of this occurs in his 1908 paper ‘Mathematical Logic as based on the Theory of Types’. Russell argues here that our intuitive understanding of the role of ‘All’ in ‘All men’ as restricting the range of the variables it binds to men leads to logical mistakes; but, as the title of the paper implies, he also argues that this role needs to be understood in a way which accommodates the restrictions inherent in his theory of types. Russell begins by asking: How are we to interpret the word all in such propositions as ‘all men are mortal’? (Russell 1908: 69) and responds with a suggestion that, on the face of it, might be taken as an informal proposal to construe ‘all’ as part of a restricted quantifier ‘all men’:

Restricted Quantifi ers and Logical Theory 37 At fi rst sight, it might be thought that there could be no difficulty, that ‘all men’ is a perfectly clear idea, and that we say of all men that they are mortal. (Russell 1908: 69) Russell now continues, however: But to this view there are many objections. (Russell 1908: 69) and this leads him to lay out eleven numbered points in which he sets out these objections and presents his own alternative position (Russell 1908: 70–2). He fi rst objects that a universal proposition such as ‘all men are mortal’ does not imply the existence of men, an implication which he (mistakenly) takes to be required by the response he has suggested. The next two objections assume that the suggestion to which he is responding holds that ‘all men’ gets its meaning from the fact that it names a distinct object (presumably mankind), and to this Russell objects that ‘all men’ is a denoting phrase which has no meaning in isolation, and, further, that even if there is such an object somehow named by ‘all men’ we certainly do not attribute mortality to it when we say ‘all men are mortal’. So far the protagonist of restricted quantifiers will feel that Russell has not really addressed his position, but in the next, fourth, objection he does come much closer: (4) It seems obvious that, if we meet something which may be a man or may be an angel in disguise, it comes within the scope of ‘all men are mortal’ to assert ‘if this is a man, it is mortal’. Thus . . . it seems plain that we are really saying ‘if anything is a man, it is mortal’, and that the question whether this or that is a man does not fall within the scope of our assertion, as it would do if the all really referred to ‘all men’. (Russell 1908: 70–1) Russell’s argument here seems to be that if (to put the opposing position in my own words) we construe the quantifier in ‘all men are mortal’ as restricted, we have always to settle the question whether this or that ‘falls within the scope of our assertion’, i.e. is a man, before we can decide whether the question of this thing’s mortality is relevant to whether or not all men are mortal; whereas if we construe the quantifier as unrestricted, we can set aside this preliminary question and just consider, as ‘seems obvious’, whether anything at all we encounter, including an angel in disguise, falsifies ‘if this is a man, it is mortal’. On the face of it, this is not persuasive: for what matters to the truth or falsehood of ‘all men are mortal’ is the mortality or not of men, and not the mortality of anything else, including angels. Russell makes it appear otherwise by introducing the case of something which, he says, ‘may be an angel in disguise’, that is, concerning which we cannot readily tell whether it is an immortal man or an immortal angel. In this situation, until we have determined what this being is, we cannot tell

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whether it is a putative counterexample to the thesis that all men are mortal; and thus, from an epistemological perspective, it is certainly relevant to the assertibility of ‘all men are mortal’. Nonetheless, what we have to decide in this situation is whether ‘this is a man’, precisely because if it is not a man, then whether or not it is immortal is irrelevant to whether all men are mortal. So, as the use of a restricted quantifier makes explicit, the question whether this is a man does ‘fall within the scope’ of the assertion that all men are mortal in the sense that the answer to this question determines whether the further question of this being’s immortality is relevant to the truth of our assertion. Russell does not see things this way and takes it that ‘all men are mortal’ means ‘it is always true that if x is a man, x is mortal’. So at this point it appears that he wants to construe the quantifier as unrestricted. But he now adds that he needs to say more about the range here of ‘always’ since in his reaction to the logical paradoxes with which he has started the paper he rejects ‘illegitimate totalities’ such as ‘all values’. Hence he concludes here that the range of ‘always’ ‘must be somehow restricted within some legitimate totality’; and yet, he continues, ‘it is quite essential that we should have some meaning of always which does not have to be expressed in a restrictive hypothesis’ (Russell 1908: 71). Russell’s reasoning here repeats that which he applied to ‘all men are mortal’. Just as, he thinks, we cannot construe assertion of this proposition as the restricted assertion ‘it is always true of men that they are mortal’ but must take it as the relatively unrestricted assertion ‘it is always true that if x is a man, x is mortal’, similarly, he thinks, we cannot express the restriction inherent here in ‘always’ as, say, ‘it is always true of members x of class i that if x is a man, x is mortal’, since this has to be construed as ‘it is always true that if x belongs to class i, then, if x is a man, x is mortal’ (Russell 1908: 71), which simply invites the question as to the scope here of ‘always’. Thus the restriction inherent in ‘always’ has to be ineffable, something which cannot be expressed; and Russell finds this in his conception of the ‘range of significance’ (Russell 1908: 72) of a propositional function. This is the totality of values for which the function is either true or false, and it falls short of all ‘imaginable values’, since it excludes those things which are such that insertion of the thing’s name into a phrase which expresses the propositional function yields a sentence which is meaningless and therefore expresses no proposition. The point of this conception is that because it arises as ‘an internal limitation upon x, given the nature of the function’ (Russell 1908: 72), it does not need to be expressed as the kind of restrictive hypothesis which gives rise to the regress argument Russell used to reject the use of explicit restrictive clauses. So in the end Russell’s quantifiers are indeed restricted, but always only tacitly, by means of ‘internal limitations’ arising from the range of significance of the propositional functions occurring in a proposition. Hence, despite the restrictions inherent in his theory of types, Russell’s conclusion here is that the overt expression of generality by means of quantifiers and variables should always be unrestricted. Notoriously, however,

Restricted Quantifi ers and Logical Theory

39

within Russell’s ramified theory of types, this is not how the use of quantifiers and variables is then developed. For the hierarchy of orders he associates with impredicative definitions makes explicit use of restricted quantifiers and variables to obstruct the derivation of contradictions such as his own paradox. Not surprisingly these restrictions also obstruct the logical reconstruction of mathematical analysis, leading Russell to introduce the axiom of reducibility in order to legitimate the introduction of predicative functions with unrestricted variables to make it possible to express the generalizations which his ramified theory with its restricted variables had not previously allowed him to express. This was not a convincing or stable position, and the outcome confirms the conclusion suggested earlier—mathematical logic is best formulated with as few restricted quantifiers as possible. What was nonetheless significant was the reaction to it, in particular the belief that the paradoxes whose resolution Russell sought through the use of the ramified theory of types are better addressed in other ways. As a result it was reasonable to infer that logical theory is best formulated without restricted quantifiers at all, and this is what then happened. The result was that as Russell’s logical theory was assimilated into philosophy during the fi rst part of the twentieth century, fi rst-order logic with unrestricted quantifiers, U, was conceived not only as a basic framework for mathematical logic but, quite generally, as the basic theory of reasoning.

5

RESTRICTED QUANTIFIERS AND BINARY QUANTIFIERS

So far I have presented R as a challenge to U on the grounds that it is for some purposes a preferable logical theory. But there is a different way in which the approaches inherent in R and U can be compared, namely by considering how readily they can be generalized to other natural language quantifiers. The starting point here is the fact that, on the face of it, natural language quantifiers are typically restricted. The claim which is characteristic of the protagonist of U, as Russell’s argument considered above shows, is that the ‘meaning’ (or logical form) of sentences with these natural language restricted quantifiers is best displayed by transforming them into sentences with unrestricted quantifiers; so ‘all men are mortal’ becomes ‘for all x, if x is a man, then x is mortal’, and so on. The issue to be considered now is how far this type of transformation can be extended to natural language quantifiers other than those familiar from classical logic. One large group of quantifiers where this transformation is possible are the numerical quantifiers, such as ‘At least five’ and ‘Exactly five’: for these quantifiers are formally similar to the familiar existential quantifier ‘some’, understood as ‘at least one’. Thus ‘At least five children were late’ is equivalent to ‘for at least five x, x is a child and x was late’, and so on. But if one turns from numerical claims of this kind to ones in which a proportionality claim in made, such as ‘Exactly half the children were late’,

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no similar equivalent transformation to an unrestricted quantifier is available. For this sentence is not equivalent to ‘for exactly half the domain x, x is a child and x was late’, nor to anything similar with a different connective (conditional, disjunctive, etc). So at this point the protagonist of R can press the merits of his position which captures this sentence as ‘(Exactly half x: x is a child) (x was late)’, and thus permits the formalization of obvious inferences such as: (Exactly half x: Fx) (Gx)

(Exactly half x: Fx) (∼Gx)

Of course one can think of other ways of formalizing this inference, for example by introducing reference to sets and their cardinality. But all these take one away from the manifest quantificational structure of the sentence, and it seems clear that insofar as one wants to respect this one has to stick with restricted quantifiers. But even this has been disputed. In a discussion of the logic of ‘Most men are immortal’, having observed that this proposition is not equivalent to any proposition which just uses the unrestricted quantifier ‘(Most x) (. . .x. . .)’, David Wiggins argued (Wiggins 1980) that one should not infer that the proposition is to be thought of as having the form ‘(Most x: x is a man) (x is immortal)’ with a restricted quantifier. Instead, he argues, we should see here a binary structure of the form: (Most x) (x is a man, x is immortal) Wiggins is, I think, right to see that there is a distinction here, which can be drawn in the Fregean terms I used earlier: binary quantifiers express second-level relations which just take a pair of fi rst-level concepts as arguments, whereas restricted quantifiers express second-level functions which typically take a single fi rst-level concept as argument and whose value is a second-level concept that typically takes another fi rst-level concept as argument. But why should one not accept that ‘Most’ expresses a restricted quantifier? Wiggins has the proponent of restricted quantifiers interpret the role of ‘man’ in ‘(Most x: x is a man) (x is immortal)’ as ‘defi ning a universe of discourse or range of entities to which the quantifier “most x” is restricted’ (Wiggins 1980: 332); but, he continues, the statement that an entity only counts as part of the range of the quantifier if it is a man is itself an instance of quantification; and it is a quantification either unrestricted or less restricted than the quantifier it elucidates . . . But then it seems that, however we limit our quantifiers at any point, we always need to be possessed of a less restricted device of quantification . . . I conclude that it would be a great pity for us to talk ourselves out of the unrestricted natural language quantifier (Wiggins 1980: 332)

Restricted Quantifi ers and Logical Theory 41 Wiggins’s argument here seems to be that because it only makes sense to say that a term such as ‘man’ restricts the ‘range’ of a quantifier such as ‘most x’ if the quantifier itself is not intrinsically restricted in this way, we should regard the quantifier as unrestricted, introducing what he calls ‘a preexisting unspecified and unrestricted mass of entities’. Indeed, he infers, we should not really think of the role of ‘man’ as restricting the quantifier at all, but instead as ‘delimiting some defi nite portion of this [mass of entities], which is what the sentence is about.’ (Wiggins 1980: 332). This comment indicates that there is in fact little substantial disagreement here: for on the conception of restricted quantifiers I have employed (as the semantics set out in §3 indicate), the role of the restriction is precisely to ‘delimit’ a subset of the domain (Wiggins’s ‘pre-existing unspecified and unrestricted mass of entities’) by reference to which the proposition is evaluated. In the light of this point Wiggins’s objection to restricted quantifiers is seen to depend on his assumption that such quantifiers are to be understood as themselves ‘defi ning a universe of discourse or range of entities to which the quantifier . . . is restricted’, as if each restricted quantifier ‘defi nes’ a new domain for the variables it binds. But that assumption is to be rejected: although the truth of a proposition with a restricted quantifier is evaluated by reference to the relevant restriction of the domain, the domain itself remains unchanged by the restrictions. Wiggins’s objection to restricted quantifiers is, therefore, misconceived. But his discussion does nonetheless raise the question as to the relationship between the restricted quantifiers I have been discussing and binary quantifiers. Binary quantifiers were introduced by Mostowski (1957) and Lindström (1966) in the context of a theory of generalized quantifiers, whose basic idea is the generalization of the Fregean thought that quantifiers are second-level concepts: a unary quantifier is a quantifier with just one fi rstlevel concept as argument, a binary quantifier has two concepts as arguments, a ternary quantifier has three concepts as arguments and so on. I have emphasized that the structure of restricted quantifiers is, on the face of it, rather different from that of a binary quantifier since they are complex unary quantifiers. Nonetheless, because two fi rst-level concepts standardly occur as arguments in propositions which employ a restricted quantifier, it follows that such quantifiers do in fact meet the condition for being a binary quantifier. They are binary quantifiers, but they have a distinctive structure in which the fi rst-order concepts which occur as arguments have different roles: one concept restricts the quantifier, and thereby forms with it a unary quantifier such as ‘(Most x: x is a man) (. . .x. . .)’, the other concept then completes a thought such as ‘(Most x: x is a man) (x is immortal)’. A question which can now be asked is whether binary quantifiers which have this distinctive restrictive structure satisfy some other distinctive logical conditions which fit them to be restrictive, and this is a matter that has been much discussed by linguists since the 1980s (see especially Keenan and Stavi 1986, and Keenan and Westerståhl 1997). Before entering into

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this discussion, however, it is worth identifying some clear cases of binary quantifiers which are not restricted. A good case is ‘There are more As than Bs’, i.e. (More x) (Ax, Bx) For it is not that the domain is here restricted to the As and that members of this subset are then assessed, somehow, for whether or not they are Bs. Indeed the natural idiom for a restricted quantifier is not here available: ‘More As are Bs’ is, as it stands, incomplete. There are complete sentences such as ‘More As than Cs are Bs’, but here a ternary quantifier is used which is indeed restricted to the Bs, of which it is said that more are As than are Cs. But the fact that this ternary quantifier is restricted does not show that there is some tacit restriction in the use of the simple binary quantifier ‘There are more As than Bs’; and it is easy to generalize this case to other comparative quantifiers such as ‘There are just/twice/half as many As as Bs’, etc. Thus the question posed above as to what distinguishes restricted quantifiers among binary quantifiers can be thought of informally as the search for logical conditions which are satisfied by ‘All’, ‘Most’, ‘Many’ etc, but not by comparatives such as ‘More’. A starting point is provided by the moral from Russell’s discussion of ‘All men are mortal’, namely the truth of this proposition is dependent on the mortality or not of men and is not affected by the immortality of an angel, even in disguise; similarly, the truth of ‘Most As are Bs’ depends on the proportion of Bs among the As and is insensitive to the proportion of Bs among things which are not As. By contrast the truth of ‘There are more As than Bs’ depends simply on the relative number of As and Bs, and the question of whether any of these Bs are As is irrelevant. A plausible formalization of this intuition is the test for ‘conservativity’ (see Keenan and Stavi 1986: 274ff): CONS

‘(Qx) (Ax, Bx)’ is a restricted quantifier only if ‘(Qx) (Ax, Bx)’ is equivalent to ‘(Qx) (Ax, Ax & Bx)’

The thought here is that the intuitive irrelevance of the Bs which are not-As in the case where ‘(Qx) (Ax, Bx)’ is a restricted quantifier is captured by the requirement that the contribution made by the condition ‘Bx’ to the truth-conditions of ‘(Qx) (Ax, Bx)’ is one which could equally be made by the conjunctive condition ‘Ax & Bx’ which explicitly excludes Bs which are not As. This test seems to separate quantifiers as one would want: ‘Most As are Bs’ is equivalent to ‘Most As are ABs’ but ‘There are more As than Bs’ is not equivalent to ‘There are more As than ABs’. So is conservativity sufficient as well as necessary for restrictedness? Linguists have argued that it is not: their favorite counterexample is ‘blik’ (see Keenan and Westerståhl 1997: 855), which is to be interpreted as follows:

Restricted Quantifi ers and Logical Theory 43 ‘(blik x) (Ax, Bx)’ is true iff the number of not-As = 3 (so blik cats are dark just where there are three things which are not cats). It is clear that ‘blik’ is, intuitively, not a case of a restricted quantifier since the truth of ‘(blik x) (Ax, Bx)’ depends only on the things which are not As; thus so far from restricting the domain to the As, the As are here taken to be irrelevant. But, technically, ‘blik’ does satisfy the test of conservativity: we do get ‘(blik x) (Ax, Bx)’ is true iff ‘(blik x) (Ax, Ax & Bx)’ is true However, on reflection, this result is an artifice; it reflects the fact that ‘(blik x)’ is not a substantive binary quantifier at all since the role of the condition ‘Bx’ in ‘(blik x) (Ax, Bx)’ is vacuous. To eliminate cases of this type one should hold that a quantifier is binary only where there is no equivalent unary quantifier that can be obtained from it just by omitting one of the arguments. ‘blik’ fails to satisfy this condition. But it is easy to construct a related quantifier which does satisfy this condition and conservativity, but which is still not, intuitively, a restricted quantifier. Thus consider ‘blikk’, interpreted as follows: ‘(blikk x) (Ax, Bx)’ is true iff the number of not-As = 3 & {x: Ax} ∩ {x: Bx} ≠ ∅ One might propose that the reason that ‘blikk’ is not a restricted quantifier is not that the truth of the clause ‘the number of not-As = 3’ in its truthcondition depends on the members of the domain which are not As rather than those which are As, but is simply the fact that this clause does not concern the relation between the As and the Bs at all. But consider the similar binary quantifier ‘clikk’, defi ned as follows: ‘(clikk x) (Ax, Bx)’ is true iff the number of As = 3 & {x: Ax} ∩ {x: Bx} ≠ ∅ Here too there is a clause which does not concern the relation between the As and the Bs, but in this case it does concern the As, rather than the not-As. And I think this is sufficient to entitle this quantifier, ‘clikk’, to count as restricted; indeed ‘(clikk x) (Ax, Bx)’ can be interpreted as ‘Some of the three As are Bs’. ‘(blikk x) (Ax, Bx)’, however, cannot be similarly interpreted as ‘Some of the three not-As are Bs’ (that would be ‘(clikk x) (not-Ax, Bx)’); and this point does now show clearly that what blocks the interpretation of ‘blikk’ as a restricted quantifier is the way in which the truth of ‘(blikk x) (Ax, Bx)’ depends, in part, not on the members of the domain which are As, but on those which are not-As.

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Thus if ‘(Qx) (Ax, Bx)’ is to be a restricted quantifier, what is needed is a way of ensuring that the truth of ‘(Qx) (Ax, Bx)’ depends just on the members of the domain which satisfy ‘Ax’ so that conditions such as that the number of not-As = 3 are eliminated. The standard way of attempting to secure this result is by means of a condition on domains: the proposal is that extending the domain by adding extra members in a way that does not affect the set of As should not affect the truth-value of ‘(Qx) (Ax, Bx)’ (Keenan and Westerståhl 1997: 855). It is easy to see that ‘(blikk x) (Ax, Bx)’ fails to satisfy this condition. But there are going to be other unwanted quantifiers which satisfy even this condition, e.g. ‘(brikk x) (Ax, Bx)’ is true iff the number of not-As ≥ 3 & {x: Ax} ∩ {x: Bx} ≠ ∅ To eliminate this type of case one also needs to allow oneself to shrink the domain while still leaving the As unaffected. Thus one needs to require that the truth-value of ‘(Qx) (Ax, Bx)’ should not be affected by any changes in the domain as long as these changes leave the set of As unchanged. This ‘domainindependence’ strategy is, in effect, a possible world strategy: the requirement is that the truth-value in the actual world of ‘(Qx) (Ax, Bx)’ should be maintained in all worlds which differ from the actual world only by extending or reducing the domain but where there is no change to the set of As. To formulate this ‘domain-independence’ (DI) requirement, say that where one world w′ differs from another world w only by adding extra objects to the domain, or removing some objects from it, while preserving the membership in w of the set of As, w′ ≈D wA. Then, taking simple truth to be truth in the actual world, w@ , the domain-independence requirement is as follows: DI ‘(Qx) (Ax, Bx)’ is a restricted quantifier only if ‘(Qx) (Ax, Bx)’ is true iff (∀w′: w′ ≈D w @ A) (‘(Qx) (Ax, Bx)’ is true in w′) Thus the fi nal suggestion is that while conservativity (CONS) and domainindependence (DI) are separately necessary conditions for a binary quantifier ‘(Qx) (Ax, Bx)’ being a restricted quantifier, they are jointly sufficient. Domain-independence ensures that only the members of the domain which satisfy ‘Ax’ are relevant to the truth of ‘(Qx) (Ax, Bx)’ and conservativity ensures that the application of the condition ‘Bx’ is restricted to things which satisfy ‘Ax’. This proposal provides an account of what is distinctive about restricted quantifiers among binary quantifiers. But it does not follow that restricted binary quantifiers are not equivalent to quantifiers which are not restricted. The comparison between R and U already shows that for each classical restricted binary quantifier, there is an equivalent unrestricted unary quantifier. The case of ‘Most’ shows that this result does not generalize;5

Restricted Quantifi ers and Logical Theory 45 nonetheless one can still ask whether there is an unrestricted binary quantifier equivalent to the restricted quantifier ‘Most’. In this case the answer is surely affi rmative: ‘Most As are Bs’ is equivalent to ‘There are more ABs than A not-Bs’, i.e. (More x) (Ax & Bx, Ax & not-Bx) As we have seen this latter binary quantifier is not restricted; it fails the test of conservativity. So now the question is how far this kind of equivalence generalizes. A typical case to think about is ‘Many As are Bs’. Although this is vague, its content can be expressed as ‘The proportion of the As which are ABs is significant’ i.e. greater than some contextually specified threshold, e.g. 10%. Here the quantifier remains restricted; but the proposition is equivalent to one to the effect that the ratio between the ABs and the A not-Bs is similarly significant (though the threshold will be different), and this is a proposition which one might express by means of an unrestricted binary quantifier formally similar to ‘More’: (Significant ratio x) (Ax & Bx, Ax & not-Bx) We do not have a natural language quantifier to express this thought, but there seems no reason of principle why there could not be one. This suggests that wherever a restricted quantifier is used to make even a vague claim of proportionality, it is likely that its content can also be expressed by means of an unrestricted binary quantifier which makes explicit what is being compared, although the quantifier may well be somewhat artificial. Since the obvious cases of restricted quantifiers are of this kind, the implication seems to be that in each such case there is an equivalent proposition employing only unrestricted quantifiers.

NOTES 1. Tim Smiley himself opened up this topic in Smiley (1962); see also Stevenson (1975). 2. The only exception I am aware of is McKay (1989). 3. It is then rather odd that at the end of Begriffsschrift §12 he introduces Aristotle’s square of opposition in a way which suggests that his new logical theory captures the logical relations of Aristotle’s theory, including A/E ‘subalternation’, the thesis that universal judgments have existential import. For that implication was ruled out by his treatment in §5 of A-type universal judgments in terms of a material conditional which is true when its antecedent is false. 4. Russell was of course familiar with Frege’s work but he never explicitly acknowledged or discussed Frege’s contribution to the logic of quantifiers. Instead his approach to the logic of quantifiers was initially stimulated by his encounter in 1900 with Peano’s work on the formalization of arithmetic, though this then

46

Thomas Baldwin became a topic which he developed on his own as a theory about ‘denoting phrases’ and which he first clearly expounded in his 1905 paper ‘On Denoting’. The debt to Peano is acknowledged in Principia Mathematica where Whitehead and Russell remark that ‘In the matter of notation, we have as far as possible followed Peano, supplementing his notation, when necessary, by that of Frege or by that of Schröder’ (Whitehead and Russell 1910: viii). Russell never alludes to Peirce’s contributions to logic, including his theory of quantifiers, probably because of his general hostility to the Boolean algebraic approach to logic within which Peirce worked. One might speculate that there is an indirect connection between Peirce and Russell via Schröder, since in vol. III of Algebra der Logik Schröder introduces and uses, with due acknowledgment, Peirce’s notation of quantifiers and bound variables (Schröder 1895: 36–7). But I see no evidence that Whitehead and Russell drew on this aspect of Schröder’s work. 5. The intuitive argument for this claim was that ‘Most As are Bs’ is not equivalent to any proposition of the form ‘(Most x) (Fx)’ where ‘Most x’ occurs as an unrestricted unary quantifier. Some years ago Barwise and Cooper proved a stronger claim: there is no fi rst-order representation of ‘Most As are Bs’. See Barwise and Cooper (1981: 214ff), esp. theorems C12, C13.

REFERENCES Barwise, J. and Cooper, R. (1981) ‘Generalised quantifiers and natural language’, Linguistics and Philosophy 4: 159–219. Davidson, D. (1980) ‘Criticism, comment and defense’ in his Essay on Actions and Events, Clarendon Press: Oxford, pp. 122–48. Frege, G. (1879) Begriffsschrift, translation in J. van Heijenoort (ed.) Frege and Gödel, Cambridge, MA: Harvard University Press, 1970, pp. 1–82. . (1893) Grundgesetze der Arithmetik, vol. 1, partial translation by M. Furth as Basic Laws of Arithmetic, Berkeley, CA: University of California, 1967. . (1903) Grundgesetze der Arithmetik, vol. 2, page references are to the partial translation in P. Geach and M. Black (eds) Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, 1952. Keenan, E.J. and Stavi, J. (1986) ‘A semantic characterisation of natural language determiners’, Linguistics and Philosophy 9: 253–326. Keenan, E.J. and Westerståhl, D. (1997) ‘Generalised quantifiers in linguistics and logic’ in J. van Bentham and A. ter Meulen (eds) Handbook of Logic and Language, Cambridge, MA: MIT Press, pp. 837–94. Lindström, P. (1966) ‘First-order predicate logic with generalized quantifiers’, Theoria 32: 186–95. McCawley, J. (1981) Everything Linguists have Always Wanted to Know about Logic, Oxford: Blackwell. McKay, T.J. (1989) Modern Formal Logic, New York: Macmillan. Mostowski, A. (1957) ‘On a generalization of quantifiers’, Fundamenta Mathematicae 44: 12–36. Peirce, C.S. (1884) ‘On the algebra of logic (second paper)’, published in C. Kloesel and N. Hourser (eds) Writings of Charles S. Peirce, Bloomington, IN: Indiana University Press, 1993, pp. 111–5. . (1885) ‘On the algebra of logic: a contribution to the philosophy of notation’ in C. Kloesel and N. Hourser (eds) Writings of Charles S. Peirce, Bloomington, IN: Indiana University Press, 1993, pp. 162–203. Russell, B. (1905) ‘On denoting’, reprinted in R. Marsh (ed.) Logic and Knowledge, London: Allen and Unwin, 1956, pp. 39–56.

Restricted Quantifi ers and Logical Theory 47 . (1908) ‘Mathematical logic as based on the theory of types’, reprinted in R. Marsh (ed.) Logic and Knowledge, London: Allen and Unwin, 1956, pp. 57–102. Smiley, T. (1962) ‘Syllogism and quantification’, Journal of Symbolic Logic 27: 58–72. . (1981) ‘The theory of descriptions’, Proceedings of the British Academy 67: 321–37. Schröder, E. (1895) Vorlesungen über die Algebra der Logik, vol. 3, Lepizig: Teubner. Stevenson, L. (1975) ‘A formal theory of sortal quantification’, Notre Dame Journal of Formal Logic 16: 185–207. Whitehead, A.N. and Russell, B. (1910) Principia Mathematica, vol. 1, Cambridge: Cambridge University Press. Wiggins, D. (1980) ‘“Most” and “all”: some comments on a familiar programme, and on the logical form of quantified sentences’ in M. Platts (ed.) Reference, Truth and Reality, London: Routledge, pp. 318–46.

3

Logical Form James Cargile

1

DEFINING ‘LOGICAL FORM’

Alonzo Church says that the validity of an argument is a matter of the logical ‘form alone, independently of the matter’ (Church 1956: 2–3). He offers an example (Booth)

I have seen a portrait of John Wilkes Booth; John Wilkes Booth assassinated Abraham Lincoln; thus I have seen a portrait of an assassin of Abraham Lincoln.

which he says ‘will be recognized as valid, and presumably from the logical form alone’. He follows it with another argument (Somebody) I have seen a portrait of somebody; somebody invented the wheeled vehicle; thus I have seen a portrait of an inventor of the wheeled vehicle. which is not valid, saying that ‘The superficial linguistic analogy of the two arguments as stated is deceptive’. He says that this is easy to see in this case, but other cases can be more difficult to recognize, which makes it ‘desirable or practically necessary for purposes of logic to employ a specially devised language, a formalized language . . . which shall reverse the tendency of the natural languages and shall follow or reproduce the logical form—at the expense, where necessary, of brevity and facility of communication’. It is noteworthy that Church does not explain what the logical form of either (Booth) or (Somebody) might be. He later presents some formalized languages, in which the notion of a form is clear. One example is the result of replacing a constituent name ni in a complex name n with a variable x. The resulting expression n(x) with the variable, is then a form and the original n can be seen as the result of substituting the name ni for the variable x in that form n(x), which shows that n is of the form n(x) (Church 1956: 10). We will call this the substitutional notion of logical form. However, Church says in a footnote (Church 1956: 10, fn. 26) to this account ‘this is

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a different use of the word form from that which appeared . . . in the discussion of form and matter’. That ‘discussion of form and matter’ was the one cited above, where (Booth) was an example of an argument the validity of which will be ‘recognized from the logical form alone’. But in fn. 26 Church goes on to say, of his newly introduced notion of form, which we have called ‘substitutional’ and which he says is different from the notion fi rst invoked, ‘We shall distinguish the latter use [the notion first invoked], when necessary, by speaking more explicitly of logical form’. It is hard to see how one use of ‘logical form’ could be distinguished from another by ‘speaking more explicitly of logical form’. Our position is that Church’s ‘latter use’ is highly important for logic but has not been made clear. By contrast, the substitutional notion of logical form, in application to such a standard system of formal logic as the lower predicate calculus (LPC) is perfectly clear, definable with mathematical precision. This is not to claim any personal prowess in the area of mathematical precision,1 and no such precise definition will be given here. Still, a (reluctant) effort at an example may be helpful. Church’s substitutional logical form is based on the difference between exhibiting a well formed formula of a formal system (wff) as a substitution instance of a scheme, and exhibiting it as a substitution instance of another wff. For example, the wff (∀x)(∃y)Rxy→(Rab∨Rba) relative to a careful formulation, can be exhibited as of the schematic forms: φ; φ→ψ; φ→(ψ∨χ); (∀α)φ→(ψ∨χ) and (∀α)(∃β)φ→(ψ∨χ), where α and β are individual variables and φ, ψ, and χ are wffs, or atomic wffs or atomic wffs with a two place predicate etc. Or it can be exhibited as a substitution instance of e.g. (∀x) (∃y)Rxy→(Rzb∨Rbz). These are things allowing of mathematical precision. One way to ground this precision is with orthographic characterizations of the basic symbols. Equations can be given for the symbols in the way that a circle can be defi ned. The tokens will then be like geometric figures—invisible point sets in space discussed with the help of our crude outlines on blackboards. The substitutions then come down to geometric facts. Alternatively, the tokens could be sound sequences and temporal considerations would be important. Identifying symbols with their Gödel numbers would give another basis. The situation with respect to natural languages, such as English, the language of (Booth) and (Somebody) is markedly less clear. Not that there is any obstacle to giving a similarly precise representation of an English sentence as a substitution instance of a form. The problem is with assigning logical significance to such substitutional facts. Philosophers have debated whether the grammar of natural language is ‘misleading as to the logical form’. 2 If validity is entirely based on logical form, then to the extent to which the logical form of the argument is unclear, its validity is unclear. But the validity status of (Booth) and (Somebody) is perfectly clear. What is unclear is what their ‘logical form’ might be. In LPC identifying the (or a) ‘logical form’ of a wff, either as some scheme or some other wff, can be an elementary exercise for beginning students. The assumption that this is

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just an improvement on something less clearly marked in natural language, should be regarded with suspicion. The substitutional notion for a formal system suggests extrapolation to the idea that the logical form of an English sentence is a substitutional form from which the sentence is derived, and which brings out the logical features of the sentence relative to its contribution to the validity or invalidity of arguments. It must then appear that the substitution forms of LPC serve this function nicely, while the forms for English are harder to fi nd and less reliable. This is a further elaboration of the target of our suspicion. Church says ‘English . . . evolved over a long period of history to serve practical purposes of facility of communication, and these are not always compatible with soundness and precision of logical analysis’ (Church 1956: 2). It is not clear that serving practical purposes is ever incompatible with sound and precise logical analysis, but for the examples given, we can see as soundly and precisely the validity of (Booth) and the invalidity of (Somebody) as we can any argument in LPC. What the practicality of English is incompatible with is mathematically precise pairing of syntactic structure with logical functions. English has general grammatical rules, of course, and this is no doubt fundamental to learning the language. But the rules were ‘made to be broken’ by poets, or philosophers, or simple people needing charitable hearing. This is not a merely superficial flexibility. It is the basis of dialogue, and dialogue makes possible access to more than a denumerable range of properties. This is why humans, who can only keep in mind a small fi nite number of properties, are nonetheless able to grasp properties beyond the infi nite but orderly store of fi rst-order arithmetic. This flexibility in rules, for example, governing what constitutes a sentence and the interpretation of sentences, is incompatible with the sort of logically secure generalizations about validity possible with respect to LPC wffs. Whether ‘validity’ in application to LPC even has the same meaning as for English is questionable. Many formal logicians are deeply resistant to this idea, holding that a natural language can be regulated as closely as a formal system. For example, Saul Kripke says I propose the following test for any alleged counterexample to a linguistic proposal: If someone alleges that a certain linguistic phenomenon in English is a counterexample to a given analysis, consider a hypothetical language which (as much as possible) is like English except that the analysis is stipulated to be correct. Imagine such a hypothetical language introduced into a community and spoken by it. If the phenomenon in question would still arise in a community that spoke such a hypothetical language (which may not be English), then the fact that it arises in English cannot disprove the hypothesis that the analysis is correct for English. (Kripke 1979: 16)

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In (a relevant formulation of) LPC, such an expression as (∀F)(∀x) (Fx∨~Fx), is, by defi nition, an expression of that formal system which is not a wff of that system. Similarly, it is by defi nition a wff of (a relevant formulation of) ‘second-order predicate calculus’. Those necessary truths about the formal systems have some connection with empirical facts about stipulations made by historical formulators. It is an essential feature of natural language that it does not bear a similar relation to stipulations by formulators. One formal system may have significant advantages or disadvantages for conveying ideas, raising cost-benefit problems for choosing systems. Natural language users may blithely pursue any expressive advantage that comes in view, limited primarily by the need for an audience. If the fortuitous expressive techniques turn out to be exploitable for production of logical bewilderment, this can be frustrating for someone trying to control what is said (‘for purpose of logical precision’). Plato despaired of achieving such control in (generally distributed) writing. Insofar as it can be achieved with a formal system, that system is not a genuine language. It can be diffi cult to repair a formal system which has turned out unsatisfactorily. With natural language, there is nothing to repair. Someone asks ‘Is Bill honest?’ We reply ‘He is and he isn’t’. An ‘analysis’ is offered: we have asserted a contradiction of the form ‘P and not-P’. Not in English, we haven’t! Can it be stipulated for some related language? Why would that language have to follow that rule? Kripke holds that there could be a community which speaks the ‘strong Russell language’ . . . : defi nite descriptions are actually banned from the language and Russellian paraphrases are used in their place. Instead of saying ‘Her husband is kind to her’, a speaker of this language must say ‘Exactly one man is married to her, and he is kind to her’, or even (better), ‘There is a unique man who is married to her, and every man who is married to her is kind to her’, or the like. (Kripke 1979: 16) This might be called a ‘hypothetical relative’ of English (surely no one would claim it is ‘utterly foreign’?). What is the force of ‘a speaker of this language must say’? Someone says ‘Her husband is kind to her’. So what? It can be ruled ‘That’s not Variant8.7a-ese!’ But what if everyone understands just fi ne (as one might expect, given their understanding ‘man who is married to her’)? The identity of natural languages is quite vague. Their boundaries are not determined by stipulations (in general use—it might be settled in a law suit by arbitrary ruling), but by what works among users of the language with sufficient freedom from special advance arrangements (such as codes or universal gestures and signs, etc). We can of course ban the use of any identifiable form of speech, leaving the violators hanging in the public square for motivation. Such exercises of arbitrary power do not

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create language rules (though they may, admittedly, influence the evolution of rules). Leading to his discussion of (Booth) Church cited two arguments (‘I and II’) valid by virtue of logical form, saying Verbal similarity in the statements of I and II, arranged at some slight cost of naturalness in phraseology, serves to highlight the sameness of form. But, at least in the natural languages, such linguistic parallelism is not in general a safe guide to sameness of logical form. (Church 1956: 2) It is plausible to say that ‘linguistic parallelism’ in English is not a safe guide to validity. Whether it is a guide to ‘logical form’ requires a better explanation of what that would be. Applying the same rephrasal technique Church used to show sameness of form for his preceding examples, we can see (Booth) and (Somebody) have the common form F1: ‘I have seen a portrait of ___1; ____1 stands in the _____2 relation to _____3; thus I have seen a portrait of a thing that stands in the ____2 relation to _____3’. Thus it cannot be having the form F1 that shows the validity of (Booth). It may seem easy enough to further qualify F1. ‘John Wilkes Booth’ and ‘Abraham Lincoln’ are ‘proper names’ and ‘somebody’ and ‘the wheeled vehicle’ are not. (Booth) is thus an instance of a more restricted form F 2 which requires a ‘proper name’ in blanks 1 and 3. ‘Proper name’ or ‘name’ or ‘individual constant’ can mark a precisely defi nable idea in LPC (or a different one in ‘Free Logic’, etc.) In application to English, these are vague notions, enmired in Epistemology, a branch of Metaphysics. The goal of fi nding genuine logically proper names in English has been elusive, leading to such candidates as ‘this’ and ‘that’. Admittedly, it makes sense to class these pronouns as ‘singular terms’, but that sense is not precise for English either. ‘Passenger pigeon’, classed as a general term, went from applying to millions, to applying to one and then none, without becoming a ‘singular term’ during its one instance period. One response to the perceived scarcity of logically proper names in ordinary English is the suggestion that grammatical proper names are really just abbreviations for definite descriptions. Debate about this suggestion should be limited by the consideration that, insofar as there are no English ‘logically proper names’, there are no ‘definite descriptions’ either. What logical function does a ‘name’ serve? It may in some case ‘call to mind’ a named thing. A defi nite description may in some case express a property uniquely possessed (relative to a ‘context’) by some thing. In English, these expressions may do other things as well, but in any case, what is the logical use of doing them? The ruling that ‘the whale’ is a ‘defi nite description’ in ‘The whale struck our ship’ but not in ‘The whale is a mammal’ seems to make sense. But it is easy to describe a case in which it is equally a ‘defi nite description’ in ‘The whale is a mammal’. It is no doubt significant that it

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is not as easy to describe a case in which it is not a ‘defi nite description’ in ‘The whale struck our ship’. But it is still possible.3 Assigning the arbitrary title ‘defi nite description’ can be done equally for a string of simple symbols of LPC or letters of English. The LPC arbitration has the distinct advantage that the expression cannot be given different functions, while this cannot be regulated for English. In English, the very same token can serve different logical functions (by cutting and pasting). It may well be objected that the fact that utterances violating rules of English may nonetheless be well understood, thanks to kindness and insight into the speaker’s intentions, etc, should not count against the fi rmness of these rules. There may indeed be ‘fi rm’ rules of English, but they must yield to ‘practical purposes of facility of communication’. That does not prevent sound and precise logical analysis, but it does make those rules unsuitable for the formulation of logical generalizations about validity. This difficulty in formulating fi rm logical general rules about substitution forms of English should not be mistaken for a generic difficulty in the way of determining the validity of particular arguments in English.

2

SI VALIDITY

There is a notion of validity which makes no appeal to ‘logical form’, based on the idea of the associated (material) conditional of an argument. That is the conditional whose antecedent is the conjunction of the premises and whose consequent is the conclusion. An argument with fi nitely many premises is valid if and only if the associated conditional is necessarily true. Necessarily true material implication is often called ‘strict implication’. We may thus call this the strict implication (SI) definition of ‘valid’ (see e.g. Mates 1972: 5). Whenever a conjunction strictly implies a proposition, there is an argument in the logical sense, with the conjuncts as the premises, without regard to whether anyone has or even could advance those premises as providing support. SI validity is not an evaluative notion—attributing it to an argument entails nothing about whether the inference from the premises to the conclusion would be good or bad. Being defi ned in terms of necessity may make it questionable to a philosophical naturalist, but in not being evaluative it is naturalistic. Naturalists should agree that material implication is a relation that holds between things that are true or false. For those who hold that some propositions are neither true nor false (or that some are both) a simple procedure is to screen candidates and take only ones that are either true or false and not both. That will leave room for disagreement about the logical properties of the things disqualified. Sticks and stones should break no logical bones, but if the rejects get involved in valid arguments that makes SI validity an incomplete account of validity. With strict implication, things are worse, since even with commendable resolve about the necessary truth of ‘classical

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logic’, there is disagreement about necessary truth itself, and consequently about strict implication (see Hacking 1963). Whether what is necessary is necessarily necessary (iterated modality) or what is possible is necessarily possible is not a matter of consensus. There are different formal systems for strict implication, corresponding to differences about iterated modality, which mark potential sources of substantial philosophical disagreement. This relates to a problem about the nature of the things that are premises and conclusions in argument. A popular candidate is sentences: ‘By an argument we mean a system of declarative sentences (of a single language), one of which is designated as the conclusion and the others as premises.’ (Mates 1972: 5). In defending this proposal ‘to take sentences as the objects with which logic deals’, Mates considers ‘proposals to talk instead about statements, propositions, thoughts or judgments’. His response is that ‘as described by their advocates, however, these latter items appear on sober consideration to share a rather serious drawback, which, to put it in the most severe manner, is this: they do not exist’ (Mates 1972: 10). For any set of sentences4, it is possible (and necessarily possible) for all but one to be true and the remaining one false. To borrow an example, (Soc): ‘Socrates died in 399 B.C. Therefore, Socrates died in 399 B.C. or Socrates did not die in 399 B.C.’ It is impossible for (Soc) to mean what it does to the intended audience of this writing and have the premise sentence true and the conclusion sentence false. But it is perfectly possible that all meaning in (Soc) should stay the same except for ‘or’ meaning what ‘and’ now does for us, in which case the premise would be true and the conclusion false. Mates says, of the conclusion sentence, ‘it is well to remember that the question is: given the sense of this sentence, is there a possible world of which it would, in that sense, be false? The circumstance that it would be false if, e.g. the word “or” meant what “and” means, is irrelevant’ (Mates 1972: 13). That is an excellent observation (assuming that ‘is there a possible world?’ is a mere figure of speech for ‘is it possible?’), and mentioning the possibility of ‘or’ coming to mean what ‘and’ now does could properly be dismissed as an annoying distraction—in many discussions. But that is because we usually assume without discussion that we are ‘given the sense of’ the sentences (and we do not simply identify the premises with the sentences). What is a sentence? Mates says, of written sentences: A sentence . . . is an object having a shape accessible to sensory perception, or at worst, it is a set of such objects . . . Only reasonably good eyesight, as contrasted with metaphysical acuity, is required to decide whether a sentence is simple or complex, affi rmative or negative, or whether one sentence contains another as a part . . . whether the conclusion of a sound argument is always somehow contained in the premises is not hard to settle if we take it as referring to sentences . . . matters are quite otherwise when we try to answer the same kinds

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of questions about propositions . . . Propositions, we are told, are the senses or meanings of sentences . . . (Mates 1972: 10–11) We agree that written sentences are material objects. For our purposes it is helpful to consider tokens of letters, words, sentences, paragraphs, etc as written in bronze on plaques or inscribed in stone or cut out as stencils. Some philosophers deny there are such things as plaques or stones (or baseballs or chairs, etc),5 but we strongly disagree and heartily accept the existence of tokens. (But not ‘types’, which we regard as a misleading figure of speech.) Orthographic versions of tokens can be defined mathematically, like geometric figures. Like those figures, these will be invisible to us, but we can use visible figures to help us visualize them to some extent.6 Mere ocular inspection, indeed, suffices to determine whether one such token is part of another.7 (Simple or complex, affirmative or negative, is another matter.) An orthographic token is a spatial object and any meaning it has is never a necessary feature. Whether a token of, for example, ‘i’ is a letter is not a mere matter of inspection. Individuating the letters as names of speech sounds (or factors in sounds), we get tokens other than the orthographic, but the point is the same. (Soc) could represent a one-word assent, like our ‘yes’. Some southerners pronounce ‘yes’ in three syllables. 35 is, admittedly, a bit much (especially since we do not concede this could not be ‘English’, as if that could form a logical boundary), but it is perfectly possible logically. It is inconsistent to demand that the sense of (Soc) be kept fi xed for its logical assessment and then respond to those who call the senses of sentences, propositions, that there are no such things as propositions. (Of course some philosophers will say that senses do not have to exist to serve all their logically important functions.) We can often see clearly that sentence S as used by agent X in situation Y expresses the proposition that P. (Thus placed, the sentence may be called a ‘statement’, but if this is not a term for the proposition thereby expressed, this is at best a convenient shorthand.) Thus we know perfectly well that the concluding sentence of (Soc) as used here by me expresses the proposition that etc. But this is still a contingent fact. Under the requirement of necessary connection, mere sentence structure cannot defi ne logical connections. This is a crucial issue between nominalism and platonism— whether a logical form could be a contingently existing linguistic form, i.e. an orthographic token. One motive for rejecting iterated necessity is the analysis of necessity in terms of analyticity. ‘Analytic’ makes best sense as a predicate of sentences. Applied to propositions it is quite obscure. ‘All bachelors are unmarried’ is analytic if anything is, and that is clearly relative. In ‘At this year’s graduation, all doctors are married, some masters are married and some unmarried, and all bachelors are unmarried’ it is not analytic. Therefore, ‘analytic’ does not iterate. ‘“All bachelors are unmarried” as used by X in Y, is analytic’ is clearly not analytic.8

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One strategy for preserving SI validity as applicable to sentences is to explain the necessity in terms of sentence analyticity. Thus it could be said that the associated conditional for (Soc), as used by me in this essay, is analytic. On this account, the possibility of the premise sentence being true and conclusion false is irrelevant. This strategy is a way of fi nessing the question as to what it is to be given the sense which guarantees the validity. Not only does it avoid propositions and other senses, it would provide a reductive account of necessity, fitting the SI account of validity into naturalist Empiricism. When there is a series of arguments beginning with premise P and leading to conclusion Q, with each step warranted by a good general rule of inference, then one who knows this can ‘validly’ deduce Q from P and the possibility of a person doing this means that it is possible to deduce Q from P, that is, Q is deducible correctly from P, which is to say that P entails Q. There is a famous question whether the relations of strict implication and entailment coincide. It seems to be assumed (barring those who reject the notion of necessary truth altogether) that if P entails Q then P strictly implies Q, but the converse is controversial.9 The sentence analytic version of SI validity makes the identification of entailment with strict implication plausible. The necessity of the material implication P→Q would consist in its analyticity. There would be an ‘analysis’ exhibiting this and that should count as a proof of Q from the hypothesis P. This strategy is working with a very weak version of strict implication. If S4 (or S5) is right, the strategy is wrong. It makes the validity of (Soc) (construed as consisting of sentences) relative to a certain audience and a contingent matter of fact. Quine famously criticized philosophers’ use of ‘analytic’ as being no genuine improvement on their use of ‘necessary’. One can agree with that without giving up on the use of ‘necessary’. Quine disparaged appeal to defi nability in natural language as an account of analyticity, while fi nding defi nability in a formal system acceptable for logic. That is based on the control a formal system gives over the uses of its formulas as parts of the system. This control allows a variety of systems, with the choice of one system rather than another being a matter of pragmatic interests. This leads to a logical relativism which even Quine found unattractive. Strict implication cannot be made out as a relation between sentences without weakening the relation so as to make it inadequate to express the full power and authority of logic. However we may settle the ontology of premises and conclusions, SI validity is, notoriously, possessed by any argument with such a premise as 2+2=5567!, or such a conclusion as 2+2=4. The abundance of utterly trivial and worthless arguments that are thus SI valid (many are also sound) undercuts the value of ‘valid’ as an evaluative term for arguments. It is a feature which is neither necessary10 nor sufficient for being a good and conclusive argument. In ordinary use, ‘valid’ is commonly applied to arguments (or complaints or claims for compensation, etc), evaluatively. This .

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is no reason to reject the SI defi nition. But it might motivate looking for something evaluative.

3

VALIDITY BY VIRTUE OF LOGICAL FORM

A significantly different conception of validity is the one offered by Church. This validity is not a matter of any kind of necessity, but of the form of premises and conclusion. Sentences do have forms, such as ‘subject-predicate’ etc. So it is natural to look for the relevant logical forms among sentence forms. This leads to the earlier complaint about the logical unreliability of sentence forms of natural language and to resorting to formal systems with mathematically definable syntax. However, this defi nition, whether geometric, or numerical, or whatever, describes objects which are not necessarily connected with logical properties and is not a proper basis for an account of validity. The ‘forms’ P→Q, ~~P, etc in LPC have an advantage over the English ‘forms’ ‘If P then Q’ or ‘It is not the case that it is not the case that P’. They can be tied by defi nition to certain specific functions, such as expressing the relation of material implication or double material negation. The possible functions of the English forms are not so subject to restriction. They may be fi xed in a dialogue, and the fi xing of interpretation for formal systems by logicians does resemble a kind of dialogue between the practitioners. This leads many formal logicians to regard the difference between formal and natural languages as philosophically insignificant. But the ‘language’ of a formal system plays a very different role in dialogue from that of natural language. It is misleading to speak, for example, of ‘the logical form of a (material) conditional’. Forms characterized by geometric-orthographic items or by Gödel numbers are different things. Then there is Polish notation with the orthographic form →PQ, or the Arabic version writing backwards to show P comes fi rst, etc, etc. The important thing is the logical property of being a material conditional, a property which can be expressed in different forms and different kinds of form. It is not necessarily true that a token of P→Q, a material object, is a conditional. It is true by defi nition of LPC that two wffs connected by a conditional symbol constitute a conditional wff. That a specified shape and size, or sound or color, etc is a conditional symbol is a contingent fact. The expression P→Q is no more necessarily connected with material implication than is the English ‘If P then Q’. This could encourage the view that there is no essential difference between English and some formal system. But that would be a mistake. We can say of some (written or spoken) utterance ‘That’s not English’ just as we can say ‘That’s not LPC’, but we are open to correction in a very different way. Dialogue in English may lead us to understand the utterance and bring it into English. LPC does not work in that way. But in either case, logical significance requires identifying

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a logical function which is independent of linguistic expression. It is only expressing the language independent logical functions of material negation and material implication that endows ~~P→P with a ‘logical form’. If the same wff in intuitionist systems has a different logical form, then such form is not a matter of substitution in an orthographic form. It would be typical of some nominalists to say that ~~P→P is a different wff in intuitionist systems. They appeal to the idea of homophony, or better, homography. There are letters and written words that are homophones in the straightforward sense of being orthographically different things that stand for the same sounds. But there are no homographs, that is, orthographically identical words differing in history and meaning. The very same token of ‘bank’ can be used with different meanings at the same time. Bill and Bob may post the same note ‘I am at the bank’, knowing that Bill’s wife and Bob’s wife will each get the right message. It is true that Mrs. Bill’s understanding of the word is derived from a different history of usage from Mrs. Bob’s. This does not license such a mystery as that the identical token is distinct words. Alfred Tarski said of ‘natural language’ that this language is not something fi nished, closed or bounded by clear limits. It is not laid down what words can be added to this language and thus in a certain sense already belong to it potentially. We are not able to specify structurally those expressions of the language which we call sentences . . . (Tarski 1956: 164) English cannot be defi ned by rules of sentence construction because exceptions (inclusive or exclusive) can always grow into the evolving language. That is not an obstacle to specifying the structure of a particular sentence. But what is ‘specifying structure’? (Cook): ‘Only Bob (among these people) likes his cooking’ is ambiguous. That is, ‘it could mean’ (Cook1) ‘Bob is the only person (among these) who likes Bob’s cooking’ or (Cook 2) ‘Bob is the only person (among these) who likes his own cooking’. For an assertion of (Cook), it could be said that we need a better specification of the structure. Then, for most of us, (Cook1) or (Cook 2) would qualify as an adequate specification. For what purposes? General purposes of understanding, notably including logical purposes. If someone infers, from (Cook), ‘Ben does not like his own cooking’ (Ben being among the relevant people), he is confused if his basis was (Cook1), correct if it was (Cook 2). We might say that the logical form of (Cook) is unclear, while the logical forms of (Cook1) and (Cook 2) are clear. ‘The logical form of sentence S is clear’ no more licenses the conclusion ‘There is a unique x such that x is the logical form of S’ than does ‘It is clear what follows from sentence S in this use’. It has been said that the logical form of an English sentence is just its translation into LPC. Such a translation would ‘specify the structure’. But in translating ‘Neither George nor

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Bob passed’ there is no basis for calling ~(G∨B) better than ~G&~B or G↓B. What is the logical form of ~(G∨B)? It may seem that, at least, is clear. So what is it? The (material) negation of a (material) disjunction? A disjunction of what? Perhaps two ‘atomic propositions’? Is ‘George passed’ an ‘atomic proposition’ which happens to entail that George has been a student in a course (and not that George died, etc)? There is no problem in recognizing atomic wffs of LPC or simple subject-predicate sentences of English. Recognizing ‘atomic sentences’ of English could be easy too, once we are told what the classification means. ‘Expressing an atomic proposition’ would be a false start. ‘Atomic’ does not mark a property of propositions. There is such a logical property of a proposition as being the material negation of a material disjunction of two propositions, and an instance of this property is the proposition which is the material negation of the material disjunction of the proposition that George passed and the proposition that Bob passed. That proposition is the same proposition as the material conjunction of the negation of the proposition that George passed and etc. It is the joint denial (material of course) of the proposition that etc. The proposition itself is not a negation or a conjunction, etc. A sentence is, say, a conjunction, partly due to its orthographic form, but also to the use made of it. The role of the orthographic form is complex. But a proposition has no orthographic form. It has logical properties, such as entailing the proposition that George did not pass, or being the material conjunction of certain propositions, which is very different from being a conjunction marked orthographically. Logical properties of a proposition may be made clear, or clearer, to us by sentences which express the propositions. They may express them in such a way as to make the logical properties clear. Formulating such sentences may be called giving the logical form. That way of speaking can cause confusion as to what logical form is. Different formulations can bring out different properties. A particular property of the proposition may be emphasized in one formulation and obscured in another. This is in opposition to the idea that the sense of a linguistic expression is compositionally derived from the senses of its linguistic parts. The ideal of a sentence which shows its true logical form is associated with the thought that then at least compositionality would be true. But sense can be formulated differently depending on the direction of the dialogue. If ‘Shem kicked Shaun’ abbreviates (misleadingly perhaps) ‘There is an x such that Kicking(x) and By(x, Shem) and To(x, Shaun)’, that fits nicely with ‘and he did it with his left foot at noon’. But ‘Shem played the active role in the kicking relation, to Shaun’ fits better with ‘and he did it repeatedly’. ‘Socrates is wise’, ‘The property of being wise is instantiated by Socrates’, ‘Socrates, the property of being wise and the relation of instance to property are in the three-place instantiation relation of instance, property, and instantiation relata, respectively’, etc are linguistic items with different linguistic parts identifying different logical features. They may all nonetheless express the same proposition.11

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This is obscured by the idea that a proposition has parts. A sentence has parts and the parts hold together thanks to linguistic features. Frege was concerned with how the parts of a ‘thought’ (proposition, sense of a sentence) hold together. But the proposition expressed by ‘The property of being Socrates and the property of being wise are coinstantiated’ is not constituted by the property of being Socrates and the property of being wise and the relation of coinstantiation being held together. Rather, the proposition is to the effect that those two properties and the relation hold together in a certain way—the properties being in that relation. Frege wisely observes that ‘a thought can be split up in many ways, so that now one thing, now another, appears as subject or predicate’ (Frege 1892: 49). This shows that the thought itself does not have parts in the way that its linguistic representations do. Here are some passages in English which might be said to be misleading as to their logical form. (Beer)

If you want one, there is a beer in the refrigerator.12

This could be used to express a material conditional or even (in a peculiar set-up) a ‘causal conditional’. We might say it is usually not a genuine conditional, except that we do not have any agreement on what it is to be a ‘genuine conditional’. It is not that the true logical function of (Beer) is not brought out by its ‘grammatical form’. Its function in common use is no logical problem. It is usually not connected with a logically interesting property. (Hit)

Bill cursed Bob, Bob punched Bill, therefore, Bob punched Bill and Bill cursed Bob.

(Hit) could be used to express a true claim (material conjunction), but it could also be used to express a false one, and for that reason we can say that (Hit) is not of the ‘form’ (Conjunction Introduction) because the ‘and’ is not (guaranteed) material conjunction. (Any)

It is not the case that it is not the case that any applicant from Utah was accepted; therefore, any applicant from Utah was accepted.

‘Any applicant from Utah was accepted’, in a standard use, would be conveying a false claim if two applicants from Utah were accepted and three were rejected. But in that situation, ‘It is not the case that any applicant from Utah was accepted’ would be taken most naturally as also false. The claim that it is not the case that it is not the case that any applicant from Utah was accepted would be true. So the argument would have a true premise and false conclusion.

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Intuitionists would not cite (Any) in support of their case against double negation elimination. Some philosophers might say that (Negation Elimination) is not a valid form in English. That would be true for a crude (but precise!) substitutional account, but it is better to note that the English form ‘It is not the case that P’ does not reliably express the negation of P. We should not give up on the elimination of double negation just because English allows cases like (Any). (Isn’t)

It is and it isn’t.

(Isn’t) is common and its uses are not usually false. Even Dialethiests do not appeal to such uses to support their claim that some contradictions are true. They want a substantial position about genuine contradictions. But conceding that a use of (Isn’t) may not express a contradiction creates a prima facie conflict with the position that being a contradiction is a matter of form. It is no help to observe that (Isn’t) should not be translated into LPC as P&~P. The tendency of English to allow such cases is probably an example of the tendency Church said needs to be reversed. But contra Church, serving practical purposes is not incompatible with sound and precise logical analysis. We can see as soundly and precisely the invalidity of (Any) as we can any invalid argument in LPC. What the practicality of English is incompatible with is the view that validity is entirely due to a logical form inherent in the sentences of the argument. In English, any standard formal device is liable to being overridden for ‘practical purposes of communication’. But the idea that LPC is more successful at making logical form clear to the eye should be cast in doubt by the example of the intuitionist versions of PC and LPC. They have the same wffs. The difference as to (Negation Elimination) is not about a ‘logical form’ detectable by ‘reasonably good eyesight, as contrasted with metaphysical acuity’ (Mates 1972: 11). It is a matter of seeing that different ideas are expressed. That is equally possible for arguments in English.

4

PLATONIC LOGICAL FORM

Whether an argument is valid or not is a property of the argument and the property is possessed by virtue of possessing other properties. (Conjunction Introduction) is associated with the property of being an argument based on asserting two propositions, P and Q, and on that basis inferring the proposition which is their (material) conjunction. (Negation Elimination) is associated with being an argument from the (classical) negation of the negation of a proposition to the proposition. By far the easiest way for us to grasp the idea of arguing in these ways is to see it as being done in some language or other, but there is no relativity to language and it is not

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a linguistic feature of the argument. Familiarity with a language may lead us to associate certain linguistic forms with presenting an argument with a certain property, and recognition of those linguistic forms may be (for properly programmed brains) a matter of simple ocular inspection, but this is no guarantee of a necessary connection between the linguistic form and the property. All properties13 are platonic forms and so we can call logically important properties platonic logical forms. (Booth) is an instance of the following platonic form: an argument with a premise which (somehow) identifies a thing x as bearing a relation R1 to a thing y, and a premise reporting that y bears a relation R 2 to a thing z, and a conclusion to the effect that x bears R1 to a thing that bears R 2 to z. Any argument of that form is valid. That form, though here described in English, has no essential connection to English and could be equally well described in many other languages. The role of the ‘name’, or a ‘definite description’ in the argument, does not require a general account. For x to bear R1 to a thing that bears R 2 to z is for x to bear the composite R1oR 2 relation to z. That would allow a more abstract description of a valid form. We could name the form, say ‘Simple Relation Composition’. Church’s stress on seeing an argument to be ‘valid from its form alone’ is salutary if this is seeing the logical properties invoked by the argument. The linguistic formulation should then be recognized as one of countless possible ways of exhibiting those properties. Even if, for a certain form, it is impossible for premises to be true with conclusion false, this would not be needed to defi ne the form or its validity, which is a matter of its being a good form. For example, (Conjunction Introduction) is a good form of inference in the way that a certain kind of action is good. General laws of validity could then be seen as normative for inference in the style of general moral laws. There would be a similar possibility of disagreement about what those laws are, which possibility might be obscured by the relatively lower frequency and intensity of disagreement. This ideal cannot be realized with linguistic forms.14 Of course Church did not deny that. Perhaps we are merely offering an interpretation of what he said. No ‘argument’ construed strictly as a set of sentences without regard to the propositions they express, can be valid. It is by virtue of the logical relations a formulation happens to present, rather than the symbols as objects, that an instance is valid. Validity itself is not something dependent on any particular language. Any valid argument that can be expressed in English can be expressed in some language with different forms—not merely different orthographically, but different also in logical details. Church says To adopt a particular formalized language . . . involves adopting a particular theory or system of logical analysis. (This must be regarded as the essential feature of a formalized language, not the more conspicuous but theoretically less important feature that it is found convenient

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to replace the spelled words of most (written) natural languages by single letters and various special symbols.) (Church 1956: 3) A ‘formalized language’ has no need of adoption. An intuitionist mathematical logician knows far more about classical LPC than I do, in spite of all my devotion to classical logic. He may teach it for years as his main source of income without any inclination to adopt the associated theory of logical analysis. One essential feature of such a formalized language as LPC is that its connection with such logical relations as material implication or negation can be correctly made a criterion for qualifying as operating within that system regardless of whether such operation is ‘adoption’. The primary problem with logical form is identifying genuine languageindependent logical features. Dialogue in natural language is our best hope. Merely appealing to English expressions, as in: ‘A proposition has the form P&R if it could be expressed by two sentences linked with an “and”’.15 Or in general, appealing to an English connective (‘. . . and . . .’, ‘if . . . then . . .’, ‘. . . unless . . .’, etc) to identify a logical one is like the man lost in New York City calling for a cab to get him at the intersection of ‘Walk’ and ‘Don’t Walk’. Experiences with orthographic tokens can lead to associations describable as ‘meanings’ of the tokens. Tokens with that sort of meaning can then be used as tools in dialogue to express proper meanings, properties and propositions and arguments. The orthographic forms can then come to have complex connections to platonic forms. They should not be confused with them. When we see (Booth) to be valid ‘by virtue of form alone’, this ‘seeing’ is very far from simply scanning sentences. It requires understanding relations as distinct from mere ‘relation expressions’ of English. A formal language may be helpful for keeping track of such things, because it allows forestalling use of ‘relation expressions’ for tasks other than expressing relations, by providing standard symbols rather in the way mathematics commonly does. This does not make the relations linguistic items. Instantiation of a platonic logical form should not be confused with substitution in an orthographic linguistic form or with facts about what transformations are in compliance with purely formal transformation rules. It is reasonable to hold that anyone who understands material conjunction and knows the premises P, Q can justify concluding that P&Q. That requires no particular language. To be able to proceed in that way is what it is to understand the form (Conjunction Introduction).16 Such an inference could plausibly be held to be ‘valid’ in an evaluative meaning. We can also say that the inference is valid by virtue of the nature of the languageindependent relation of material conjunction, and thus, in that sense, valid by the nature of a form. This general warrant for conclusions appealing to (Conjunction Introduction) does not require a prior idea of necessity and such an inference has merit marked by calling it ‘valid’. Although this is not a general account of warranted inference, an easy generalization is to

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all arguments based on forms which are ‘as good’ as (Conjunction Introduction). Recognizing this common form, the form of the Good, is not a matter of simple good eyesight but does not deserve the charge of requiring ‘metaphysical acuity’.

5

STRICT IMPLICATION AND ENTAILMENT

Since I believe that conjunctive simplification, disjunctive addition, and disjunctive syllogism for the material connectives are good rules and that being deducible from a good rule is transitive, I believe that (EFQ)—P&~P, therefore Q—is a good rule. P&~P both strictly implies, and entails, Q. That is, the combination P&~P true, Q false, is impossible, and there is an argument (form) proceeding from P&~P and appealing to good rules, with Q as conclusion. But it is reasonable to doubt that the relations of strict implication and entailment are the same, because it is reasonable to doubt that every necessary falsehood can be formulated by us humans as the material conjunction P&~P. I believe that (Choice)—For every set of nonempty mutually disjoint sets, there exists a choice set—is necessarily false and thus every argument with it as a premise is SI valid. How could anyone prove that (Choice) entails P&~P? What would it mean to express that proposition in the form P&~P? We should be able to see that the proposition (Beethoven was a married bachelor) is the same proposition as (Beethoven was a man (perhaps of marital age and status not previously married) and Beethoven was both married and not married, at the same time). That is of the form R&P&~P, and we can get from there to any Q. But such insight (and especially getting credit for it) about expressing (Choice) is at best far less accessible. I could translate (Choice) into NF and then prove (with a little help from my friends) P&~P from that hypothesis. A defender of (Choice) could say that this, at most, refutes NF. A philosophical response would require an analysis of what it is to be a set, where being a set is a feature we grasp more or less well, and arguing that a better grasp shows the falsity of (Choice). Whether or not this can be done, it would be a proof about forms and in that sense formal. The principle that every true proposition is knowable is notoriously problematic. That every necessary proposition is knowable by analysis into analytic sentences is equally questionable. It is one thing to assume that, for any proposition P, it is possible to represent it in dialogue so that its logical relation to a proposition Q is made perspicuous. That may be a source of inspiration and celebration of our God-given powers of mind. It is another matter to suppose that any such representation of P could be an all purpose guide to its relation to all other propositions. It is a plausible rule that the only way to qualify as directly knowing (as opposed to knowing on the basis of trusting a reliable authority) that P

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strictly implies Q is to know that P entails Q, where the latter knowledge is displayed by producing a proof. However, unless we happen to believe that there is some ‘evidentiary relation’ between propositions, constituting facts about when P is good reason for Q even when it is logically possible for P to be true and Q false, there is the problem that what is an acceptable proof to some is unacceptable to others, even where it seems that the parties understand the purported proof. Having to grant that provability is relative to background features of the audience, is not the same as having to grant that strict implication is similarly relative. Consider the proposition (Cor) that there is a 1–1 correspondence between the points in the open interval ((0,0), (1,0)) on the X-axis (Xint) and ((0,0), (0,1)) on the Y-axis (Yint). There can be disagreement, of course, but it is reasonable to believe this. (The perpendiculars to a line bisecting the axes make the correspondence obvious, if we are not put off by the invisibility of the lines.) Then there is the proposition (Num) that the number of points in Xint is identical to the number of points in Yint. On one plausible reading of the ‘definite description’ ‘the number of points in Xint’, (Num) entails that there exists something which is a number, the number of points in Xint and in Yint, which could be called the Cardinality of the Continuum. It might be argued that the number of members of a set S (NumS) is defi nable as the set of sets in 1–1 correspondence to S. That would provide a basis for arguing that (Cor) is analytically equivalent to (Num). But even for, say, a 3-membered S, it seems that NumS is ‘too big’—that it would be 1–1 with the notorious Universal Set, the villain in Cantor’s Paradox. One response is to look for a different defi nition of ‘number’, perhaps in a consistent formal system in which (Cor) and (Num) can be proven, not only ‘equivalent’ in the way that all theorems of LPC are equivalent, but somehow analytically equivalent, so that (Cor) and (Num) are just two ways of expressing (‘carvings’ of) the same proposition.17 Here is a different response. (Cor) is necessarily true and (Num), read as entailing, rather than presupposing, existence, is necessarily false. There is a set of all sets 1–1 with Xint and it is as big as the Universal Set and Cantor’s Paradox is the result of a big misunderstanding (the power set axiom being false for ‘big’ sets). But that set does not qualify as a number (or as having a number) because a number has to be something that is possible to arrive at as the result of counting, numbering. That counting leads only to the series of alephs, never to 2 to the alephanything. It may reasonably be objected that these ‘responses’ are hopeless unless backed up by detailed ‘formal’ analysis which allows computer checkable ‘proofs’. LPC has been good for set theory in that it identifies some basic logical functions which allow reliable computations about complex constructions from them. But it is the identifying, rather than the certainty of its symbols having the connection, that is the value. That the symbols can be connected to logical functions should not lead to confusing the logical functions with symbols.18

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NOTES 1. For which Timothy Smiley is justly renowned. 2. In his (1950: 323), Strawson attributes to Russell the view that one important way that sentences seeming from their grammatical structure to be about an individual may be ‘significant . . . is that their grammatical form should be misleading as to their logical form’. This a well founded attribution, summarizing something Russell stressed repeatedly. 3. An acute reader pointed out to me that this is also true of the bare plural ‘whales’ in ‘whales are mammals’ and ‘whales are surfacing offshore’, etc. 4. The qualification ‘distinct sentences’ is not truly necessary, but is acceptable here to avoid needless complexity. 5. Perhaps these philosophers can accommodate stencils, as only requiring patterned breaks in the clouds of ‘atoms’. 6. Visible tokens are ‘tokens’ in a different sense from the invisible geometric ones. But this need not be discussed here. 7. Though for stencils, ‘parthood’ might be a somewhat different relation. We will avoid mereology. 8. ‘Analytic’ as a predicate of judgments as introspectible mental acts, does iterate (to a limited degree). On Kant’s picture, one (somehow) fi nds the predicate concept ‘unmarried’ upon inspecting the subject concept ‘bachelor’ and this fi nding constitutes the analyticity of the judgment. Assuming this makes sense, it should also be possible to make this act of judgment as a whole a subject of scrutiny and thus fi nd in it the feature of being a fi nding of its predicate in its subject. This is a logically significant difference between Kant’s conceptualist view of analyticity, applicable to judgments, and the Fregean notion applicable to sentences. 9. It has been well said that Timothy Smiley has made the most logically incisive contribution on this topic; see his (1959). 10. Consider such an argument as the following, offered in response to a challenge (issued by someone who has been away for a few months) to the claim that Lazarus will not be at a certain gathering: ‘Lazarus has been dead and in a tomb for several days—the stench of death is heavy there. Therefore, he will not be participating in the dinner gathering tonight.’ This is as good and conclusive as human argument can be. But it is not valid. 11. To the right audience. Some good person might express an admiration for Socrates with ‘Socrates is wise’ without grasping a proposition. 12. As distinct from ‘If you want one then there is a beer in the refrigerator’. 13. Properly understood. Being red is a property. The color red is, it would seem, a color, whatever that may be. The distinction of platonic forms from alternatives is not possible here. But a key feature is being closed under (truthfunctional) logical operations. The logical property of being a contradiction has instances. But being both a contradiction and not a contradiction is equally a property though having no instances. 14. We may, alas, need ‘genuinely linguistic’—forms from which systems of sentences arise by substitution based on orthographic qualifications. Placing logical restrictions on acceptable substitutions, as with: ( sentence expressing a proposition ), therefore, ( sentence expressing a proposition which follows logically from the earlier proposition ), (perhaps with a restriction insuring that ‘therefore’ means what it should) is a powerful resource for champions of linguistic forms. 15. Besides conferring the form P&R on the proposition expressed by the conclusion of (Hit), it confers the same form on any proposition P, by virtue

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of P’s being the same proposition as (P ∨ Q)&(P ∨ ~Q). And any proposition would be both of the form P&R and the form P ∨ R since P can be (P&Q) ∨ (P&~Q). 16. It must be conceded that (Conjunction Introduction) as a purely syntactic rule allows great clarity about its relation to other rules, such as: P&Q, so Q&P; P&Q, so P; P&Q, so Q. The relation between (Conjunction Introduction) as a platonic form of argument and other patterns is notably less clear. Can someone be credited with inferring P&Q from P, Q if he is unsure whether it follows from Q&P? A purely syntactic approach can ‘leave such questions to another subject’. Hopefully we have made it plausible here that that subject is logic. 17. There is a powerful critique of some attempts to formally analyze such ‘sameness of content’ in Potter and Smiley (2001). 18. I am indebted to Jonathan Lear and Alex Oliver for helpful criticism of an earlier draft of this essay, and to Gene Mills for helpful discussions.

REFERENCES Church, A. (1956) Introduction to Mathematical Logic, Princeton: Princeton University Press. Frege, G. (1892) ‘On concept and object’, page references are to M. Black and P. Geach (eds) Translations from the Philosophical Writings of Gottlob Frege, 2nd edn, Oxford: Basil Blackwell, 1960, pp. 42–55. Hacking, I. (1963) ‘What is strict implication?’, Journal of Symbolic Logic 28: 51–71. Kripke, S. (1979) ‘Speaker’s reference and semantic reference’ in P.A. French, T.E. Uehling Jr. and H.K. Wettstein (eds) Contemporary Perspectives in the Philosophy of Language, Minneapolis: University of Minnesota Press, pp. 6–27. Mates, B. (1972) Elementary Logic, 2nd edn, New York: Oxford University Press. Potter, M. and Smiley, T. (2001) ‘Abstraction by recarving’, Proceedings of the Aristotelian Society 101: 327–38. Smiley, T. (1959) ‘Entailment and deducibility’, Proceedings of the Aristotelian Society 59: 233–54. Strawson, P.F. (1950) ‘On referring’, Mind 59: 320–44. Tarski, A. (1956) ‘The concept of truth in formalized languages’ in his Logic, Semantics and Metamathematics, J.H. Woodger (trans.), Oxford: Clarendon Press, pp. 152–278.

4

The Socratic Elenchus No Problem James Doyle

1

‘THE PROBLEM OF THE ELENCHUS’

In the examination of Socrates’ method, a great deal of attention has been paid to Vlastos’s ‘problem of the elenchus’. I shall argue that there is no such problem. I say that there is no such problem, and not that I have solved it, since the very supposition that there is such a problem betrays a number of confusions about Socrates’ method. The elenchus (interrogation or refutation) is Socrates’ method of argument in a number of dialogues which, partly for this reason, tend to be grouped together and called ‘Socratic’. Typically, Socrates takes a thesis put forward by an interlocutor—often an attempt to define a key ethical term—and shows that its negation follows from other propositions the interlocutor assents to, which I shall refer to as premises. I shall not be much concerned with the details of the many instances of the elenchus in the various Socratic dialogues; my focus is more on concepts than on texts. I shall assume that this much of Vlastos’s account of the elenchus is true enough, of enough instances of the elenchus, to capture something important about the shape of Socrates’ usual procedure.1 At any rate, if Vlastos is wrong to suppose that Socrates takes the elenchus to establish anything at all (Benson 1995), Vlastos’s problem certainly doesn’t arise. My claim is that it doesn’t arise even if he is right. Vlastos complained that Socrates does not by this method demonstrate the falsity of the original thesis. His complaint merits extended quotation: What Socrates . . . does in any given elenchus is to convict p [the interlocutor’s thesis] of being a member of an inconsistent premise-set; and to do this is not to show that p is false but only that either p is false or that some or all of the premises are false. The question then becomes how Socrates can claim, as I shall be arguing he does claim in ‘standard elenchus’, to have proved that the refutand is false, when all he has established is its inconsistency with premises whose truth he has not tried to establish in that argument: they have entered the argument simply as propositions on which he and the interlocutor are agreed. This is the problem of the Socratic elenchus. (Vlastos 1994: 3–4)

The Socratic Elenchus 69 [H]ow is it that Socrates claims to have proved a thesis false when, in point of logic, all he has proved is that the thesis is inconsistent with the conjunction of agreed-upon premises for which no reason has been given in that argument? Could he be blind to the fact that logic does not warrant that claim? . . . Suppose the following were to happen: a witness gives testimony p on his own initiative and then, under prodding from the prosecuting attorney, concedes q and r, whereupon the attorney points out to him that q and r entail not-p, and the witness agrees that they do. Has he then been compelled to testify that p is false? He has not. Confronted with the confl ict in his testimony, it is still up to him to decide which of the confl icting statements he wants to retract. So Polus, if he had had his wits about him, might have retorted: I see the inconsistency in what I have conceded, and I must do something to clean up the mess. But I don’t have to do it your way. I don’t have to concede that p is false. I have other options. For example, I could decide that p is true and q false. Nothing you have proved denies me this alternative. And why shouldn’t Polus in that crunch decide to throw q instead of p to the lions? How strongly he believes in p we have already seen: he thinks it absurd of Socrates to deny it when almost all the world affi rms it. For q, on the other hand, he has no enthusiasm. He may have conceded it only, as Callicles observes later, ‘because he was ashamed to speak his mind’ (Gorgias 482e2). Why shouldn’t Polus then jettison q with the feeling ‘good riddance’? He would then have come out of the elenchus believing that doing injustice is better and nobler than suffering it; his latter state would have been worse than his fi rst. Couldn’t this always happen? Whenever Socrates proved to his interlocutors that the premises they had conceded entailed the negation of their thesis, why couldn’t they hang onto their thesis by welshing on one or more of the conceded premises? (Vlastos 1994: 21–2) The ‘problem of the elenchus’, then, is that Socrates claims to have refuted the interlocutor’s thesis, when all he has really done at most is show that the thesis is inconsistent with the premises the interlocutor agreed to. So although Socrates insists that the interlocutor must reject his own thesis (or, equivalently, accept its negation [Smiley 1996]), he is really only entitled to insist that the interlocutor choose between that and going back on one or more of the premises from which the falsity of the thesis has supposedly been shown to follow. Vlastos thinks that the problem requires a drastic solution. Socrates’ insistence on rejecting the thesis can be vindicated only if we ascribe two

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outlandish-sounding assumptions to him, namely ‘[A] Whoever has a false moral belief will always have at the same time true beliefs entailing the negation of that false belief’ and ‘[B] The set of elenctically tested moral beliefs held by Socrates at any given time is consistent’ (Vlastos 1994: 25 and 28). 2 He further says that ‘from [A] and [B] Socrates would naturally infer that his own set of beliefs is the true set’ (Vlastos 1982: 714). He seems not to have noticed that, on the conception of proof implicit in his account of the elenchus, Socrates could not take it as established that his set of beliefs is the true set, but only that the denial of that would be inconsistent with [A] and [B]. Setting that aside, the idea that Socrates took it as established that all his ethical beliefs are true looks to be in tension with his famously skeptical and agnostic stance on many ethical issues.

2

ASSERTION IN THE ELENCHUS

At fi rst sight, it may look as though Vlastos sees a problem only because he ignores the distinction between asserted and unasserted contents. He supposes that ‘in point of logic’ all Socrates has proved is that the interlocutor’s thesis is inconsistent with the premises. Such a claim of inconsistency is equivalent to a conditional statement of the form: if the premises are true, the thesis is false. On this account, Socrates is not entitled to reject the thesis, nor to require the interlocutor to reject it, but only to maintain that it must be rejected if the premises are true. This suggests that Vlastos thinks that Socrates is only entitled to assert what he has proven with no undischarged assumptions. One might object that on the contrary, it is precisely because the premises have been asserted by all concerned that the acquisition of the conditional licenses, via modus ponens, Socrates’ full-blooded rejection of the thesis, and requires the interlocutor to follow suit. This infringes the rule about only being entitled to assert what you prove without assumption, but why accept that? Vlastos’s ‘problem’ only seems to arise because, as his courtroom analogy makes clear, he takes the original thesis and the premises that form the basis of its refutation to be on a par with respect to the admissibility of rejecting them. Yet Socrates puts not just persons but propositions to the test: the inquiry is conceived as taking the original thesis as its object: the question is, is it true?—that is, should we assert it, as the interlocutor wants to? It would clearly be premature to assert it in the course of the inquiry.3 The premises, by contrast, are asserted there. Vlastos writes as though what gets asserted as the conclusion of the inquiry were a conditional of the form: if the premises are true, then the thesis must be rejected. The antecedent of an asserted conditional is not itself asserted, so that on this construal the interlocutor may, consistently with the results of the inquiry, reject a premise and persist in asserting the thesis.

The Socratic Elenchus 71 If Vlastos does think that Socrates is entitled to assert no more than what he has established without assumption, this is only because he sees that if the proof does depend on unproven premises, the elenchus is not then conclusive. For the interlocutor, like the witness in Vlastos’s courtroom analogy, could choose to go back on one or more premises, so that a new refutation of his thesis would be needed. Essentially the same point could be made by mentioning that someone else might take the interlocutor’s place, who did not accept all the premises; again, Socrates would have to come up with a new refutation if he is going to compel the new man to reject the thesis. I shall return to this possibility later. Vlastos is quite right that the interlocutor may (in one sense of ‘may’, but see below) respond to Socrates’ refutation by changing his mind about a premise, so that the rejection of his thesis no longer follows from what he is prepared to assert. But if this is a reason to impugn Socrates’ argument as a demonstration to the interlocutor that the thesis is false, it is probable that no-one demonstrates any categorical proposition to anyone, ever (Kraut 1983: 62). Shouldn’t we say instead that as far as this inquiry is concerned, the falsity of the thesis has been established? It is plausible to identify an inquiry in part by its assumptions and its object. Suppose that the interlocutor withdraws his assent from one of the premises of the original refutation, challenging Socrates to come up with a new one that doesn’t rely on that premise. This would not be a move within the inquiry, as Vlastos implies, but rather the initiation of a new inquiry all together. As a criticism of Vlastos, this is still unsatisfying. It makes it sound as though everything depended in the end on how we individuate inquiries. Why should that matter so much? In any case, why suppose that an inquiry must be thought of in this way? There seems to be a ‘second-order’ way of thinking about it, which promises to vindicate Vlastos’s account. On such a view, the real ‘result of the inquiry’ is pieced together out of ‘subinquiries’: conversational episodes defi ned by their asserted premises. These are thought of as experiments, of a sort, to determine ‘what happens’ if we take certain premises for granted. After a number of such sub-inquiries, one could construct a picture of some of the inferential relations among the various propositions involved or, equivalently, a menu of some of the admissible combinations of belief. This sort of account is suggested by Vlastos’s talk of interlocutors deciding to abandon premises when they are shown to entail the negations of their favored theses. One combination of beliefs is ruled out, but it is up to the interlocutor which as-yet-unimpugned combination to take up instead: ‘it is still up to him to decide which of the conflicting statements he wants to retract’ (Vlastos 1994: 21). The ‘problem of the elenchus’ is then that this inquirer’s prerogative is not acknowledged by Socrates when he insists that the thesis be rejected. The fi rst thing to notice about this line of thought is that it doesn’t actually show that Socrates is wrong to ‘reject’ the thesis in the context of the sub-inquiry, provided that the ‘rejection’, like the ‘assertion’ of the premises,

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is understood as a piece of make-believe in the spirit of ‘What would happen if we were committed to these propositions?’ Vlastos’s ‘problem’ now amounts to Socrates’ assumption that the assertions made in the context of the elenchus are in earnest, in a way that precludes taking them up and setting them aside at will. Vlastos is certainly right that Socrates rules out this way of thinking about the elenchus, but why does he? Although the premises are usually expressed with the conventional marks of assertoric force, it does not follow that the purpose of stating them is to put them forward as true (Davidson 1979: 113); it may instead be (roughly) to test them for their consequences, as one asks rhetorical questions in context with a purpose other than getting information.4 Attention to the purpose of assertion in the language, then, will not enable us to decide between Socrates’ construal of the elenchus and Vlastos’s, although it does enable us to see that Socrates’ construal is a perfectly admissible and natural one, aligning as it does the purpose in context of asserting the premises with the function of assertion in the language: putting forward as true. It is not a matter of logic that Vlastos’s construal is the right one; but no more does logic tell us that he cannot construe the elenchus as a kind of inferential laboratory, whereby the purpose in context of assertion is to test for consequences, as Vlastos effectively supposes. In order to see why Socrates’ construal of his own procedure is the right one—to see why there is no problem of the elenchus—we need to look beyond the logic of assertion, to the question of what purpose the participants might coherently be supposed to have, and to intend others to recognize them as having (etc), 5 in making assertions in the context of an elenchus. This in turn will require us to look more closely at what Socrates means by examining or testing a person.

3

TESTING PERSONS AND SAYING WHAT YOU BELIEVE

Vlastos’s account now requires Socrates to think of the elenchus in a way that is interpretatively extremely implausible. There is no evidence that Socrates does not take the interlocutor to be expressing assent to the premises. On the contrary, Socrates emphasizes repeatedly that he does conceive of the elenchus as involving genuine assent to the premises. The assertionrule for the elenchus is an especially stringent version of say what you believe. Here I appeal to Vlastos’s (1994: 7–10) own excellent treatment of Socrates’ rationale for this rule, in the very paper in which he purports to identify ‘the problem of the elenchus’. Socrates tests not just propositions but persons; part of what this means is that he wants to discern the implications of what the interlocutor will own up to believing: ‘What shall we say in response [to our imaginary interlocutor] if he asks us “So piety isn’t a just sort of thing, nor justice a pious one . . . ?”?

The Socratic Elenchus 73 Speaking on my own behalf, I’d say myself that justice is a pious thing and piety a just one . . . Does it seem this way to you, too, or would you forbid me from giving this answer?’ ‘It doesn’t seem to me to be quite so simple that I should concede that justice is a pious thing and piety a just one, Socrates’, said [Protagoras]; ‘there seems to me to be some sort of difference here. But what of it?’ he said. ‘If you want, let’s say that justice is a pious thing and piety a just one’. ‘Don’t treat me like that,’ I said. ‘I have no interest in putting this “if you want” and “if you think so” to the test: I want to test you and me. I say “you and me” because I think that the argument (logos) will be tested most effectively if the “if” is taken out of it’. (Protagoras 331a6–9, b1–3 and b7–d1, my emphases) Vlastos also cites Socrates’ similar insistence in the Gorgias: S OCRATES : By the god of friendship, Callicles, don’t you think you can joke with me, or answer with whatever comes into your head, contrary to what you think, or take what I say as a joke. For you see that the subject we are discussing is one to which even a man of small intelligence should give the highest priority: it is the subject of how to live. (Gorgias 500b6–c4; cf. Crito 49c11–d1) Socrates thinks that the seriousness of the subject requires the interlocutor to give an honest account of his beliefs. It is extraordinary that in the same paper in which he discusses this strict say what you believe requirement, Vlastos asks why Polus couldn’t have avoided Socrates’ refutation by simply reneging on a premise, and sees in this question ‘the problem of the elenchus’. Clearly, if the interlocutor is prepared to take up and set aside premises in the casual manner Vlastos envisions in the Polus example, it cannot make sense for Socrates to care very much whether the interlocutor is prepared at the outset to assert a premise rather than, say, its negation, since their asserting a premise cannot after all indicate that they are prepared to associate themselves with it in any serious way. If an interlocutor fl ip-flopped on premises in this way, Socrates would object, with justice, that he was making a mockery, not so much of the ‘say what you believe’ rule, as of the seriousness with which Socrates laid it down. It is even more extraordinary that Vlastos wonders how Socrates could have objected if Polus had tried to maintain consistency by going back on something he asserted earlier. Vlastos writes as though Socrates’ interlocutors never take up this option, but it does happen. In fact, in the very

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next conversation in the very same dialogue, Callicles tries to do exactly this, twice: SOCRATES: . . . Tell me now too whether you say that the pleasant and the good are the same or whether there is some pleasure that isn’t good. C ALLICLES: Well, my argument will be inconsistent if I say that they’re different, so to prevent this I say they’re the same. SOC: You’re wrecking your earlier statements, Callicles, and you’ll no longer be adequately inquiring into the truth of the matter with me if you speak contrary to what you really think. C AL: You do it too, Socrates. SOC: In that case, it isn’t right for me to do it, if it’s what I do, or for you either. (Gorgias 495a2–b3) Callicles is here going back on an earlier assertion: that the pleasant and the good are the same. Socrates’ reaction is exactly what I claimed, on the basis of Vlastos’s own account of the ‘say what you believe’ rule, it would be if Polus had taken Vlastos’s proffered ‘way out’: he accuses him of bad faith. A little later Callicles suddenly pretends he had been only joking and switches premises when his hedonism is under threat: C ALLICLES: I’ve been listening to you for quite a while, Socrates, and as I’ve been going along with what you say, I’ve been thinking to myself that, if anyone makes any concession to you, even in jest, you seize upon it gleefully, as boys do. As if you could really imagine that I or anyone else didn’t believe that some pleasures were better and others worse! SOCRATES: Oh Callicles, how unscrupulous you are! You’re treating me like a child. First you say things are one way, then that they are another, and so you deceive me. But I didn’t think at the outset that I’d be deceived intentionally by you, because you were my friend. But now, I have been misled, and it seems I must make the best of what I have, in the words of the old proverb, and accept what I’ve been given by you. (Gorgias 499b9–c6, my emphasis) Socrates supposes that Callicles has ‘hung onto his thesis by welshing on one or more of the conceded premises’, in Vlastos’s phrase, and as a consequence accuses him of deceit. When Callicles initially ‘conceded’ the premise, Socrates took him to be committed to such a degree that he would not simply abandon it if it looked like causing trouble for a thesis he had a fondness for. So when Callicles does abandon it Socrates sees that he had been mistaken. But for a charge of intentional deceit to make sense, he must have supposed that Callicles bore responsibility for the impression he gave

The Socratic Elenchus 75 Socrates; that is, that Callicles knew that Socrates would get an impression he also knew to be false. Callicles knew this because Socrates has insisted all along on the interlocutor answering according to his real view. But this can’t matter if one’s ‘real view’ can shift around according to how one feels about the implications being drawn from it. Callicles’ knowledge that he has deceived Socrates is indicated by his disingenuous style of admission. He makes a lame pretence of having been joking up to this point (foreshadowed in the question to Chaerephon with which he initiates his long conversation with Socrates [481b6–7]). Why? Because he knows full well that that is the only remotely plausible way of explaining his earlier assertions as not culpably making Socrates believe something false about his (Callicles’) degree of commitment to the premise he had ‘asserted’. Callicles was well aware of the rules of the game. Vlastos says that Socrates’ elenchi do no more than ‘convict p [the interlocutor’s thesis] of being a member of an inconsistent premise-set’, and at this point someone might object on his behalf that while the interlocutor is committed to the premises, he is also committed to the thesis refuted on their basis—perhaps even more committed, as Vlastos says about the Polus case. So why should he be expected to reject the thesis rather than any of the premises? But this is to regress to an earlier stage of our dialectic with Vlastos. The interlocutor would simply be ignoring another constitutive rule of elenchus: that the thesis is being tested, so that it would be premature to assert it, and any interlocutor who tries to hold onto it by going back on a premise is no more serious about scrutinizing the thesis’s credentials in order to decide whether to believe it, than he is about committing himself to the truth of the premises. In my account of what the elenchus is, I have appealed to what Socrates takes the elenchus to be. This may look circular, because in supposing Socrates to take it to work that way, I’m presupposing that that’s a coherent way to take it. But isn’t that exactly what Vlastos denies: that Socrates has any right to reject the thesis? My reply to this has two stages. The fi rst was to show that the way Socrates understands the elenchus is a perfectly plausible way for it to be; this can be shown on the basis of facts about what is involved in assertion, belief and so on, which facts nevertheless do not decide between Socrates’ way and the way Vlastos understands it. Second stage: what decides this further issue is the simple fact that Socrates makes it clear to his interlocutors that this is how he understands the procedure. Socrates’ avowed conception of the elenchus is now an explanans and not, as Vlastos supposed, an explanandum. Socrates’ own affi rmation of the elenchus’s significance is not undermined by the ‘problem of the elenchus’. On the contrary, it is precisely this affi rmation that enables him to ensure that the ‘problem’ never arises. All it takes for the interlocutor to be committed to rejecting the thesis in the face of a refutation on the basis of the premises, is that he have no excuse for not realizing that Socrates intends him to understand the procedure as compelling that rejection in those

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circumstances (since it is the thesis that is being tested), and as presupposing a commitment to the premises consistent with the procedure having this compulsive power (since the interlocutor must say what he really thinks).

4

ELENCHUS AND PROOF

Now suppose the partisan of Vlastos says something along the following lines: ‘I concede that the interlocutor is barred by Socrates’ rules from going back on a premise, and if Socrates has made the rules clear he is entitled to expect the interlocutor to abide by them. But this just shows up the difference between what Socrates can compel an interlocutor to say in the course of a procedure whose rules he himself imposes, and what he is entitled to claim as having been established “in point of logic”. So while the interlocutor may be compelled to reject the thesis according to the rules of the elenchus, “in point of logic” Socrates has only demonstrated that the thesis is false if the premises are true’. I think this is where we get to the heart of Vlastos’s misunderstanding. For he wants to say that Socrates doesn’t succeed in doing something he takes himself to do. Vlastos only thinks it is a criticism of Socrates to point out that he doesn’t establish the falsity of the thesis ‘in point of logic’ because he thinks that Socrates takes himself to have established it in this way. And now we have to ask whether Socrates thinks anything like this at all. When Vlastos talks of Socrates rejecting the thesis, he is referring to the last stage of the elenchus, where Socrates claims that the thesis is false and calls upon the interlocutor to reject it too. What does this move amount to? What does Socrates take himself to be doing, and with what justification? Well, he is rejecting the thesis, or asserting its negation. And part of the point of his doing so is to get the interlocutor to go along with this. Given what the rules of the elenchus say about what it is to test a proposition and what the interlocutor is committed to in asserting a premise, Socrates supposes that the interlocutor cannot deny that the thesis is false and must be rejected. If the argument is valid, he is surely correct. The interlocutor cannot go back on a premise without making his earlier assertions deceitful, and he has been unable to fi nd a flaw in the argument to the negation of the thesis. Socrates feels entitled to reject the thesis because the interlocutor can make no objection, which is to say that the interlocutor must reject it too. As to the question of how Socrates can be justified ‘in point of logic’ in rejecting the thesis, the simple answer is that he never claims to have done any more than justify his rejection to the interlocutor, so that it is only the interlocutor (and by implication anyone who is committed to the same premises) who is required to reject it in the face of Socrates’ argument. If by talking of what Socrates has demonstrated ‘in point of logic’, Vlastos means what Socrates has demonstrated, period, as opposed to what he has demonstrated to a given

The Socratic Elenchus 77 interlocutor, we have no reason to suppose that Socrates makes any claim at all about what he has demonstrated ‘in point of logic’. The elenchus is essentially relative to an interlocutor: in his conversation with Polus, Socrates is quite explicit that what is demonstrated to one interlocutor is not necessarily thereby demonstrated to any other: SOCRATES: . . . And if I can’t procure your own agreement as the single witness to what I say, I don’t think I will have accomplished anything worthwhile with regard to the subject of our discussion; and the same goes for you, I think: if I don’t bear witness to what you say, even though I’m only one person, the testimony of all these others is neither here nor there. (Gorgias 472b6–c2) My technique is to procure a single witness to what I’m saying, the person toward whom my argument is directed, and to count his vote alone. I disregard everyone else; I’m not talking to them. (Gorgias 474a5–b1) Sometimes Vlastos suggests that what has been proved unconditionally (on the basis of no undischarged assumptions) is conditional in form, as when he says that the only thing that has been demonstrated is the inconsistency of the premises and the thesis. At other times he says that the negation of the thesis has been proven, but only on the basis of premises which have been assumed without proof. The deduction theorem tells us that these formulations are anyway equivalent. It is as if Vlastos thought that the proof-type of the elenchus could be represented as a metalogical formula, and in such a way as to settle the issue of what gets proved. But the resulting conception of proof is one we cannot coherently ascribe to Socrates. The only things Socrates ever says in an elenchus about whether a proposition has been proved explicitly concern what he takes the interlocutor to be compelled to agree to. Finding out what the interlocutor is compelled to agree to is really the purpose of the enterprise. There is not much point in judging it unfavorably compared with some other conception of proof. It never entered the competition: calling it a proof at all can amount to no more than what Socrates understands by testing a person and testing a proposition. Now much of what the interlocutor must concede and what he may consistently reject obviously depends upon what else he believes, and the elenchus by its nature identifies which of the interlocutor’s other beliefs are functioning as relevant premises. All that Socrates could mean by proof is what he takes the interlocutor to be required to assent to; what this amounts to will obviously depend on what the interlocutor has conceded by way of premises. Since a different person may refuse to concede one or more of those premises, there is no guarantee that a given elenchus would be successful against anyone else. There is an unmistakable illustration of this in the Gorgias: Socrates has compelled Polus to concede that it is worse to inflict injustice than to

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suffer it, a crucial premise being that it is more shameful. The whole of the long and eventful conversation with Callicles takes its philosophical impetus from Callicles’ immediate rejection of that premise: C ALLICLES: By nature, you see, the more shameful thing is always just what’s worse for one: in this case,6 suffering injustice, whereas committing injustice is only conventionally more shameful. This isn’t the sort of thing that happens to a man—suffering injustice, I mean—but to some slave who’d be better off dead than alive, since he can’t protect himself or his dependents against injustice and abuse. (Gorgias 483a7–b4) Does Socrates retort that, since the preferability of suffering injustice has been established absolutely, Callicles’ claim is inadmissible? Of course not: Socrates immediately sees that Callicles’ dissent from a crucial premise of the elenchus that worked on Polus means that it is powerless to compel Callicles’ assent. He effectively acknowledges that the compulsive power of the elenchus is entirely dependent upon the specific beliefs the interlocutor in question is prepared to own up to, by embarking on a completely new elenctic investigation with Callicles, trying to show that the theses Polus could not avoid consenting to also follow inexorably from the different premises Callicles is prepared to commit himself to. Socrates’ conception of proof, insofar as he has one, is strictly personrelative: he never so much as speculates about whether a given elenchus has established its conclusion in any sense other than that his interlocutor is required to assent to it. When Vlastos tries to specify what Socrates ‘claims’ to have established, by unflattering contrast with his, Vlastos’s, specification of all that has really been established, in both cases he is clearly working with a conception of proof that is not relative in that way, and in fact it is this feature of his conception that makes it, he thinks, more rigorous. And yet his fi rst specification, of what Socrates claims to have established, is baseless; and the second, of all that has really been established, is irrelevant. First: we cannot read off from any given elenchus what, if anything, Socrates takes himself to have proven in Vlastos’s nonperson-relative sense, because he never says anything about the question, and what he does say is (as far as I can tell) consistent with very many views on the matter, including Vlastos’s. Furthermore, what Socrates does shows that he does not take himself to have established the conclusion absolutely, because if a new interlocutor rejects premises the fi rst interlocutor accepted, Socrates immediately sees that a new elenchus is in order. The second specification, of what Socrates has ‘really’ proven, presupposing this same conception of proof, to which Socrates has expressed no aspiration and concerning which he makes no claims, is clearly irrelevant as a criterion for judging the accuracy of Socrates’ own ideas about what he has ‘proven’ to this interlocutor.

The Socratic Elenchus 79 The thing about Socrates is that he’s always talking to someone and he thinks that everything is always up for grabs. It is as though Vlastos were really trying to express the complaint that Socrates hasn’t refuted the thesis to the reader’s satisfaction, because the reader may not assent to the premises. But if Socrates is going to be convicted of not having done something he claims to have done, it had better be something he can be coherently supposed to have claimed to have done. When Socrates says what he thinks has been established, he’s never ‘breaking the fourth wall’ to let the reader know. For this too is a claim addressed to one or more interlocutors, and so subject to the usual elenctic testing: if an interlocutor disputes what he says, we are back to the business of proof which is, as always, proof to an interlocutor. Although Socrates never ‘breaks the fourth wall’, if we can somehow imagine him doing so after an elenchus and discovering that the reader is unmoved by his demonstration because he won’t sign up to one of the premises, he would not insist that the reader must reject the thesis because its negation has just been demonstrated. He would be well aware that this hasn’t been demonstrated to the reader, and he would begin another elenchus. In this respect such a reader of an elenchus is in the same boat as Callicles as he looks on while Socrates defeats Polus’s thesis on the basis of a premise he would never assent to. ‘I disregard everyone else’, says Socrates; ‘I’m not talking to them’. But we must be careful not to misconstrue his disregard. As we have seen, he certainly means, at least, that his argument essentially depends upon such premises as the interlocutor is committed to, so that he makes no claim that it need compel anyone who does not share those commitments. But it does not follow that he is indifferent to the impression made on the onlookers (if there are any) by the elenctic spectacle.7 After all, Plato cannot intend these dialogues to benefit only those readers who would concede Socrates’ premises, and Socrates always treats onlookers as potential participants, whose interventions are prompted by the elenchi they observe. And so our criticism of Vlastos is mitigated: although Socrates’ dialectical attention is focused exclusively on the interlocutor, he must yet intend the elenchus to be instructive to any who witness it. Vlastos is right that, if the argument is valid, readers and onlookers must concede only that the interlocutor’s thesis is inconsistent with premises which he has conceded but they need not.8 But this point is itself consistent with Socrates’ conception of what the elenchus does for onlookers and, we may suppose, with Plato’s conception of what it does for the reader: not to demonstrate the truth of its conclusion, but to provoke new philosophical activity.9 NOTES 1. So that we know where we are, here are a few examples that I take to fit the sketch: Charmides 160e2–161b4 (thesis: self-control [sōphrosunē] is a sense of shame [aidōs] [160e2–5]; premises: self-control is fi ne and good [160e13]; Homer was right to say that a sense of shame is not always a good

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2.

3. 4. 5. 6. 7.

thing [161a2–4]); Laches 192b9–193d10 (thesis: courage is a kind of endurance of the soul [192b9–c1]; premises: courage is fi ne [192c5–7]; ignorant endurance is bad and harmful [192d1–3]; people ignorant of a skill may be on that account more courageous than those who have knowledge [192e1– 193c11]); Lysis 212a8–d5 (thesis: two people are friends whenever one loves the other [212a8–b5]; premises: it is possible to be hated by a person one loves [212b7–c3]; if one person hates another, neither can be called a friend of the other [212c4–d1]); I don’t claim to have listed all premises, and I acknowledge that it may be a tricky matter to identify them. I shall also assume that Socrates usually believes the premises (the prominent exception is Euthyphro 7b2–4), so that when I talk of ‘what gets asserted’ in an elenchus I mean what Socrates asserts and the interlocutor assents to. My assumption, that Vlastos’s account fits enough of the elenchi in the dialogues plausibly called Socratic to tell us something important about the shape of Socrates’ procedure, is made for the sake of argument and falls far short of the guiding assumption of the Vlastos-inspired ‘strong programme’ in Socratic studies, whereby the Socratic dialogues are treated as windows on a single philosophical world. Vlastos and the other main commentators on the Socratic dialogues have been strangely impervious to Anscombe’s (1958) and Williams’s (1985) warnings about the dangers of anachronism in talking about the category of the moral in connection with the ancient Greeks. Vlastos acknowledges that ‘for “moral” Socrates has no special word’ (1994: 6), but insists that ‘in expounding Socratic doctrine [Socrates] uses aretē to mean “moral virtue”’ (n. 25), referring the reader to his (1991), where he states ‘Any lingering doubt . . . in my readers’ mind [concerning Socrates’ use of aretē to designate precisely what we understand by moral virtue] may be resolved by referring them to the fact that whenever he brings the general concept under scrutiny—as when he debates the teachability of aretē in the Protagoras and the Meno—he assumes without argument that its sole constituents or “parts” . . . are five qualities which are, incontestably, the Greek terms of moral commendation par excellence: andreia (“manliness”, “courage”), sōphrosynē (“temperance”, “moderation”), dikaiosynē (“justice”, “righteousness”), hosiotēs (“piety”, “holiness”), sophia (“wisdom”).’ This just pushes the problem back. We can agree that ‘justice’ translates dikaiosynē without conceding that the Greek term denotes a ‘moral virtue’. Even if it is not anachronistic to talk of morality here, this cannot be how Socrates understood the concept. If it were, then when Callicles gave his account of what true (‘natural’) justice really amounts to (Gorgias 482e5–484c3), which is clearly not a moral virtue at all, Socrates would have convicted him immediately of self-contradiction. The core meaning of aretē, for Socrates as for the Greeks generally, is quality that makes one good at being a human being; in the case of dikaiosynē this leaves it open whether one ought to be prepared to harm no-one (Socrates’ account at Crito 49c2–6), harm one’s enemies (standard Greek account) or harm pretty much everyone weaker than oneself (Callicles’ account). Unless for purposes of reductio, as in what Robinson (1953: ch. 3) calls indirect elenchus. This is Austin’s (1975: lecture VIII) distinction between illocution and perlocution. On the constitutive role in linguistic communication of the speaker’s intention to affect the hearer by inducing a recognition in the hearer that the speaker has the intention so to affect her, see Grice (1957). Reading hoion with Dodds. I am grateful to Jonathan Lear for pressing the question of what we get out of looking on.

The Socratic Elenchus 81 8. What competent readers and onlookers are brought to see as a matter of logic is still not fully captured by a metalogical formula, however. The point is not just that one proposition has been proven on the basis of others, but also that the interlocutor, in the context of this living conversation, if he understands the rules of elenchus, must reject his own thesis, on pain of rendering his own behavior uninterpretable. If he continues to insist on his thesis, it can no longer be clear that we are dealing with genuine assertions and inferences on his part, and our ascriptions of belief to him are undermined. For the relevant conception of interpretation, see Davidson (1973 and 1974); for its application to a specifi c Socratic conversation, see Doyle (2010). Yet this too may be thought of as a logical matter: the ‘logic of assertion’ determines the limits of what an interlocutor may purport to assert if he is to remain in ‘good standing in logos’, (a phrase of Jonathan Lear’s, whose suggestion I am here following up). Vlastos was a great pioneer in the study of ancient philosophy partly because he brought to it a new emphasis on the evaluation of argument characteristic of ‘analytic’ philosophy. Yet his own understanding of that style of philosophy perhaps brought with it a too narrow conception of logic as essentially concerned with formal calculi. 9. Timothy Smiley was my fi rst teacher of logic, and I am very grateful to the editors for inviting me to contribute to this volume in his honor. I would also like to thank them, and Danielle Allen, Dan Devereux, James Ladyman, and Giles Pearson, for helpful comments on earlier versions of this essay.

REFERENCES Anscombe, G.E.M. (1958) ‘Modern moral philosophy’, Philosophy 33: 1–19. Austin, J.L. (1975) How to Do Things with Words, J.O. Urmson and M. Sbisà (eds), 2nd edn, Cambridge, MA: Harvard University Press. Benson, H. (1995) ‘The dissolution of the problem of the elenchus’, Oxford Studies in Ancient Philosophy 13: 45–112. Davidson, D. (1973) ‘Radical interpretation’, reprinted in his Inquiries into Truth and Interpretation, Oxford: Oxford University Press, 1984, pp. 125–40. . (1974) ‘Belief and the basis of meaning’, reprinted in his Inquiries into Truth and Interpretation, Oxford: Oxford University Press, 1984, pp. 141–54. . (1979) ‘Moods and performances’, reprinted in his Inquiries into Truth and Interpretation, Oxford: Oxford University Press, 1984, pp. 109–21. Doyle, J. (2010) ‘Socrates and Gorgias’, Phronesis, forthcoming. Grice, H.P. (1957) ‘Meaning’, reprinted in P.F. Strawson (ed.) Philosophical Logic, Oxford: Oxford University Press, 1967, pp. 39–48. Kraut, R. (1983) ‘Comments on Gregory Vlastos, “The Socratic elenchus”’, Oxford Studies in Ancient Philosophy 1: 59–70. Robinson, R. (1953) Plato’s Earlier Dialectic, 2nd edn, Oxford: Clarendon Press. Smiley, T. (1996) ‘Rejection’, Analysis 56: 1–9. Vlastos, G. (1982) ‘The Socratic elenchus’ (abstract), The Journal of Philosophy 79: 711–4. . (1991) ‘Happiness and virtue in Socrates’ moral theory’ in his Socrates: Ironist and Moral Philosopher, Ithaca, NY: Cornell University Press, pp. 200–32. . (1994) ‘The Socratic elenchus: method is all’ in his Socratic Studies, M. Burnyeat (ed.), Cambridge: Cambridge University Press, pp. 1–28. Williams, B. (1985) Ethics and the Limits of Philosophy, London: Fontana.

5

What Makes Mathematics Mathematics? Ian Hacking

mathematics. Originally, the collective name for geometry, arithmetic, and certain physical sciences (as astronomy and optics) involving geometrical reasoning. In modern use applied, (a) in a strict sense, to the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and (b) in a wider sense, so as to include those branches of physical or other research which consist in the application of this abstract science to concrete data. When the word is used in its wider sense, the abstract science is distinguished as pure mathematics, and its concrete applications (e.g. in astronomy, various branches of physics, the theory of probabilities) as applied or mixed mathematics. (The Oxford English Dictionary)

Many students of Timothy Smiley have been interested in the philosophy of mathematics. We have seldom paused to ask what counts as mathematics. Certainly we have thought a good deal about the nature of mathematics—in my case, for example, about the logicist claim that mathematics is logic.1 But we took ‘mathematics’ for granted, and seldom reflected on why we so readily recognize a conjecture, a fact, a proof idea, a piece of reasoning, or a subdiscipline, as mathematical. We asked sophisticated questions about which parts of mathematics are constructive, or about set theory. But we shied away from the naïve question of why so many diverse topics addressed by real-life mathematicians are immediately recognized as ‘mathematics’. The Oxford English Dictionary (OED) definition is an excellent one. Why not stop right here, and answer our question by quoting the dictionary? Because the kinds of things we call mathematics are, in a word, so curiously miscellaneous. The dictionary already hints at that, in part by its implication that the concept of mathematics itself has a history, with the name applying in different ways to different categories over the course of time. 1

A MATHEMATICIAN’S MISCELLANY2

The arithmetic that all of us learned when we were children is very different from the proof of Pythagoras’ theorem that many of us learned as adolescents. When we began to read Plato, we saw in the Meno how to construct

What Makes Mathematics Mathematics? 83 a square double the size of a given square, and realized that the argument is connected to ‘Pythagoras’. But that is totally unlike the rote skill of doubling a small integer at sight, or a large one by pencil. Both types of examples are unlike the idea that Fermat had when he wrote down what came to be called his last theorem. We nevertheless seem immediately to understand his question about the integers. The situation is very different from the proof ideas that lie behind Andrew Wiles’ discovery of a way to prove the theorem. Few of us have mastered even a sketch of that argument. Is it ‘the same sort of thing’ as the proof that there is no greatest prime? I am not at all sure. The mathematics of theoretical physics will seem a different type of thing again, but we should not restrict ourselves to theory. Papers in experimental physics are rich in mathematical reasoning. Take a recent very successful field, very cold atoms, virtually at absolute zero (cf. Hacking 2006). There are two fundamentally different types of entity, bosons and fermions. All elements but one have isotopes that are bosons. Ions of any chosen species of boson go into the same ground state when they lose enough energy. Then they all have the same wave function, which leads some experimenters to speak of macroscopic wave functions, which would once have seemed to be a contradiction in terms. This all started in 1924 when Einstein read a letter about photons from S. N. Bose, and saw from the equations that something very strange should happen near 0ºK: there would be a new kind of matter. Only in 1995 was it possible to get a few thousand trapped ions of rubidium-47 cold enough to create what we now call Bose-Einstein condensate. Although the experimental results in this field are brand new, the mathematics that Einstein took for granted was mostly old-fashioned, taken from what has been compared to a physicist’s toolkit. (Krieger 1987, 1992). Much of it had been around for more than a century when Einstein made use of it. Many of Professor Smiley’s students could have picked it up after a couple of hours to refresh their memories. The mathematics in the toolbox—and the way it is used—is very different from that of the geometer or number theorist. The mathematical part of the physicist’s toolbox is mostly old, but something entirely new has been added. We have powerful computational techniques to make approximate solutions to complex equations that cannot be solved exactly. They enable practitioners to construct simulations that establish intimate relations between theory and experiment. Today, most experimental work in physics is run alongside simulations. Is the simulation of nature by powerful computers (applied) mathematics, in the same way that modeling nature using Lagrangians or Hamiltonians is called applied mathematics? Modern condensed matter physics, of which the theory and practice of cold atoms is a part, employs sophisticated mathematical models of physical situations. Economists also construct complicated models. They run computer simulations of gigantic structures they call ‘the economy’ to try

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to guess what will happen next. The economists are as incapable of understanding the reasoning of the physicist as most physicists are in making sense of modern econometrics. Are they both using mathematics? We are not really sure whether to say that programmers writing hundreds of metres of code are doing mathematics or not. We need the programmers to design the programs on which we solve, by simulation and approximation, the problems in physics or economics. What part is mathematics and what part not? We do not dignify as mathematics the solving of chess problems, white to mate in three. Few people will call programming a computer to play chess an instance of mathematics. Arithmetic for carpentry or commerce seems very different from the theory of numbers. What, then, makes mathematics mathematics?

2

ONLY WITTGENSTEIN SEEMS TO HAVE BEEN TROUBLED

It is curious that Wittgenstein seems to have been the fi rst notable philosopher ever to emphasize the differences between the miscellaneous activities that we file away as mathematics. ‘I should like to say’ he wrote in his Remarks on the Foundations of Mathematics, that ‘mathematics is a MOTLEY of techniques of proof’ (Wittgenstein 1978: III §46, 176). When he repeated this idea, saying that he wanted ‘to give an account of the motley of mathematics’, he went out of his way to emphasize it. Where the translators use a single capitalized word, Wittgenstein’s German has a capitalized adjective followed by an italicized noun: ‘ein BUNTES Gemisch von Beweistechniken’. In Luther’s Bible, bunt is the word for Jacob’s coat of many colors, and the word in general means parti-colored, and, by metaphor, miscellaneous. ‘Motley’ is an apt translation of Wittgenstein’s double-barrelled phrase, ‘buntes Gemisch’, for it implies a disorderly variety within a group. The German noun ‘Treiben’ denotes bustling activity; ‘ein buntes Treiben’ is emphatic, meaning a real hustle and bustle with all sorts of different things going on. Likewise, ‘ein buntes Gemisch’ is not just a mixture, but rather a mixture of all sorts of different kinds of things. When the adjective is in capital letters, and the noun in italics, BUNTES Gemisch, Wow! Thus the metaphor captures an aspect of grouping different from Wittgenstein’s well known family resemblances. This is not to deny that there are family resemblances between the motley examples of mathematics that I have just given; it is instead to suggest that mere family resemblance is not enough to collect them together. There are of course disagreements about how to use Wittgenstein’s notion of family resemblance, and about what he himself intended. If, as writers as diverse as Renford Bambrough (1961) and Eleanor Rosch (1973) have suggested, virtually all general terms are family resemblance terms, then the notion gives no help with the identity of mathematics in particular. I

What Makes Mathematics Mathematics? 85 favor the ‘minimalist’ interpretation fi rmly advanced by Baker and Hacker (1980: 320–343). They argue that Wittgenstein’s use of the notion is critical rather than constructive, and that he used it chiefly for what they call psychological and formal concepts. The latter includes language, number, Satz. He also used the idea implicitly in his discussion of names. Baker and Hacker also mentioned passages where the notion of a family resemblance occurs in accounts of ‘the concepts of proof, mathematics, and applications of a calculus, and of the ways in which mathematics forms concepts’ (Baker and Hacker 1980: 340; Hacker deleted this observation in his revised edition, 2005). None of the citations bears on the motley of mathematics, or the question of what makes mathematics mathematics. That phrase, ‘BUNTES Gemisch’, was making a positive point different from Wittgenstein’s remarks about family resemblance. In an essay that Smiley invited for a British Academy symposium on Mathematics and Philosophy, I connected Wittgenstein’s ‘motley of mathematics’ with some of his other thoughts about mathematics (Hacking 2000). I shall not pursue those ramifications here. I wanted only to notice that one philosopher had taken my question seriously. I shall not mention him again. We do take for granted that the answer to the title question of what makes mathematics mathematics, will not be a set of necessary and sufficient conditions for being mathematics.

3

THE PHILOSOPHY OF MATHEMATICS

An innocent abroad, who consulted the online Stanford Encyclopedia of Philosophy, and then the Routledge Encyclopedia of Philosophy, would conclude that there is no question about what counts as mathematics. She might be left wondering about what counts as the philosophy of mathematics itself. Stanford has a heading, ‘Philosophy of Mathematics’ (Horsten 2007). Routledge does not. Instead it has ‘Mathematics, Foundations of’, which covers something of the same waterfront (Detlefsen 1998). This is not about what cognoscenti would call Foundations of Mathematics, as represented say by the excellent and very active online FOM: ‘a closed, moderated, e-mail list for discussing Foundations of Mathematics’. It is about the philosophy of mathematics. Many of the topics discussed in the one article are discussed in the other. Yet although between them they list some 170 articles and books in their bibliographies of classic contributions, only 13 of these items occur in both lists. Only eight items cited by Stanford were published after Routledge appeared, so that does not explain the discrepancy. A complete accounting of all related entries in the two encyclopedias generates a little more overlap, but the initial contrast is striking. The prudent innocent will judge that the philosophy of mathematics covers a lot of

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topics, and that different philosophers have different opinions of what is most central or interesting. Perhaps the discrepancy between the two encyclopedias is due in part to different ideas about what mathematics ‘is’. Only in part, for sure, but that is food for thought.

4 A DIFFERENT TYPE OF DEMARCATION PROBLEM: WHAT COUNTS AS A PROOF? The title question, ‘What makes mathematics mathematics?’ looks like a demarcation problem. Our explorer will not fi nd the boundaries of mathematics discussed in the two encyclopedias, but she will encounter some other boundary questions. This is because there are different opinions about what mathematical arguments count as sound proofs. Only some mathematics is constructive; for brevity I group intuitionist criticism as constructivist, although the motivations are not identical. Constructivists of various stripes can tell a classically sound proof when they see one. They seem not to deny that classical arguments are the product of mathematical insight, or that they are produced by mathematicians. Perhaps there have been iconoclasts, somewhere, sometime, arrogant enough to say, ‘That is not mathematics at all’, but that does not seem to be a common reaction. We may call the constructivist critiques of classical proofs retroactive, because they apply to proofs, inter alia, that had been on the books long before the criticisms were made. It does not seem that these retroactive critiques address the larger question of what makes mathematics mathematics. There are also what I shall call conservative criticisms of new mathematical techniques. One is described in the fi nal paragraphs of the Stanford essay, which in turn refers to its fi rst citation, to Appel, Haken, and Koch (1977). That was a fi rst publication of the proof of the conjecture that any map can be colored using only four colors. The proof relies on a computer to check that some 1,936 graphs have a certain property, and relies on dozens of pages of written argument to show that any map falls into one of these 1,936 categories. ‘The proof of the four-color theorem gave rise to a debate about the question to what extent computer-assisted proofs count as proofs in the true sense of the word’ (Horsten 2007: end of third-to-last paragraph). The need for computers focused a general debate. Topologists, struggling with the map problem itself, seem to have been more vexed by the length and imperspicuity of the hand-checking that was also needed. The problem is not about using a computer. Many topologists now use computers to generate counterexamples and eliminate red herrings. Conservatives insist only that in the end, no lemma in a proof should call for a computer confi rmation. Perhaps some conservatives say that computer-assisted proofs that never mature into perspicuous proofs are not mathematics. The issue is, nevertheless, not about the boundaries of mathematics, but about the boundaries of

What Makes Mathematics Mathematics? 87 admissible proofs. Indeed we count mistaken proofs as mathematics. Thus Sir Alfred Kempe’s 1879 and P. G. Tait’s 1880 non-proofs of the four-color conjecture stood for a decade until flaws were found. Even today they are on any common understanding, ‘mathematics’. There are live current questions, about what makes a proof a proof, but they are prima facie distinct from my question, of what makes mathematics mathematics.

5

THREE KINDS OF ANSWER

There are three inviting answers to my question. They represent different attitudes, perhaps three different casts of mind: (1) Mathematics has a peculiar subject matter, which people versed in the discipline simply recognize. (2) Mathematics is a cognitive field ultimately determined by a domainspecific faculty or faculties of the human mind. It is a task of cognitive science and of neurology to investigate the faculty(ies) or ‘module(s)’ in question. (3) Mathematics is constituted less by its content than by disciplinary boundaries that have emerged in the course of contingent historical practices. These three answers are compatible. It will occur to any unsophisticated person that mathematics obviously has a peculiar subject matter, which is investigated by means of one or more mental faculties. Perhaps, as Kant thought, there is a distinct faculty for arithmetical reasoning, and another for geometrical reasoning. The disciplinary boundaries in our teaching and our professions mark our present grasp of that subject matter. There seems to remain, in this terminus of temporary good-will, a chicken-and-egg question about (1) and (2) that tacitly ignores (3). Is the peculiar subject matter of mathematics a consequence of our mental faculties, so that in some curious sense there is no mathematics without the human brain to process it? Or are the mental faculties simply honed to accord with a human-independent body of fact? Here we get two fundamentally opposed attitudes to mathematics, both of which take present disciplinary boundaries as irrelevant.

6

THE TWO ATTITUDES OR VIEWPOINTS

These attitudes are better represented not by the philosophers’ distinctions between ‘realism’ and ‘anti-realism’ but by a classic debate between a mathematician and a neurologist. One author, Alain Connes, won a Fields Medal for mathematics. The other, Jean-Pierre Changeux, directed the research

88 Ian Hacking laboratory in Molecular Neurobiology at the Institut Pasteur. Both were colleagues at the Collège de France, and so were able to be frank and direct without the rancour that sometimes attends such discussions. Connes mentions one phenomenon that impresses him deeply. Here we come upon a characteristic peculiar to mathematics that is very difficult to explain. Often it’s possible, though only after considerable effort, to compile a list of mathematical objects defi ned by very simple conditions. Intuitively one believes that the list is complete, and searches for the general proof of its exhaustiveness. New objects are frequently discovered in just this way, as a result of trying to show that the list is exhausted. Take the example of fi nite groups. (Connes and Changeux 1995: 19) Mathematicians fi rst thought that there are just six types of fi nite groups, a list complete by the end of the nineteenth century. During a period quite late in the twentieth century, exactly 20 more types were discovered—the sporadic groups. And that is all there are: end of the story. The last fi nite group to be found is called the monster, for such it is, with more than (8)×(10!) elements. Assuming that there is no deep underlying mistake in the proof, it feels as if this last idiotic group was just there all the time, laying in wait for us, with a monstrous grin on its face. And if human beings had not been smart enough to figure this out, the monster would still have been there, happy as a clam, indifferent to our stupidity. The neurobiologist agrees that phenomena like this are astonishing. But they are the consequence of cognitive procedures that are formed within the human genetic envelope of possibilities. It is a contingent fact that human beings devised group theory, within which certain structures would form, ultimately based on combinatorial practices to which our minds are given. 3

7

REALISM AND ANTIREALISM

The Routledge Encyclopedia addresses closely related issues in its entries for ‘Antirealism in the philosophy of mathematics’ (Moore 1998), and for ‘Realism in the philosophy of mathematics’ (Blanchette 1998). The difference between these two philosophies is represented in terms of a question: Are there mathematical truths that we could never know? Realists are defined as those who answer ‘Yes’, and antirealists as those who answer ‘No’. Among philosophers, the classic realist stance is platonism. I use a lowercase ‘p’, because the name, which was invented by Bernays in 1936, derives more from folk-knowledge of Plato than from historical texts. Given these two limited options, Plato would prefer realist platonism to antirealism. Connes is a platonist, but not a Platonist.

What Makes Mathematics Mathematics? 89 Among British philosophers, the antirealist analysis due to Michael Dummett has become classic, although he is not even mentioned in the Stanford article. The ‘antirealist’ neurologist fi nds Dummett’s version of antirealism uninteresting and perhaps unintelligible. Even to most cognitive scientists it is just so much conversation about language. It does not, they think, get to the heart of the matter, namely the fact that mathematics is a by-product of our brain and its ‘genetic envelope’, namely the field of possible structures that it can construct. To which the platonist makes the obvious retort: ‘Exactly so, “structures”! Structures that we reconstruct, perhaps, but the form of the structure was there for us to discover’. A third quiet voice might be heard here, that of the under-represented ‘applied’ mathematician. These structures were mostly discovered when investigating nature. They are not merely useful representations of nature, says an heir to Galileo: they are found in nature. At any rate, even though (1) and (2) appear to be compatible, they stand for very different ways of thinking about mathematics. Neither has much truck with (3), the idea that what counts as mathematics is the product of a contingent history of human endeavors and the emergence of disciplinary boundaries. The three answers betoken different interests and, in the case of (3), a research program that befits the new discipline of Science Studies. I shall connect this third perspective with the fi rst two, not because I profess Science Studies, but because it widens our understanding of the title question. I shall do this by a route that is anathema to most sociologists of science, for I proceed through Immanuel Kant.

8

KANT

An unexpected paragraph comes right at the start of Detlefsen’s survey in the Routledge Encyclopedia. It follows his assertion that Greek and medieval thinkers ‘continue to influence foundational thinking to the present day’: During the nineteenth and twentieth centuries, however, the most influential ideas [in the philosophy of mathematics] have been those of Kant. In one way or another and to a greater or lesser extent, the main currents of foundational thinking during this period—the most active and fertile period in the entire history of the subject—are nearly all attempts to reconcile Kant’s foundational ideas with various later developments in mathematics and logic. (Detlefsen 1998: 181) Kant does not loom so large in most other introductions to the subject. He is not even mentioned in Horsten’s ‘Philosophy of Mathematics’ in the Stanford Encyclopedia. I accept that there is something absolutely right in Detlefsen’s stage-setting. For among Kant’s innumerable legacies was the conviction that there

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is a specific body of knowledge, mathematics, of striking importance to any metaphysics and epistemology. In Bertrand Russell’s words of 1912: ‘The question which Kant put at the beginning of his philosophy, namely “How is pure mathematics possible?” is an interesting and difficult one, to which every philosophy which is not purely sceptical must fi nd some answer’ (Russell 1912: 85). We know what troubled him. ‘This apparent power of anticipating facts about things of which we have no experience is certainly surprising’. Detlefsen singles out two other problems, namely mathematics’ ‘richness of content and its necessity’. These are among the mathematical phenomena that have made mathematics loom so large in the work of some, but only some, figures in the canon of Western philosophy. In Hacking (2000) I emphasized the alleged a priori character of mathematical knowledge, and the alleged necessity of mathematical truths. Richness of content should, of course, be added. Alain Connes’s reaction to the finite group called ‘the monster’ is a fine example. In an axiomatic account of mathematics, we start with what seem to be rather trivial assertions, of the sort that John Stuart Mill called ‘merely verbal’, and proceed by leaps and bounds to all sorts of astounding discoveries. Richness of content beyond belief. Kant seldom mentions this, but he surely had it in mind (see e.g. Kant 1787: B 16f). On this occasion I shall hardly touch on these three topics of richness, necessity, and the a priori character of mathematics. They are not why I pick up on Kant. Instead I focus on the clause to which we never pay attention, the conviction that there is a specific body of knowledge, mathematics, of striking importance. Perhaps Kant helped to lodge that proposition in our heads, so that mathematics is just a given, a domain that makes some philosophers curious. Of course mathematics has mattered to philosophers all the way back at least to Plato, but, as we shall see, Plato’s own demarcation of mathematics is different from our own.

9

KANT’S VISION OF THE UR-HISTORY OF MATHEMATICS

Kant had already published the Critique of Pure Reason when he sat back, and reflected, that even reason has a history. That pivotal moment, between the fi rst and second editions of the Critique, took place when Europe turned from the timeless reason of the Enlightenment to the historicist world that we still to some extent inhabit. In his new Introduction for the second edition, Kant betrayed a wonderful enthusiasm for a defi ning moment in the history of human reason (as he saw it). Kant, of all people, has become a historicist. He used such purple prose that I quote it in full: In the earliest times to which the history of human reason extends, mathematics, among that wonderful people, the Greeks, had already entered upon the sure path of science. But it must not be supposed that

What Makes Mathematics Mathematics? 91 it was as easy for mathematics as it was for logic—in which reason has to deal with itself alone—to light upon, or rather to construct for itself, that royal road. On the contrary, I believe that it long remained, especially among the Egyptians, in the groping stage, and that the transformation must have been due to a revolution brought about by the happy thought of a single man, the experiment which he devised marking out the path upon which the science must enter, and by following which, secure progress throughout all time and in endless expansion is infallibly secured. The history of this intellectual revolution—far more important than the discovery of the passage round the celebrated Cape of Good Hope—and of its fortunate author, has not been preserved. But the fact that Diogenes Laertius, in handing down an account of these matters, names the reputed author of even the least important among the geometrical demonstrations, even of those which, for ordinary consciousness, stand in need of no such proof, does at least show that the memory of the revolution, brought about by the fi rst glimpse of this new path, must have seemed to mathematicians of such outstanding importance as to cause it to survive the tide of oblivion. A new light flashed upon the mind of the fi rst man (be he Thales or some other) who demonstrated the properties of the isosceles triangle. The true method, so he found, was not to inspect what he discerned either in the figure, or in the bare concept of it, and from this, as it were, to read off its properties; but to bring out what was necessarily implied in the concepts that he had himself formed a priori, and had put into the figure in the construction by which he presented it to himself. If he is to know anything with a priori certainty he must not ascribe to the figure anything save what necessarily follows from what he has himself set into it in accordance with his concept. (Kant 1787: B x.–xii., 19) We no longer countenance the hero in history, ‘be he Thales or some other’. Even if there was a historical Thales in whose head the legendary penny dropped, there had to be uptake, there had to be people to talk to, to correspond with, to turn a transient thought into knowledge that endured. And thanks to the labors of generations of scholars, we can no longer dismiss the Egyptians and the peoples of Mesopotamia as in ‘the groping stage’. We can now turn Kant’s prose into something closer to the historical facts, thanks to Reviel Netz (1999). He would prefer Eudoxus to Kant’s Thales, but the important point is that there was a moment of radical change in the human mastery of mathematics. Kant got that right, by present lights. Using the metaphor recently favoured in palaeontology, Netz suggests ‘that the early history of Greek mathematics was catastrophic’—a sudden change in the very ‘feel’ of mathematical thinking. In a lower key: ‘A relatively large number of interesting results would have been discovered practically simultaneously’ (Netz 1999: 273). Netz suggests a period of at most eighty years. We have no need to dismiss the Babylonians, Egyptians,

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and others who taught mathematics to the Greeks, in order to see that at the time of ‘Thales or some other’ a revolution in reason was wrought.

10

WHAT WAS SO REVOLUTIONARY?

In the modern spirit of iconoclasm called post-modern, post-colonial, and so forth, the Greeks are no longer ‘that wonderful people’. Their elevation to a central role in Western history, as the only begetters of all things wise and beautiful, was (it is now said) an act of European imperialism. It was all the more important in German thought in the epoch before the Germanspeaking lands had become sufficiently united to exert their power outside of central Europe. Let us not argue the point. Greeks may have been glorified in the era of European triumphalism, but there really was something revolutionary, ‘catastrophic’, that happened in those 80 years of Mediterranean history. Some Greeks, including Ionians, discovered new mathematical facts and structures, but that, from Kant’s point of view, was not what counted most. The revolutionary discovery that made it all possible was proof. Greeks uncovered what may well be an innate capacity of all human beings, the ability to make demonstrative proofs.

11

A NEW METAPHOR: CRYSTALLIZATION

Kant was absolutely right. The discovery of proof was revolutionary. But because the idea of scientific revolution has been so over-worked since the days of Thomas Kuhn, it is wise to choose another metaphor, saying that a crystallization of mathematics occurred at the time of Eudoxus (‘or some other’).4 This metaphor is intended to capture several aspects of what happened. First, the new method of demonstrative proof did not lack precursors or anticipations. As in the case of literal crystallization, there was a great deal going on before the new structure appeared. Second, this event inaugurated a whole new way of doing things, stabilized within its own local and historically contingent practices, and yet capable of transfer to new civilizations, where it would be stabilized in very different social environments. Like many a crystal of purest ray serene, it could be thrust into darkness before being uncovered again in a period of reopening and rebirth.

12

A STYLE OF SCIENTIFIC THINKING

Demonstrative proof is a distinctive style of scientific thinking. Because the very word ‘style’ is so evocative, the expression, ‘style of scientific thinking’

What Makes Mathematics Mathematics? 93 can be used in many ways. I use it in an artificially narrow sense that I acquired from the historian of science Alistair Crombie (1994). He proposed some six distinct fundamental ‘styles of scientific thinking in the European tradition’ that emerged and matured at distinct times and places. Brief tags for his six styles are: mathematical, hypothetical modeling, experimental exploration, taxonomic, statistical, and genetic. I believe he taught an important insight, but I shall not argue the case here. I mention it because my argument below is part of a larger analysis of the history of scientific reason. It can be seen as a continuation of Kant’s original ‘historicist’ thought stated in the long paragraph quoted above. Indeed in the very next paragraph after that, Kant turns to natural science (Naturwissenschaft) which, he tells us, ‘was very much longer in entering upon the highway of science (Wissenschaft)’. He takes us to the world of Galileo and of Torricelli, and speaks of a ‘discovery [that] can be explained as being the sudden outcome of an intellectual revolution’, another crystallization (Kant 1787: B xii.). Following Crombie, I file Galileo under the style of hypothetical modeling and Torricelli under the style of experimental exploration. Then I proceed to the synthesis that Kant probably intended, what I call the laboratory style of scientific thinking. That is all a matter of footnotes to Kant’s recognition of these ‘intellectual revolutions’. They were not only intellectual; they were also a matter of new things we could do with hand and eye. They briefly changed the role of Europe in world history, and permanently changed the role of our species on our planet. The metaphor of ‘crystallization’ may suggest something too rigid, too mineralogical, and too fi xed. Styles of scientific thinking evolve, do they not? Perhaps we should extend the metaphor to the science of near-life. It is often observed that viruses are equivocal: some of the time they are inanimate, but when they fi nd a host they are alive. They become alive by exploiting molecules in the host cell to create a kind of metabolism that serves them well. They are not parasites, which are autonomous organisms, but individuals that thrive by incorporating themselves into the lives of their hosts. When alive, they evolve far more rapidly than their hosts do. When inanimate, they are said to be like crystals. I seize upon the analogy. A style of scientific thinking is like a virus, a crystallization that can evolve in a host, a community, a network of human beings. Enough of metaphor. To return to real people, I emphasize that what Eudoxus and company did, was not only to establish some new mathematical facts, techniques, and proof ideas. They also discovered a new way to fi nd things out, namely by reasoning and proof. This was not a mathematical discovery, but the discovery of a human capacity of which our species had, in earlier times, only glimmerings here and there. It was the discovery and then exploitation of a mental faculty or faculties, of precisely the sort that cognitive science and neurobiology is now investigating. Netz’s book is widely admired for its reconstruction of the diagrams that are notoriously missing from surviving ancient texts. Few readers attend to

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his subtitle, A Study in Cognitive History, yet that aspect is exactly what is fundamental and wholly original. It is the fi rst detailed analysis of the cultural history of the discovery of a cognitive capacity. Some cognitive scientists conjecture that mental modules—one or more—enable us to engage in mathematical reasoning. Netz himself argues against excessive modularity, and sides with Jerry Fodor in favor of more general processing devices. (Netz 1999: 4–7.) He urges never to forget Fodor’s ‘First Law of the Nonexistence of Cognitive Science’. Fodor’s fi rst law preaches humility. ‘The more global . . . a cognitive process is, the less anybody understands it’ (Fodor 1983: 107). The cognitive processes needed for mathematical demonstration are pretty global. Mention of cognition returns us to the second answer to our title question: (2) Mathematics is a cognitive domain ultimately determined by a domain-specific faculty or faculties of the human mind. The above discussion enriches answer (2), but is wholly compatible with the ‘platonist’ answer (1). Now we pass to reflections that depend on a wholly fortuitous aspect of cultural history, thus directing us towards answer (3).

13

WHY SHOULD GREEKS HAVE CARED?

Why did some Greek thinkers think that the newly discovered capacity for demonstrative proof was so important? (This is different from the question, of why future Europeans such as Kant and Bertrand Russell thought that it was important.) Netz (1999: 209ff), following Geoffrey Lloyd (1990), suggests an answer. City-states were organized in many ways, but Athens is of central importance. It was a democracy of citizens, all of whom were male and none of whom were slaves. It was a democracy for the few; but within those few, there was no ruler. Argument ruled. If you could make the weaker argument appear the stronger, you won. Athenians were the most consistently argumentative bunch of self-governors of whom we have any knowledge. We read Aristotle for his logic and not for his rhetoric. Greeks read him for his rhetoric; his logic was strictly for the Academy. The trouble with arguments about how to administer the city and fight its battles is that no arguments are decisive. Or they are decisive only thanks to the skill of the orator, or the cupidity of the audience. But there was one kind of argument to which oratory seemed irrelevant. Any citizen, and indeed any young slave who was encouraged to take the time, and to think under critical guidance, could follow an argument in geometry. He could come to see for himself, perhaps with a little instruction, that an argument was sound. He could even create the argument, fi nd it out for himself. In geometry, arguments speak for themselves to the inquisitive mind. Cynics will say that this is a lie from the start. A ‘little’ instruction? The instruction is just a kind of rhetoric, mere oratory. Look at the classic,

What Makes Mathematics Mathematics? 95 the demonstration to be found in the Meno, of how to double a square. The slave boy is said to discover the technique by himself, unaided. He is coached by leading questions from Socrates. But in addition to prompting there is something else: the extraordinary phenomenon, accessible to almost any thoughtful reader of the Meno, of seeing that the square on the diagonal is twice that of the given square. In company with the diagram, and talk about the diagram, there is a new kind of experience, of conviction based solely on the perception of a new truth. Geoffrey Lloyd remarked that this phenomenon is truly impressive to members of an argumentative society that has no recourse to a ruler, and whose fi nal criterion is nothing more than talk and persuasion.

14

PLATO, THE KIDNAPPER

‘So what?’ asks the politician in the public arena. You can prove only recondite or useless facts. Quite aside from the uselessness of proof in political debate, it is not even useful to the architect. It is no good saying that geometrical theorems could be useful to a surveyor. (a) The surveyors already knew most of the practical facts required, for they had been acquired from empirical Egyptian mathematicians. (b) Netz reports that not a single surviving text suggests a connection between the problems and solutions of the geometers, and the practical interests of architects. Only later did mathematics become ‘useful’. Maybe Archimedes used it in the famous problem of burning mirrors. Military mathematics came into its own only in the age of Napoleon. The great mathematicians such as Laplace solved problems connected with artillery, but they were interested not in proofs but in solutions to problems of motion. Nobody debating military strategy or the tax on corn in the Agora was able to use geometrical proof. So why should Greeks have cared about proof? An answer may be that hardly anyone did. Netz’s book is about an epistolary tradition involving a small band of mathematicians exchanging letters around the Mediterranean Sea. They cared about new discoveries and new proofs, but not about the very idea of proof. Enter Plato, kidnapper. I take the label from Bruno Latour’s brilliant critical exposition of Netz’s book. His opening sentence reads: This is, without contest, the most important book of science studies to appear since Shapin and Schaffer’s Leviathan and the Air-Pump. (Latour 2008: 441) Alongside Netz, Latour is referring to Shapin and Schaffer (1985): the book subtitled Hobbes, Boyle, and the Experimental Life. I completely agree with Latour’s judgement, but for reasons quite opposite to his. Latour sees

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both books as magnificent illustrations of his network theory of knowledge—which they certainly are. But I also see Shapin and Schaffer as having presented a decisive crystallization of the laboratory style of scientific thinking, and Netz as presenting a decisive crystallization of the mathematical style of scientific thinking which we call demonstrative proof. In both cases a new style of scientific thinking became established in the practices of discovery, of creating knowledge, or, to be more colloquial but more exact, in the human repertoire of finding out. Latour rightly takes Netz’s analysis as a compelling example of knowledge sustained by a network of creators and distributors of that knowledge. Nowhere is that better illustrated than by Archimedes, who, working out of Syracuse in Italy, created and maintained an unparalleled body of new understanding, and yet had only a handful of disciples and correspondents around the Mediterranean. But what specially fascinates Latour is the isolation of this network from the rest of the ancient world, be it learned, political, or vernacular. To the great surprise of those who believe in the Greek Miracle, the striking feature of Greek mathematics, according to Netz, is that it was completely peripheral to the culture, even to the highly literate one. Medicine, law, rhetoric, political sciences, ethics, history, yes; mathematics, no. (Latour 2008: 445) The Greek and Hellenistic mathematicians were a handful of specialists talking with and writing to each other around the Mediterranean basin, and no one else cared: —with one exception: the Plato-Aristotelian tradition. But what did this tradition (itself very small at the time) take from mathematicians? . . . Only one crucial feature: that there might exist one way to convince which is apodictic and not rhetoric or sophistic. The philosophy extracted from mathematicians was not a fully fledged practice. It was only a way to radically differentiate itself through the right manner of achieving persuasion. (Latour 2008: 445) Latour overstates his case. The philosophical tradition took a good deal from mathematics: what about the golden mean, for example, or the profound role of proportion in ethical theory? That is irrelevant to Latour’s case. He proposes that the philosophers focussed on proof in order to differentiate themselves from the common herd. Thus their use of proof as above rhetoric was nothing more than a rhetorical trick. Latour pays little heed to the ways in which the philosophers were profoundly impressed by the human capacity to prove. They were, as I like to put it, bowled over by demonstrative proofs. In consequence they vastly exaggerated the potential of proof. It is easy to argue that the ensuing theory

What Makes Mathematics Mathematics? 97 of knowledge impeded the growth of scientific knowledge from the time of Archimedes to the time of Galileo. To defeat the lust for demonstrative proof, we needed another crystallization. Or in Kant’s phrase, we needed the ‘intellectual revolution’ that he associated with Galileo and Torricelli. That was the discovery of other human talents—not purely intellectual ones—and it led to the laboratory style of scientific thinking. The defi nitive history of that crystallization was Leviathan and the Air-Pump, the very book that Latour rightly pairs with Netz’s Shaping of Deduction. Let us here agree with Latour: Plato kidnapped a certain idea of proof and made it a dominant theme in Western philosophical thought. But let us not grant to Latour the idea that proof is unimportant. Let us not allow Latour to kidnap Netz, that is, to allow us to forget Netz’s own fundamental concern, cognitive history (which Latour barely mentions). On the other hand, let us extend Latour’s insight. Kant codified, for the modern world, Plato’s kidnapping of mathematics. He made the a priori, the apodictic, and the necessary the hallmarks of mathematics, even though they are noticeable only here and there in the motley of mathematical activity. This leads us, for a fi nal observation about ancient times, back to Plato’s own demarcation of mathematics.

15 PLATO ON THE DIFFERENCE BETWEEN PHILOSOPHICAL AND PRACTICAL MATHEMATICS An important tradition in reading Plato on mathematics derives from Jacob Klein (1968). He argued that Plato made a fundamental distinction between the theory of numbers and calculating procedures. Here is a brief summary of the idea, due to one of Klein’s students: Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who ‘must learn the art of numbers or he will not know how to array his troops.’ The philosopher, on the other hand, must be an arithmetician ‘because he has to arise out of the sea of change and lay hold of true being’. (Boyer 1991: 86) We need not subscribe either to the terminology or the details of the interpretation to propose that (‘real’) mathematics, for Plato, did not include the arithmetic we learned in school, and later applied in business transactions, or ordering supplies for the troops. A redescription, owing more to Netz than to Klein, would be that Euclid’s Elements made a decision, to emphasize diagrammatic proofs rather than numerical examples. Hence

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despite the depth of some work on numbers to be found in Apollonius and Archimedes, there was no tradition of ‘advanced arithmetic’ in antiquity, in the way in which there was ‘advanced geometry’.5 Plato, then, put to one side the daily uses of arithmetic in technologically and commercially advanced societies such as those of Greece or Persia. Those uses are what Klein for his own reasons called ‘logistic’. In my opinion we should avoid the notion that computation is for practical affairs in ‘the sea of change’. That is the philosophical gloss of appearance and reality all over again. A primary point, closer to the experience of doing or using mathematics, is that computation is algorithmic. It proceeds by set rules. One does not understand a calculation: one checks that one has not made a slip. There is no experience of proof as in the theory of numbers or geometry. Quite possibly there were manuals that taught how to calculate, complete with shortcuts. They would be comparable to what we sophisticates dismiss as ‘cookbooks’, mere ‘how to do it’ instructions that do not convey insight or understanding of ‘how it works’. We may conjecture that that sort of text has not been preserved, partly because on Platonic and then Aristotelian authority, it did not present ‘science’, scientia. Just possibly, if there had been a classical text of advanced arithmetic in antiquity, the questions of a priori knowledge, apodictic certainty, and necessity, would have been posed in that context. I like to imagine that those questions might, instead, have been exposed as pseudo-questions, at least in the context of the theory of numbers. These cursory remarks suggest that Plato (or his heirs) created a disciplinary boundary between mathematics, the science that every philosopher must master, and computation, the technique of commerce and the military. This bears some relationship to the recent distinction between pure and applied mathematics, but the fundamental difference is that the one involves perspicuous proof, insight, and understanding, while the other involves routine. It is important to add that, from the perspective of twentieth century British analytic philosophy, with its talk of ‘puzzlement’ in philosophy, the results of calculation are not ‘puzzling’, in the way that proof can be experienced as puzzling (cf. Hacking 2000: §4.3). Notice also that in Plato’s vision (version Klein), there is far less of a motley of mathematics than in ours—because the routine computational side of mathematics is not (real) mathematics at all.

16

PURE AND MIXED MATHEMATICS

Francis Bacon was his usual prescient self when he devised that now abandoned term, ‘mixed mathematics’, which appears at the end of the OED defi nition.6 He was captured by the powerful image of the Tree of Knowledge. Every ‘branch’ (as we still say) of knowledge had to have its place on a

What Makes Mathematics Mathematics? 99 tree. He needed a branch of the main limb of mathematics on which optics and mathematical astronomy could flourish, for they were mathematics in the older sense of the term, as noticed by the OED. These he called mixed, which also included music, architecture, and engineering. They were not mixed, I think, because they mixed deduction and observation. It was rather a matter of the sphere to which they applied. Mixed mathematics was not pure mathematics ‘applied’ to nature, but an investigation of the sphere in which the ideal and the mundane were intermingled. Both the mixed and the pure were that part of natural philosophy that fell under metaphysics, viz. the study of fi xed and unchanging relations. The Enlightenment was an era of classification, where Natural History had a proud place as the science of nature observed. Hierarchical classifications, which we now conceive as branching trees, were the model for all knowledge, be it of minerals or diseases; the tree was also the model for the presentation of knowledge itself. Bacon’s Tree of Knowledge was hardly new; Raymond Lull’s was more graphic. We now think of tree-diagrams as one of the most efficient ways to represent certain kinds of information, but preserved treediagrams begin to appear surprisingly late in human history (Hacking 2007). By Bacon’s time, however, they flourished in genealogy, logic, and many other fields, and were cast in wonderful glass on cathedral windows. That was the past: Bacon’s own Tree of Knowledge was a benchmark for the future. The image of a branching Tree of Knowledge was to persist for centuries after Lull and Bacon. It is most notably incarnate in D’Alembert’s preface to La grande encyclopédie. It is a prominent pull-out page of Auguste Comte’s Cours, the massive 12-year production that continued the encyclopedic project of systematizing all knowledge and representing its growth in both historical and conceptual terms. The Tree of Knowledge that was planted so fi rmly in the early modern world by Francis Bacon has long been institutionalized in the structure of our universities with their departments and faculties.

17

PROBABILITY—SWINGING FROM BRANCH TO BRANCH

Many a new inquiry had to be forced on to the tree. Where would the Doctrine of Chances, a.k.a. the Art of Conjecturing, fit? It was by defi nition not about the actual world, nor about an ideal world. It was about action and conjecture; it was the successor to a non-theory of luck. There was no branch on a Tree of Knowledge on which to hang it. Probability was uneasily declared a branch of mixed mathematics, less because of its content than because of its practitioners, such as the Bernoullis, who were mathematicians par excellence. The mixed, as we shall see, morphed into the applied. Hence the residual place for the ‘theory of probabilities’ as ‘mixed or applied’ mathematics alongside astronomy and physics in the OED entry.

100 Ian Hacking The Tree of Knowledge became the tree of disciplines. This may have a somewhat rational underlying structure, of the sort at which Bacon or D’Alembert aimed, but it is largely the product of a series of contingent decisions. This can be nicely illustrated by the location of probability theory in various sorts of institutions around the world. It was once a paradigm of the mixed, so you would expect it to continue as applied mathematics. That is certainly not what happened in Cambridge, where the Faculty of Mathematics is divided into two primary departments. One is Applied Mathematics and Theoretical Physics, the home of Newton’s Lucasian chair. The other is the Department of Pure Mathematics and Mathematical Statistics. Probability appears to have jumped from branch to branch of the Tree of Knowledge. In truth, to continue the ancient arboreal metaphor, it is an epiphyte. It can lodge and prosper anywhere in a tree of knowledge, but is not part of its organic structure at all.

18

PURE AND APPLIED

There is no space, here, to adumbrate the transition of nomenclature from ‘mixed’ mathematics to ‘applied’ mathematics. Perhaps the switch was from an idea of mixing mathematics and the study of nature, to one of applying mathematics to nature. That picture may well be too anachronistic or at least too simple a vision. Galileo’s own famous image is a compelling alternative. The Book of Nature is written in the Language of Mathematics. Galileo did not apply abstract structures to nature. He found the structures in nature, and articulated their properties, thereby reading the Book of Nature itself. Husserl (1936) rightly seized upon what Galileo was doing as radically new, and said that Galileo mathematized nature. Galileo might have retorted that Husserl had things upside down: ‘I did not mathematize Nature, for she is already mathematical, and waiting to be read’. Galileo’s contribution may have been, as Netz puts it, a footnote to Archimedes (Netz and Noel 2007: 26). It was certainly not a footnote to Plato. Galileo had no truck with Plato’s conception of mathematics as outside this world. I realize that this statement fl ies in the face of a received tradition established by Alexandre Koyré, according to which Galileo was permeated by Platonism. Let us compromise, and say that in the world of Galileo, mathematics had an entirely new role. The situation looks more straightforward half a century later. Newton distinguished practical from rational mechanics. He took geometry to be a limiting case of practical mechanics, important to builders and architects. Geometry, the very possibility of which so astonished Plato, was placed alongside the practical arts, which Plato did not count as mathematics at all. There is little reason to think that Newton, the greatest mathematician of his age, cared much about the phenomenon or experience of proof

What Makes Mathematics Mathematics? 101 which Plato had made central to his fetishism of mathematics. To continue Latour’s metaphor, slightly tongue-in-cheek, we may venture that Galileo and Newton liberated mathematics from the philosophical bonds in which kidnapper Plato had enslaved it. Newton’s rational mechanics was among other things the general theory of motion, and hence of what is constant underneath ever-changing Nature. That can be presented as conforming to Plato’s imperative, to discover the reality behind appearance. Whatever it was, it was mathematics, set out in a book with an unambiguous title: Philosophiæ Naturalis Principia Mathematica, the mathematical principles of natural philosophy.

19

PURE KANT

‘Pure’—rein—evidently plays an immense role in Kant’s fi rst Critique, starting with its title. The primary contrast for both the English and the German adjectives is ‘mixed’.8 Hence Bacon’s branching of mathematics into pure and mixed. The next, moralistic sense of being free from corruption or defilement, especially of a sexual sort, comes a close second. At the start of his rewritten Introduction for the second edition of the fi rst Critique, Kant emphasizes what, for him, was the primary contrast: ‘The Distinction Between Pure and Empirical Knowledge’ (Kant 1787: B 1). Kant’s question, which Russell repeated with such enthusiasm, was ‘How is pure mathematics possible?’ What contrasts with ‘pure’ on this occasion? We hear ‘applied’. Galileo and Newton did not speak of applied mathematics. Kant’s opposite of pure mathematics was empirical.9 In fact Kant asked a pair of questions, one after the other: How is pure mathematics possible? How is pure science of nature (Naturwissenschaft) possible? (Kant 1787: B 20) Today we are puzzled, and some are baffled, by the idea of a pure science of nature. In his footnote Kant clearly contrasts it to ‘(empirical) physics’.10 Kant cites, as an example of pure science of nature, primary propositions ‘relating to the permanence in the quantity of matter, to inertia, to the equality of action and reaction, etc’. On one reading, Kant is talking about Newtonian mechanics. Kant embedded pure mathematics in the Transcendental Aesthetic, the launching pad for his entire theory of knowledge. Plato had made mathematics a matter of Ideas in a realm other than that of appearance. Kant made it part of transcendental idealism, and arithmetic and geometry conditions of all possible experience. This was a radical innovation, and yet a continuation of Plato’s leitmotif. Kant was restoring a Platonic vision of pure mathematics as something utterly separate and absolutely fundamental to the nature of

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knowledge. Kant was a kidnapper too. He kidnapped mathematics from the mathematicians by insisting that some of what they did was pure. The great mathematicians of the generation that flourished in the era of the fi rst Critique, men such as Lagrange, Legendre, and Laplace, did not see things that way. They were mathematicians. In general the scientists of that era made no difference between pure and applied. It was the tidy Kant who put a category of pure mathematics up front. That is the core truth behind Detlefsen’s starting point in the Routledge Encyclopedia, the statement that during the nineteenth and twentieth centuries, the most influential ideas in the philosophy of mathematics have been those of Kant. The philosophy of mathematics, as many of us understand it, starts from an unquestioned assumption that there is pure mathematics. We then proceed with Plato’s and Kant’s vision as something to accept, to modify, to explain, or to reject. The philosophy of mathematics is implicitly about the philosophy of pure mathematics, with a coda, asking how some of it is so applicable to nature. The separation of the pure from the applied could not happen on Kant’s say-so alone. It also called for some highly contingent events in disciplinary organization. 20

APPLIED MATHEMATICS

Our idea of applying pure mathematics to nature should not be read back into Kant or Newton. We do have a convenient benchmark for the distinction between pure and applied. In 1810 Joseph Gergonne (1771–1859) founded what is usually regarded as the first mathematics journal, Annales de mathématiques pures et appliquées. Most of the articles were contributions to geometry, Gergonne’s own field of expertise. German followed suit in 1826, when A. L. Crelle (1780–1855) founded the Journal für die reine und angewandte Mathematik. The focus was quite different. Crelle published most of Niels Henrik Abel’s (1802–1829) papers that transformed analysis. The nominal distinction between pure and applied did not take hold for some time. I very much doubt that readers of Gergonne’s and Crelle’s journals knew which articles were pure and which were applied, if, indeed, they asked the question at all. Lagrange inventing Lagrangians, and even Hamilton inventing Hamiltonians, did not think they were applying mathematics to nature. They were investigating nature mathematically. Only after the fact do we abstract the mathematics from nature and ‘purify’ it, and then, retroactively, speak of applying the pure mathematics to scientific problems. 21

PURE MATHEMATICS

Here I should like to be not just insular but local. The analytic tradition in the philosophy of mathematics is properly traced back to Frege, but a lot of the stage-setting is grace of Whitehead and Russell. Their opus is a

What Makes Mathematics Mathematics? 103 third benchmark. Compare their book with Newton’s. Two great works are titled Principia Mathematica. They are entirely different in content and project. Perhaps nobody really believed in Whitehead and Russell’s great book. Possibly only the authors read all three volumes. But even the great German set-theorists set themselves up with that work as a monument, even if it turned out, to everyone’s surprise, to be essentially incomplete. So it is worth the time to consider the mathematical milieu that Whitehead and Russell took for granted—and their conception of pure mathematics for which they hoped to lay the foundations. At least in British curricula we can locate the point at which ‘Pure Mathematics’ became a specific institutionalized discipline. In 1701, Lady Sadleir had founded several college lectureships for the teaching of Algebra at Cambridge University. In 1863, the endowment was transformed into the Sadleirian Chair of Pure Mathematics, whose fi rst tenant was Arthur Cayley. This coincided with an important shift in the teaching of mathematics. The old Smith’s prize, founded in 1768, was the way in which a young Cambridge mathematician could establish his genius. In the old days, up until 1885, it was awarded after a stiff examination in what would now be called applied mathematics. One Wrangler who went on to tie for the Smith’s Prize became the greatest British mathematician of the nineteenth century. But what we call mathematics has changed. We do not call him a mathematician but a physicist. I mean James Clerk Maxwell. Many names hallowed in the annals of physics, such as Stokes, Kelvin, Tait, Rayleigh, Larmor, J. J. Thomson, and Eddington won the Smith’s prize for mathematics.11 Yet after 1863, what was called mathematics at Cambridge was increasingly pure mathematics rather than Natural Philosophy. It was within this conception of mathematics that Russell came of age. Likewise it was in this milieu that G. H. Hardy became the preeminent local mathematician, whose text, Pure Mathematics, became a sort of official handbook of what mathematics is, or how it should be studied, taught, examined, and professed at Cambridge. Russell’s vision of mathematics was not determined by Hardy’s, or vice-versa, but the two visions are coeval, a product of a disciplinary accident in the conception of mathematics.

22

CONTINGENCY, NECESSITY, AND NEUROLOGY

I have sketched only the beginning of an argument, that what is counted as mathematics depends in part on a complex and very contingent history. I do not mean to imply that the history could have gone any way whatsoever. It was constrained by its content, and by human capacities. They no longer constrain in the same way. The advance of fast computation is changing the entire landscape of human knowledge, including that of mathematics. That is a topic for the future. Here I have been concerned with the past.

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Our picture of the philosophy of mathematics is of philosophical reflection on a defi nite and predetermined subject matter. I suggest that the subject matter itself is much less determinate than we have imagined. This does not undercut the debate between the two attitudes I have mentioned, (1) platonic and (2) neurobiological, or the traditional philosophical debate between ‘realists’ and ‘antirealists’ of mathematics. It will surely go on as before. I urge only that the more difficult but perhaps more answerable question should now become: how have the platonic and neurobiological constraints jointly interacted with the contingent history of mathematics from ‘Thales’ to now?

NOTES 1. My own exploration of that topic owes a great deal to discussions with Smiley when we were colleagues. The upshot was Hacking (1979). This essay was a long-delayed philosophical spin-off from a research paper (Hacking 1963) that profited from Smiley’s constant advice. 2. This phrase is the title of J. E. Littlewood’s (1953) charming potpourri of mathematical anecdote and examples. 3. I got the phrase ‘genetic envelope’ from a conversation with Changeux; I do not think he has used it in print. 4. The idea of a crystallization of a style of scientific thinking was introduced in Hacking (2009), and will be developed in a continuation of that book. My Crombian theme of styles of scientific thinking was launched long ago (Hacking 1982, 1992). 5. These remarks are my version of a short discussion with Netz in March 2009. 6. The assertion that the term ‘mixed mathematics’ is original to Bacon is due to Brown (1991). 7. Two late quotations from the OED speak of applying, but differ in their meaning. In 1706: ‘Mixt Mathematicks, are those Arts and Sciences which treat of the Properties of Quantity, apply’d to material Beings, or sensible Objects; as Astronomy, Geography, Navigation, Dialling [sundials], Surveying, Gauging &c.’ In 1834, in Coleridge: ‘We call those [sciences] mixed in which certain ideas of the mind are applied to the general properties of bodies’. 8. OED: Without foreign or extraneous admixture: free from anything not properly pertaining to it; simple, homogeneous, unmixed, unalloyed. Grimm’s Deutsches Wörterbuch: frei von fremdartigem, das entweder auf der Oberfläche haftet oder dem Stoffe beigemischt ist, die eigenart trübend. 9. I know of only one occasion where he spoke of applied mathematics, namely in his Lectures on Metaphysics (delivered from the 1760s to the 1790s). ‘Philosophy, like mathematics as well, can be divided into two parts, namely into the pure and into the applied’ (Kant 1790–1: 307). 10. Kant wrote ‘eigentilichen (empirischen) Physik’, which Kemp Smith renders ‘(empirical) physics, properly so called’. Like the English noun ‘physics’, Physik in Kant’s time still meant natural science in general. Kant might have meant something more like ‘real (empirical) physics’. 11. Partly thanks to T. J. Smiley, by 1961 an essay in modal logic could win a Smith’s prize. Robert Smith, Plumian Professor in Astronomy, doubtless turned over in his grave. He intended the encouragement of Mathematics and Natural

What Makes Mathematics Mathematics? 105 Philosophy. It may amuse students of today’s economy to know that Smith endowed the prize with profits from the South Sea Bubble. The structure of the competition for the Smith’s and related prizes was reorganized in 1998.

REFERENCES Baker, G.P. and Hacker, P.M.S. (1980) Wittgenstein: Understanding and Meaning, An Analytical Commentary of the Philosophical Investigations, vol. 1, Oxford: Blackwell. 2nd edn, extensively revised by P.M.S. Hacker (2005). Bambrough, R. (1961) ‘Universals and family resemblances’, Proceedings of the Aristotelian Society 61: 207–22. Blanchette, P.A. (1998) ‘Realism in the philosophy of mathematics’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 8, London and New York: Routledge, pp. 119–24. Boyer, C.B. (1991) A History of Mathematics, 2nd edn, revised by U.C. Merzbach, New York: Wiley. Brown, G.I. (1991) ‘The evolution of the term “mixed mathematics”’, Journal of the History of Ideas 52: 81–102. Connes, A. and Changeux, J.P. (1995) Conversations on Mind, Matter and Mathematics, Princeton: Princeton University Press. Originally published as Matière à pensée, Paris: Odile Jacob, 1989. Crombie, A.C. (1994) Styles of Scientific Thinking in the European Tradition: The history of argument and explanation especially in the mathematical and biomedical sciences and arts, 3 vols, London: Duckworth. Detlefsen, M. (1998) ‘Mathematics, Foundations of’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 6, London and New York: Routledge, pp. 181–92. Fodor, J. (1983) The Modularity of Mind: An Essay on Faculty Psychology, Cambridge, MA: MIT Press. Hacking, I. (1963) ‘What is strict implication?’, Journal of Symbolic Logic 28: 51–71. . (1979) ‘What is logic?’, The Journal of Philosophy 76: 285–319. . (1982) ‘Language, truth and reason’ in M. Hollis and S. Lukes (eds) Rationality and Relativism, Oxford: Blackwell, pp. 48–66. . (1992) ‘“Style” for historians and philosophers’, Studies in History and Philosophy of Science 23: 1–20. . (2000) ‘What mathematics has done to some and only some philosophers’, in T.J. Smiley (ed.) Mathematics and Necessity, Oxford: Oxford University Press for the British Academy, pp. 83–138. . (2006) Another New World is Being Constructed Right Now: The Ultracold, preprint 316, Berlin: Max-Planck Institut für Wissenschaftsgechichte. . (2007) ‘Trees of logic, trees of porphyry’, in J. Heilbron (ed.) Advancements of Learning: Essays in Honour of Paolo Rossi, Florence: Olshki, pp. 146–197. . (2009) Scientific Reason, Taipei: National Taiwan University Press. Horsten, L. (2007) ‘Philosophy of Mathematics’, in E.N. Zalta (ed.) The Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/archives/win2007/ entries/philosophy-mathematics/. Husserl, E. (1936) ‘Der Ursprung der Geometrie als intentional-historisches Problem’, an appendix in D. Carr (trans.) The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy, Evanston: Northwestern University Press, 1970, pp. 353–78.

106 Ian Hacking Kant, I. (1787) Critique of Pure Reason, page references are to the translation by N. Kemp Smith, London: Macmillan, 1929. . (1790–1) ‘Metaphysic L 2’ in his Lectures on Metaphysics, K. Ameriks and S. Naragon (trans), Cambridge: Cambridge University Press, 1997, pp. 299– 354. Klein, J. (1968) Greek Mathematical Thought and the Origin of Algebra, E. Brann (trans.), Cambridge, MA: MIT Press. Krieger, M.H. (1987) ‘The physicist’s toolkit’, American Journal of Physics 55: 1033–8. . (1992) Doing Physics: How Physicists Take Hold of the World, Bloomington and Indianapolis: Indiana University Press. Latour, B. (2008) ‘The Netz-works of Greek deductions’, Social Studies of Science 38: 441–59. Littlewood, J.E. (1953) A Mathematician’s Miscellany, London: Methuen. Lloyd, G. (1990) Demystifying Mentalities, Cambridge: Cambridge University Press. Moore, A.W. (1998) ‘Antirealism in the philosophy of mathematics’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 1, London and New York: Routledge, pp. 307–11. Netz, R. (1999) The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History, Cambridge: Cambridge University Press. Netz, R. and Noel, W. (2007) The Archimedes Codex: Revealing the Secrets of the World’s Greatest Palimpsest, London: Wiedenfeld and Nicolson. Russell, B. (1912) The Problems of Philosophy, page references are to the 1946 edn, London: The Home University Library. Rosch, E. (1973) ‘Natural categories’, Cognitive Psychology 4: 328–50. Shapin, S. and Schaffer, S. (1985) Leviathan and the Air-Pump: Hobbes, Boyle and the Experimental Life, Princeton: Princeton University Press. Wittgenstein, L. (1978) Remarks on the Foundations of Mathematics, 3rd edn, G. H. von Wright, R. Rhees and G.E.M. Anscombe (eds), G.E.M. Anscombe (trans.), Oxford: Blackwell.

6

Smiley’s Distinction Between Rules of Inference and Rules of Proof Lloyd Humberstone

0

PREAMBLE

Since my days as an undergraduate and then a graduate student in England in the period 1968–1974 I have been an appreciative consumer of Timothy Smiley’s work, though the fi rst time I heard him referred to as Tim (or had a chance to meet him personally) was on the occasion of an Australasian Association for Logic conference organized by Graham Priest at the University of Western Australia in 1983. At that conference, I presented a version of the paper abstracted as Humberstone (1984), on the subject of the unique characterization of connectives, which is intimately connected to a theme of Smiley (1962), pursued also from a somewhat different angle in another classic paper of the same vintage, Belnap (1962). For a BPhil thesis at the University of York, I worked on a theme from Smiley (1963), trying to see how far his idea could be taken, of reducing the number of modal operators (broadly understood) to one—which we could loosely think of as expressing a kind of absolute necessity—in terms of which other modal notions could be processed as forms of relative necessity (absolute necessity given this or that statement, formally represented by a sentential constant). A descendant of this work appeared as Humberstone (1981), fi xing a problem (pointed out by Kit Fine) in my early efforts with the aid of a suggestion from Dana Scott (my supervisor for yet another BPhil thesis, this time at Oxford); related and subsequent developments are surveyed in Humberstone (2004). Some ideas from Smiley (1996) on negation and rejection are taken up in Humberstone (2000b). Footnote 1 of Smiley (1962), concerning substitution and replacement, inspired much of Humberstone (in preparation). The topic for what follows—a contrast between two kinds of rules, especially as rules appear in the axiomatic approach to logic—was (to my knowledge) fi rst aired in Smiley (1963). While it has occasionally been alluded to in subsequent years, it has never received the sustained attention it deserves. Here I express my appreciation for Smiley’s rewarding logical work by giving it at least some of that attention.

108 1

Lloyd Humberstone

SMILEY ON RULES AND THE DEDUCTION THEOREM

For a proof system in the axiomatic or ‘Hilbert’ style for a given logic, in which certain formulas are laid down as axioms and certain rules are given for deriving theorems from the given axioms, an ancillary consequence relation is defi ned,1 often called deducibility and associated with the Deduction Theorem. In the end, ‘deducibility’ may in the end turn out to be the wrong term (see Note 20 below). In any case, the direction of the relation is wrong here: we should say, more accurately, that the consequence relation in question holds between a set of formulas and an individual formula when the latter is deducible from the former. In the simplest case, illustrated here for the implicational fragment of intuitionistic propositional logic, one takes as axioms all formulas instantiating the following schemata. (A1) (A2)

A → (B → A) (A → (B → C)) → ((A → B) → (A → C))

and as the sole rule, Modus Ponens: A, A → B / B. (Premises before the slash, conclusion after it.) One then defi nes the consequence relation alluded to above, IL , say, thus: Γ IL B if and only if B can be obtained by a series of applications of the rule Modus Ponens starting from formulas which are either axioms or elements of Γ. Finally, then the Deduction Theorem for IL says that for any set of formulas Γ and any formulas A, B: If Γ, A

IL

B then Γ

IL

A→B

This result (originally due to Herbrand and Tarski) is proved by induction on the number of applications of Modus Ponens used to obtain B from Γ ∪ {A} together with the axioms. 2 Strangely enough (A2), though originally used as an axiom by Frege well before the Deduction Theorem was explicitly contemplated (in Frege 1879), seems exactly tailor-made for enabling the inductive step of this proof to go through. If we abbreviate ‘∅ IL B’ to ‘ IL B’ (as suggested in Note 1), then the above defi nition of ‘Γ IL B ’ has the expected effect that ‘ IL B’ amounts to the claim that B is provable on the basis of the above axiomatization, and indeed can be taken as defi ning provability, since it asserts that B can be obtained by some number of applications of Modus Ponens from the axioms (there being nothing on the left of the ‘ ’ to appeal to). On the other hand, if we fi rst defi ne provability in these terms, we might go on to characterize a consequence relation, *IL , say, in the following terms, in * which ‘theorem’ is used in place of ‘provable formula’ for brevity: Γ IL B if and only if B can be obtained by a series of applications of the rule Modus Ponens starting from formulas which are either theorems or elements of Γ. This differs from the earlier definition only in that the word ‘axioms’ has

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been replaced by the word ‘theorems’. One easily sees that for all formulas B and sets of formulas Γ: Γ

IL B

if and only if Γ

* IL

B,

so the above formulation of the Deduction Theorem (for this system) could equally well be formulated with ‘ *IL ’ in place of ‘ IL ’. Now, as is well known, the presence of additional logical vocabulary (such as connectives ∧, ∨, ¬) in the language, or of additional axioms—for instance giving the vocabulary concerned its intuitionistic properties, or giving that vocabulary (→ included) the properties associated with it in classical logic—does not present any obstacle to conducting the proof of the Deduction Theorem for the richer logic. We will soon be considering the addition of a modal operator ̞ to the language. But when it comes to adding further rules some care is needed, as Smiley (1963) observed. Alternatively, since for many purposes it is convenient to consider axiom schemata such as (A1) and (A2) above as 0-premise rules, one could put this point in terms of adding further proper rules, where a proper rule is understood as an n-premise rule with n ≥ 1. (On this way of speaking, an axiom schema is a 0-premise sequential rule, with sequentiality as defined in the following section.) For what follows, however, it is more convenient to exclude 0-premise rules and the word ‘proper’ is occasionally used for emphasis only. Where we have an axiomatization using (proper) rules in addition to Modus Ponens the question arises as to what it would be for the Deduction Theorem to be satisfied. One way of responding would be to keep the definition exactly as above, and say the Deduction Theorem for the envisaged system requires that when B can be obtained by applications of Modus Ponens from the axioms and the formulas in some set Γ ∪ {A}, then A → B can be similarly obtained from the axioms and the formulas in Γ. A second reaction would be to focus not on the rule Modus Ponens itself in the original case but on the fact that this rule exhausted the set of proper rules used in the axiomatization, which gives rise to a second abstraction from the original case. Now we say that Deduction Theorem holds for the envisaged extended system provided that whenever a formula B can be obtained by applications of any of the primitive rules of the system from the axioms and the formulas in some set Γ ∪ {A}, then A → B can be obtained, by applying some of those rules, from the axioms and the formulas in Γ. Let us put this another way. Say that a consequence relation satisfies the Deduction Theorem just in case for all sets of formulas Γ and formulas A, B, in the language of , we have: (DT)

If Γ, A

B then Γ

A→B

For an axiomatic system S to satisfy the Deduction Theorem is for a certain consequence relation associated with S to satisfy (DT); however, we still

110

Lloyd Humberstone

use the phrase ‘Deduction Theorem’ even when dealing with a consequence relation presented otherwise, as long as the condition (DT) is satisfied.3 The question is how to pass from an axiomatic system to the ‘associated’ consequence relation of interest. According to the fi rst of the two reactions just described, even when S has primitive rules other than Modus Ponens, the relevant consequence relation fi xes the consequences of a set as the results of applying only Modus Ponens to theorems of the logic and formulas in the set. According to the second reaction, the crucial consequence relation is characterized instead by replacing the privileged position of Modus Ponens and allowing derivability using any of the primitive rules of S. Smiley (1963: 115) discusses the modal logic S2, axiomatized using various schemata and, alongside Modus Ponens (there called R1), the further rule, R2: ̞(A → B) / ̞(̞A → ̞B). In defi ning the associated consequence relation, Smiley allows only the use of Modus Ponens and not also of this modal rule, in the defi nition of what it is for a formula to be deducible from a set of formulas. He makes the following remarks, in which the phrase ‘for material implication’ is a reminder of the fact that the Deduction Theorem makes reference to the connective ‘→’ (actually notated differently in Smiley 1963, as ‘⊃’), and the phrase ‘both systems’ is occasioned by the fact that Smiley is discussing not only S2 but another system, called by him OS2, the details of which do not matter for present purposes. An almost immediate consequence of these defi nitions is that the deduction theorem for material implication holds in both systems. There are asser tions in the literature that the deduction theorem fails for S2, but they are the result either of treating R2 as a straightforward rule of inference (as in Moh 1950, p. 61) or else of a mistake.4 (Smiley 1963: 115) At fi rst sight, this may seem to be nothing but an endorsement of the fi rst of the two reactions distinguished above: we give a privileged position to Modus Ponens in fi xing the consequence relation required to satisfy (DT). This would not be very satisfying as a general account, placing, as it does, so much emphasis on a particular rule. Nor is it in fact Smiley’s position, as the words ‘treating R2 as a straightforward rule of inference’ in the above quotation betray. So we have to back up a little in Smiley (1963) and see what this phrase means. On the page before that on which the above passage occurs, Smiley writes as follows: In formulating OS2 and S2 in this way it is intended that the rule R2 is not to be used unrestrictedly, but only in the generation of further theorems from theorems. In this it resembles the rule of substitution for propositional variables, the rule ‘from A infer A’ in S4, or indeed the rule of generalisation in the predicate calculus. These might all be

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called ‘rules of proof’ as opposed to proper ‘rules of inference’ like R1.5 (Smiley 1963: 114) As Smiley proceeds to explain, the difference between the two kinds of rules emerges when the deducibility relation—the consequence relation that is pertinent to the Deduction Theorem, that is—associated with the axiomatic system is considered. In his own words: ‘B is deducible from A1, . . . , An if there is a sequence of formulae ending in B, of which every member either is one of A1, . . . , An or else is a theorem or else follows from preceding formulae by R1’. Incorporating the rules of proof/rules of inference distinction to give a general formulation of the idea, we have an axiomatic system as a triple 〈Ax, R, Rinf〉 with Ax a set of formulas (the axioms) of some language, and R and Rinf two sets of rules whose premises and conclusions are formulas of that language, and with R inf ⊆ R . The subscript on ‘Rinf ’ is of course intended to suggest ‘(rules of) inference’; you may call the rules in R ‘rules of proof’, or alternatively reserve this label for, as it were, rules of proof proper (‘mere rules of proof’, we might equally well say): those in R but not in Rinf, or more formally, those in R \ Rinf. The former course is adopted here. For S = 〈Ax, R, R inf〉, the theorems of S are the formulas derivable from Ax by means of the rules in R, while the deducibility consequence relation associated with S, S , is defi ned by: Γ S B just in case B can be derived from theorems of S together with formulas in Γ by the application * of rules in Rinf. This is like the earlier defi nition of ‘ IL ’, in that it appeals to an already available defi nition of theoremhood or provability. But if a formulation along the lines given for IL above is preferred, it can proceed in the following terms: Γ S B if and only if B can be obtained from formulas in Γ ∪ Ax by applying the rules in R subject to the condition that when premises figure in an application of a rule in R \ Rinf, none of those premises should be an element of Γ or have been derived by earlier applications of the rules to a formula in Γ. As Schurz (1994: 387) puts it: the rules of proof proper should be ‘applied only to those members of the proof which do not depend on premises in Γ ’.6 While the conceptualization of axiomatic systems along the 〈Ax, R, R inf〉 lines here distilled from Smiley’s discussion is occasionally seen in the literature—for example, we fi nd it in Gabbay (1981: 9), as the distinction between ‘consequence rules’ (rules of inference) and ‘provability rules’ (rules of proof)—by far the more prevalent approach, evidenced especially in one stream of Polish logical work, is quite different.7 In this work, such a system is conceived of as a pair 〈Ax, R〉 and the induced consequence relation matches the second way of characterizing the Deduction Theorem described above: use of any of the rules in R, together with the formulas in Ax, is permitted for deriving consequences according to this consequence relation. We will take up the contrast between the Smileyinspired and the more widely prevalent conception of axiomatization in the next two sections.

112 2

Lloyd Humberstone

DERIVABLE/ADMISSIBLE: ANOTHER DISTINCTION

Anderson and Belnap (1975), discussing the addition of a modal operator (̞) for necessity to the then favoured system E of entailment (which is the intended reading of ‘→’ for the quotation below), address the rule of necessitation (A / ̞A) in the following terms: In the fi rst place we are led by a strong tradition to believe that the necessity of any theorem (of a formal system designed to handle the notion of logical necessity at all) should also be a theorem; unless this requirement is met, the system simply has no theory of its own logical necessities. For this reason we would like to have it be true that, whenever A is provable, then necessity of A is also provable. This condition could be satisfied by incoherent brute force, as it is for example in systems like M (Feys–von Wright), where a rule of necessitation is taken as primitive. It could equally well be satisfied by taking A → ̞A as an axiom. Both courses are equally odious, the latter because it destroys the notion of necessity, and the former because, if A → ̞A is neither true nor a theorem, then we ought not to have—in a coherent formal account of the matter—a primitive rule to the effect that ̞A does after all follow from A. This constraint would not bother us if we were simply trying to defi ne the set of theorems of E recursively in such a way that a digital computer, or some equally intelligent being, could grind them out. But our ambitions are greater than this; we would like to have our theorems and our primitive rules dovetail in such a way that if E says or fails to say something, we don’t contradict it or violate its spirit. (Note that neither →E nor &I does so). Nevertheless it should be true, as a lucky accident, so to speak, that whenever A is a theorem, ̞A is likewise.8 (Anderson and Belnap 1975: 235f) It doesn’t strictly make sense to say, as here, that ‘A → ̞A is neither true nor a theorem’, even when a particular formula A is specified, but presumably the authors intend the (informal) claim that if a statement does not in general entail its necessitation—see the start of §3—then, to pick up the quotation verbatim ‘we ought not to have—in a coherent formal account of the matter—a primitive rule to the effect that ̞A does after all follow from A.’ Well, perhaps we ought not to have a rule of inference to that effect, since we are supposing that the necessitation of a statement cannot in general be inferred from that statement, but how is this an objection to taking necessitation as a rule of proof? Could this passage from Anderson and Belnap be rewritten so as to take account of Smiley’s distinction? Would the principle behind it, given that distinction, be something along the following lines: one should not use rules of proof one does not endorse as rules of inference? This would mean a ban on the use of all the examples in Smiley’s list from the previous section: not just Necessitation, but also

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Uniform Substitution, and generalization-like rules in predicate logic. These last raise numerous complications of their own (issues about free variables and individual parameters) which are best avoided. Indeed in Smiley’s own more recent thoughts on the matter there is considerable distaste shown for doing propositional logic with propositional variables and a rule of Uniform Substitution.9 Here we ignore this change of heart and proceed with some reminders as to the semantic side of these rules (though Uniform Substitution will tend to take a back seat in the discussion). This will be attended to in §3, which presumes a basic familiarity with the (Kripke) semantics for normal modal logics. Before that, there is more to say about the passage from Anderson and Belnap quoted above. The concluding sentence of the quotation says that despite not wanting Necessitation as a primitive rule, ‘Nevertheless it should be true, as a lucky accident, so to speak, that whenever A is a theorem, ̞A is likewise.’ In other words, the rule should be admissible (i.e. the set of theorems should be closed under application of the rule).10 All the attention to whether the rule is primitive is completely beside the point: the contrast that counts here is between rules that are derivable (primitive or derived) on the one hand, and rules which are merely admissible (admissible but not derivable) on the other.11 Anderson and Belnap’s antipathy to the rule of Necessitation should be expressed not as an objection to its being primitive in an axiomatization, but to its being derivable at all. (We return briefly in §3 to the question of whether this objection is well-founded.) Let us review the notion of derivability as it applies to the simple conception of axiomatizations as pairs 〈Ax, R〉. We can think of a rule as the set of all its applications and, for an n-premise rule, take these applications to be (n + 1)-tuples of formulas of the language under consideration, the fi rst n positions occupied by premise-formulas and the fi nal position by the conclusion-formula.12 Such a rule is derivable on the basis of 〈Ax, R〉 just in case for any application 〈B1, . . . , Bn , C 〉 of the rule, we can obtain C from {B1, . . . , Bn} ∪ Ax by means of the rules in R. (This is just to say that B1, . . . , Bn C, for the consequence relation determined by 〈Ax, R〉 in the manner described in §1.) What becomes of this notion on the Smiley conception of axiomatic systems in the form 〈Ax, R, Rinf〉, in which a distinguished subset of the rules are deemed to be rules of inference and not just rules of proof? A derivable rule of proof should be, as in the simpler set-up just reviewed, any rule derivable on the basis of the reduct 〈Ax, R〉 of 〈Ax, R, Rinf〉. One might think that correspondingly, a derivable rule of inference should be a rule derivable on the basis of the reduct which discards rules of proof proper, i.e. on the basis of 〈Ax, Rinf〉. The idea is that we want to allow chaining together of inference steps without any admixture of proof steps which are not sanctioned as licensing inferences (by the given partition of R into Rinf and R \ Rinf). But the proposed defi nition does not implement the idea correctly, since it excludes the use of rules of proof proper in yielding theorems from which, together with the premises of an application of

114 Lloyd Humberstone the would-be derived rule of inference, yield the conclusion with the aid of primitive rules of inference. In this case, the rule of proof is not itself applied to those premises and should not count against the derivability of the envisaged rule. Accordingly, a correct defi nition of the derivability of a rule of inference ρ should instead have it that the conclusion of any application of ρ can be obtained from its premises by successive applications of rules in Rinf together with theorems provable on the basis of 〈Ax, R, Rinf〉. We could equally well say: provable on the basis of 〈Ax, R〉, since Rinf ⊆ R (and the selection of a particular subset of R as Rinf has no bearing on the set of theorems—only on the induced consequence relation). Before passing to a consideration of the semantic side of these matters, let us review and streamline the terminology with which we have been working. Let us describe an axiom system or axiomatization 〈Ax, R〉 of the simple-minded kind as representing the undifferentiated (more explicitly: ‘rule-undifferentiated’) approach to the subject, and a system 〈Ax, R, Rinf〉 in the Smiley-inspired style as representing the differentiated approach. Anything that can be done on the former approach can be done on the latter, since we allow the possibility that R = Rinf ; but the latter is more flexible in the manner already described as to how to associate a consequence relation with an axiomatic system. Although one usually speaks of a logic (or more generally a theory), understood as a set of formulas, as being axiomatized by an axiomatic basis, whether of the undifferentiated 〈Ax, R〉 type or of the differentiated type, it does no harm to speak of what we have been calling the associated consequence relation in either case as itself axiomatized by the basis in question.13 Reviewing the contrast, we say on the undifferentiated approach that the consequence relation 〈Ax, R〉 axiomatized by 〈Ax, R〉 is the least consequence relation (on the language in question) ′ such that: (1)

′ B for all B ∈ Ax, and

(2) Γ ′ B whenever for some A1, . . . , An ∈ Γ, 〈A1, . . . , An , B〉 ∈ ρ, for some ρ ∈ R, while on the differentiated approach we say that the consequence relation 〈Ax, R, R inf 〉 axiomatized by 〈Ax, R, R inf〉 is the least consequence relation (on the language in question) ′ such that: (1)*

′ B for all B such that

〈Ax, R〉

B, and

(2)* Γ ′ B whenever for some A1, . . . , An ∈ Γ, 〈A1, . . . , An , B〉 ∈ ρ, for some ρ ∈ Rinf. Note that (1)* here invokes the defi nition provided by (1) and (2) for the 〈Ax, R〉 reduct of the given 〈Ax, R, Rinf〉. It provides the theorems of the logic * axiomatized; compare the characterization of IL in §1. (Occasionally below,

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the set of such theorems will be referred to as the ‘formula logic’ concerned, for contrast with logics conceived of as themselves being consequence relations.) An n-premise rule ρ is derivable on the basis of 〈Ax, R〉 when for any 〈A1, . . . , An , B〉 ∈ ρ, we have A1, . . . , An 〈Ax, R〉 B. For the differentiated approach we have two notions, depending on whether the derivability of ρ as a rule of proof or the derivability of ρ as a rule of inference is at issue. In the former case, the defi nition is as before, using the reduct 〈Ax, R〉 of the axiomatization 〈Ax, R, Rinf〉 in question: we simply require that A1, . . . , An 〈Ax, R〉 B, whenever 〈A1, . . . , An , B〉 ∈ ρ. For the latter case, we have ρ derivable as a rule of inference provided that A1, . . . , An 〈Ax, R, Rinf 〉 B whenever 〈A1, . . . , An , B〉 ∈ ρ. Digression. Let us pause to consider the question of admissibility. Since the admissibility of a rule is a matter of the set of provable formulas being closed under the rule, one naturally associates admissibility with the ‘rule of proof’ side of the picture. Could separate sense be made of something’s being admissible as a rule of inference on the differentiated approach? Well, an alternative (though equivalent) defi nition of the admissibility of ρ on the basis of the undifferentiated axiomatization 〈Ax, R〉 is that 〈Ax, R〉 and 〈Ax, R ∪ {ρ}〉 have the same theorems. On the differentiated approach one says the same thing, with Rinf as an idle parameter: 〈Ax, R, Rinf〉 and 〈Ax, R ∪ {ρ}, Rinf〉 have the same theorems. This suggests a defi nition for the admissibility of ρ on the basis of 〈Ax, R, Rinf〉, namely that 〈Ax, R, Rinf〉 and 〈Ax, R ∪ {ρ}, Rinf ∪ {ρ}〉 axiomatize the same consequence relation.14 We could make such a defi nition, but it does not lead to anything of interest, since the admissible rules, so defi ned, would coincide with the derivable rules. End of Digression. The rules of interest here, still keeping Uniform Substitution to one side, are all of them sequential rules in the sense of Łoś and Suszko (1958), which means that for each rule, ρ, say, there is a sequence of formulas— sometimes called the skeleton of ρ—〈A1, . . . , An , B〉 such that the applications of ρ are precisely those 〈C1, . . . , Cn , D〉 for which there is some substitution s with 〈C1, . . . , Cn , D〉 = 〈s(A1), . . . , s(An), s(B)〉. For example, Modus Ponens and Necessitation have skeletons 〈p, p → q, q〉 and 〈p, ̞p〉, respectively.15 Note that for an axiomatization on either the differentiated or the undifferentiated approach, even when all the primitive rules (those in R or just those in Rinf) are sequential, the derivable rules are typically not sequential, since the union of any two n-premise derivable rules is a derivable rule. However, the derivable rules will still in this case be substitutioninvariant in the sense that every substitution instance of an application of the rule is an application of the rule.16 Even when the rules R are all sequential, so that it is effectively decidable whether or not a putative application of one of them is indeed such an application, the additional requirement that the set R, as well as the set Ax of axioms, should also be recursive, is often imposed to do justice to the idea that it should be effectively decidable whether or not a putative proof is a proof. An

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especially simple way of satisfying this demand arises from various notions of finiteness for an axiomatization. Say that a differentiated axiomatization 〈Ax, R, Rinf 〉 is finite if the sets Ax and R (and therefore also Rinf) are finite, and that 〈Ax, R, Rinf〉 is schematically finite if R is a finite set of sequential rules and Ax is the set of all substitution instances of a finite subset Ax0 of Ax. The idea in the latter case is that we avoid the use of uniform substitution by describing the (in general) infinite set Ax by means of finitely many axiom-schemata, arising from Ax0 by replacing distinct propositional variables occurring in the formulas thereof by distinct schematic letters. Since these definitions appeal only to Ax and R, we may take them over intact for undifferentiated axiomatizations 〈Ax, R〉. The notion of schematically finite axiomatizations will be employed in the following section.

3

SEMANTIC CONSIDERATIONS

Although the distinction between rules of proof and rules of inference was introduced by Smiley in discussion of what one might naturally consider to be a purely syntactical matter, the Deduction Theorem, there is a clear semantical motivation revealed by the choice of terminology. Informally, this can be illustrated in the case of Necessitation by considering an argument with premise, ‘John F. Kennedy was shot in Dallas’, say, and conclusion ‘It is (logically or metaphysically) necessary that John F. Kennedy was shot in Dallas’. This contrasts with an argument having as premises a conditional and its antecedent, and as conclusion its consequent. In the latter case we think of the conclusion as something which can legitimately be inferred from the premises, and in the former case of the conclusion as not being something to be inferred from the premise. This difference makes it natural to describe Modus Ponens as being—and Necessitation as not being—a rule of inference. It is equally natural to want to continue with a gloss on the difference which invokes some notion of truth-preservation as characterizing transitions in the former case and not in the latter. Indeed a main focus of interest in connection with rules has always been over differences in which semantic features they preserve,17 and the preservation behavior of the two rules mentioned in the Kripke semantics for normal modal logics is well known: Modus Ponens is locally truthpreserving in any model (preserves truth at any point—or ‘world’—in the model) while Necessitation is only globally truth-preserving (preserving the property of being truth-at-all-points in the model, for any model). Thus we can refine the connection between inference and truth-preservation implicit in Smiley’s classification of Modus Ponens and Necessitation (sometimes abbreviated to ‘Nec’ below) as positive and negative instances, respectively, of the ‘rule of inference’ category, by saying that it is local truth-preservation that matters for the intuitive idea of inferrability. It is well known18 that since the local/global distinction applies not only to preservation of truth but also to preservation of validity, where a formula is valid at a point in a (Kripke) frame

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if it is true at that point in every model on the frame, and valid on a frame if it is valid at every point therein, what we end up with here is a fourfold distinction. Uniform Substitution preserves validity at a point, and a fortiori validity on a frame, but unlike Modus Ponens it does not preserve truth at a point, and unlike Necessitation it does not preserve truth throughout a model. So, lacking the local truth-preservation characteristic, it too gets classified in the second passage quoted from Smiley in §1 as a rule of proof which should not be counted as a rule of inference. In what follows attention will be specifically on the truth-based rather than the validity-based incarnation of the local/global contrast. (Note also that this summary has ignored further preservation characteristics arising for models with a distinguished point or set of points used in modal actuality logic, various non-normal modal logics, etc.) Such verdicts are made available, but are not made inevitable, by the differentiated approach to axiomatization. After all, given R as our set of primitive rules, we can select as Rinf , consistently with the differentiated approach, any subset of R, even if on the intuitive grounds just rehearsed, the rules in question would not naturally be called rules (or principles) of inference. For example, we may want to axiomatize (in the sense explained at the end of §2) the ‘model consequence’ relation of global truth-preservation, which we will denote by superscripting a turnstile with the fi rst three letters of ‘global’, and subscripting it with ‘K’, the name of the smallest normal modal logic (considered as a set of formulas), to indicate that no restriction is imposed as to the models of interest. That is, we defi ne: glo

Γ K B iff for every Kripke model M, if every formula in Γ is true at all points in M, then so is B. (Since this is a semantic characterization of the consequence relation in question, some may prefer to see ‘ ’ in place of ‘ ’.) A mild variation on the canonical model method (see e.g. Kracht 1999: Proposition 3.1.3) shows that Kglo has a schematically fi nite axiomatizaton in the shape of 〈Ax, R, Rinf〉 with Ax the set of instances of any fi nite set of schemata which together with Modus Ponens (or MP for short) yield all truth-functional tautologies along with all instances of the K-schema:

̞(A → B) → (̞A → ̞B) and R = Rinf = {MP, Nec}. We could also supply a straightforwardly fi nite axiomatization of the same consequence relation by taking as Ax a fi nite set of axioms which together with Modus Ponens and US (Uniform Substitution) suffice for the truth-functional tautologies, and R = {MP, Nec, US}, Rinf = {MP, Nec}. The result(s) just given can be formulated in terms of rule-soundness and rule-completeness: the derivable rules of inference of the two axiomatizations described are all and only those rules which preserve truth

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throughout an arbitrary model.19 If we took the second axiomatization and pushed US into R inf we should have a correct rule-soundness statement for arbitrary frames: all derivable inference rules preserve validity on an arbitrary frame. But the corresponding rule-completeness statement in this case would not be correct, in view of the existence of Kripke-incomplete normal modal logics. glo The consequence relation K does not satisfy the Deduction Theorem, of course, since for example p → ̞p is not true throughout every model, even though ̞p is true throughout every model throughout which p is true, reflecting the distance we have traveled here from Smiley’s original conception of the rule of inference/rule of proof distinction. A rapprochement is possible, however, if we repackage the above points in a more sophisticated version of the differentiated approach. Instead of adding what were rules of proof proper (i.e. rules in R \ R inf) into the ‘rule of inference’ compartment R inf, we could drop the idea of a single consequence relation associated with (‘axiomatized by’) a rule-differentiated basis 〈Ax, R, R inf〉, and instead distinguish the induced inferential consequence relation—formerly just the consequence relation thereby axiomatized—from the induced probatory consequence relation, defi ned similarly but allowing arbitrary rules from R, rather than just those in R inf , when working out the consequences of a set of formulas. 20 Then we could say, concerning the fi rst (schematically fi nite) axiomatization described above, that it has as its probatory consequence relation precisely the model consequence relation Kglo . (This would not be the case for the second axiomatization, since US does not preserve truth throughout a model. In the alternative terminology, this would make for a failure of rule-soundness with respect to the class of all models.) Whichever version, naive or sophisticated, of the differentiated approach is taken—and from now on we revert to the naive version—we should certainly be wary of the claim of Anderson and Belnap quoted in §2. According to these authors, we should not have ‘a primitive rule to the effect that ̞A does after all follow from A’, since even if (as the sophisticated version allows us to say) ̞A cannot be inferred from A, there is a perfectly good sense—given by global truth-preservation—in which ̞A does indeed follow from A. That is, the truth of ̞A throughout a model follows from the truth of A throughout the model. The situation in this respect is quite different from the well-known converse of Necessitation, Denecessitation, i.e. the sequential rule with skeleton 〈̞p, p〉, under which the set of theorems of K is also closed. 21 (Of course Anderson and Belnap were not discussing K, but the present point shows the weakness of their suggestion that Necessitation itself should, ‘in a coherent formal account of the matter’, have no higher status than that of an admissible rule.) If we threw this rule in with Necessitation, it would destroy the soundness half of the rule-soundness and rule-completeness result mentioned, since the truth throughout a model of ̞A does not imply that A has this same property.

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To conclude this section we illustrate the convenience of the differentiglo ated approach in connection with the local analogue of K , which we loc naturally call K . T hat is, we defi ne: loc

Γ K B iff for every Kripke model M, and any point in M, if every formula in Γ is true at that point in M, then so is B. Note that unlike its global counterpart, with which it agrees for Γ = ∅, this consequence relation does satisfy the Deduction Theorem, 22 and it relates Γ to B just in case the conjunction of fi nitely many formulas in Γ provably implies B in the formula logic K. We have the following striking contrast: (1)

(2)

loc K

has no schematically fi nite undifferentiated axiomatization, while loc does K

have a schematically fi nite differentiated axiomatization.

Part (2) of this assertion is clear enough, since we have, for example, 〈Ax, R, R inf〉 with Ax as above (apropos of which the K schema was mentioned) and R comprising Modus Ponens and Necessitation, and R inf consisting of just Modus Ponens. For (1) we give just a sketch of the proof. If we have 〈Ax, R〉 as a schematically fi nite axiomatization of Kloc, with R = {ρ1, . . . , ρm} (each ρi sequential), then in view of the superscripted ‘loc’, each ρi must be locally truth-preserving (in every model). Where ρi has skeleton 〈C1, . . . , C n , D〉, we can then add all the formulas s(C1) → (s(C 2) → . . . → (s(Cn) → s(D) . . .)) to Ax, for every substitution s, calling the result Ax +. This amounts to adding all instances of a schema with distinct schematic letters replacing distinct propositional variables in the implication with successive antecedents the C j and consequent D. So 〈Ax +, {MP}〉 would also be a scheloc matically fi nite axiomatization of K . But that would mean that this new basis would provide a schematically fi nite axiomatization of the formula logic K, and a minor adaptation of a proof in Lemmon (1965: 302, Theorem 2)23 shows that there is no such axiomatization with Modus Ponens as the sole proper rule. 24

4

CLOSING COMMENTS

The prevalence of the undifferentiated approach, with its attendant conflation of rules of proof with rules of inference, has had some pernicious effects, the most recent of which is perhaps the interpretation of the term normal as it applies to consequence relations in modal logic. Since traditionally the

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main consequence relation associated with a modal logic has been the local (i.e. locally truth-preserving) consequence relation (as is remarked in the opening sentence of Kracht 1999: ch. 3), it has been customary to regard a consequence relation on the language of modal logic as normal if whenever Γ A, we have ̞Γ ̞A (where ̞Γ is {̞C : C ∈ Γ}). The corresponding sequent-to-sequent rule is sometimes called Scott’s rule and Scott (1974) specifically inveighed against confusing the special case of this rule in which Γ is empty—a vertical transition, in his terminology (think of a rule display with sequent-premise(s) above and sequent conclusion below a line)—with the condition that for all formulas A, we should have A ̞A: a horizontal transition. Yet just this last condition is imposed in work by Blok and Pigozzi and others since 1989 in abstract algebraic logic (or ‘AAL’) and has come to be regarded as defi ning the normality of a consequence relation. 25 I take this contrast between the acceptable vertical transition and the unacceptable (in the local case) horizontal transition to be Smiley’s point again: Necessitation is fi ne as a rule of proof but not as a rule of inference. Like everyone else on the undifferentiated side of the fence, the AAL community sees a (formula-to-formula) rule in the axiomatization of a logic and only knows one thing to do with it: use it to obtain consequences of arbitrary sets of formulas. Giving this inappropriate defi nition of normality (for consequence relations) has conspired with the fact that the local consequence relations concerned are not (in general) algebraizable by the standards of Blok and Pigozzi (1989) to make them very much second class citizens of the AAL world, by comparison with their global counterparts. 26 There is no reason to tie normality to the global side of the local/global division in this way. Scott’s horizontal/vertical contrast suggests a notation which would decorate the skeleton of a sequential rule with information as to the status of the premises of its applications (as in Humberstone 2008: 443): a superscripted downward arrow indicates a vertical transition from that premise and a superscripted rightward arrow, a horizontal transition. Thus Necessitation as a rule of proof would have the decorated skeleton 〈p⇓, ̞p〉 while for Necessitation as a rule of inference (if one were interested in such a thing—as in the naive implementation of the differentiated strategy in §3) the skeleton would be 〈p ⇒, ̞p〉. When we consider rules with more than one premise, however, the binary division into rules of proof and rules of inference loses its apparent exhaustiveness, since the premise positions may be differently tagged. For example, suppose that ∗ is a binary connective and consider the four possible decorations of the rule-skeleton 〈p, q, p ∗ q〉: 〈p ⇒, q ⇒, p ∗ q〉

〈p ⇒, q⇓, p ∗ q〉

〈p⇓, q ⇒, p ∗ q〉

〈p⇓, q⇓, p ∗ q〉.

The fi rst and last of these are straightforwardly a rule of inference and a rule of proof respectively (essentially what we might think of as andintroduction and Adjunction, respectively, regarding ∗ as representing

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conjunction), 27 but the second and third pair defy straightforward classifi cation in either category, indicating sequent-to-sequent rules as given respectively below, with ‘ ’ here used as though it were a sequent-separator, so as to avoid having to introduce further notation: fB  AfA B

fA  BfA B

The whole area of sequent-to-sequent rules—as in natural deduction and sequent calculus—lies well outside the boundaries of the present discussion. There is an interesting question as to how (or indeed whether) to apply the rule of inference/rule of proof distinction to them, now thinking of the premises for an application of a rule as being sequents rather than formulas. Another question that would deserve attention, returning to formulato-formula rules, concerns the closeness of the link between the Deduction Theorem and the intuitive idea of a rule of inference, suggested by the discussion in Smiley (1963) (and also—implicitly—by that of Schurz 1994). When 〈A, B〉 ∈ ρ ∈ R inf we have A B for the associated consequence relation (the associated inferential consequence relation, on the sophisticated version of the differentiated approach), but should we always have A → B? The supervaluational semantics of Thomason contains examples suggesting a negative answer, taking B as ‘it is true that A’ (1970: 273) and ‘it is inevitable that A’ (275), though what they raise most acutely is the question of the status of what the inference to a certain conclusion is an inference from: does it represent a mere supposition, an envisaged piece of new information, or what?28 In the end the phrase ‘rule of inference’ may itself turn into something of an umbrella term subsuming interestingly different cases, much as the term ‘rule’ itself did before Smiley articulated the distinction between rules of proof and rules of inference.

NOTES 1. The notion of a consequence relation is taken to be familiar here; see p. 15 of Shoesmith and Smiley (1978) for the defi ning conditions (called there Overlap, Dilution, and Cut for Sets). The usual notational liberties will be taken in connection with a consequence relation : for example ‘Γ, A B’ and ‘ B’ abbreviate ‘Γ ∪ {A} B’ and ‘∅ B’, respectively. 2. For the detailed proof, see Kleene (1952), §21. Kleene’s exposition is especially clear on the precise inductive structure of the argument. Despite the sentence which follows, a version of (A2) with its two antecedents permuted would do equally well, and this is the version appearing in Kleene’s discussion. (References to the original Herbrand and Tarski sources can be found in Kleene’s 1952: 98.) In recent years many generalizations of the Deduction

122 Lloyd Humberstone

3. 4. 5.

6.

7.

8. 9. 10.

11.

Theorem have been considered—see Czelakowski (2001: ch. 2) for an extensive sampling—but in what follows we have in mind just the simple traditional version of the result. The terminology is not wholly felicitous in the general case (as remarked in Humberstone 2006a: 46) but is nonetheless convenient. Fn. 3 in Smiley (1963) at this point cites Barcan (1946) as confusing the claim that A implies B with the claim that A B. For conformity with our notation, Smiley’s ‘L’ has been replaced by ‘̞’. The last rule, A / ̞A, mentioned by Smiley in this passage is the rule of Necessitation. Note that in view of the terminological proposal made in the passage, this should not really be glossed (as Smiley does) as ‘from A infer ̞A’. (‘From A to ̞A’ is better.) Nor should one formulate the rule as ‘if A then ̞A’: this is rather the statement that the rule in question is admissible (and no rule should be confused with a statement about rules or indeed about anything else). For ‘proof’ here, read ‘deduction’, this being the term (after which the Deduction Theorem is named) for a record of the derivation of the formula on the right of the ‘ ’ from those on the left (with the help of axioms). Rather than alluding to rules of proof proper, Schurz speaks of rules different from Modus Ponens, since he is considering the (typical) case in which that is the only rule of inference (in Smiley’s sense). Although deductions in the present sense are often described as sequences of formulas, they are better visualized as trees with the deduced formula at the root and axioms and elements of Γ at their leaves. Each non-leaf node is immediately dominated by nodes labeled with formulas which are premises for an application of a rule whose conclusion formula labels that node. A node (or the occurrence of a formula labeling it) is Γ-dependent if the subtree with it as root has some node labeled with an element of Γ. (Alternatively, see the definition of dependence in Kleene 1952: 99.) See e.g. Pogorzelski (1971), Prucnal (1972), Wojtylak (1983), and Pogorzelski and Wojtylak (2005). It should be added that the ordered pairs considered in this literature come as 〈R, Ax〉 rather than in the reverse order given in the following sentence of the main text, with the axioms first (which is more convenient for present purposes). Such pairs are referred to in various ways in the papers just cited, including ‘systems of propositional calculus’ and ‘logical systems’. Even those who do not make explicit use of the ordered pair theme for identifying axiomatizations implicitly operate with the same conception (in §2 we will call it the undifferentiated approach) of how such an axiomatization is to induce a consequence relation, and this includes a great many authors, from Łoś and Suszko (1958) to Blok and Pigozzi (1989) and beyond. The fi nal sentence in the quotation actually starts a new paragraph in the source text. This seems to be at least part of the drift of Smiley (1982). Curiously Anderson and Belnap do not employ the notion of admissibility in the discussion on p. 236, even though they have explained the notion on p. 54; instead we get this nonsense about it being a ‘lucky accident’ that ̞A is provable whenever A is. Many respectable logicians, especially writing before the mid-1960s, simply ignore this distinction. For example Kleene (1952) calls all admissible rules derivable (or derived), and when he wants to talk about derivable rules (in the present sense) uses the phrase ‘directly derivable’. Kripke (1965) uses the terms ‘admissible’ and ‘derivable’ interchangeably (again, simply to mean admissible). Kleene (1952: 92) says ‘A metamathematical theorem of the simple form Δ E is a derived rule of the direct type’, thereby confl ating (contrary to the recommendation of Note 5 above) rules with statements. It is also worth noting that his indifference to the rule of

Smiley’s Distinction Between Rules of Inference and Rules of Proof

12.

13.

14. 15.

16.

17.

18. 19. 20.

123

proof/rule of inference distinction has led to cloudy formulations in this and other works—see Kielkopf (1972). Since the order of the premises is immaterial, a cleaner treatment—though rather fussier than would here be desirable—might take an application of an n-premise rule to be not 〈B1, . . . , Bn , C〉 but rather 〈[B1, . . . , Bn ], C〉 in which [B1,. . . ,B n ] is the multiset of the formulas concerned. Thus if n = 2 and B1 is the same formula as B2 , this is a multiset in which the formula in question occurs twice. This usage is not unheard of in the literature. The only danger is that of confusion with something more general, as when a proof system with initial sequents and sequent-to-sequent rules (as in the case of the sequent calculus approach) is referred to as an axiomatization of the sequent logic—and thus (except in the substuctural logics case) of the obviously associated consequence relation (and the initial sequents are similarly referred to as ‘axioms’). In the present discussion, only formulas count as axioms, and sequent-tosequent rules come up for discussion only in passing in the fi nal section. Note that we have to add ρ to the set of rules of proof as well as to the set of rules of inference, since we require the latter to be a subset of the former. We suppose that all (sentential) languages under consideration have p1, p2 , . . . , pn , . . . as propositional variables (sentence letters), and abbreviate the fi rst two in this list to p, q. Note that as defi ned, there is really no unique skeleton for a given rule, since the propositional variables could be relettered—to say nothing of the arbitrariness of the order in which the premise formulas A1, . . . , An in a skeleton 〈A1, . . . , An , B〉 appear (commented on in Note 12 above). Note that we speak of the substitution instances of a formula, but the instances of a schema. (Derivatively, we also speak of the substitution instances of an n-tuple of formulas, when the same substitution is applied to all of the formulas in the n-tuple.) Note that the sequential rules are those susceptible of schematic representation with the slash notation (in the ‘A / ̞A’ style). Łoś and Suszko (1958) called substitution-invariant rules structural, but this makes for an unfortunate collision with the terminology of ‘structural rules’ as deployed, for example by Gentzen, in connection with sequent-to-sequent rules (namely for rules whose formulation does not involve any particular logical vocabulary). The same authors pointed out that uniform substitution itself is not (in the present terminology) a substitution-invariant rule. Examples of the literature in this vein include Fagin et al. (1992), Brady (1994), Humberstone (1996). The last reference concerns sequent-to-sequent rules, however, rather than formula-to-formula rules, our main focus here. The fi rst reference describes some interesting modal examples; see also the fi rst new paragraph on p. 386 of Schurz (1994). See Fitting (1983) or van Benthem (1985), for instance. Substitution-invariant or ‘schematic’ versions of the validity-preserving relations are described in Fagin et al. (1992). This terminology, though not quite with the present understanding, can be found in Belnap and Thomason (1963). In fact it is this consequence relation, rather than its inferential cousin, which is most commonly meant by talk of deducibility in practice, even though this is not the relation relevant to the Deduction Theorem and mentioned under the heading of ‘deducibility’ in §1. For example, the two occurrences of ‘interdeducible (in the field of K)’ on pp. 577 and 581 of Hughes (1980), appear in connection with pairs of formulas each of which can be derived from the other only with the aid of US, meaning that these derivations do not constitute suitable ‘deductions’ in the sense of the Deduction Theorem.

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21. Schurz (1994: 386, top paragraph) also gives this example, as do Font and Jansana (2001: 438). Schurz’s use of the example is an indication of how misleading his title is: ‘Admissible versus Valid Rules’. By ‘valid’ is here meant locally truth-preserving (in the class of models for the logic in question): emphasis on the contrast between this and admissibility is exactly the mistake Anderson and Belnap make. But Schurz’s discussion, its title notwithstanding, is alert to the significance of derivability as opposed to admissibility (and suggests that he would have been happy, had he been aware of it, to embrace Smiley’s distinction between rules of proof and rules of inference). 22. This point does not depend on the ‘K’ subscript, but holds for the local and global consequence relations corresponding to any normal modal logic. An extended discussion of the relations between these two consequence relations for the case of (an expressively impoverished fragment of) S5 appears in Humberstone (2006b). For more general considerations on the current local/ global contrast, see the fi rst section of ch. 3 of Kracht (1999). 23. Or see the Corollary to Theorem 7 of Kripke (1965: 219). Lemmon (amongst others) calls what are here called schematically fi nite axiomatizations simply fi nite axiomatizations (intending, further, that the sole rule employed is Modus Ponens). 24. While the contrast between (1) and (2) may initially seem impressive, it must be conceded to depend in large part on the specific identification of axiom schemata with zero-premise sequential rules, and the way ‘schematically finite’ captures the idea of a finitude of schemata, so understood. On this reading, something like ‘̞n (p → p)’ understood as summarizing the prefixing of any number of occurrences of ‘̞’ to the formula ‘p → p’, would not count as an axiom schema, let alone the ‘doubly schematic’ version with ‘A’ in place of ‘p’. Relaxing the requirement so as to admit these last as axiom schemata would allow the inferential consequence relation associated with the class of all models to have a schematically finite axiomatization with Modus Ponens as the sole rule, destroying the contrast between (1) and (2) in the text, because just as the use of schematic formula-letters exploits the fact that all applications of US can be made to precede any applications of MP, so this numerical schematicity exploits the fact that all applications of Nec can be made to precede any applications of MP. This device (essentially) was employed in Fitch (1973). 25. See the references given in Note 40 of Humberstone (2006a), where I have complained about this before. (The text to which that note is appended contains a misprint: ‘Scott’s own recommendations on score’ should read ‘Scott’s own recommendations on this score’.) 26. Help is on the way—indeed, has already arrived—as far as this latter consideration is concerned: see Font and Jansana (2001). 27. Compare the ‘Four forms of Modus Ponens’ in Scott (1974), as refi ned by the discussion in §6 of Humberstone (2000a) as well as in Humberstone (2008). ‘Adjunction’ here is the Hilbert-style rule of that name in Anderson and Belnap (1975) and elsewhere. 28. For more on the supposition/update contrast just drawn, see Humberstone (2002: §3).

REFERENCES Anderson, A.R., and Belnap, N.D. (1975) Entailment: the Logic of Relevance and Necessity, vol. 1, Princeton, NJ: Princeton University Press. Barcan, R.C. (1946) ‘The Deduction Theorem in a functional calculus based on strict implication’, Journal of Symbolic Logic 11: 115–7.

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Belnap, N.D. (1962) ‘Tonk, plonk and plink’, Analysis 22: 130–4. Belnap, N.D. and Thomason, R.H. (1963) ‘A rule-completeness theorem’, Notre Dame Journal of Formal Logic 4: 39–43. van Benthem, J. (1985) Modal Logic and Classical Logic, Naples: Bibliopolis. Blok, W.J. and Pigozzi, D. (1989) ‘Algebraizable logics’, Memoirs of the American Math Soc. 77, no. 396. Brady, R.T. (1994) ‘Rules in relevant logic—I: semantic classification’, Journal of Philosophical Logic 23: 111–37. Czelakowski, J. (2001) Protoalgebraic Logics, Dordrecht: Kluwer. Fagin, R., Halpern J.Y., and Vardi, M.Y. (1992) ‘What is an inference rule?’, Journal of Symbolic Logic 57: 1018–45. Fitch, F.B. (1973) ‘A correlation between modal reduction principles and properties of relations’, Journal of Philosophical Logic 2: 97–101. Fitting, M. (1983) Proof Methods for Modal and Intuitionistic Logics, Dordrecht: Reidel. Font, J.M. and Jansana, R. (2001) ‘Leibniz fi lters and the strong version of a protoalgebraic logic’, Archive for Mathematical Logic 40: 437–65. Frege, G. (1879) Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle an der Saale: Louis Nebert. Gabbay, D.M. (1981) Semantical Investigations in Heyting’s Intuitionistic Logic, Dordrecht: Reidel. Hughes, G.E. (1980) ‘Equivalence relations and S5’, Notre Dame Journal of Formal Logic 21: 577–84. Humberstone, L. (1981) ‘Relative necessity revisited’, Reports on Mathematical Logic 13: 33–42. . (1984) ‘Unique characterization of connectives’ (Abstract), Journal of Symbolic Logic 49: 1426–7. . (1996) ‘Valuational semantics of rule derivability’, Journal of Philosophical Logic 25, 451–61. . (2000a) ‘An intriguing logic with two implicational connectives’, Notre Dame Journal of Formal Logic 41: 1–40. . (2000b) ‘The revival of rejective negation’, Journal of Philosophical Logic 29: 331–81. . (2002) ‘Invitation to autoepistemology’, Theoria 68: 13–51. . (2004) ‘Two-dimensional adventures’, Philosophical Studies 118: 17–65. . (2006a) ‘Extensions of intuitionistic logic without the Deduction Theorem’, Reports on Mathematical Logic 40: 45–82. . (2006b) ‘Identical twins, deduction theorems, and pattern functions: exploring the implicative BCSK fragment of S5’, Journal of Philosophical Logic 35: 435–87; Erratum, ibid. 36 (2007): 249. . (2008) ‘Replacing Modus Ponens with one-premiss rules’, Logic Journal of the IGPL 16, 431–51. . (in preparation) ‘Replacement in logic’. Kielkopf, C.F. (1972) ‘Premisses are not axioms’, Notre Dame Journal of Formal Logic 13: 129–30. Kleene, S.C. (1952) Introduction to Metamathematics, New York: van Nostrand. Kracht, M. (1999) Tools and Techniques in Modal Logic, Amsterdam: NorthHolland (Elsevier). Kripke, S.A. (1965) ‘Semantical analysis of modal logic II. Non-normal modal propositional calculi’ in J.W. Addison, L. Henkin, and A. Tarski (eds) The Theory of Models, Amsterdam: North-Holland, pp. 206–20. Lemmon, E.J. (1965) ‘Some results on fi nite axiomatizability in modal logic’, Notre Dame Journal of Formal Logic 6: 301–8. Łoś, J. and Suszko, R. (1958) ‘Remarks on sentential logics’, Indagationes Math. 20: 177–83.

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Moh, S.-K. (1950) ‘The Deduction Theorem and two new logical systems’, Methodos 2: 56–75. Pogorzelski, W.A. (1971) ‘Structural completeness of the propositional calculus’, Bull de l’Académie Polonaise des Sciences, ser. des sciences math., astr., et phys. 19: 349–51. Pogorzelski, W.A. and Wojtylak, P. (2005) ‘A proof system for classical logic’, Studia Logica 80: 95–104. Prucnal, T. (1972) ‘Structural completeness of Lewis’s system S5’, Bull. de l’Académie Polonaise des Sciences, ser. des sciences math., astr., et phys. 20: 101–3. Schurz, G. (1994) ‘Admissible versus valid rules: a case study of the modal fallacy’, Monist 77: 376–88. Scott, D.S. (1974) ‘Rules and derived rules’ in S. Stenlund (ed.) Logical Theory and Semantic Analysis, Dordrecht: Reidel, pp. 147–61. Shoesmith, D.J. and Smiley, T.J. (1978) Multiple-Conclusion Logic, Cambridge: Cambridge University Press. Smiley, T.J. (1962) ‘The independence of connectives’, Journal of Symbolic Logic 27: 426–36. . (1963) ‘Relative necessity’, Journal of Symbolic Logic 28: 113–34. . (1982) ‘The schematic fallacy’, Proceedings of the Aristotelian Society 83: 1–17. . (1996) ‘Rejection’, Analysis 56: 1–9. Thomason, R.H. (1970) ‘Indeterminist time and truth-value gaps’, Theoria 36: 264–81. Wojtylak, P. (1983) ‘Corrections to Prucnal (1972)’, Reports on Mathematical Logic 15: 67–70.

7

Relative Validity and Vagueness Rosanna Keefe

1

SUPPRESSED RULES AND RELATIVE VALIDITY

Everyday English arguments come in all sorts of forms and are frequently casually expressed, with crucial elements remaining merely implicit. Logicians attempt to formalize these arguments. But it can be a mistake to project too much of those formalizations onto the everyday arguments themselves. Timothy Smiley has repeatedly drawn our attention to the subtleties of ordinary language reasoning and some of the mistaken assumptions and simplifications that logicians and philosophers often make in connection with it. At one time, logicians sought to treat all arguments as syllogisms, but that now seems clearly too restrictive. But it may also not be right to assume that arguments nonetheless take one of the wider range of forms that logicians now recognize. Perhaps there is no fact of the matter about the real underlying form of an argument in many cases. In this essay I consider the possibility of arguments that are taken to be good arguments in an informal setting, despite not being strictly or absolutely valid. In his ‘A tale of two tortoises’ (1995), Smiley provides an attractive framework for assessing arguments as relatively valid. He defends the appealing thought that everyday arguments can rely on rules of inference that are not formally valid: such rules can be presumed to be good ones in the context, just as context can justify suppressed premises relied upon in other arguments. In §§2 and 3, I draw on Smiley’s discussion of suppressed rules and relative consequence in considering arguments involving vague predicates. In particular, I examine the treatment Delia Graff Fara proposes to deal with relative validity within the framework of a supervaluationist theory of vagueness. And I argue that Smiley’s discussion suggests objections to Fara’s account and allows us to respond to the criticisms that she levels at the supervaluationist theory of vagueness. Further features and merits of Smiley’s framework are thereby revealed. First, in this section, I examine Smiley’s treatment of relative validity and draw out some consequences for the assessment of different sorts of arguments. Suppose we are assessing the argument that a subject uses on some occasion. On Smiley’s account, it is often appropriate to describe the argument

128 Rosanna Keefe as valid relative to a system R. R may be an established logical system of axioms and rules or a ‘casual, isolated rule’ (Smiley 1995: 730), or something in between, and it is a system that is, in some sense, reasonably assumed in the context. So, for example, Smiley may be relying on a timetable that justifies the argument ‘it’s Tuesday, so this is Paris’ and this rule may be (part of) the system R that is employed in the context. If R contains a rule that is not itself necessarily truth-preserving, then an argument can be valid relative to R, without being strictly valid. The premises of an argument imply the conclusion by R—i.e. the argument is valid relative to R—iff there is no way to falsify that inference without falsifying R. An inference counts as being falsified if there is an assignment of the truth-values T and F to sentences such that the premises are assigned T and the conclusion is assigned F; a rule is falsified if some instance of it is falsified. Note that when considering what can be falsified, ‘the only restriction [on the assignment of Ts and Fs] is that all occurrences of the same sentence should be assigned the same truth-value’ (Smiley 1995: 730). This means that there could be a system relative to which ‘A&B, so A’ does not come out as valid, for there is a way to falsify it (assigning F to A and T to A&B). Of course, it is valid relative to any system containing the standard elimination rules for conjunction, and it is fair to expect that these rules are assumed in almost every context. When it comes to assessing an everyday argument—deciding whether it establishes its conclusion—part of the task is often to uncover the assumptions on which it rests, where, in a successful argument, these assumptions will be justified in the context. One strategy is the familiar one of uncovering a suppressed premise in an argument. Suppose the argument is explicitly stated as merely ‘A, so B’. As Smiley captures it, the missing premise P must be true, must need no further justification in the circumstances and must be such that ‘A, P; so B’ is valid (Smiley 1995: 731).1 Smiley’s alternative suppressed rule strategy is to seek a rule, R, which is truth-preserving, needs no further justification in the circumstances and which results in ‘A; so B’, being valid by R. So, for example, take the argument ‘Alan is taller than Benji, Benji is taller than Carmine; so Alan is taller than Carmine’. The suppressed rule strategy can see this argument as employing a rule capturing the transitivity of ‘taller than’—‘from x is taller than y and y is taller than z, infer x is taller than z’, whereas the suppressed premise strategy needs to call upon a suppressed premise such as ‘for all x, y, and z, if x is taller than y and y is taller than z, then x is taller than z’ with the employment of the rule of universal instantiation (to yield the appropriate instance of the premise) and modus ponens (to yield the conclusion). The argument revealed by the suppressed rule strategy may seem more appealing and more faithful to the argument intended than the alternative with the suppressed premise. Clearly, it would be implausible to defend the suppressed rule strategy if it meant that arguments could never be diagnosed as involving suppressed

Relative Validity and Vagueness 129 premises, but fortunately this is not so. The strategy involves finding a contextually appropriate bunch of rules and axioms relative to which the argument is valid, and so any argument that simply leaves an obvious premise unmentioned can be formalized with that missing premise as an axiom which is assumed in the circumstances. 2 Smiley’s strategy thus encompasses the suppressed premise strategy, while providing much greater flexibility as to how an argument is formalized. Just as, he argues, the traditional logician’s attempt to see all arguments as disguised syllogisms was much too restrictive, so the refusal to admit merely relatively valid rules also overly regiments everyday reasoning. Sometimes we employ reasonable rules that don’t neatly match the logician’s chosen formalization. Just because standard formal systems don’t involve rules corresponding to the transitivity of ‘taller than’ but do have elimination rules for the logical connectives, this does not mean that our arguing mirrors this division and incorporates premises rather than rules to reflect the one but not the other. Should we see the suppressed premise and suppressed rule strategies as ways to uncover the real argument that the subject was employing? Is it plausible to think that there must be a fact of the matter about what the real form of argument was, even though the subject only expressed part of it? Or might it be that several filled-out arguments are equally faithful to his intentions—he would endorse all of them—and there is no unique argument he had in mind? If there were a unique formalized argument behind each everyday argument, then it could be a good question whether we need to call upon suppressed rules as well as suppressed premises to capture the intended argument. So, Smiley’s framework would be appropriate there. But, Smiley does not think that this is the way things are with respect to our everyday arguments, and we may well agree with him here. He proposes that the point of formalizing arguments is for some theoretical purpose (Smiley 1995: 735): so we should not see the project as uncovering the defi nitive argument within. But if we are not uncovering the real form of the arguments behind the everyday arguments with the two strategies, is there any point in distinguishing them? Isn’t it all then a matter of imposing structure in whatever way works for the logician’s purposes, in which case, the suppressed premise strategy may be best for the logician? As we will discuss in §2, typically there is a possible premise available to bridge the gap between premises and conclusion whenever there is a rule for the job.3 One opponent here is someone who advocates the suppressed premise strategy and assumes that the formalized arguments it yields are the genuine filled-out arguments and that the only way to establish a conclusion is for one’s argument to fit that form. Smiley gives us a way to reject this stance. Good arguments can be merely relatively valid and the suppressed rule strategy can reveal a better form for some such arguments. Smiley summarizes: ‘the rule strategy thus takes the argument as it comes’ (Smiley 1995: 731). Unlike a rule, a premise is internal to the structure of the

130 Rosanna Keefe argument and thus adding one alters the structure of the argument. There may be no fact of the matter what form the argument really takes, yet we can formalize as is useful, where uncovering suppressed rules may yield an argument more like the original, which would surely be an advantage. Traditional logicians who forced everything into the form of a syllogism were wrong to do so, and wrong to think all reasoning is really in that form. And this is so, even if they ended up with arguments that would not be rejected by those offering the original arguments (at least if rejection here amounts to declaring one of the premises false or the argument invalid). In assessing arguments, the logician could follow the suppressed premise strategy for the convenience of regimenting everything to fit their chosen formal system, but they would need to recognize that this convenience is the justification for adopting that strategy over the suppressed rule strategy and to acknowledge the limitations of the chosen system. Let us explore a little further what may be opened up to us by recognizing relative validity. If we adopt the suppressed rule strategy and allow positive evaluation of arguments on the basis of relative validity, this opens up a range of ways in which relative validity and formal validity can come apart. First, there is the kind of case Smiley emphasizes, where an argument is relatively valid but falls short of being absolutely valid because it uses a rule which is justified in the circumstances, but is not absolutely valid. Smiley’s timetable case is of this type. Sometimes it can be reasonable to rely on rules that are only contingently reliable, as long as they need no further justification in the circumstances. A different kind of case is one where the argument is relatively valid and absolutely valid (i.e. necessarily truth-preserving), but not formally valid (by the chosen system). The ‘taller than’ case is of this form, as the formal systems that the logician uses to formalize our arguments do not capture the transitivity of ‘taller than’. The suppressed premise strategy needs to include an extra premise stating transitivity and this is what Smiley objects to as misrepresenting the argument. The rule we appeal to with the suppressed rule strategy is, however, necessarily truth-preserving, so the resulting argument is still absolutely valid. Finally, the account of relative validity allows for notions of relatively validity that are stronger than the notion of validity assumed by the classical logician. For, recall that to falsify a rule, all that is needed is an assignment of Ts and Fs to sentences—restricted only by the requirement that all occurrences of the same sentence are assigned the same truth-value—such that for some instance of the rule, the premises are true and the conclusion false. This allows for the falsification of, for example, ‘A; so A or B’, and so there could be systems relative to which that rule is not valid. So, for example, we can consider validity relative to a relevance logic and there may be contexts in which these stricter conditions are justified. An argument comes out valid relative to all systems if it satisfies the condition that there is no way to falsify the inference without falsifying R, for any R.

Relative Validity and Vagueness 131 Although arguments such as ‘A; so A’ and ‘A, B; so A’ are of this type, the absolute validity of an argument does not guarantee that it is valid relative to any system. The fi rst type of case offers a new sort of perspective on an interesting asymmetry between the demands that, according to many logicians, we place on premises of our arguments and demands we place on our rules—an issue that Smiley has also highlighted elsewhere (1998). On the standard story, rules must be necessarily truth-preserving, while premises need be merely true, not necessarily true. Why the different standards?4 There are, of course, many things we can say about why something more than mere truth-preservation is needed for the rules we employ in our arguments. For example, we need to know that the argument is truth-preserving without knowing the truth-value of the conclusion: knowing that the rule necessarily preserves truth guarantees that the conclusion is true if the premises are. But something less than necessary truth-preservation could satisfy the extra requirements here and, in particular, relative validity—with rules that are justified in the circumstances—may be enough. If a rule is justified in the circumstances, we may have enough guarantee that the conclusion is true given that the premises are. There will be contexts where necessary truthpreservation is required, but since this is a special case of relative validity, these can be straightforwardly accommodated.

2

RULES, CONDITIONALS, AND CONDITIONAL PROOF

It might be thought that any justification for a rule that would bridge the gap between premises and conclusion will equally justify a corresponding universally quantified conditional, which may be taken as a suppressed premise. For example, if the rule allows the move from x is F to x is G, it will be justified if and only if we are justified in accepting the premise ‘for all x, if x is F, x is G’. Does this cast doubt on the significance of the distinction between the two strategies? In particular, if we shouldn’t assume there is a fact of the matter about the exact form of argument behind an everyday argument and if the background assumptions in the context would justify either the rule or the corresponding missing premise, then the choice between them might seem pointless. First, why think that both the rule and corresponding premise would be equally justified in the context? Consider some examples, starting with our argument turning on the transitivity of ‘taller than’. Anyone who endorses the rule that from x is taller than y and y is taller than z, we can infer that x is taller than z, will equally endorse the corresponding generalization that states transitivity with a universally quantified conditional. The justification for both of these is the same. Similarly, it is Smiley’s timetable that justifies the move from ‘it’s Tuesday’ to ‘this is Paris’ and that timetable would justify both the rule and the conditional premise ‘if it’s Tuesday, this

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is Paris’. But the fact, when it is one, that the subject would endorse both conditional and rule when offered them, does not show that either rule or conditional is what is involved in the argument. It does not show that the suppressed rule strategy is unnecessary or that the distinction is pointless, for the argument may be particularly well captured with the rule. If I have good reason to endorse the argument ‘A; so B’ and you ask me whether I accept ‘if A, B’, I will say ‘yes’, but that does not mean that this conditional is part of my argument. In this section, I ask whether there are further reasons to distinguish the strategies by considering a framework in which rules and corresponding conditionals can come apart. Conditional proof tells us that from A, B C, we can infer B A ⊃ C. It implies that having a reason to accept a rule allowing you to infer C from A gives you reason to accept A ⊃ C within an argument. Conditional proof is a compelling rule of inference and often presented as an introduction rule for the conditional. Regarding the closeness of the suppressed rules and suppressed premises strategies, it may also seem to back up the thought that if we are justified in employing a rule allowing us to infer B from A, then we are justified in employing the conditional premise A ⊃ B. But this doesn’t yet cover the wide range of cases in which there is a suppressed element of the argument, for conditional proof concerns individual arguments, involving particular sentences as premises and conclusion, whereas the kinds of rules mooted to be suppressed rules are inherently general. Now, a particular argument will typically only use or rely on one instance of the rule (or, at most, a fi nite number of instances), and from that instance (or those instances) of the rule, we can use conditional proof to deduce the corresponding conditional(s). But a formalization involving the conditional(s) would misrepresent the subject’s argument, even assuming it involves a suppressed premise, for the subject should be seen as appealing to a general premise—a universal generalization. (For example, in the case invoking the transitivity of ‘taller than’, the subject must be taken as appealing to this as a general feature of ‘taller than’—as captured by a universally generalized conditional statement—rather than the particular conditional that delivers the particular conclusion about Alan et al.) And, though we can present our rule as a schema, allowing any substitution for the schematic letters, a suppressed premise must be a sentence, not a schema, so we cannot take a schematic rule inferring instances of B from corresponding instances of A and convert it to the schematic conditional ‘if A, B’. In cases such as the transitivity one, it is easy to generate a generalized premise by quantification over the relata of the ‘taller than’ relation. In cases where the schematic letters are sentence letters, however, this is not so straightforward. Indeed, Smiley raises a similar point about the generality of the two approaches as an advantage of the suppressed rule strategy over the suppressed premise one. As he explains, to match the rule of modus ponens, a premise would either need to be formulated in the metalanguage

Relative Validity and Vagueness 133 or as the generalization ‘For all sentences P and Q, if P and If P, Q are true, Q is true’, with its problematic employment of ‘true’ applied to its own sentences (Smiley 1995: 732). For our purposes, this is enough to suggest that the availability of conditional proof does not show that a case of the suppressed rule strategy can be converted to a reasonable case of the suppressed premise strategy. But I put these cases aside and ask the following question: if conditional proof were to fail, would that cast further doubt on the claim that suppressed rules can be replaced by suppressed premises? Plausibly it would: it would imply a special role for rules that cannot be assumed to be replaceable by appeal to conditionals, and this would be highly significant for an advocate of the suppressed rule strategy. Compelling though conditional proof is, there is at least one framework within which it has been questioned for certain distinctive types of arguments. Within the supervaluationist theory of vagueness, conditional proof can fail for arguments involving the ‘defi nitely’ operator, D (DFa states that a is defi nitely F or a clear case of F; ¬DFa&¬D¬Fa states that a is not definitely F and not defi nitely not F, i.e. it is a borderline case of F). So, perhaps within a supervaluationist theory of vagueness we may have further reason to defend the suppressed rule strategy for formalizing informal arguments. Another reason to consider this theory of vagueness here is in relation to the treatment of relative validity, for it has been argued—by Delia Graff Fara—that a supervaluationist’s treatment of relative validity has undesirable consequences. The supervaluationist theory of vagueness is a theory that determines truth-values for a vague sentence by appeal to the different ways in which it could be made precise: roughly, P is true (false) iff it is true (false) on all ways of making the language precise, where we can call those ways precisifications. One characteristic feature of vague predicates is that they have borderline cases, and on the supervaluationist theory, the corresponding predications come out neither true nor false. If Tek is a borderline case of tall, then ‘Tek is tall’ is true on some ways of making it precise and false on others, so it is neither true nor false simpliciter. But if Bruno is clearly tall then, however you draw the boundary to the tall people to make ‘tall’ precise, he falls above the boundary, so ‘Bruno is tall’ is true on all precisifications and so true simpliciter for the supervaluationist. In such a case, we can say that Bruno is defi nitely tall, where ‘defi nitely P’ or DP is true iff P is true on all precisifications. I defended a supervaluationist theory of vagueness in my (2000), and the classic source is Kit Fine’s (1975). It is frequently advocated by philosophers, sometimes in passing, for it promises many logical and/or intuitive advantages over alternative theories of vagueness (see my 2000 for further discussion). Now, on the usual general characterization of validity, an argument is valid iff it is necessarily truth-preserving. Since truth, for the supervaluationist, is truth on all precisifications, validity is necessary preservation of

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truth-on-all-precisifications. It can straightforwardly be shown that, in the language without the ‘defi nitely’ operator, arguments are valid according to supervaluationism if and only if they are valid according to classical logic (see Keefe 2000: 175–6). So, although the semantics is non-classical (truthvalues are determined by quantification over models), the logic comes out classical, which is often seen as a significant advantage of the theory. For example, an instance of the Law of Excluded Middle such as ‘Either Fred is bald or he is not bald’ comes out true even if Fred is borderline bald, because however ‘bald’ is made precise, one or other of the disjuncts—so also the disjunction—will be true. And this is mirrored for other classical logical truths and classically valid inferences since the precisifications quantified over are themselves classical models. But when the D operator is added to the language, the truth-value on a precisification of a sentence involving D depends on what is true at other precisifications. The nonclassicality of the semantics then gets reflected in the logic. For example, the argument from P to DP is supervaluationarily valid: if P is true (on all precisifications), then DP is true (on all precisifications). But, whereas conditional proof would allow us to conclude that P ⊃ DP is valid (i.e. follows from no premises), this is not a legitimate conclusion in the framework. For P ⊃ DP can fail to be true: if P is a borderline case predication, then DP is false, but on some precisifications P is true and so on those precisifications, P ⊃ DP is false. P ⊃ DP thus fails to be true simpliciter, demonstrating that conditional proof has failed. Other classical rules of inference, including reductio ad absurdum, can be shown to fail with similar examples involving the D operator. Since these are compelling and widely used rules of inference, this has been raised as an objection to the supervaluationist theory (see Machina 1976; Williamson 1994: 151–2). The supervaluationist’s typical response is to stress that the rules only fail in the presence of the defi nitely operator and that everyday reasoning is thus not under threat. As Williamson summarizes the position, ‘if we had to exercise caution only when using this special operator, then our deductive style might not be very much cramped’ (1994: 152).5 As we will see below, Fara argues that the failures are more widespread. Before considering her argument, however, I ask whether someone endorsing occasional failure of conditional proof has additional reason to recognize the suppressed rule strategy as an important new approach to assessing arguments. Let us construct an argument in which a rule is acceptable but the corresponding conditional is not. Suppose someone, S, was to offer an argument of the form ‘A, B; so DA&DB’. According to the supervaluationist, this argument is valid and we might formalize it as employing two instances of the rule ‘from P infer DP’, plus &-introduction. A strategy that seeks to fi nd missing premises rather than rules may instead appeal to the conditional premises ‘if A, DA’ and ‘if B, DB’. But, according to the supervaluationist, these premises will fail to be true if A is borderline. Now, premises needn’t

Relative Validity and Vagueness 135 be necessarily true, so the falsity of such a conditional premise in certain circumstances doesn’t yet undermine the fi lled-out argument. But, nonetheless, to have grounds to endorse the premise ‘if A, DA’, S would need to rule out the possibility that A is borderline, i.e. S would need to assume that DA. (D¬A is ruled out given the premise A.) But then, this assumption, which could be offered as a further premise, clearly trivializes the argument. So, a supervaluationist is better off uncovering the missing rule ‘A therefore DA’ rather than a missing premise of the form A ⊃ DA. Of course, the above example is hardly a worked example of ordinary reasoning. It isn’t clear that in ordinary arguing we would be prepared to endorse arguments from A to DA any more (or less) than we would assent to the corresponding conditional, ‘if A, DA’. Either seems fi ne if we crudely and wrongly assume that ‘a is red’ means that a is defi nitely red. But if we attend to borderline cases of ‘is red’, there is likely to be puzzlement over whether to infer ‘a is defi nitely red’ from ‘a is red’ and over whether to accept ‘if a is red, a is defi nitely red’—a puzzlement that mirrors the difficulty we have in assessing ‘a is red’ at all in such cases. So, it may be that there are no cases in which someone’s argument is well captured by invoking a suppressed rule such as ‘P; so DP’, but cannot be captured with a suppressed premise. And if in ordinary contexts people are strongly inclined to infer a conditional from the corresponding hypothetical reasoning, then we don’t want a story about the assessment of ordinary arguments that relies on the failure of this natural mode of reasoning. Otherwise in trying to capture the ordinary argument, we will end up drawing distinctions that arguers themselves would not recognize. Moreover, the original argument in the example, ‘A, B; so DA&DB’ is valid according to the supervaluationist anyway; for if A and B are both true (i.e. true on all precisifications) then DA&DB will be too. So no suppressed premise is needed to reveal how the premises imply the conclusion. If the suppressed premise strategy employs formalization within the supervaluationary system, then it can appeal to the rule ‘from P infer DP’ in the formalization—a rule which is absolutely (not merely relatively) valid. So, we do not yet have a reason to employ the suppressed rule strategy rather than the suppressed premise strategy in the light of the supervaluationist failure of conditional proof. In the next section, I turn to the kinds of case that Fara considers, where absolute validity does fail.

3 FARA ON RELATIVE VALIDITY FOR THE SUPERVALUATIONIST In a classical framework, an argument is valid iff it preserves truth in all models; supervaluationary validity requires preservation of truth in all supervaluationary models. Now, the intended supervaluationist model

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consists of a set of classical models corresponding to the precisifications of our vague predicates and truth in that model is equated with truth in all those precisifications. Just as unintended classical models can be thought of as assigning different (precise) meanings to predicates, unintended supervaluationary models assign different vague meanings to predicates, each modeled by a set of classical models corresponding to the precisifications of those vague meanings. So, supervaluationary models each consist of a set of classical models (precisifications) and truth in that supervaluationary model is equated with truth in all the precisifications in the corresponding set. Validity is thus preservation of truth (i.e. truth in all precisifications) in all supervaluationary models. So, an argument ‘A, B; so S’ is valid if and only if, for every supervaluationist model, if A and B are true on all precisifications in that model, S is true in all those precisifications too. Now, just as the classical models quantified over in the classical scheme include models that don’t respect candidates for analytic truths such as ‘noone is both short and tall’ or ‘someone with no hairs on his head is bald’, so (unintended) supervaluationist models can fail to respect such truths too. They do, however, come out true in the intended supervaluationist model (because true in all the precisifications in that model) and are labeled penumbral truths. Other penumbral truths include ‘this is red or pink’, said of a borderline red–pink blob and ‘if Tim is tall, Tek is tall’, when Tim and Tek are both borderline tall, but Tek is taller than Tim. The fact that the supervaluationist can accommodate such penumbral truths is trumpeted as an advantage of the theory—theories involving truth-functional threevalued logics, for example, cannot boast this feature (see e.g. Keefe 2000: 96–8). Fara, though, considers corresponding arguments, such as ‘Al is tall, therefore he is not short’, and maintains that the theory should also respect these as, in some sense, good arguments. As long as there are (unintended) supervaluationist models in which someone counts as both tall and short, such an argument will not count as valid, for there will be models in which truth is not preserved. Now, Fara offers the supervaluationist a technique by which arguments that are not strictly valid can be treated as relatively valid, by allowing restrictions on the models over which truth must be preserved. So ‘A, B; so S’ is valid relative to a class of supervaluationary models, C, iff in every model in C, if A and B are true in the model, S is true in it too. We may fi nd that the arguments we somewhat casually endorse in some particular context are the arguments that are truth-preserving in some given set of models. This, of course, is a technique that could also be used within a classical framework, or any other framework. For example, we could limit the classical models to those that respect all analytic truths. We would thereby allow the relative validity of ‘John is a bachelor; so John is unmarried’, despite it not being formalized as valid on the classical scheme. The interest

Relative Validity and Vagueness 137 in the supervaluationist case, for Fara’s purposes, is the way in which certain rules of inference can fail given a restricted set of models, even for arguments not involving the D operator. To demonstrate the failure of conditional proof within a restricted class of models, consider an argument with a premise, A, that is not true in any models in the class, though it is neither true nor false in some of those models. It is trivially true that ‘A; so B’ is truth-preserving in all of the relevant models whatever B is, for there is no model in the set with A true, so none with A true and B false. But the conditional A ⊃ B may not be true in some of the models, if there are precisifications in those models in which A is true and B is false. Fara’s own main example concerns a sorites series, where we can consider a series of men, each 1mm shorter than the previous one, where the fi rst is 7ft tall and the last is 4ft. On the supervaluationist’s picture, it is true that the first members of the series are tall, false that the last ones are, but there are borderline cases in between of which it is neither true nor false that they are tall. According to Fara, ‘from the supervaluationist’s perspective . . . it’s not possible for it to be true that X is tall and Y isn’t unless it’s true that they’re not adjacent members of a sorites series of “tall”—it follows from X’s being tall and Y’s not being tall that they differ in height by more than a millimeter’ (Fara 2003: 214). Thus, according to Fara, the supervaluationist should endorse appropriate instances of the inference, (A)

Fx&¬Fy; therefore ¬Rxy

(where F is the sorites predicate and R is the relation between adjacent members of a plausible sorites series for that predicate). The conditional that would follow by conditional proof is of the form (⊃)

(Fx&¬Fy) ⊃ ¬Rxy

e.g. ‘If x is tall and y isn’t, then x is more than a millimeter taller than y’. But this conditional may not be true for the supervaluationist, for, when x and y are adjacent members of the series and at least one of them is a borderline case of F, there are precisifications on which (⊃) is false and so that sentence is not true simpliciter. To model the supposedly good argument here as relatively valid on Fara’s scheme—i.e. truth-preserving in some restricted class of models—she suggests the class, C, of models containing ‘those models that faithfully represent everything the supervaluationist takes herself to know a priori’ (Fara 2003: 215). Since the case for ¬Rxy following from Fx&¬Fy is surely taken by the supervaluationist as ‘the kind of thing you can figure out while sitting thinking in your office’ (Fara 2003: 213), C is restricted to models satisfying this condition and thus the argument (A) is valid relative to C. Fara is particularly targeting supervaluationists, such as myself, who defend the supervaluationist failure of some classical rules of inference by

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emphasising the very limited range of cases in which they fail—only those involving the D operator—and regarding normal arguments as untouched. Fara’s framework for relative validity provides a wider range of cases in which those rules fail, including cases that do not involve the D operator. But for the objection to take hold, we need to be convinced that Fara’s account of relative validity successfully captures cases that may plausibly be taken as good arguments despite their falling short of strict validity. I will argue that it does not. First, I question whether she has succeeded in identifying an appropriate class of models relative to which a consequence relation can be defi ned. For the particular relativized consequence relation on which Fara focuses, the relevant class of models consists of those that ‘faithfully represent everything the supervaluationist takes herself to know a priori about a sorites series’ (Fara 2003: 215). Let us fi rst consider, more generally, the notion of models representing what is known a priori. There may be no set of models faithfully representing everything an ordinary arguer takes as a priori, for the ordinary arguer may well take a sorites premise to be true a priori; after all its truth seems to follow simply from the vague meaning of the sorites predicate. Now, hopefully most subjects will notice the threat of paradox and so not accept a typical sorites premise (whether as a priori or not), but, nonetheless, there may be more subtle yet paradoxical principles that a typical subject would regard themselves as knowing a priori.6 We may hope that the theorist is better off than this and that, in particular, what the supervaluationist takes herself to know a priori yields a unique set of models. But we may question this too. An ordinary supervaluationist may be subject to inconsistencies like any other subject, so we will, at the least, assume this defi nition involves the idealized supervaluationist. But, even then there are problems. Now, Fara starts from an informal presentation of what she describes as ‘a supervaluationist conception of what follows from what’ (Fara 2003: 213). This leans on cases such as ‘Al is tall; so he is not short’ which is an argument corresponding to the penumbral truth, ‘if Al is tall he is not short’. It is not implausible to claim that someone who accepts the penumbral truth is likely to see the corresponding argument as in some way a good argument. At this informal point in the discussion, we are told to focus on ‘the class of models that verify every penumbral sentence’ (Fara 2003: 213). The issues here are very close to discussions we might have about restricting the set of classical models to those respecting analytical truths. And, of course, delineating such a class of truths is not an easy matter and not only in practical ways. Putting these issues aside, we should question Fara’s shift from initial informal talk of ‘penumbral consequence’ and ‘the class of models that verify every penumbral sentence’ to the class of ‘models that faithfully represent everything that the supervaluationist takes herself to know a priori’ (Fara 2003: 215). This latter notion includes things the supervaluationist knows by pondering aspects of the intended supervaluationary model and

Relative Validity and Vagueness 139 the distribution of truth values along a sorites series. And, most notably, they are things that cannot be captured in a single (penumbral) truth: the only way to express what is supposedly known in the relevant cases is metalinguistically—as Fara does in saying ‘it is not possible for it to be true that X is tall and Y isn’t unless it’s true that they’re not adjacent members of a sorites series for “tall”’ (Fara 2003: 213–14)—or with the D operator as DFx&D¬Fy ⊃ ¬Rxy. This makes this kind of case very different from the original arguments such as ‘Al is tall; so he is not short’ and describing them as cases of penumbral consequence is surely misleading. (Note, additionally, such cases do not involve the penumbra of borderline cases, but rather trade on facts about definite cases.) An account that includes the argument (A) as a case of penumbral consequence is as questionable as one that includes (⊃) as a penumbral truth. Supervaluationists deny that the latter is a penumbral truth (and their model formalizes this denial): it would, of course yield a sorites paradox. Similarly, then, they should reject an account of penumbral consequence that yields this problem argument as among its instances.7 I will not here offer an alternative account of penumbral consequence, but there is no reason to think that any such account would yield the failures of conditional proof that we see in Fara’s account any more than modeling typical unproblematic conceptual truths yields problematic arguments in classical logic. So, we should not accept Fara’s identification of her class of models as providing an account of penumbral consequence or of ‘the supervaluationist conception of what constitutes a good argument’. Nonetheless, we might accept that Fara can delineate a class of models in which the argument (A) does hold and conditional proof fails. Could there be other reasons to take this to constitute an interesting relativized consequence relation? I will argue that there are not: this is not a class of models that we should be interested in and we have no reason to classify as relatively valid the arguments it declares to be valid. First, it is at best only a relation that supervaluationists themselves would call upon. For the things that need to be assumed—i.e. the features of models that follow from the controversial supervaluationist theory—are not generally assumed to be the case, either by the everyday subject or by other theorists in general. We cannot assume that preserving truth in those models is of any significance to the ordinary reasoner. But it isn’t even clear that Fara has captured a notion that the theorist advocating supervaluationism should be interested in. As supervaluationists, they will typically maintain that it is the non-relativized notion of consequence that best captures the appropriate consequence relation. They defend the various features attached to such a relation (including the classicality and the limited failure of conditional proof) and offer it as providing the logic of vague language. Why would they reason using Fara’s relativized notion? Even if they assumed they could deduce some features of the intended model by considering just those models that can’t be ruled out

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a priori (the class of models Fara is singling out), they should not and would not represent this as a consequence relation that can be straightforwardly stated in the object language. Let us examine this in more detail by asking whether we should be seeking to model Fara’s argument ‘Fx&¬Fy; so ¬Rxy’ as relatively valid. I argue that there is no case for doing so. The supervaluationist can agree that if Fa and ¬Fb are true in one of the models Fara describes, then ¬Rab must be true in that model, but this should not be formulated as an entailment from Fa and ¬Fb to ¬Rab any more than as the conditional ‘if Fa&¬Fb, then ¬Rab’. The inference from Fa and ¬Fb to ¬Rab is clearly one that we should be very wary of, for otherwise a sorites paradox looms. If someone were to argue ‘Al is tall and Bill is not tall, so Al is more than a millimeter taller than Bill’, it would be reasonable to respond as follows ‘that inference is questionable because if it held, you could move from x is 1mm taller than y and x is not tall to y is not tall, and this clearly leads us down a sorites series’. Assuming they didn’t retract their original argument, the arguer would be likely to reply that their conclusion did follow in this case because Al is definitely tall and Bill is definitely not tall. But then, the real premise of the argument is this statement, in which the ‘defi nitely’ operator is crucially involved.8 There is no reason to think that reflective reasoners would endorse the argument (A) while rejecting (⊃). They might accept both (problematically) or neither, but, in modeling ordinary reasoning, there is no role for a model that captures the possible but idiosyncratic combination of rule without conditional. So, I maintain that there is no more need to uncover a sense in which ‘Fx&¬Fy; so ¬Rxy’ is a good argument than there is to uncover a sense in which the sorites premise is true. This is perhaps good news for the Epistemic Theorist such as Williamson, who maintains bivalence and a sharp, unknown, boundary between the Fs and not-Fs even for a vague F (see e.g. Williamson 1994). For such a theorist, the sorites premise is straightforwardly false and argument (A) is invalid since the premises will be true and the conclusion false for a pair of things that straddle the boundary of F-ness. But, for the epistemic theorist, there is no set of models containing the intended one in which the argument preserves truth. If we did need to accommodate a sense in which (A) was valid, then this would be an objection to the epistemic theory. But we cannot dismiss the epistemic theory so easily, for, as I have argued, there is no such sense.9 Even if there is no reason to highlight the particular relativized consequence relation that Fara describes, could it nonetheless be the case that Fara’s framework provides a successful model of relative validity, suggesting that relativizing to any class of models is legitimate? Again, I argue not. Whenever you have a truth-preserving argument, you have a class of models relative to which the argument is valid (the singleton of the intended

Relative Validity and Vagueness 141 model, at least). But this is not sufficient to regard the argument as relatively valid: it may be merely accidentally truth-preserving. Clearly, being relatively valid in this sense is not a way of being a good argument. Is there any interest in the idea of relative validity outside the context of assessing particular arguments? It may be of theoretical interest and the generalization to cases that wouldn’t correspond to any everyday arguments may be instructive, but if this is the role for relative validity, then we should not see it as modeling ordinary reasoning. To return to the specifics of Fara’s objection to supervaluationism, recall that the aim was to show that ‘the sense in which [supervaluationists] endorse classical logic is more qualified than typically advertised’ (Fara 2003: 196). But providing theoretical constructions within the supervaluationist framework in which classical logic fails does not establish this objection, and, as I have argued, Fara has not provided a type of inference the supervaluationist should seek to model that does fail to be classical. So, the fact that some classical forms of inference such as conditional proof fail with the relation identified by Fara as a relative consequence relation does not show that there is any threat to such reasoning in any real (even theoretical) context. These points can be helpfully illuminated by contrasting Fara’s treatment of relative consequence with Smiley’s treatment. A general account of relative validity will allow for different instances or cases relative to which arguments are or aren’t valid, whether the instances are identified in terms of systems of rules or sets of models or whatever. We need to provide conditions that must be satisfied in order for something to be an appropriate instance of relative validity in that context. And they should be conditions that are in some sense accessible to the ordinary subject. Smiley has presented a strategy for assessing arguments, where an argument can be judged acceptable because it is valid relative to a system of rules and axioms that are justifi able in the context. Any relevant rules must be such that they are truth-preserving and need no further justification in the context. The relation Fara focuses on as the supervaluationist’s a priori entailment will not satisfy this or comparable conditions and Fara’s account of relative validity is less satisfying in not providing alternative conditions. According to Smiley, when we are told that an argument is relatively valid, we can ask ‘relative to what’ and expect an answer that reflects what the arguing subject knows or believes. We do not have to maintain that reasoning subjects can formulate the system they are assuming, but we can require that they would endorse the rules or axioms that the theorist identifies as forming the relevant system, otherwise it will not plausibly be counted as one that needs no further justification in the circumstances. There is no comparable test in relation to a set of models (at least none that doesn’t reduce it to the kind of picture on which axioms and rules are what matter after all). There may be theoretical interest in an account of relative

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validity that is abstracted away from actual applications and from the type of conditions mentioned above, but an argument’s being relatively valid cannot then be assumed to be any reason to think it is a good argument in any sense. For the project of assessing everyday arguments, we would do well to employ Smiley’s framework.10

NOTES 1. Or, as Smiley actually puts it, it must be ‘such that the argument “A, P; so B” has a sporting chance of being valid’ (my emphasis), where we test whether it actually is valid by formalization. This formulation allows the logician to give the subject credit for presenting an argument that at least seems cogent, even if, in fact, it might fail to be valid, just as a premise might fail to be true. 2. Although Smiley says that the suppressed rule strategy consists of looking for a rule R satisfying his conditions (regarding truth-preservation and justification in the circumstances) (Smiley: 1995 731), he is explicit earlier and later in the article that the system relative to which an argument may be valid can involve axioms. The account only needs a little adjustment, however. We can state that the strategy for an argument ‘A; so B’ is to fi nd a system R of rules and axioms, such that, fi rst, the axioms are true and the rules truthpreserving; second, the axioms and rules need no further justification in the circumstances; third, the argument ‘A; so B’ is valid by R. An argument is valid by R if it is impossible to falsify the inference without falsifying at least one of the rules or axioms. 3. Makinson provides an elegant formal introduction to ‘three different ways of getting more than is authorized by classical consequence out a set of premises’ (Makinson 2005: 18) and the differences between them. His three ways involve using additional background assumptions (as in the suppressed premise strategy), adding rules (as with the suppressed rule strategy) or restricting the set of valuations over which truth must be preserved (a method corresponding to Fara’s treatment of relative validity discussed in §3 below). 4. Smiley (1998) emphasizes that this apparent orthodoxy has not always been assumed. For example, Philo’s account of logical consequence requires only that it is not the case that the premises are true and the conclusion false. 5. Williamson goes on to argue that the same rules can fail with locutions such as ‘. . .ish’, as in ‘baldish’ and ‘tallish’, where the ‘. . .ish’ modifier may naturally be treated by consideration of precisifications besides that in question— e.g. someone is baldish on a precisification if they are bald on some but not all precisifications. In my (2000: 180–1), I argue that, as with arguments involving the D operator, such exceptions are defensible and to be expected. 6. Problems with basing an account of consequence on what subjects assume they know a priori mirror problems with taking the penumbral truths as truths we take ourselves to know a priori. The sorites premise threatens to undermine this latter approach, see my (2000: 183, fn. 14). The supervaluationist can identify the penumbral truths without making this assumption. 7. Fara singles out the premises of phenomenal sorites paradoxes as most plausibly providing cases of arguments that express penumbral connections, considering examples such as ‘this looks red and that doesn’t; therefore this does not look the same as that’, and others that involve ‘looks the same’ or ‘tastes

Relative Validity and Vagueness 143 the same’ etc. Phenomenal sorites paradoxes raise interesting issues, some of which Fara discusses in her (2001), to which I reply in my (manuscript). The corresponding phenomenal sorites premises—e.g. ‘if a looks red and b looks the same as a, then b looks red’—are false for the supervaluationist and so are not penumbral truths. Given this, the cited arguments will equally not count as cases of penumbral consequence either. 8. Now, Fara does consider the supervaluationist response that her case is really one involving an implicit premise such as (DFx&D¬Fy) ⊃ ¬Rxy and replies that this strategy would mean that ‘argumentation involving the D operator is much more common than it would at fi rst appear’ (Fara 2003: 218). But, her claim here relies on the assumption that she is dealing with a common form of reasoning in ordinary circumstances and I have argued that this is not so. Insofar as the argument is compelling, it is indeed an argument that can only be captured with the D operator. 9. We may demand that the epistemic theorist explains why the inference can sometimes seem like a good one, and the answer is likely to mirror the explanation of why the sorites premise seems true—e.g. because of its confusion with some principle about defi nite cases of the predicate. Any such explanations are equally available for the supervaluationist. 10. For helpful advice and comments, I would like to thank Dominic Gregory, Alex Oliver, Jonathan Lear, and those who attended my talk at the Popper Seminar at the LSE. It should be very clear from the essay that it was Timothy Smiley’s work that inspired me to write on this topic. Timothy also supervised my PhD thesis in Cambridge: he continues to provide me with an exemplary model of how philosophy should be done.

REFERENCES Fara, D.G. (2001) ‘Phenomenal continua and the sorites’, Mind 110: 905–35. Originally published under the name ‘Delia Graff’. . (2003) ‘Gap principles, penumbral consequences and infi nitely higherorder vagueness’ in J.C. Beall (ed.) Liars and Heaps: New Essays on Paradox, Oxford: Oxford University Press, pp. 195–221. Originally published under the name ‘Delia Graff’. Fine, K. (1975) ‘Vagueness, truth and logic’, Synthese 30: 265–300. Keefe, R. (2000) Theories of Vagueness, Cambridge: Cambridge University Press. . (manuscript) ‘Phenomenal sorites paradoxes and looking the same’. Machina, K. (1976) ‘Truth, belief and vagueness’, Journal of Philosophical Logic 5: 47–78. Makinson, D. (2005) Bridges from Classical to Nonmonotonic Logic, London: King’s College London Publications. Smiley, T. (1995) ‘A tale of two tortoises’, Mind 104: 725–36. . (1998) ‘Consequence, Conceptions of’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 2, London: Routledge, pp. 599–603. Williamson, T. (1994) Vagueness, London: Routledge.

8

The Force of Irony Jonathan Lear

More than any logician of his time, Timothy Smiley taught us that there are more good arguments in heaven and earth than are dreamt of in our formalizations. Indeed, the attempt to formalize a good argument can lead to straight-jacketing and distortion: In the grip of the misguided belief that all correct reasoning is really syllogistic, traditional logicians took arguments that were in perfectly good order as they stood and forced them into the straitjacket of the syllogism. In the grip of a similar but I think equally misguided belief, their successors force the straightforward inference ‘Al is older than Bill, Bill is older than Charlie; therefore Al is older than Charlie’ into the form of a universal instantiation followed by modus ponens. It is a truism that we constantly rely on all sorts of things—timetables, geometry, casual items of knowledge or information—to sustain inferences as well as to sustain assertions. When one talks of the practical value of logic, of logic as something that ordinary people use, it is logic in this extended sense, not the textbook sense, and similarly with the language of ‘proof’, ‘follows from’, ‘deduction’ and ‘must’. (Smiley 1995: 731–2) I am interested in a form of practical reason that can be extraordinarily powerful—those who experience its force experience it as a practical syllogism— but if one were to try to formalize it, it is hard to see how it would look like anything other than some kind of tautology: perhaps, ‘If you are an A, then be an A’. More puzzling, the power of the injunction does not flow from the transition from the indicative in the antecedent to the imperative so much as from a developing understanding of what it means to be an A.

1

1.1

IRONY

Kierkegaardian Irony

Kierkegaard called this form of practical reasoning irony. But it has become particularly difficult to understand what this form of persuasion

The Force of Irony 145 is for three reasons. First, in the contemporary world ‘irony’ is used to mean pretty much the opposite of what Kierkegaard meant. In the contemporary world, irony is used to express detachment from any particular values, whereas for Kierkegaard irony is a paradigmatic expression of commitment to them. In the contemporary world, irony is used to say the opposite of what one means (perhaps with the intention of being recognized as doing so), whereas for Kierkegaard irony is a vehicle for saying exactly what one does mean. In the contemporary world, irony is the opposite of earnestness, whereas for Kierkegaard, irony is a form of intense earnestness. Thus if we take Kierkegaard to be speaking our lingo, the result is confusion. Yet though these two uses are polar opposites, they are related. Second, Kierkegaard’s views on irony developed over time. When philosophers in the English-speaking world want to consult Kierkegaard’s views on irony, they naturally tend to go to The Concept of Irony (1841).1 However, Kierkegaard himself came to regard that text—his Magister’s thesis—as very much a young man’s work. In a later work, Concluding Unscientific Postscript, a pseudonymous author, Johannes Climacus, makes fun of Kierkegaard’s early views: ‘As can be inferred from his dissertation,’ Climacus tells us, ‘Magister Kierkegaard’ has ‘scarcely understood’ Socrates’ teasing manner (Kierkegaard 1846: 503 n. 90). Thus there has been a tendency to treat as ‘Kierkegaard’s view of irony’ aspects of his early treatment that Climacus subjected to ironic undermining. Which brings us to the third reason. When it comes to irony, Kierkegaard wants to be ironic: that is, he doesn’t want to talk about this form of persuasion, he wants to be persuasive in this way. He wants to trigger in his reader the experience of irony. And he is justifiably concerned that a direct and didactic account of what irony is will only get in the way. He shudders at the thought of academics expounding on his theory of irony—what could do more to flatten irony’s power?—and he sets out to explode that possibility with his irony. His best work on irony is itself a stunning ironic performance in which a pseudonymous author not only pours scorn on the Kierkegaard’s own efforts to explain irony; he turns the idea of an academic discourse on irony into an occasion for hilarity (see e.g. the appendix of Kierkegaard 1846, ‘A glance at a contemporary effort in Danish literature’, especially the footnote that runs pp. 274n–277n). Still, a good joke requires an audience that is more or less in a position to get it. And I am concerned that, with the reversal of meaning that has occurred in the contemporary world, we could use more stage-setting than we get in his published work. Thus for all the dangers in trying to approach Kierkegaard’s irony directly, I think it is worth the risk to play the role of straight man. So in this essay I want fi rst to give an account of at least one important thread of what Kierkegaard meant by irony; second, I want at least to indicate how this peculiar form of persuasion got lost in modernity; fi nally, I want to explain why it matters to recover it.

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If one reads through his corpus, one will not find a more straightforward account of irony than an entry Kierkegaard made in his journal on December 3, 1854. I quote here the fi rst part: My entire existence is really the deepest irony. To travel to South America, to descend into subterranean caves to excavate the remains of extinct animal types and antediluvian fossils—in this there is nothing ironic, for the animals extant there now do not pretend to be the same animals. But to excavate in the middle of ‘Christendom’ the types of being a Christian, which in relation to present Christians are somewhat like the bones of extinct animals to animals living now—this is the most intense irony—the irony of assuming Christianity exists at the same time that there are one thousand preachers robed in velvet and silk and millions of Christians who beget Christians, and so on. In what did Socrates’ irony really lie? In expressions and turns of speech, etc? No, such trivialities, even his virtuosity in talking ironically, such things do not make a Socrates. No, his whole existence is and was irony; whereas the entire contemporary population of farm hands and business men and so on, all those thousands, were perfectly sure of being human and knowing what it means to be a human being, Socrates was beneath them (ironically) and occupied himself with the problem—what does it mean to be a human being? He thereby expressed that actually the Trieben of those thousands was a hallucination, tomfoolery, a ruckus, a hubbub, busyness . . . Socrates doubted that one is a human being by birth; to become human or to learn what it means to be human does not come that easily . . . (Kierkegaard 1854: 277–8) Irony in the deepest sense is not for Kierkegaard a play of words, but a form of life. It is a form of life which encompasses his ‘entire existence’. But existence for Kierkegaard is not mere biological life; it is the human way of living in the light of self-understandings, tasks, ideals, and narratives that are accepted and rejected. For my entire existence to be irony, I’ve got to be able to live my entire life as an exemplification of my understanding of irony. What kind of living is this? Kierkegaard uses an archeological metaphor, but we don’t need to go on an imagined trip or even carry out an imagined dig: if, say, a duck is waddling across the courtyard, it simply does not matter whether there are bones of its ancient ancestors buried in the ground. The reason is that the duck, in its waddle, is not thereby pretending to be the same as its ancestors. Here is the pivot-point of irony: it becomes possible when you encounter animals who pretend. This is a distinctively human form of life.

The Force of Irony 147 When Kierkegaard says that other animals don’t pretend, he is not making a point about make-believe. Rather, he is using ‘pretend’ in the older sense of put oneself forward or make a claim. 2 Think of the pretender to a throne: she is someone putting herself forward as the legitimate heir. Now in the most elemental sense, pretense goes to the heart of human agency. Even in our simplest acts, pretense is there, at least as a potentiality. You see me bent over and ask, ‘What are you doing?’ and I say, ‘Tying my shoes.’ Right there in that simple answer for which I have non-observational, fi rst-person authority, I am making a claim about what I am up to (see Anscombe 1957: 8–9). Human self-consciousness is essentially constituted by our capacity to pretend in this literal and non-pejorative sense: in general we can say what we are doing; and in doing that we are making a claim about what we are up to. So far, so good. The possibility of irony opens up around a species of pretense that I will call identity-claims: putting oneself forward as a certain kind of person, acting in ways that one takes to be exemplifications of a certain kind of life. There are two salient points about this form of pretense. First, in general, we tend to make identity claims by latching on to some socially accepted role or socially received understanding of an identity. And, second, in joining that social role, there open up all sorts of non-explicit ways of putting oneself forward. So, for instance, the role of professor in the United States at the beginning of the twenty-fi rst century is fairly well-established: there’s a range of dress you can expect a professor to wear, a way of being in front of a lectern and delivering a paper. And there are even socially acceptable ways of demurring from the role: special ways of not wearing the right clothes, not giving a standard talk. That too can be part of the social pretense. But in this variety of socially recognized ways, I put myself forward as a professor. In this way a whole range of activity— including dress, mannerisms, a sense of pride and shame—can all count as pretense in that they are all ways of putting oneself forward as a professor. Since even our simplest acts are regularly embedded in our sense of who we are, the possibility of irony is pervasive. The possibility of irony arises when a gap opens between pretense and an aspiration or ideal which, on the one hand, is embedded in the pretense— indeed, which expresses what the pretense is all about—but which, on the other hand, transcends the life in which that pretense is made. The cases that primarily concerned Kierkegaard were not of individual hypocrisy, but ones in which the individual was an able representative of a social practice that itself fell short. Let ‘Christendom’ refer to the understanding of Christianity as it was embedded in the social rituals, customs, and practices.3 To grasp the power of Kierkegaard’s critique it is crucial not to caricature Christendom. Obviously, there were vain priests within Christendom who cut a ridiculous figure. But Christendom also included a received history of the Church that included the Reformation, a division of the various Protestant sects; a tradition of reflecting on various approaches to Christianity and branding some as heretical; and so on. It is a mistake—and it diminishes Kierkegaard’s

148 Jonathan Lear point—to think of Christendom as an unreflective or unselfcritical practice: Christendom certainly contained priests and theologians who wrote books on Christian theology and history. The problem would not be so difficult and irony would not be so important if reflection and criticism were not already part of the social practice, in this case Christendom. Christendom is the shared, social pretense of Christianity—the myriad ways the social world (and its inhabitants) put itself forward as Christian—and whether we are Christian or not, religious or not, we can agree that serious claims are being made. Kierkegaard’s fundamental ironic question is: (1) In all of Christendom is there a Christian? Or, to put it more bluntly, (2) Among all Christians, is there a Christian? It is a striking fact about us that we can immediately hear there is a question being asked, rather than a meaningless repetition. The form of the question is a tautology; yet we do not hear it as a tautology: and it is, I think, a revealing fact that this should be so. The question is asking whether people who understand themselves as Christian are living up to the ideals of Christian life. If, by contrast, we were to ask, (*) Among all ducks, is there a duck? we would have no idea what question, if any, was being asked. Unlike human life, duck life does not involve pretense: ducks do not make claims for themselves, they do not put themselves forward as anything at all. Thus there is no room for a gap opening between pretense and aspiration.4 By contrast, it is characteristic of human life—either explicitly or in our behavior—that we do make claims about who we are and the shape of our lives. This quintessentially human activity of putting oneself forward as a certain kind of person can, in certain circumstances, set us up for the fall: this can occur when the pretense simultaneously expresses and falls short of its own aspiration. Irony, for Kierkegaard, is the activity of bringing this falling short to light in a way that is meant to grab us. Note that this Kierkegaardian irony is a form of intense earnestness. In asking whether there is a Christian in Christendom, Kierkegaard is asking what he takes to be the most important question he could be asking both of himself and of his fellow Europeans; and he is saying exactly what he means, not its opposite. As Johannes Climacus put it, From the fact that irony is present, it does not follow that earnestness is excluded. That is something only assistant professors assume. (Kierkegaard 1846: 277n.)

The Force of Irony 149 Obviously, Kierkegaardian irony is very different from the contemporary understanding of ironic detachment, in which one’s ‘irony’ mocks earnestness, perhaps by saying the opposite of what one means. Nevertheless, I think the two uses are intimately related, and thus it is a mistake to conclude that there are just two different uses of the same word. Later in the essay I shall try to indicate how this could be. Let us continue with Kierkegaard’s example. Suppose that a hallmark of Christian life is loving one’s neighbor as oneself. The tricky part—the reason irony is needed—is that Christendom ostensibly already contains this teaching as well as an understanding of what it is to fall short. Indeed, Christendom even contains its own version of irony. I spend Sunday morning listening to a sermon about ways we fail to live up to that ideal. I leave the Church and pass a beggar on the street; he irritates me; then I remember the priest’s sermon. I turn around and give the beggar a dollar. He says, ‘You must be listening to your priest’. What is he saying? We’ll never know. But I may understand him in a number of different ways. I may, fi rst, take him simply to be remarking that it is a memory of the priest’s words that pricked my conscience. Or I may take him to be speaking ironically in the familiar sense of exuding sarcasm about the paltry nature of my donation. He’s telling me in his ‘ironic’ way—saying the opposite of what he means in a way that I can recognize—that I should have given him a twenty. So far, we haven’t left Christendom: my sense of falling short, my sense of his ‘irony’ all fall within received social understandings. But suppose now that it occurs to me that I have learned from my priest and that is my problem! The thought now is that even when I am pricked by conscience and experience myself falling short of my own ideal—that entire package I learned in Christendom falls abysmally short of Jesus’s teaching. I have a dawning sense that if I were actually going to love my neighbor, I’d have to change the way I live—not merely in terms of the way I treat others, but in the steps I undertake to change myself into a person capable of such love. It is hard to imagine a stronger experience of the practical necessity of a form of persuasion. And so, there is room to ask the ironic question, (3) Among all those who love their neighbors, is there anyone who loves his neighbor? The falling short at issue itself transcends our normal and received understanding of what it is to fall short. It is as though an abyss opens between our previous understanding of the ideal and our dawning sense of what it now calls us to. This is the strangeness of irony: we seem to be called to an ideal that, on the one hand, transcends our ordinary understanding and our ordinary lives—as yet we only have an enigmatic, uncanny sense of what we are called to—but, on the other hand, seems to be an ideal to which we are already committed.

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This is what makes irony compelling. It is the mirror-image of an oracle. An oracle begins with an outside source telling a person who he is in terms he at fi rst fi nds alien and enigmatic. There is then an unsettling process of familiarization: the person comes to understand what the oracle means as he comes to recognize that he is its embodiment. With this robust form of irony, the movement is in the opposite direction: a person gives a familiar designation to himself. As the irony unfolds, not only does the designation become unfamiliar, the person experiences himself as falling abysmally short of what he had previously taken himself to be. And while I do not know what the proper metaphysical account of this experience is, the language that suggests itself to describe the experience is Platonic: a person comes to recognize that the ideal he has taken himself to be living with is merely an appearance, a shadow, an imitation of the ideal to which he is in fact committed.5

1.2

Kierkegaard’s Socratic Irony

Kierkegaard’s irony is a special instance of the Socratic ‘What is it?’ question. It begins with a question whether anyone lives up to the conditions of the identity with which they put themselves forward. That provides an occasion for inquiring into what the identity really consists in. And, like Socrates, the answer is not supposed to be a contribution to theoretical reason, but a practical change in how one lives. So, consider the following Socratic questions and their Platonic answers: (4) Among all politicians (in Athens), is there a single politician? (Gorgias 513e–521e) His answer is that no one in the entire cohort of those who put themselves forward as politicians qualify, nor do those who Athenians have assumed to have been great politicians, notably Pericles. For none of them have genuinely been concerned with making the citizens better. ‘I am one of the few Athenians,’ Socrates says, ‘not to say the only one—who understands the real political craft and practice politics—the only one among people now’ (Gorgias 521d). Similarly, Socrates asks (5) Among all rhetoricians, is there a single rhetorician? (Gorgias 502d– 504a; see also Phaedrus 259e–274b) His answer again is that no one who puts himself forward, or anyone so reputed from earlier times, has been engaged in anything more than shameful flattery and gratification (Gorgias 503a–d). The true rhetorician looks to the structure and form of the soul, and crafts his speech so as to lead souls toward virtue and away from vice (Gorgias 504d–e, 503e–504a). Plato’s implication is that if there is a single rhetorician in all of Athens it is Socrates. And again,

The Force of Irony 151 (6) Among all doctors (in Athens) is there a doctor? Plato’s answer: there is Socrates, for he is the one genuinely concerned with promoting health (Charmides 156e–157b, 170e–171c; Gorgias 521a; Republic III 405a–408e, 409e–410e, VIII 563e–564c and X 599b–c). Those who put themselves forward as doctors are in effect gratifiers: fifthcentury-BC equivalents of the prescribers of Valium, Viagra and the purple pill. In Plato’s view they are only helping the dissolute extend their lives. (7) Among all shepherds, is there a shepherd? (Republic I 345b–e) Plato: there is Socrates because he alone understands that the true shepherd looks to the good of his flock, not to those who feed off them. (8) Among all the wise, is there a wise person? There is Socrates, for he alone knows that he does not know (Apology 23a–b). And so on. These questions all have the same form—and in each case the possibility for irony arises by showing that the pretense falls short of its own aspiration. It is a movement that exposes a pretense in the nonpejorative sense to be pretense in the pejorative sense. This is not the place for a detailed defense of this interpretation of Socrates—though I should mention that this account goes against the received accounts of Socratic irony (as deception, putting on a mask, saying the opposite of what one means, perhaps with the intention of being recognized as doing so, and so on); and it even goes against received accounts of what Kierkegaard meant by Socratic irony. I intend to show that these accounts are mistaken—but not in this essay! For the moment, I will only mention in passing that this account of Socratic irony provides an overarching unity to Socrates’ method that would otherwise go unnoticed. On the standard interpretation, the Socratic method is identified as refutation, the elenchus, which is characterized formally as an attempt to elicit a contradiction—p and not-p—from an interlocutor. When the figure of Socrates in the dialogues abandons the elenchus, he is portrayed as having given up on his own method. There is then the famous charge that he has just become a mouthpiece for Plato (see notably Vlastos 1991: 45–6 and 53). But if one takes Socratic irony to be exposing the gap between pretense and aspiration, then the elenchus can be seen as one instance of this method. For in each case, the interlocutor is someone who puts himself forward—as knowing what justice or piety or courage is—and in each case the interlocutor is shown to fall short of aspirations that it has seemed to him he already fulfills. By concentrating on the formal feature of contradiction, commentators have ignored an essential non-formal feature: that it brings out the gap between pretense and aspiration. Thus when Socrates shifts from the elenchus to other ways of bringing out this gap, he need not been seen

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as giving up on Socratic method, nor as having become a mouthpiece for Plato; rather, he is taking up myriad forms of one method, Socratic irony.

1.3

Transcendence

I have already said that I do not know the proper metaphysical account of the kind of transcendence that is at stake in irony. (It seems to me that it is compatible with both Platonic and constructivist ontology.) But the important point right now is that the transcendence at issue is of available social pretenses, and this is a possibility that can be realized in human life. We are not talking about transcendence of the human realm altogether. Kierkegaard thought it was extraordinarily difficult to become (and be) a Christian, but he thought it was possible; Plato thought it was difficult to become (and be) a doctor properly understood, but he thought it possible. These genuine human possibilities of transcendence tend to escape our notice. In part this is because the social pretense puts itself forward as an adequate understanding of, say, what an identity consists in. But it is also true that the social sciences tend to overlook this possibility of transcending social pretense. This is because, in general, the social sciences want to collect data that are measurable, repeatable, and statistically analyzable. Now if we look at the ironic questions, we can see they establish two columns. Left

Right

Christian

Christian

Politician

Politician

Doctor

Doctor

Rhetorician

Rhetorician

Shepherd

Shepherd

Sophist

Sophist

...

...

The left-hand column is formed from the fi rst occurrence of the relevant term that expresses the social pretense; in the right-hand column, there is the second occurrence of the same term, which invokes the aspiration. Roughly speaking, the left-hand column gives us the domain of the social sciences. It gives us the domain that is straightforwardly accessible when it comes to collecting data that is objectively measurable. The perennial challenge for social scientists is to figure out ways to operationalize a question: and that will inevitably tend one’s research in the direction of the left-hand column. So, for example, if one wanted to understand religion in America, one might try to establish reliable statistics for what percentage of the population attends church each week, what percentage self-describe as religious, and so on. This is all data that comes from the left-hand column. Though

The Force of Irony 153 a life exemplifying any of the categories in the right-hand column is neither mysterious nor supernatural, it is difficult to capture and it certainly does not lend itself to straightforward data-collection or measurement. There is no statistically reliable way to answer the ironic question, ‘Among the millions who worship on Sunday, has anyone ever worshipped?’ So what, then, is the transcendence of the right-hand column? It is difficult to say, not because it is an ineffable mystery, but because almost everything one wants to say admits of an interpretation that is appropriate to the left-hand column. It is as though one already has to have some capacity for irony to grasp what it is about. Let us use an example that is close to home, the category of student or teacher. The left-hand column is easy enough to establish: a student is someone who is enrolled in a recognized school. Now we might be tempted to think that if we add on a few conditions we can move on over to the right. But everything is going to depend on how those conditions are themselves understood. So, the conditions that seem to me relevant are that a student properly understood needs to be someone who takes on the life-task of becoming a person who is open to the lessons that the world, nature, and others have to teach her. In so doing, she recognizes that the task is as never-ending as it is voracious. That is, one is never done trying to shape oneself into a person who is open to the world in these ways; and no experience can be ruled out as irrelevant. Obviously, satisfying these conditions takes one well beyond the run-of-the-mill student; but there are ways of doing it that remain within received understandings. Ditto if one tries to nail it down by adding that one needs to take individual responsibility for what all this consists in. These statements need not take one out of the realm of social pretense. And yet they also seem to me to be the right sort of statements to make about what the right-hand understanding of student is. The difference lies in what we understand these conditions to consist in. In particular, the ideal of openness must include an openness to the possibility that all previously received understandings of what openness consists in themselves fall short of what openness really demands.6 And taking responsibility must consist in a willingness to orient oneself according to this revised understanding, regardless of what the social pretense recognizes or demands. Note that it is possible to be open to this possibility and come to the conclusion that at least some version of the social practices are adequate. None of this requires a perfect instantiation of a student or teacher. If one were to ask, (9)

Among all the students, is there a student?

I think the answer is ‘yes’, but not because there is some perfect form of a student walking in our midst. It is that the person has allowed in her life the possibility that the ideal may open up in ways that transcend the received understandings of what the ideal consists in, and is committed in living

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in relation to that ideal. Because it does require such a life-commitment, Kierkegaard called it subjectivity (Kierkegaard 1846: ch. 3, 301–60). This is again an ironic use of the term that stands at a significant distance from what we normally mean by ‘subjectivity’ today. In contemporary parlance, a person’s subjectivity simply is an aspect of her pretense—how she understands what she takes to be her point of view; whereas for Kierkegaard, subjectivity is a life-task that requires putting one’s own pretense into question. The right-hand column carves out the domain of Kierkegaardian subjectivity: and that is the domain of irony as way of life.

1.4

The Disappearing Ironist

In comparison to Kierkegaard, contemporary accounts of irony look insipid. Richard Rorty, for example, defi nes an ironist as someone who, fi rst, ‘has radical and continuing doubts about the fi nal vocabulary she uses, because she has been impressed by other vocabularies or books she has encountered’ (Rorty 1989: 73). (A fi nal vocabulary is that which one uses to formulate basic projects, important hopes, doubts, praise, and blame.) Second, she has ‘realized that the arguments phrased in her current vocabulary can neither underwrite nor dissolve these doubts’ (73); and, third, ‘insofar as she philosophizes about her situation, she does not think that her vocabulary is closer to reality than others, that it is in touch with a power not herself’ (73). This last condition implies that Rorty’s ironist, in direct opposition to Kierkegaard, can never be in earnest about any fi nal vocabulary. Rorty’s ironist responds to doubt by looking elsewhere; and she remains equally remote from all fi nal vocabularies she considers or uses. Note that Rorty’s ironist need never leave the left-hand lane of life. To continue with an example we have been using, imagine an inhabitant of Christendom who starts to have doubts about the institutionalized practices. Something about it she experiences as routine, hollowed out. She then looks sideways over at other fi nal vocabularies. So, she reads books about Judaism and Islam, reads about Confucianism and Buddhism, she even tries out some New Age spirituality. Perhaps she visits temples, mosques, and other shrines. The temptation to caricature her is of course enormous; but I’d like to refrain from doing so. The point is that in investigating these other fi nal vocabularies, there is no pressure thereby generated to question the various social pretenses other than in terms of other pretenses. Rorty’s ironist is compatible with a decadent world-weariness in which any choice is as good as any other because none can really resolve doubts. From Kierkegaard’s perspective, Rorty’s ironist is not an ironist at all, but someone who is living a run-down version of an aesthetic life. Different fi nal vocabularies are treated as though they were objects of disinterested choice: one could choose them on the basis of being ‘struck by doubt’ (an aesthetic phenomenon if ever there was one) with one’s own fi nal vocabulary.

The Force of Irony 155 But the point here is not to criticize Rorty. In fact, Rorty’s irony is what irony would look like if the right-hand column—the column of subjective aspiration to a pretense-transcendent ideal—made no sense. Then there would only be the disenchantment with a given social pretense (and its fi nal vocabulary) while the only alternatives on offer were other social pretenses (and their fi nal vocabularies). That is why there is reason to think that there are not simply two different uses of the word ‘irony’, but that contemporary use is a diminished version of what Kierkegaard meant. For if our ears suddenly became deaf to the meanings in the right-hand column, then irony would inevitably come to be an expression of detachment and lack of commitment rather than an expression of earnestness and commitment. One might think of Kierkegaardian and Socratic irony as a twopart movement of detachment and attachment: detachment from the social pretense in order to facilitate attachment to the more robust version of the ideal. But if one obliterates the second part of the two-part movement all that remains is irony as a form of detachment. And it would make sense to experience the ironist as saying something other than he means; because there would, in these diminished circumstances, be no available right-hand meaning with which he could say what he does mean.7 It seems to me that Rorty’s account of irony is symptomatic of something that has happened in modernity that has made it difficult to hear the resonances of the righthand meanings, which Kierkegaard called subjectivity.

2

MODERNITY

So in this second section I would like to try work out at least a preliminary understanding of what the change has been that has obscured the possibility of real irony.8 There do seem to be a number of currents that may have arisen contingently but together render current forms of social pretense resistant to irony. First, identity. Our contemporary paradigms of identity have arisen out of histories of discrimination, oppression and victimization. In the waves of immigration to the United States in the nineteenth and twentieth centuries, people were labeled as Italian, Irish, Jew in part to tag them and keep them separate from dominant culture. (Similar tagging occurred in the United Kingdom and Europe as immigration from former colonies increased.) There would then be little social room for an ironic question whether anyone fits the category. The point of the tag was to keep people inside the category and not let them out. Another route has been where the identity has been formed in conscious response to a history of oppression. Prime examples are black and African-American—formed in response to pejorative terms that had previously been used. Another would be gay, a term self-consciously dignified by homosexuals themselves, as in ‘gay pride’ (see Appiah 2005, especially pp. 62–113). These formulations are

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self-consciously part of a process aimed at encouraging self-esteem in a group that historically has been demeaned by the dominant culture. It would run counter to one of the aims of such formulations if, in the conceptualization, it left open the possibility that almost no one in the hitherto oppressed group lived up to the demands of the category. Of course, it is possible to formulate an ironic question in any of these cases. And there have certainly been debates within each of these groups as to what their central ideals should be and how they should be understood. However, given the various histories of discrimination and oppression, the focus has been on how the social pretense should be understood. And this has provided a paradigm for our contemporary conception of identity. Second, identification. Perhaps in relation to this paradigm, identification is treated as a discrete psychological act: a conscious or unconscious choice or acceptance of a particular identity. It is a conceptualization of psychological activity appropriate for the left-hand column of our ironic questions. For in the paradigm case, the social identity simply applies and the only question is whether one sees oneself in those terms. By contrast, identification in a Kierkegaardian spirit would be a subjective life-task. It wouldn’t merely be a matter of psychic choice, but of life-time commitment to living in certain ways. Third, inner cosmopolitanism. It is a striking feature of modernity that we regularly are born into, inherit, are given, or take on a plethora of identities, and it is often unclear how they could all fit together. This is a marvelous feature of modernity, but it does provoke challenges; and this can give rise to a peculiar defense. Just as a country might hold its wealth in a basket of currencies, so a prudent inhabitant of modernity invests in a basket of identities—to protect against the vicissitudes of any one. While we may celebrate them all, we don’t have to be trapped into any one. Thus there will be a tendency to become somewhat tentative with any particular identity; and this tentativeness will keep us in the realm of identity as social pretense. Fourth, the liberal value of letting each person set his or her own end makes us understandably reluctant to question the identities people use to structure their ends, and their sense of self. These are, by and large, social pretenses.

3

WHY BOTHER?

This leads to the final question I want to begin to tackle in this essay: why not just go with the flow? Why not simply accept the Rortian celebration of modernity, and treat Socratic-Kierkegaardian irony as a relic? The answer, I think, is that this is not really a viable option. As human, we are creatures who are susceptible to the force of irony—and, on certain occasions at least, irony does have a legitimate claim on us. We may be motivated to

The Force of Irony 157 ignore this, historical circumstances may contingently cover over this possibility, but a proper self-understanding requires that we come to grips with the role of irony in human life. I will close with brief remarks on how this self-understanding manifests itself in the case of the individual and in the case of culture.

3.1

Ironic Soul

It is only when one grasps the structure of irony in terms of pretense and aspiration that one can see that the drama of irony is recreated inside the human soul. Ever since Plato’s Republic it has been a philosophical theme that human happiness requires harmonious integration of disparate psychic parts. In the Republic’s account, the human soul has three parts: appetite (epithumeia), which is multifaceted and capable of desiring an indeterminate range of things, but of which hunger, thirst, and sexual desire are paradigms; spirit (thumos) which seeks honor, recognition from others, victory; and reason (logistikon) which desires to know how things really are and which seeks the truth of the whole. But this psychic structure sets us up for irony. For spirit is an internal voice of pretense: it is the organized part of the soul through which we put ourselves forward as wanting something or as taking up a certain role or position that deserves the recognition of others. And reason is custom-made to be the part of the soul that expresses our aspiration to know how things really are—and not simply as a theoretical matter, but so that we can live in its light. Thus from this Platonic perspective the psychic drama of trying to integrate the soul simply is the internal task of trying to get the voices of pretense and aspiration into accord.9 Indeed, from a Platonic perspective, it is precisely because we have this psychic structure that we can hear the ironic question as a genuine question, rather than a mere tautology. The fi rst occurrence of the term will tend to trigger the thumoeidic, spirited part of our soul, while the second occurrence will trigger reason; and the confl ict between these parts will be experienced as some form of falling short. As is well known, Plato thought there were different personality types, and that these types determined by which part of the soul ruled over the other parts (Republic IX 581b–c). In the healthy personality, it is reason that rules. Spirit is a motivated part of the soul seeking recognition, say, as a hero. Reason in effect ‘says’ to spirit that the only way really to be a hero is to be courageous, and true courage is something like this . . . . Reason in its proper function is thus engaged in the projects of 1) trying to figure out what the proper ends of human life are; and 2) trying to encourage the other parts of the soul to see that these proper ends are really its ends. This activity basically has the shape of internal irony. The fi rst movement is trying to figure out what, say, courage is really all about: this is, in effect, trying to grasp the right-hand column. The second movement is trying to wean spirit, the pretending part of the soul, away from its training in social

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pretense. Pulled away from the roar of the crowd, spirit is taught (by reason) that if it really seeks honor then it must orient itself toward true courage. By contrast, with the spirited personality, spirit rules over the soul and reason is enslaved to its desires. Spirit then simply adopts some socially recognized way of winning honor; and reason is reduced to an instrumental, calculative function of figuring out how to get it. This is a soul in which it would be difficult for real irony to get a grip. However, Plato thinks that only the soul in which reason rules is both happy and stable. Thus not only must the spirited person be unhappy, the parts of his soul must be in some conflict. Thus there is potentially room for irony to be a force for psychic change.10 One might well think that using Plato’s conceptualization is too easy a response to Rorty’s challenge, because a bourgeois modernist like Rorty would deny we have a faculty of reason of the Platonic sort. For him, assuming that we have reason begs the question in favor of the robust conception of irony, rather than his pale alternative. So, by way of contrast, let us consider a non-philosophical conceptualization of psychic structure, that of Freud. A follower of Darwin, Freud did not think we were put on earth for any particular task; and, given how we emerged and how we must live in society, Freud did not think it was given to us to be happy. Still, insofar as happiness was possible, he thought it required psychic integration. His conceptualization grew out of a study of neurotic conflict. Famously, it is in terms of id, ego, and superego. But it is possible to conceptualize these parts also in terms of pretense and aspiration. The ego is the voice of pretense par excellence: it is that part of the soul by which we put ourselves forward in the social world, make claims about who we are or what we are doing. And the superego can be conceptualized as a voice of aspiration. In its way it is trying to state how things really are and trying to cajole the ego into taking up the task. It is a voice in relation to which we either experience ourselves as falling short or are inspired to take up the challenge to live up to an ideal. Again, we can see in this non-philosophical characterization of the human soul that happiness requires psychic integration; and psychic integration is the harmonization of voices of pretense and aspiration.11 Now one might object, in a Rortian spirit, that in the Freudian case it needn’t really be a case of irony, because the superego might be scolding merely in terms appropriate to what we have called the left-hand column of life. Typically, this is the case. And, as Freud so brilliantly described, the superego regularly turns a person’s own aggression inward, directed on the ego, cruelly keeping the ego within the rigid demands of civilization (see Freud 1930). So, to continue with the examples we have been using, a full-fledged member of Christendom might feel guilty because he skipped attending church, or a doctor might feel guilty because he cheated on his medical exam—this is the stuff of social pretense. The aspirations of the superego, one might object, will typically not take us beyond the realm of pretense. This is a good objection. The question though is whether it has to be the case. The fact that psychoanalysis is not basically a theoretical science

The Force of Irony 159 but rather a form of therapeutic engagement suggests that the answer is no. And all my clinical experience suggests that it does not. People typically enter psychoanalysis with specific and sometimes bizarre complaints—(a person is turned off by her partner’s slight paunch; it brings her stepfather to mind)—but if the analysis is allowed to continue patients come to recognize that they are striving to inhabit a subjective category. Of course, that is not how they would put it, but what they want to know is whether they will ever love or be loved, whether they will ever have or be a friend, whether they will ever be able to create, contribute, or appreciate anything. When these questions come alive, the social pretense is never an ultimately acceptable answer. The analysand may already be part of what for all the world looks like a successful marriage; what he wants to know is: what is it to be married? It is never true that, say, the analysand has a clear and unshakeable sense of what marriage is; and his only question is whether he can live up to it. Rather, marriage itself becomes a puzzle; and he needs to settle simultaneously what it is and whether he can or even wants to live it. When a psychoanalysis is allowed to unfold it is a process by which a person deepens him- or herself as a subject (properly understood). And this suggests that, appearances to the contrary, the superego can be (slowly) eased out of dominating surveillance-and-punishments functions of left-handed life. Indeed, in a successful treatment, psychoanalytic therapy seems to be a process by which a punitive superego, rigidly attached to social pretense is transformed into a capacity for more playful and creative questioning of what one’s ideals are. Rather than being a policeman for civilization’s ideals, a transformed superego expresses itself in a creative ability to disrupt its own ideals in the name of its ideals. It would facilitate a process by which the ego is cajoled, in less punitive and more humorous and erotically enticing ways, into taking these ideals with ironic seriousness. Irony, properly deployed, is a form of therapy for the superego, and facilitates its transformation into an internalized capacity for irony.

3.2

Ironic Culture

If we consider human self-understanding at the level of culture—of the fi rstperson plural—there is a crucial aspect of our life with concepts—in particular, the concepts with which we understand ourselves—that we ignore if we pass over the force of irony. The possibility of irony brings to light that we are fi nite creatures in this sense: we are creatures who understand ourselves and our world with concepts that are themselves vulnerable. How are we to live well with that possibility? This is a question of practical selfunderstanding. The only way I have found to think about this is by looking at a particular case. The Crow Nation is a tribe of the northwest plains of (what is now) the United States that had to suffer a catastrophic transition into the modern world.12 Their traditional way of life became impossible more or less

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overnight, when they moved onto the reservation in the spring of 1884. From that impossibility flowed a peculiar kind of unintelligibility—one might call it practical unintelligibility. That is, the traditional concepts of Crow life continued to make theoretical sense as something one could remember and discuss, but they ceased to make sense as ways one could understand oneself or others as going forward in life. To take a paradigmatic example, how could one any longer understand oneself or others as being a warrior if it was no longer possible to go into battle? And so there opened up the possibility of an ironic question that earlier would have hardly made sense: (10) Among all the Crow, is there a Crow? Although this question has the same form as the previous ones, there is a significant difference. With the Socratic and Kierkegaardian questions (1–9), the ideal expressed on the right was perhaps enigmatic, hard-toreach, disruptive, but it was assumed to be in good shape. We may grasp the ideal only dimly but there was no question for Socrates or Kierkegaard, that the ideal was there to be apprehended. The issues then were to get a better grasp of ideals that were themselves in good shape and to come to a better practical understanding of how the pretense fell short; and what kinds of disruption were appropriate. With the Crow, by contrast, both the pretense and the ideal have simultaneously fallen into disrepair. In this case, the ironic question is not asking whether the social pretense lives up to the ideal, but whether social pretense and the ideal any longer make sense. Now what would it be to face up to this human possibility well? I have seen basically three different responses among members of the tribe: (1) fall into despair; (2) transfer over to the social pretense of another culture; (3) try to recreate and reinvent Crow culture. There is something sympathetic one can say about each of these options; but it is option (3) that seems more like an embrace of human fi niteness: for it must inevitably ask, ‘What would it be like to live well now as a Crow?’ And, a fortiori, one cannot give the answer in terms of any available social pretense. One needs to be asking questions which are ironic, but which express immediately pressing needs of the culture, for example: (11) Among our wise people, who any longer counts as wise? What would it be any longer to be wise? (12) What would it be to face courageously, with dignity and integrity the collapse of courage, dignity, and integrity as they have traditionally been understood? So: courageously to face the collapse of courage requires a fortiori that one transcends earlier social understandings of what courage consists in. This isn’t just a transformation of theoretical understanding; it requires a

The Force of Irony 161 transformation of how one practically engages with life’s possibilities and challenges. Thus this transcendence needs to involve not merely re-thinking what courage requires in this altered world but also the possibility of psychic reorientation. This, I think, is what has been going on in dreamvisions, in a renewed outpouring of Crow poetry and hip hop music, in reinventions of religious celebrations, as well as in continuing discussions among Crow leaders about how to lead the tribe. They have been trying to work out what courage, dignity, and integrity is really all about so that they can formulate distinctively Crow ways of going forward in radically new circumstances. What is striking about this process is how it has manifestly declined to go down the Rortian path of adopting the fi nal vocabulary of another culture. There is a lesson here that has nothing to do with valorizing an alternative non-Western form of life; or pitying a victimized culture for being deprived of its fi nal vocabulary. Rather, we can now see two different ways of responding to the peculiar fi niteness of the human condition. One response, for which Rorty has been emblematic, is to celebrate the myriad forms of life, of socially available identities, fi nal vocabularies, diversity. In terms of toleration, there is much to be said in favor of this celebration. But it seems more a defense against human fi niteness than an embrace of it. By contrast, irony embraces this form of human fi niteness: for it is itself the activity of bringing to light values embedded in a social practice that transcend the vulnerable forms that may well go under. This, I think, manifests a better practical understanding of who we are.

3.3

Conclusion

Irony is a peculiar form of practical argument by which one is called—or one calls oneself—to be oneself. It is not simply a standard case of reflective criticism by which I fi nd that I fall short of an ideal; it is more like an enigmatic, but nevertheless persuasive disruption in which even the ideal is seen to fall short of itself. For those who have felt its grip, there is no more powerful form of persuasion. I hope I have at least begun to show how it is that the persuasion might be legitimate.13

NOTES 1. I am not here concerned with Kierkegaard scholars per se; but with philosophers who wish to make reference to Kierkegaard in the development of their own philosophical argument. As notable examples, see Vlastos (1991: 42–4 and 132n) and Nehamas (1998: 52–3, 69–72 and 86). 2. The Danish is ‘udgive sig . . . for’, literally ‘give themselves out to be’. On its own ‘udgive’ is ‘publish’, ‘to put something out there’. With ‘for’ the meaning is ‘present themselves as’. Thus ‘udgive sig for’ is not normally predicated of animals. My thanks to David Possen for help with the Danish.

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3. This would include what Charles Taylor has called the social imaginary of Christianity: shared images, fantasies, and myths that are embedded in and elaborate those rituals and customs. See his (2004), especially pp. 23–67. 4. To be sure, one can imagine circumstances in which this form of question does make sense in non-human animal life. For instance, if all the remaining tigers in the world existed in captivity one might ask, (**)

5.

6. 7. 8.

Among all tigers, is there a tiger?

One would be asking whether the actually existing tigers were able to live a life appropriate to their species. In this case there would be no pretense, but captivity would impose conditions about which it would make sense to ask whether it was any longer possible to live a tiger’s life. A gap would have opened between mere biological life and the possibility of flourishing as a tiger. But the extraordinary conditions needed for this question to make sense shows that in general it doesn’t make sense to think of non-human animals as falling short in their own lives. This is because the identity of other species doesn’t depend on their making claims about their identity; it doesn’t depend on them making any claims at all. If we can make sense of their falling short via such extraordinary circumstances, it is due to our pretenses: the way we put ourselves forward in dealing with rest of the animal and natural world. We are now in a position to see not only how Christendom can fall short of Christianity properly understood; but how it actually serves as a defense against it. For, as we have just seen, Christendom uses the very same language as Christianity. Take any hallmark of Christian commitment and the inhabitant of Christendom believes that he already subscribes to it. Even if he is failing at the task, he takes himself to understand what he is failing at. Thus one way to avoid being a Christian is to take oneself already to be one. This need not be rank hypocrisy: it can flow from a sincere attempt to be a Christian by following received understandings of what it is to be one. The power of Christendom flows from the fact that it uses the language of Christianity to put itself forward as an adequate understanding. Indeed, it thereby purveys a sense that when it comes to Christianity, nothing is missing. The aim of irony is to blow open this illusion of a world. But it must do so by using the same language that Christendom uses. Let’s go back to that beggar’s remark. If I am stung by its irony, it doesn’t really matter what he meant or what effect he intended to produce: for me, it is as though a new task has opened up that was hitherto unavailable, though I would use the same words to describe it as I have used before. But it is at least imaginable that a person might think that a Christian way to love one’s neighbor is to occasion in them a deeper understanding of the Christian ideal of loving one’s neighbor. Thus can irony itself be a manifestation of that Christian commitment. I believe Kierkegaard regarded his own ironic activity in such a light. More work needs to be done on what this openness to possibility consists in; but that lies beyond the scope of this essay. To put it in a nutshell: Socrates’ accusers do accuse him of deception, but that is because they are deaf to the right-hand meanings with which he is speaking. In this context, it is worth noting that insofar as Rorty gets close to Kierkegaardian irony, he treats it with contempt. His ironist is opposed to an oafish figure he calls a ‘metaphysician’, one who simply ‘assumes that the presence of a term in his own fi nal vocabulary ensures that it refers to something that has a real essence’ (Rorty 1989: 74–5). This is of course a caricature; and it doesn’t even have superficial plausibility unless one conflates theoretical and practical reason. For Kierkegaard’s irony is not a contemplative inquiry into

The Force of Irony 163

9.

10. 11.

12. 13.

special objects, ‘essences’; it arises when one questions the essence of one’s own commitments. So, a doctor might wonder, ‘I have spent my life treating people, but have I actually promoted anyone’s health? What is human health anyway?’ The appropriate answer here is not theoretical insight into some weird object but rather an orientation (or reorientation) of one’s life. The case of appetite is complex because in the elemental cases, appetites are for basic bodily needs: hunger, thirst, sex. However, Plato makes it clear throughout Republic that appetites are multiform and they can structure the way people put themselves forward in the world. See especially the appetitive personality and the oligarchical personality (Republic VIII 553a–555b and IX 580d–581a). I try to take this thought seriously in a modern psychoanalytic context in my (2000). Just as there is a question in the Platonic case about how to conceptualize appetite in these terms, there is similarly a question of how so to conceptualize the Freudian id. But, if we look carefully, we can see id-like forms of aspiration. Id-like forms of graspingness, hunger, desire can take on an elemental aspirational form. To take a caricatured example, think of Cookie Monster on Sesame Street: ‘Me want Cookie!’ This is not simply the expression of a discrete and satisfiable desire for an independent object; it is the expression of a way of life. This is aspiration in its id-like form. To see a less cartoonish form, consider the aspiration of our current generation of bankers. I discuss their case in detail in my (2006). I further discuss the issue of practical intelligibility in a cultural context in my (2009). I am indebted to Gabriel Lear and Candace Vogler for years of discussion on these topics, and to Timothy Smiley for a lifetime of conversation and inspiration.

REFERENCES Anscombe, E. (1957) Intention, page references are to the 2000 edn, Cambridge, MA: Harvard University Press. Appiah, K.A. (2005) The Ethics of Identity, Princeton: Princeton University Press. Freud, S. (1930) Civilization and its Discontents, reprinted in J. Strachey (ed.) The Standard Edition of the Complete Psychological Works of Sigmund Freud, vol. 21, London: Hogarth Press, 1981, pp. 57–145. Kierkegaard, S. (1841) The Concept of Irony with Continual Reference to Socrates, page references are to the 1989 edn, H.V. Hong and E.H. Hong (eds), Princeton: Princeton University Press. . (1846) Concluding Unscientific Postscript to Philosophical Fragments, page references are to the 1992 edn, H.V. Hong and E.H. Hong (eds), Princeton: Princeton University Press. . (1854) Journal entry, December 3rd 1854, published in H.V. Hong and E.H. Hong (eds) Søren Kierkegaard’s Journals and Papers, vol. 2, F–K, Bloomington: Indiana University Press, 1970, pp. 277–8. Lear, J. (2000) Therapeutic Action: An Earnest Plea for Irony, New York: Other Press. . (2006) Radical Hope: Ethics in the Face of Cultural Devastation, Cambridge, MA: Harvard University Press. . (2009) ‘Response to Dreyfus and Sherman’, Philosophical Quarterly 144: 81–93. Nehamas, A. (1998) The Art of Living: Socratic Refl ections from Plato to Foucault, Berkeley and Los Angeles: University of California Press.

164 Jonathan Lear Rorty, R. (1989) ‘Private irony and liberal hope’ in his Contingency, Irony and Solidarity, Cambridge: Cambridge University Press, pp. 73–95. Smiley, T. (1995) ‘A tale of two tortoises’, Mind 104: 725–36. Taylor, C. (2004) Modern Social Imaginaries, Chapel Hill: Duke University Press. Vlastos, G. (1991) Socrates: Ironist and Moral Philosopher, Cambridge: Cambridge University Press.

9

The Matter of Form Logic’s Beginnings Alex Oliver

The words matter and form have an ominous sound. They tend to waken echoes from unknown windings of forgotten controversy. But we mean to be deaf, and these murmurs must not stay us, now our logical voyage approaches its end. (Bradley 1928: 519)

When Timothy Smiley taught me logic, he taught me to worry about its beginnings, i.e. ‘the only part of it that can properly be called philosophical logic’ (Russell 1914: 50–1). Textbook beginnings are often short: ‘the best way to fi nd out what logic is is to do some’ (Lemmon 1965: 1), but I learned how far theoretical understanding can lag behind logical practice. When Russell spoke of logic’s beginnings, he specifically meant ‘logical forms’. He knew that logicians cannot cut themselves off from metaphysics, since they also study objects and their properties. Modern logicians, however, seem to live by the proverb ‘Take care of the Ps and their forms will take care of themselves’. For it is a curious fact that although there is plenty of (rather inconsequential) debate about the constituents of arguments— whether sentences, statements or propositions—it is rare to fi nd a proper discussion of form. Yet form is what logic is all about. What limited discussion there now is often displays all the hallmarks of primitive, inchoate conceptualization. It is riddled with toxic ambiguity and shrouded with dark sayings and oppositions: ‘grammatical form misleads as to logical form’, ‘apparent, surface, overt form vs real, deep, hidden form’, and ‘valid arguments are valid in virtue of their form, not their matter’. Here I try to clear some of the ground. I have learned much from reflecting on Smiley’s ‘The schematic fallacy’, his Presidential Address to the Aristotelian Society (Smiley 1982). Appropriately enough, that paper begins and ends with Aristotle (‘Aristotle created mathematical logic . . . Logic has travelled a long way since Aristotle, but not always for the better’). This brings me to the other sense in which he has taught me about beginnings. I shall not be deaf to episodes in the history of logic. I do not dismiss it as a history of corruptions or treat earlier logicians as naive gropers after ideas and truths that only we moderns have fully articulated and appreciated. We have much to learn about our

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concept of form from its past. In what follows I hope there may be one or two surprises.

1

SPEAKING OF FORMS

We say that a sentence has or displays or instantiates or is an instance of or is of a certain form. As these idioms suggest, forms are properties. Other common words for forms are ‘shape’, ‘pattern’, and ‘structure’, i.e. forms concern the symbolic make-up of sentences. These worldly properties can be given linguistic expression. Logicians need to express forms, since generalizations are about forms and logicians need to generalize, both in setting up and exploring a logical system. In order to speak of forms it is often convenient to use syntactic or metalinguistic variables (schematic letters)—A, B, C etc—which range over expressions of an object-language. For the most part, I shall deal with the simplest case in which the values of the variables are sentences of the propositional calculus. These variables are used in schemata to speak of forms, as when we say that the sentence P&Q is of the form A&B. The schema A&B is read: the sentence consisting of A followed by ‘&’ followed by B. (This follows Church’s convention that juxtaposition and ‘&’ stand for themselves, which is much more natural and no less correct than Quine’s quasi-quotation; see Church 1956: §08, and Quine 1951: §6.) Like its constituent variables, a schema also has values: when the values of A and B are P and Q, the value of A&B is P&Q. The variables A, B occur free in the context ‘P&Q is of the form A&B’, but they play the same role there as the existentially quantified ones in ‘for some A and B, P&Q is A&B’. To talk about form, one may mention a schema rather than use it, e.g. ‘P&Q is an instance of “A&B”’, where this means that P&Q is derivable by substitution from ‘A&B’. The rules governing substitution are that sentences replace the variables A, B, and replacement must be uniform in the sense that all occurrences of the same variable are replaced by the same sentence; different variables may—but need not—be replaced by different sentences. For some purposes, the different idioms ‘is of the form’ and ‘is an instance of’ are both serviceable. But sometimes using the schema is swifter than mentioning it: ‘if C is of the form A→B, it is true iff A is false or B is true’. Here the free C has a universally quantified role as in ‘every C of the form A→B is true etc’. We can be briefer still: ‘A→B is true iff A is false or B is true’, where now the free variables within the schema are interpreted as universally quantified.

2

THE AMBIGUITY OF ‘FORM’ AND ‘INSTANCE’

In the textbooks, schemata are variously called diagrams, templates, skeletons, blueprints, outlines, frames, frameworks. They are also commonly

The Matter of Form 167 called patterns or forms. ‘Pattern’ and ‘form’ are therefore ambiguous: they can mean either a property of a sentence or a schema (a similar ambiguity in ‘pattern’ infects the metaphysics of predicates; see Oliver 2010). The two are different. After all, replacing A in ¬A with C would do the same job. The two schemata ¬A and ¬C cannot both be the one property, and because there is no reason to favor one over the other, neither of them is it. In any case, the relevant property needn’t be expressed by using or mentioning a schema at all, at the cost of some prolixity. We could say instead that ¬P consists of ‘¬’ followed by a sentence, or else that it is identical to the result of inserting a sentence in the gap in ‘¬ ’. There is also ambiguity in ‘instance’. I have spoken of instances of forms and instances of schemata. This may encourage wrongly identifying a form with a schema. But ‘instance’ means different things in the different cases. To be an instance of a form is a special case of the general relation between a thing and a property it has. To be an instance of a schema is spelled out in terms of substitution, as above. There is, however, a tight equivalence between the two relations. A third source of ambiguity is Church’s technical sense of ‘form’, according to which a form is obtained from a complex singular term by replacing occurrences of singular terms within it by variables, e.g. ‘x’, ‘x + 3’, ‘3 + x’ and ‘x + y’ are all forms obtained from ‘2 + 3’ (see Church 1956: §02). A schema such as ¬A qualifies as one of Church’s forms, although they have nothing particularly to do with forms conceived as properties of sentences.

3

EXPRESSIVE LIMITATIONS OF SCHEMATA

Using or mentioning schemata to talk about form is natural and common because schemata are succinct and pictorial, though how to read them is obviously determined by convention and context. The schemata we have so far considered are of a limited kind. They consist of syntactic variables, possibly repeated, juxtaposed with connectives and brackets imported from the object-language, including as a limiting case a single variable standing alone. Call these schemata basic. They express forms that can equally well be expressed via substitution applied to an object-language sentence. For example, the instances of the schema A↔B may be specified by fi xing on the object-language sentence P↔Q. Their limited make-up means that the range of basic schemata is limited in its expressive power, and therefore misses out some significant forms. This is one reason why one must resist the slide from ‘logicians study forms’ to ‘the objects of their study—logical calculi—contain forms, i.e. schemata, not genuine sentences’. As we have seen, schemata aren’t forms, but only serve to express them. And it turns out that object-language schemata, like metalinguistic basic schemata, cannot express all the forms that

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the logician wishes to study (see Smiley 1982: 8–9). To see one side of the difficulty, consider ‘A&B is contingent’. It is false, since the schema lets in too much, e.g. P&¬P and (P∨¬P) & (Q∨¬Q) are both instances. One way to exclude them is to add a restricting condition: ‘A&B, where A and B are distinct and atomic, is contingent’. But this restriction cannot be expressed by any object-language schema. The second, converse difficulty is that basic schemata sometimes don’t cover enough instances. We need to enlarge, not restrict. Smiley gives a neat example: tautologies of the forms A↔B → (¬A↔¬B), A↔B → (A&C ↔ B&C), and A↔B → (A∨C ↔ B∨C) are themselves all of a common form, but no basic schema is sufficiently broad to capture all of its instances. One way to solve the problem is to enrich the notation used in schemata. For example, the schema (A↔B) → (S(A, C, D) ↔ S(B, C, D))—reading ‘S(A, C, D)’ as ‘the result of substituting A for C in D’—captures the instances of the relevant form (see Smiley 1982: 9). Alternatively, one could express this law of extensionality by coming at the problem from the other direction, e.g. by restricting the basic schema (A↔B) → (C↔D). Enriching basic schemata with notations for substitution is pervasive in descriptions of logical systems, starting—naturally—with rules of substitution. These notations may, of course, be combined with restrictions in order to fi x on the right instances, e.g. the fi rst-order axiom scheme ‘∀x A → Axt , where t is substitutable for x in A’. Another common device for expressing forms not captured by any basic schema is the use of subscripted numerals and dots, as in the definitions of various normal forms ‘B is in conjunctive normal form if it is of the form A1&A 2& . . . &An , where each Ai is of the form etc’.

4

EXTENDING THE APPLICATION OF FORM

Sets of sentences also have forms, e.g. {P, ¬P} is of the form {A, ¬A}. Repetition of variables now serves two purposes, matching repeated occurrences within and across sentences. The form of a set of sentences may depend, then, not only on the symbolic make-up of its members taken individually, but also on symbolic relations between them. As before, basic schemata may be restricted or enriched. And we can generalize even further by adopting a new sort of syntactic variable Γ ranging over sets of sentences. Any (possibly infi nite) set of sentences is of the form Γ. So one can now say: Γ is satisfiable, if each of its members is atomic. It is routine to treat implication, symbolized , as a relation holding between sets of sentences and a single sentence. Schemata for forms of sets can then be carried over in order to assist compact generalization, e.g. Γ B iff Γ ∪ {¬B} is not satisfiable. All this talk of sets, however, is obfuscatory and unnecessary. Worse, it wrecks theory. For when implication relates a set of sentences to a sentence, it ceases to be transitive. Also, logical equivalence cannot be reciprocal

The Matter of Form 169 implication, and we end up with absurdities like {P} implies P but not vice versa (see Oliver and Smiley 2001: n. 15). We should instead return to Aristotle’s original, plural definition of a valid argument in his Prior Analytics: ‘a deduction is a discourse in which, certain things having been supposed, something different from the things results of necessity because these things are so’ (A1 24b18–20, translations are from Smith 1989). No set of premises there, just premises (plural). Similarly, it is better to say that some sentences are (jointly) satisfiable, not the set of them. The preceding discussion of forms of sets can now be adapted by simply stripping away the set-theoretic braces, e.g. P, ¬P are of the form A, ¬A, where the plural schema is read ‘the sentence A and the sentence consisting of “¬” followed by A’. This schema expresses a form that two sentences have collectively; indeed, neither has it separately. Any number of sentences are of the form Γ, where Γ is now reconceived as a plural variable which takes one or more sentences under an assignment of values. Thus: if Γ are of the form A, ¬A, they are not satisfiable. Ditto for implication: Γ C iff Γ and ¬C are not satisfiable. As far as the propositional calculus goes, an argument (in a highly idealized sense) may be identified with a structured object featuring one or more sentences as its premises (we can do without sets of premises), and a sentence as its conclusion. For example, the argument with premises P, P→Q and conclusion Q may be written P, P→Q, so Q. This argument is of the form A, A→B, so B and, like any argument, it is of the form Γ, so C.

5

ARISTOTLE’S FORMS

In his theory of syllogisms, Aristotle invokes forms of a wide variety of kinds and of different levels of generality. There are forms of single sentences (i.e. premises or conclusions)—universal, universal affi rmative etc. Then forms of premises taken two at a time, such as the three figures, or more specific forms such as fi rst figure with both premises universal. Then forms of valid syllogism, such as being in one of the three figures, or being in the fi rst figure, or being in a particular mood of the fi rst figure, such as Barbara. In his statement of the validity of Barbara—‘if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C’ (A4 25b39–40)—Aristotle imports free variables from Greek geometry and uses them in a schema to frame a concise generalization about his system, although he often retreats instead to long-winded description. But where we might write e.g. ‘All As are Bs, All Bs are Cs All As are Cs’, Aristotle uses the conditional idiom to express implication. Contra Łukasiewicz (1951: ch.1 §1), Aristotle’s schematic conditional is not itself a particular syllogism, perversely conceived as a single sentence of the formalized objectlanguage (for the definitive demolition of Łukasiewicz, see Smiley 1973 and 1982). No: the conditional belongs to Aristotle’s informal metalanguage.

170 Alex Oliver The letters A, B, C are free metalinguistic variables, interpreted as universally quantified, ranging over terms of the object-language. Aristotle uses his letters in different contexts for different purposes. As the example above illustrates, he sometimes combines his letters with one of his four logical constants, thus ‘A is predicated of every B’. At other times he simply puts ‘AB’, which allows him to generalize across all kinds of sentences, universal or particular, affi rmative or negative. In still other contexts, the value of a single letter is not a term, but a whole sentence. To the best of my knowledge, it has never been remarked that he also uses his letters as plural variables (like plural Γ in §4). In fact, given his defi nition of a deduction with its plural ‘certain things having been supposed’, this is perfectly natural. If one wants to speak generally about implication, one needs to speak generally about premises in the plural, without confi ning oneself to a particular number of them. For the logician, at least, there could be no better argument for plural logic: the plural idiom, far from being redundant or an optional extra, is needed to do logic. Here, then, is Aristotle’s proof that ‘it is not possible to deduce a falsehood from truths’ (‘truths’ plural): For if it is necessary for B to be when A is, then when B is not it is necessary for A not to be. Thus, if A is true, then it is necessary for B to be true, or else it will result that the same thing both is and is not at the same time: but this is impossible. (B2 53b12–15) Although Aristotle felt no need to explain the general idea of a schematic letter—it plainly had some currency—he does explicitly help his audience understand the plural character of A—this was his innovation: But let it not be believed, because A is set out as a single term, that it is possible to result of necessity when a single thing is, for that cannot happen: for what results of necessity is a conclusion, and the fewest through which this comes about are three terms and two intervals or premises. (B2 53b16–20) When there are just two premises, ‘A is put as if a single thing, the two premises being taken together’ (B2 53b24–5). This is mistranslated by Tredennick (1938) as ‘Thus although A is posited as a single term, it represents the conjunction of two premises.’ A conjunction of two premises is a single sentence, so A would have a single value after all, contrary to Aristotle’s caution. The translator is evidently governed by a singularist reflex which reduces plural to singular, and he is therefore blind to Aristotle’s insistence that A has genuinely plural values. It is important to note that although Aristotle emphasizes that deductions cannot have a single premise, in order to make vivid the plural character of A, this restriction is a red herring. For even if A could take a single

The Matter of Form 171 premise as its value, it can also take two or more, in which case construing it as a singular variable fails. In any case, Aristotle can hardly have believed that nothing follows from a single premise. His acceptance of conversion is sufficient evidence. Even so, there is a good sense in which nothing new follows from a single premise. Since it was getting something new that fascinated him, he defi ned deductions accordingly as requiring more than one premise. In his proof that what follows from possible premises is itself possible, Aristotle again uses a plural A: if someone were to put the premises as A and the conclusion as B, it would result not only that when A is necessary altogether B is also necessary, but also that when A is possible B is possible. (A15 34a22–24) It is now Smith’s commentary that is corrupted by singularism, for he claims that Aristotle treats ‘the premises of a deduction as a single thing . . . “A is necessary altogether” . . . as I translate it, means “both the premises of which A is composed are necessary”’ (Smith 1989: 131). But this makes a nonsense of Aristotle’s plural characterisation of A, since he again warns that ‘one must not take “when A is, B is” as if it meant that B will be when some single thing A is’ (A15 34a16–17). A does not stand for one thing composed of several premises, but stands for the many premises themselves.

6

ONE OR MANY?

So far I have worked with a rough and ready notion of form. More precise characterizations issue in different specific notions of form. A notion of form determines a range of forms, i.e. properties that count as forms according to that notion. Thus one might characterize a notion of sentence-form by allowing only those forms that are expressible using basic schemata: call this basic schematic form. A different notion further constrains the range of forms by allowing only basic schemata that feature one connective in combination with the appropriate number of distinct variables, such as ¬A or A→B, or feature one variable standing alone, such as A, with the restriction in the latter case that A is atomic. Call this main connective form. A significant distinction between notions of form turns on how many forms a given notion assigns to the relevant objects. If exactly one form is assigned to each, the notion is unique; if more than one form is assigned to some, it is manifold. Whereas main connective form is designed to be unique, basic schematic form is manifold: (P&Q) ∨ (P→Q) is of the basic schematic forms A, A∨B, (A&B) ∨ C, A ∨ (B→C), (A&B) ∨ (C→D), (A&B) ∨ (A→D), (A&B) ∨ (C→B), (A&B) ∨ (A→B). The last three reflect the different ways of bringing out the recurrence of sentences.

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There is nothing in the idea of form that forces notions of form to be unique. True, we often say of a particular form, that it is the form of a sentence, but we can say this even when the sentence has many forms. For, as Russell remarked, we ‘speak of “the son of So-and-so” even when So-and-so has several sons’ (Russell 1905: 44). Strawson claims that we have a tendency to think of notions of form as unique ‘encouraged by misleading analogies; e.g. with the form of a sonnet (a sonnet cannot be both Shakespearean and Petrarchan in form), or with the shape of a vase’ (Strawson 1952: 54). Certainly, some notions of form and shape give sonnets and vases a unique form or shape. But other broader notions are manifold, so Strawson’s diagnosis is unconvincing. Anyone with a smattering of geometry or prosody knows that a vase can be both a cuboid and a cube, and that a sonnet can be both Petrarchan and Miltonic.

7

VALIDITY AS FORMAL

An argument-form is purely valid if every argument of that form is valid; a form is purely invalid if every argument of that form is invalid; an impure form is neither purely valid nor purely invalid. Beware: these labels and this way of dividing up the territory are not typical. The general run of textbooks say that a purely valid form is valid tout court. Much worse, they work with a dichotomy, not a trichotomy, so a form which is not valid is called invalid. This second, more common way of speaking means that an invalid form may have valid arguments as instances, since impure forms as well as purely invalid forms are brought under the same umbrella. Consequently there is potential for error, if the idea of invalid form is applied carelessly. For example, so-called formal fallacies cannot now be defi ned as instances of invalid forms, at least not if one wants to conclude that to commit a fallacy is to produce an invalid argument. B, A→B, so A is an invalid form on this way of speaking, but has the valid instance P, P→P, so P. Similarly, the method of refutation by logical analogy (i.e. by counter-instance) does not show that an argument is invalid if it is applied to one of its impure forms. There are two senses in which validity is formal, since a notion of argument-form may make the validity of arguments dependent on form in either a weak or a strong sense. Validity is weakly dependent on form if every valid argument has at least one purely valid form. Validity is strongly dependent on form if every form of every valid argument is purely valid. An argument’s validity may be weakly dependent on form even though the argument is also an instance of an impure form. Strong dependence, on the other hand, rules out impure forms altogether, in which case invalidity of arguments will also be strongly dependent on form in the corresponding sense. Note that the defi nitions of weak and strong dependence are neutral

The Matter of Form 173 with respect to whether the given notion of form is unique or manifold, though if a unique notion makes validity weakly dependent on form, it is bound to make it strongly dependent too. Some examples from Aristotle will help illustrate these ideas. The notion of syllogistic figure is a unique notion of form, since each syllogism is in exactly one figure. Each figure is an impure form, since it has both valid and invalid instances. So validity and invalidity are not even weakly dependent on figure. But, of course, Aristotle’s logical work relies on the more fi ne-grained notion of form that became known as the mood of a syllogism. Again, the notion is unique—each syllogism is in exactly one mood—but now each mood is either purely valid or purely invalid. Hence validity and invalidity are strongly dependent on mood. Moving to the propositional calculus, every valid argument has more than one basic schematic form, but at least one of its forms will be purely valid. Hence validity is weakly dependent on this notion of form. It is not strongly dependent, however, since any argument has an impure, basic schematic form; e.g. any valid one-premise argument has the impure form A, so B. The case is different for invalidity, since there are invalid arguments that have no purely invalid, basic schematic forms: an extreme example is P, so Q with its sole and impure form A, so B. This shows up the limitations of this notion of form. We need to restrict the schema. Merely requiring that A, B be distinct won’t do, since this still lets in the valid P, so Q∨¬Q. But if we also add that A, B are atomic as well as distinct, the resulting doubly restricted schema expresses a purely invalid form. Reflecting on Aristotle’s notion of mood suggests a design problem for the propositional calculus, viz. to characterize a unique notion of form which makes validity and invalidity strongly dependent on form. Smiley (1982: 5–8) solves it by using doubly restricted schemata of the kind just illustrated. The form of an argument is expressed by a schema obtained from the argument by substituting variables A, B, C etc for atomic sentences, different variables for different sentences, the same variables for the same sentences, and the resulting schema is restricted with the condition ‘A, B, C etc are atomic and distinct’. In other words, two arguments share the same form according to this notion of form if there is a 1–1 map between their constituent atomic sentences such that substitution according to the map turns one argument into the other. (Smiley describes a general procedure for characterizing the ‘atoms’ of any logical system, and hence for characterizing the corresponding notion of form for the system. As he notes, this notion of form also makes other logical properties strongly dependent on form: logical truth, satisfiability, etc.) The textbook accounts of the formal nature of validity naturally vary with their notions of form. For instance, Smullyan (1962: 3–4) begins with ‘Validity is a structural characteristic of argument. Arguments are valid because of their forms’ and spells this out as ‘Every valid argument is an

174 Alex Oliver instance of some validating form or other’, i.e. validity is weakly dependent on form. But since he allows for impure argument-forms, validity is not strongly dependent on form. Smullyan’s is the most common account, but there are a few exceptions. Copi, for instance, says ‘validity and invalidity are purely formal characteristics of arguments, which is to say that any two arguments having the same form are either both valid or both invalid’ (Copi 1953: 237), and he therefore employs a notion of argument-form which makes validity and invalidity strongly dependent on form. In the preceding sections I have deliberately spoken of forms, not logical forms. What makes a form a logical form? If we look to practice, the forms logicians actually exploit tend to be highly general: they fi x only certain features across their instances—an item of logical vocabulary (e.g. a connective), and possibly a pattern of recurrence or non-recurrence of non-logical vocabulary (e.g. sentences), and possibly the atomic nature of the non-logical vocabulary. None of these things need be fi xed: witness the sentence-form A and the argument-form Γ, so C. It is clear that many different notions of form are notions of logical form, in the sense that the forms they determine are all logical. Of course, there is room for further stipulation here: in the context of a particular investigation, a single notion of logical form may become salient, and its forms will then be deemed the only logical forms. Note too that, as far as actual practice is concerned, forms can be specified semantically for some purposes. Witness the deductive system for fi rst-order logic in Enderton 1972: 104, which skips over the proof-structure of the propositional calculus and relies on its decidability in order to get on to the more inclusive system. Its axioms ‘are all generalizations of wffs of the following forms’ (my italics), and the very fi rst form is ‘Tautologies’. A general form indeed!

8

IN VIRTUE OF FORM

I now come to the phrase ‘valid in virtue of form’ and its cognates. At its weakest, to say that validity is validity in virtue of form is just another way of saying that validity is formal, in which case it can be disambiguated as in §7 into weak and strong senses. One may also say of a particular argument that it is valid in virtue of its form. Correspondingly, this could simply mean either—weakly—that the given argument has a form that is purely valid (i.e. every argument of that form is valid), or—strongly—that each of its forms is purely valid (i.e. every argument sharing any form with the given argument is valid). ‘In virtue of’ may also carry an explanatory connotation: we often read that an argument is valid because of / on account of / due to its form, or that its validity depends on / is determined by its form, or that its form is responsible for its validity or makes it valid. There is certainly a kind of explanation available if validity is formal, even though there is obviously no causation, and even though a form by itself is not

The Matter of Form 175 the sort of thing that can be valid (as opposed to purely valid). We have a logical law: every argument of such-and-such form is valid. We have a particular case: this argument has such-and-such form. And we deduce: this argument is valid. This would, of course, be no explanation if the form of the argument could only be characterized in terms of its validity, or if its validity is defi ned in terms of its form. The drive to explain leads one to search for more and more general purely valid forms under which a given argument falls in order to fi nd out what is really essential to its validity. How far we go depends on the notion of form available. For example, we may escape the confi nes of basic schematic form by e.g. bringing P, P→Q, so Q under the purely valid form A, S(A→B, C, D), so S(B, C, D), which it shares with P, (P→Q) & (R→T), so Q & (R→T) and P, R ∨ (P→Q), so R∨Q, with S indicating the result of substitution as in §3. All of this discussion also applies to ‘invalid in virtue of form’. There is, however, a quite different epistemological context which leads one in precisely the opposite direction, i.e. to the most specific logical form. This is the context Smiley focuses on. Suppose one doesn’t know whether a particular argument is valid, and suppose one wants to describe its form ahead of this knowledge, but still be sure that it will be valid or invalid in virtue of its form so described. Then one must use a doubly restricted basic schema of the kind discussed in the previous section, since for all one knows, the argument is invalid, and only such a schema will be guaranteed to pick out a purely invalid form. And this is despite the fact that if the argument is valid, one could lift the restrictions, and still have a purely valid form. Given this set-up, ‘it would obviously be cheating to alter one’s account of their [the arguments’] logical form after the event [i.e. after fi nding out that they are valid]’ (Smiley 1982: 7).

9

FORM AND MATTER: THEN

Although inspired by Aristotle’s metaphysics, the contrast between form and matter was fi rst applied to arguments in something approaching the modern sense by his peripatetic commentators. Thus Alexander of Aphrodisias characterizes Aristotle’s figures as ‘a sort of common matrix: by fitting matter into them, it is possible to mould the same form in different sorts of matter’ (see §1.2.1 of his commentary; for an illuminating discussion of form and matter in ancient logic, see Barnes 1990 and 2007). Apropos Aristotle’s statement of the validity of Barbara (‘if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C’), he acutely remarks: He uses letters in his exposition in order to indicate to us that the conclusions do not depend on the matter but on the figure, on the conjunction of the premises, and on the modes. For so-and-so is deduced

176

Alex Oliver syllogistically not because the matter is of such-and-such a kind but because the combination is so-and-so. The letters, then, show that the conclusion will be such-and-such universally, always and for every assumption. (§4.4.1)

Alexander’s ‘because the combination is so-and-so’ might as well be ‘in virtue of its form’. He recognizes that when form is identified with mood, validity is formal, and that Aristotle uses his letters to express the corresponding generalizations connecting form and validity. Buridan, writing in the fourteenth century, shared Aristotle’s modal notion of consequence, but distinguished two varieties, formal and material, on the basis of a prior distinction between form and matter. As he puts it in his Treatise on Consequences: ‘a consequence which is acceptable in any terms is called formal, keeping the form the same’ (§1.4.2). A materially valid argument is valid but not formally valid. Buridan gives the example ‘A man runs; therefore, an animal runs’. It is only materially valid, because it ‘does not hold in all terms (keeping the form the same)’ (§1.4.3): witness the invalid ‘A horse walks; therefore wood walks’. By the time we reach Bolzano (1837), the notion of form does not merely happen to characterize validity or one kind of it. Form is used to define validity (or rather the corresponding relation of ‘deducibility’). Bolzano applies form not to sentences, but to extralinguistic propositions conceived as made up, in a rather language-like way, of ideas. Forms of propositions are ‘kinds’ or ‘species’, i.e. properties of them, which are identified with classes of their instances (Bolzano 1837: §§12 & 81). He expresses forms using schemata, e.g. ‘Some A are B’, and he notes the different sense of form in which the form is the schema (§§12 & 81; cp. my §2 above). As for matter or content: ‘call what is left indeterminate in such a class of propositions the content of the propositions in the class, such as A and B in the above example’ (§12). Bolzano’s leading method is to ‘envisage certain ideas contained in [propositions] as variable; we then consider the new propositions which are generated, if these ideas are replaced by any other ideas whatever, and observe what truth values they take on’ (§154). Thus his generalized version of logical truth: a proposition is universally satisfi able, relative to some choice of variable ideas, if it is true and so is every other proposition that results from it by substitution for the variable ideas. Since the resulting propositions are each in the same form as the original, ‘universally satisfiable propositions could also be said to be true by virtue of their kind or form’ (§147; note ‘true by virtue of . . .’, not ‘universally satisfiable by virtue of . . .’—the latter is an uninteresting consequence of the defi nition of ‘universally satisfiable’). Bolzano relativizes and so generalizes. There are as many specific notions of universal satisfiability as there are choices of variable ideas (i.e. forms). He acknowledges that he has gone beyond anything that might be called

The Matter of Form 177 logical truth, since the fi xed ideas may be ‘concepts alien to logic’, but he rightly despairs of giving any sharp and indisputable circumscription of the logical (§148). As for argument, ‘I say that propositions M, N, O . . . are deducible from propositions A, B, C, D, . . . with respect to variable parts i, j, . . . , if every class of ideas whose substitution for i, j, . . . , makes all of A, B, C, D, . . . true, also makes all of M, N, O, . . . true’ (§155). Note that each specific relation of deducibility is formal, since any choice of variable ideas goes along with a corresponding notion of form. In effect, Buridan’s fi xed distinction between formal and material consequence is replaced with indefi nitely many kinds of formal consequence, most of which have little to do with what we could call logical consequence. Gone too is modality (for Bolzano’s criticism of Aristotle’s ‘results of necessity’, see §155, n.1). Like Bolzano, Quine defined logical truth in terms of plain truth and form (see e.g. Quine 1980). But Quine invoked the forms of sentences, not propositions, and had none of Bolzano’s generalizing spirit. In order to tackle logical consequence, he replaces several premises with their single conjunction, treats consequence as a relation between two sentences, and then reduces it to the logical truth of the corresponding conditional. Unfortunately, this leaves out the case of infi nitely many premises, and Quine’s defi nition comes unstuck here (see Boolos 1975: 52–3). To put it in Bolzano’s terms, Quine’s trouble is that some ideas are not expressed.

10

FORM AND MATTER: NOW

Virtually every modern logic textbook differs from each of Aristotle, Buridan, Bolzano and Quine. Unlike Aristotle and Buridan, validity is nonmodal. And unlike Bolzano and Quine, validity is not defi ned in terms of form, but given a model-theoretic—often heavily set-theoretic—defi nition. ‘Necessarily truth-preserving’ is not replaced with ‘every argument of the same form is truth-preserving’ but by ‘truth-preserving in every interpretation’. The formal nature of validity is then a substantial matter of fact, not an easy consequence of a defi nition. Unlike Buridan, only formal validity is treated. Finally, unlike Bolzano, validity is only very partially relativized: there is one notion per logical system, and variation across systems (e.g. the propositional calculus and the predicate calculus) is strictly limited to a few notions of ‘logical’ validity. When modern discussions explain that an argument is valid in virtue of its form, they often invoke the contrast between form and matter (or content). This contrast is liable to mislead. Consider Lemmon: ‘it is mainly in virtue of their form, as it turns out, rather than their subject-matter that arguments are valid or invalid’ (1965: 5). The contrast between form and subject-matter sounds exclusive but isn’t, since in the propositional calculus connectives are often held fi xed across the instances of the relevant forms,

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and every connective has its distinctive content. Bolzano was well aware of the misleading dualism of form and content: ‘Actually, the parts of these propositions which are determinate have, in certain respects, the same claim to be called content as do the parts left indeterminate’ (1837: §12). A fault in the other direction is over-emphasis on connectives. Bostock, for example, recommends calling propositional calculus ‘the logic of truth-functors’ (Bostock 1997: 14), but this leaves out valid arguments which feature no connectives, such as P, so P or P, Q, so P. Ignoring such arguments is common: ‘whether or not a logical form is valid depends at bottom on the meanings of the sentential connectives which occur in it’ (Forbes 1994: 45); ‘Formal validity of an inference was validity in virtue of the molecular forms of its premises and conclusion’ (Jeffrey 1967: 103). Note too that shallow analysis of a valid argument relies on the fact that its purely valid forms may fi x only some of its connectives. P&Q, (P&Q)→R, so R, for instance, has the purely valid form A, A→B, so B. Hence its validity is not dependent on the content of all the connectives it contains. In other cases, no particular occurrence of a connective need be fi xed: e.g. P&¬P, Q↔¬Q, so R is of both the purely valid forms A&¬A, B, so C and A, B↔¬B, so C. Its validity is overdetermined, and explanation of validity by deduction from generalizations about purely valid forms therefore bifurcates. Of course, even when connectives appear in an argument, its validity is also generally dependent on the recurrence of sentences. But, as with connectives, there are cases in which not all recurrences are essential, and no particular recurrence need be. In fact, depending on the make-up of the system, it may be that no recurrence at all is needed, e.g. if the primitive * expresses the binary truth-function which always maps to truth, P, so Q*R is valid, even though no atomic sentence recurs. The evident lesson is that validity in the propositional calculus is a complex and heterogeneous affair. As we have seen in §7, there are precise senses in which validity is formal given some precise background notion of form. But summing up the form in ‘formal’ in a snappy slogan is liable to involve false dichotomies or over-generalization.

11

LOGICAL FORM VS GRAMMATICAL FORM

Another logician’s slogan is that grammatical form misleads as to logical form. Kaplan spells it out: In ordinary language, replacements which do not change the apparent grammatical form of sentences, for example replacing a proper name with ‘someone’, may well introduce or obliterate relations of logical consequence between the affected sentences, thus indicating a change in logical form. (Kaplan 1970: 235–6)

The Matter of Form 179 Kaplan fi xes on the hackneyed paradigm of the mismatch between grammar and logic: proper names vs quantifier phrases. But there is an error here from the start, as I have argued at length in Oliver (1999). Names and quantifier phrases do not in fact belong to the same grammatical category, never mind whether one looks to the multifarious classifications of so-called traditional grammars or of modern descriptive grammars, or instead uses the method—utterly foreign to grammarians—of grouping together expressions according to whether they are intersubstitutable salva congruitate (‘Witty Kaplan wrote on Russell’ is grammatical but ‘Witty someone wrote on Russell’ is not.) Logicians would do well to observe the grammarian’s slogan—‘grammar has to state facts, not desires’ (Jespersen 1924: 55). The complaint that grammatical form misleads suggests an ideal of a logically perfect language in which grammatical form and logical form coincide, a language in which there is a ‘thoroughgoing reduction of logical structure to grammatical structure’ (Massey 1970: 95). And, of course, it is commonly supposed that the familiar logical systems meet this ideal: In an artificially regimented fi rst-order language, two sentences with identical structural properties always display similar logical properties . . . in our fi rst-order language the logical properties go hand in hand with surface grammatical structure, and we know this is not the case in English. (Etchemendy 1983: 320) This should immediately strike one as a peculiarly mistaken claim about formalized languages. As far as the usual textbook presentations go, &, → and ∨ are listed as items of vocabulary and brought under the grammatical heading ‘binary connectives’ (they are intersubstitutable, since in designing formalized languages we can make our desires facts). It follows that P&P, P→P and P∨P are all of the same grammatical form, yet they are logically very different and these logical differences are typically reflected in logical forms that differ in the connective that is held fi xed. Grammar and logic are far apart even in the propositional calculus. The same point can be made about the predicate calculus. In explaining how grammatical and logical form may diverge, why not forget names altogether and instead emphasize the misleading grammatical similarity between quantifier phrases themselves—e.g. ‘everyone’ and ‘someone’— which are after all logically very different? But, of course, predicate calculus is no better than English in this regard, since the mismatch between grammar and logic persists: ∀ and ∃ are both grammatically classified as quantifiers. Logicians, unlike grammarians, are not interested in the similarities between quantifier phrases. They are interested in the differences. Of course, it is possible to redesign the grammar of formalized languages so that e.g. the connectives don’t count as items of vocabulary falling under the same grammatical category, but are each a part of a special grammatical

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operation: the general operation of applying a connective is replaced by specialized operations of applying &, applying →, etc (see Quine 1970: 28–9; his constructions do not reflect textbook practice, pace Jackson 2006: §3). But besides lacking a clear rationale (it certainly isn’t surface grammar, nor is it a descriptive grammarian’s technique), this trick backfi res. For if it is open to us to impose such a grammar on the propositional calculus, why not on English as well? But then there would be no mismatch between grammar and logic to start with. Evans attempts to bring out the oddity of equating grammatical structure with logical structure by distinguishing two kinds of validity: ‘the validity of some inferences is to be explained by reference to the meanings of the particular expressions occurring in them, while that of other inferences is due, rather, to the way in which the sentences are constructed out of their parts’ (Evans 1976: 199). He calls the second kind of inference structurally valid: in effect, it corresponds to the limiting case of Bolzano’s deducibility relation in which no idea is fi xed. Evans illustrates the fi rst kind of validity using the inference from P&Q to P. This, he says, is not structurally valid, since it depends on the particular meaning of &, as shown by the structurally identical but invalid inference from P∨Q to P. The question then arises whether any kinds of valid argument within the propositional calculus are structurally valid. Obviously, arguments like P, so P or P&Q, so P&Q would count, but Evans claims that no ‘standard’ inference involving the sentential connectives is structurally valid ‘with the exception of inferences involving substitution of sentences with the same truth value’ (Evans 1976: 214). Or as Sainsbury puts it, ‘the only inferential behavior common to all [binary connectives] is the substitution of equivalents’ (Sainsbury 2001: 362). This, however, is false. An elementary example is P, Q, P→P, so P→Q, which is structurally valid, since it remains valid never mind what truth-function → expresses, yet it clearly doesn’t involve substitution of equivalents. It is valid in virtue of the form A, B, AcA, so AcB, where c ranges over truth-functional connectives. No dependence on a particular connective there.

12

PUTTING INTO LOGICAL FORM

Philosophers are trained to put natural language sentences into logical form, i.e. to formalize them. What is this activity and what is its point? As I see it, it is an exercise in translation. One translates a sentence of English, say, into a sentence of a given logic, with the result that some of the inferential properties of the original sentence can be deduced from the logic of its translation. It is an incurably relative process, since it depends both on the choice of a logic and the particular translation into that logic. The point of the activity is to model the meanings of English sentences. The meaning of a sentence determines its inferential connections: what follows from it

The Matter of Form 181 (possibly in combination with others) and what it follows from. A given formalization highlights particular connections. Choosing to highlight different connections results in different translations into a given logic, perhaps even a change of logic. Note that to think of a logic as a model in my sense has no normative implications: formalized languages are not standards against which natural languages should be measured. They tend to be simple and regular, because simple and regular is useful—compare frictionless planes—not because simple and regular is best. But the choice of a formalized language is also determined by other factors that are more sociological in flavor. In particular, the status of fi rst-order logic (perhaps with identity) as the benchmark model has more to do with the contingent history of logic and the resulting education of philosophers than with its intrinsic suitability for formalizing English. Logicians’ urge to constrain themselves by fi xing on one preferred model is a constant theme in the history of the subject, though they naturally disagree over which is best: e.g. ‘All men are animals’ has been forced into the mould ‘a has b’, the mould ‘x(1 − y) = 0’, and even into ‘Everything is such that if it Fs, it Gs’. If we put an English sentence into logical form, what is its resulting logical form? There are two answers. First, its formal translation, i.e. a sentence of the formalized language. Second, a form of its translation, i.e. a property of it. The same ambiguity infects talk of the logical forms of English arguments. In the fi rst sense, talk of the logical form of the English original is out of place: there are as many logical forms as there are formal translations. The same point applies to logical forms in the second sense, indeed even more so if a particular formal sentence has many forms. There is a peculiar tendency to rotate the horizontal process of formalization by 90°, i.e. to conceive of the process as vertical—the uncovering or revealing of a deep, real, hidden logical form underlying the surface, apparent, overt original. But there is no point in thinking of things this way, only mystery. It is easy to understand how an archaeological dig may reveal hidden structures, or how a physical chemist may distinguish phenomenal properties of a substance from its underlying electronic configuration. But one cannot peel back an English sentence or reveal another one by inspecting it under a microscope. As James Cargile put the point apropos Davidson’s event-analysis of ‘Shem kicked Shaun’: ‘the sentence really is of a three-place relation form, with two names and an existential quantifier. An existential quantifier? Where is it?’ (Cargile 1970: 137–8; this was the original incredulous stare). The conceptual difficulties with the surface– deep contrast multiply once one adds in the relativity of formalization, for then there would have to be as many hidden forms as there are choices of translation and logic. Much better, then, to think of a formalization as illuminating the meaning of the original sentence, not by revealing a structure that the original sentence really has, but by producing an equivalent sentence whose structure makes some of its inferential relations vivid.

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Translation, of course, depends on inference, since the task is to fi nd an equivalent formalized sentence in a sense of equivalence determined by the logic at hand. That is why the torturing of English inferences into some preferred formal shape cannot show that only that shape of inference exists: the torturing itself relies on inferences of a different shape. What kinds of inferential behavior should be captured by putting sentences into logical form? Like Buridan, Davidson draws a distinction between formal and material validity in English (though he doesn’t put it in exactly those words). For example, he thinks that the inference from ‘I flew my spaceship to the Morning Star’ and ‘The Morning Star is the Evening Star’ to ‘I flew my spaceship to the Evening Star’ is formally valid, and his eventanalysis does indeed validate it (Davidson 1967: 119). On the other hand, ‘“x > y” entails “y < x”, but not as a matter of form. “x is a grandfather” entails “x is a father”, but not as a matter of form’ (Davidson 1980: 125), i.e. these inferences are materially valid. And he uses this distinction between different kinds of valid English arguments to construct a test for his formalizations: ‘see whether our paraphrases articulate the entailments we independently recognize as due to form’ (Davidson 1970: 144). In other words, all and only formally valid English arguments are to be formalized as valid arguments in his chosen logic. Thus he refuses to formalize the valid inference from ‘I flew my spaceship’ to ‘I flew’ as a valid argument, since it is not formally valid in English: witness, he says, the formally identical but invalid inference from ‘I sank the Bismarck’ to ‘I sank’ (Davidson 1970: 125). When Davidson writes that an English argument is valid ‘as a matter of form’ this at least entails that every argument of that form is valid, and if this is to be more than a triviality, he must characterize form independently of validity. But it is wildly optimistic to suppose that there are such substantial and exceptionless generalizations outside of the secure environment of a formalized language, as Cargile emphasizes in his contribution to this volume. Leaving this objection aside, how does Davidson recognize form? Take the Morning Star–Evening Star inference and replace the identity premise with ‘The Morning Star is larger than the Evening Star’. The result is invalid, but it certainly shares a form with the original. Does this show that the original, though valid, is not formally valid? Not in Davidson’s books, but then one must ask why he ignores this form, in favor of the form which keeps the ‘is’ of identity in the second premise fi xed. (The example is pertinent: although his formalization of the Morning Star–Evening Star inference as valid requires identity to be a logical constant, Davidson soon after defi ned ‘logical constant’ so as to rule it out; see his 1973: 71.) It is clear that Davidson’s choice of logic is determining what counts as a formally valid English argument, that is, the notion of formal validity in English is not acting as an independent test of correct formalization. The same lack of independence is manifest in the different ways he reacts to alleged counter-instances to formal validity. In the Bismarck inference, he is happy

The Matter of Form 183 to accept the counter-instance as real. But in cases that threaten his own match between English formal validity and validity in his preferred formalized language, he insists that the counter-instance is merely apparent: it is not really of the same form as those arguments he declares formally valid (see e.g. his remarks on the ‘superficial’ and ‘deceptive’ adverbial form of ‘deliberately’ and ‘intentionally’ in his 1967: 106 and 121). What’s driving him to say this is not an independent recognition of English inferences whose validity is or isn’t a matter of form—how did we ever get past the deceptive surface forms?—but rather the desire to protect his own formalizations by designing a matching notion of formal validity for English. As Bolzano would be the fi rst to point out against Davidson, there is an argument-form according to which Tom is a grandfather, so Tom is a father is valid as a matter of form, viz. a is a grandfather, so a is a father. Similarly, for 2>1, so 1 and i≥0} > |a 0 =an |. So the inference is truth-value-decreasing. 10. Many thanks to Jonathan Lear, Alex Oliver, and Stephen Read for comments on earlier drafts of this essay.

REFERENCES Artmann, B. (1999) Euclid—The Creation of Mathematics, New York: Springer.

A Case of Mistaken Identity? 221 Boehner, P., Gál, G., and Brown, S. (eds) (1974) Guillelmi de Ockham: Summa Logicae, St. Bonaventure, NY: Franciscan Institute Publications. Translation appearing in the main text is by Stephen Read. Couturat, L. (ed.) (1961) Opuscules et Fragments Inédits de Leibniz, Hildesheim: Georg Olms. Del Punta, F. (ed.) (1979) Guillelmi de Ockham: Expositio super Libros Elenchorum, St. Bonaventure, NY: Franciscan Institute Publications. Translation appearing in the main text is by Stephen Read. Fine, K. (1975) ‘Vagueness, truth and logic’, Synthese 30: 265–300. . (1989) ‘The problem of de re modality’ in J. Almog, J. Perry and H. Wettstein (eds) Themes from Kaplan, Oxford: Oxford University Press, pp. 197– 272. Page references are to the reprint in Fine (2005: ch. 2). . (1999) ‘Quine on quantifying in’ in C.A. Anderson and J. Owens (eds) Proceedings of the Conference on Intentional Attitudes, Stanford, CA: CSLI, pp. 1–26. Page references are to the reprint in Fine (2005: ch. 3). . (2005) Modality and Tense: Philosophical Papers, Oxford: Clarendon Press. Forrest, P. (2006) ‘The identity of indiscernibles’ in The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), E.N. Zalta (ed.), http://plato.stanford.edu/ archives/sum2009/entries/identity-indiscernible. Frege, G. (1892) ‘On sense and reference’ in Translations from the Philosophical Writings of Gottlob Frege, M. Black and P. Geach (eds), 2nd edn, Oxford: Blackwell, 1960, pp. 56–78. . (1893) Grundgesetze der Arithmetik, vol. 1, partial translation by M. Furth as Basic Laws of Arithmetic, Berkeley, CA: University of California, 1964. Gibbard, A. (1975) ‘Contingent identity’, Journal of Philosophical Logic 4: 187– 222. Hilbert, D. and Ackermann, W. (1928) Grundzüge der Theoretizchen Logik, Berlin: Springer. Kneale, W. and Kneale, M. (1962) The Development of Logic, Oxford: Clarendon Press. Kripke, S. (1980) Naming and Necessity, Cambridge, MA: Harvard University Press. Lakatos, I. (1970) ‘Falsification and the methodology of scientific research programmes’ in I. Lakatos and A. Musgrave (eds) Criticism and the Growth of Knowledge, Cambridge: Cambridge University Press, pp. 91–196. . (1976) Proofs and Refutations, Cambridge: Cambridge University Press. Leibniz, G. (1686) ‘General inquiries about the analysis of concepts and of truth’ in G.H.R. Parkinson (ed.) Leibniz: Logical Papers, Oxford: Clarendon Press, 1966, pp. 40–6. Lewis, D. (1968) ‘Counterpart theory and quantified modal logic’, Journal of Philosophy 65: 113–36. Machina, K. (1976) ‘Truth, belief, and vagueness’, Journal of Philosophical Logic 5: 47–78. Oliver, A. (2005) ‘The Reference Principle’, Analysis 65: 177–87. Priest, G. (2005) Towards Non-Being, Oxford: Oxford University Press. . (2008), An Introduction to Non-Classical Logic: From If to Is, Cambridge: Cambridge University Press. Prior, A. (1968) ‘Time, existence, and identity’ in his Papers on Time and Tense, Oxford: Clarendon Press, pp. 78–87. Quine, W.V.O. (1953) ‘Reference and modality’ in his From a Logical Point of View, Cambridge, MA: Harvard University Press, pp. 139–57. Ross, W.D. (ed. and trans.) (1928) The Works of Aristotle, vol. 1, Oxford: Clarendon Press.

222 Graham Priest van Inwagen, P. (1981) ‘The doctrine of arbitrary undetached parts’, Pacific Philosophical Quarterly 62: 123–37. Whitehead, A.N. and Russell, B. (1910) Principia Mathematica, vol. 1, Cambridge: Cambridge University Press. Williamson, T. (1994) Vagueness, London: Routledge. Wright, C. (2004) ‘Intuition, entitlement, and the epistemology of logical laws’, Dialectica 58: 155–75.

12 Inferential Semantics for First-Order Logic Motivating Rules of Inference from Rules of Evaluation Neil Tennant

I am grateful to Timothy Smiley, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from Δ’ and ‘ϕ is a logical consequence of Δ’. The notions ‘ϕ is a theorem’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verification and falsification is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘ϕ is true’ and ‘ϕ is false’ as essentially relational and inferential. A sentence’s truth-value is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truth-makers and falsity-makers are themselves proof-like objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truth-value are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph’ the former into the latter.

1

1.1

RULES OF EVALUATION

Verification and Falsification of Sentences in Models

We shall fi rst describe, in general terms, interpretations, or models, of a fi rst-order language. Then we shall provide a ‘toy example’ that will illustrate the ingredients of the defi nition. A model consists of (i) a domain of individuals; (ii) a denotation mapping for names (if there are any names in the objectlanguage); (iii) the structure that consists in primitive predicate-extensions; and (iv) the structure that consists in the mappings represented by primitive function-signs.

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The names, predicates, and function-signs make up the extra-logical vocabulary that is being interpreted by the model in question. Because we are dealing with a toy example, we shall confi ne ourselves to one-place predicates and function-signs. Towards the end of this discussion we shall have occasion to consider a two-place predicate. So our expressive resources will be quite modest. Still, we shall be painstaking in illustrating what is involved in these different components (i)–(iv) of models. We shall build up our chosen model M in stages. To the left will be a diagram, which can be thought of as the model M itself. The large dots will be the individuals; each one-place predicate extension will be represented by an enclosure; and each one-place mapping will be represented by arrows (a different style of arrow for each mapping). These diagrammatic components will be added in sequence, as the model is built up. So, for our toy example, we fi rst choose a domain of individuals (here, three). They are labeled α, β, and γ in the metalanguage. To the right of the diagram are three ‘M-relative’ rules of inference. These rules ensure that the individuals are pairwise distinct. Their conclusions are ⊥ and their premises are what we shall call saturated identity formulae (see below) involving all possible pairs of our chosen individuals. Such rules are a necessary part of an eventual set of rules that will completely capture the diagram that will be on the left once the model has been fully constructed. Indeed, the eventual set of rules on the right can be thought of as an adequate substitute for the model itself.

1.2

A Digression on Saturated Terms and Formulae

In general, terms and formulae of the object-language may contain free variables. If they do, then they are called open. Those that are not open are called closed. A closed formula is called a sentence. A closed term may be called a (simple or complex) name. The semantic value of a name, when it has one, is an individual, which the name is then said to denote. The semantic value of a sentence, when it has one, is a truth-value, and the sentence is said to be true or false according as that value is T or F. Closing an open term or formula involves substituting closed terms (of the object-language) for free occurrences of variables. Thus one could substitute the object-linguistic name j for the free occurrence of the variable x in the open term f(x), to obtain the closed term f(j) (‘the father of John’). Or, to complicate the example slightly, one could substitute the closed object-linguistic term m(j) for that free occurrence, to obtain the closed term f(m(j)) (‘John’s maternal grandfather’). Likewise, an open formula, say

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L(x, y), may be closed by substituting closed terms for its free occurrences of variables. One such closing would result in the sentence L(m(f(j)), f(j)) (‘John’s paternal grandmother loves his father’). Here we shall introduce an operation on open terms and formulae analogous to the operation of closing, but importantly different from it. The new operation will be called saturation. Like the operation of closing, the operation of saturation gets rid of all free occurrences of variables within an object-linguistic term or formula. But the way it does so is importantly different. Instead of substituting closed object-linguistic terms for free occurrences of variables, saturation is effected by substituting individuals from the domain for those free occurrences. Thus if α and β are individuals from the domain, one saturation of the open formula L(x, y) would be L(α, β). Another one would be L(m(f(α)), f(α)), where the saturation is effected by substituting the saturated terms m(f(α)) and f(α) for the free occurrences of the variables x and y respectively. When the domain D supplies all the individuals involved in a saturation operation, the resulting saturated terms are called saturated D-terms, and the resulting saturated formulae are called saturated D-formulae. Saturated terms and formulae are object-linguistic and metalinguistic hybrids. But, as mathematical objects, they are well defi ned. When one treats, in standard Tarskian semantics, of assignments of individuals to variables, one is assuming such well-defi ned status for ordered pairs of the form 〈x, α〉, where x is a free variable of the object-language, and α is an individual from the domain of discourse.1 Since standard semantics is already committed to the use of such hybrid entities, it may as well take advantage of similar hybrid entities such as saturated terms and formulae. We shall be taking advantage of them by having them feature in the rules of inference on the right in our description of models. Indeed, such rules will form a constitutive part of the model in question, as will emerge in due course.

1.3

Back to the Exposition of Models

With all subsequent displays of rules on the right, we shall omit the subscripts M next to their inference strokes, as they will always be understood from the context. (Occasionally we shall restore them, for appropriate emphases.) We shall also suppress the label M at the top and left of the diagram. The model M is not yet completely specified. For so far we have specified only its domain; we have yet to specify its predicational and functional structure. Note how we have labeled the individuals as α, β, and γ by placing these labels right next to them, within the outer box that represents the ‘boundary’ of the domain. By means of this convention we indicate that the labels are metalinguistic. As far as the object-language is concerned, the individuals could be nameless. Suppose, however, that the individuals happened to

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possess names in the object-language—say, a, b and c respectively. Then this circumstance will be conventionally indicated by placing those names outside the outer box, and indicating, with arrows, what their respective denotations are. To the right we would specify the denotation mapping d:

Our toy example will not, however, involve any names in the object-language, so we can set these details aside. Next, we add the structure of a one-place predicate P, by showing its extension in the diagram, and also by supplementing our rules of inference in order to show which individuals are, and which are not, in the extension of the predicate P. (If we failed to specify, by means of our rules, which individuals are not in the extension of P, then the rules themselves would fail, collectively, to specify the intended diagram, in which certain individuals defi nitely lie outside the enclosure indicating the extension of P.)

Finally, we add the structure of a one-place function f, supplementing again on the right with the relevant ‘axioms of function-values’:

The axioms and rules on the right re-state what one can read off from the diagram on the left. There is, however, one aspect of the diagram that is only ‘shown’, and not yet ‘said’, by the rules on the right. This is that the individuals α, β, and γ are all the individuals there are. This feature

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of the diagram will fi nd expression in the following ‘M-relative’ rules of evaluation:

The first of these rules, which allows one to evaluate as true any universal claim ∀xψ(x), requires only that ψ should hold of each of the individuals α, β, and γ. The second rule, which allows one to evaluate as false any existential claim ∃xψ(x), requires only that ψ should not hold of any of the individuals α, β, and γ. (One shows that a claim does not hold by assuming that it does, and deriving ⊥ from that assumption.) It is clear that these M-relative rules for the verification of universals and the falsification of existentials constrain the domain D to consist of exactly the individuals α, β, and γ. And so it will be with any fi nite model. Let N be such, and suppose its domain consists of exactly the n individuals α1, . . . , αn . Then the N-relative rules for the verification of universals and the falsification of existentials will be

Let us now return to our toy model M, and consider some simple sentences that can be verified, or falsified, in M. We shall choose sentences simple enough for easy, intuitive determination of their truth-values in M. The point of the exercise is to show how that intuitive determination can be represented formally as a certain kind of M-relative ‘evaluation proof’ or ‘evaluation disproof’ of the sentence in question. Here again is our model M:

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Intuitively it is easy to see that the following sentences are true-in-M, or false-in-M, as indicated:

(1)

True-in-M

False-in-M

∃xP(x)

∀xP(x)

(2)

∃xf(x) = x

∀x¬f(x) = x

(3)

∃x(P(x) ∧ ¬f(x) = x)

∀x(¬P(x) → f(x) = x)

(4)

∀x(¬P(x) ∨ ¬f(x) = x)

∀x∃yf(y) = x

(5)

∀x(f(f(x)) = x → ¬P(x))

P(f(α)) ∨ P(f(β))

The fi rst two truths are easy to verify:

In both these evaluation proofs, the fi nal step verifies an existential conclusion on the basis of an appropriate instance—α in the fi rst, γ in the second. The general form for the verification of an existential claim ∃xψ(x) is:

where α is an individual in the domain

The vertical dots indicate the presence of some verification of some instance ψ(α), for some individual α in the domain of the model. For the third truth in the list above, ∃x(P(x) ∧ ¬f(x) = x), we make use of the following rule (for the construction of verifications and/or falsifications) governing identity:

where t is any saturated D-term

The M-relative evaluation-proof, or verification, of ∃x(P(x) ∧ ¬f(x) = x) is as follows:

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Note that the penultimate step verifies the conjunction P(α) ∧ ¬f(α) = α by verifying the left conjunct P(α) and verifying the right conjunct ¬f(α) = α. The general form of such a step is

The step labeled (2) is an application of the following general rule for verifying negations:

where the vertical dots indicate the presence of some falsification of ϕ. The fourth truth on our list, ∀x(¬P(x) ∨ ¬f(x) = x), has the following M-relative verification:

Here, at the fi nal step, we see in action the rule for verifying universal claims in any model whose only individuals are α, β, and γ. One has to verify each of the three instances of the universal claim. Each of the three immediately preceding steps involved verifying a disjunction by verifying one or the other of its disjuncts. The general form of such a step is one or the other of

Even when both disjuncts are true, the most economical route to the verification of the disjunction is to focus on the disjunct with the easiest verification. Hence there is no need to add to, or alter, the form of these last two rules for the verification of disjunctions. Let us turn our attention now to the fi rst four false sentences in the list above. For these we need to construct M-relative falsifications.

230 Neil Tennant It takes only one counterinstance to render a universal claim false. To falsify ∀xP(x), we have a choice between β and γ as counterinstance. (Since α lies in the extension of P, we obviously cannot appeal to α in order to falsify ∀xP(x).) Let us take β for this purpose. The falsification is then as follows:

Any falsification of a saturated formula ϕ will use ϕ as an assumption—indeed, its sole undischarged assumption—and derive ⊥ therefrom, using as its remaining premises only the ‘basic claims’ about the model in question. (A ‘basic claim’ can of course be an inference from a primitive saturated formula to ⊥, as we have seen.) Note how the fi nal step of this last two-step falsification discharges the assumption P(β) within the subordinate (onestep) falsification, on the right, of P(β) itself. The fi nal occurrence of ⊥ is thereby made to depend only on the undischarged assumption ∀xP(x), rather than on the (false!) instance P(β) of that universal claim. The general evaluation rule being applied here for the falsification of universal claims is the following:

where the vertical dots indicate the presence of some falsification of some instance ψ(α). (In our last example, this was a one-step falsification. But in general, of course, there can be more than one step, the exact number of steps depending both on the complexity of ψ and on the chosen counterinstance α.) The second falsehood on our list above, ∀x¬f(x) = x, enjoys the following falsification:

The upper step is an application of the general evaluation rule

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for the falsification of negations. The vertical dots indicate the presence of some verification of ϕ. The third falsehood on our list, ∀x(¬P(x) → f(x) = x), can be shown to be false as follows. The falsification involves choosing β as the counterinstance to the universal claim. This choice generates the task of falsifying the conditional claim ¬P(β) → f(β) = β. A falsifi cation of a conditional θ → χ consists in a verifi cation of the antecedent θ and a falsifi cation of the consequent χ:

So we need in the case at hand to verify ¬P(β) and to falsify f(β) = β. The details are as follows:

The step labeled (2) is an application of the rule for identity explained earlier. We come to our fourth illustration of falsifications, that of the sentence ∀x∃yf(y) = x. A glance at the diagram for M reveals that α is the counterinstance we want—for it is the only individual that is not the f-value of any individual in the domain. Our task is now to fill in the dots in the falsification schema

In order to falsify the existential ∃yf(y) = α, we need to falsify each of its three instances f(α) = α, f(β) = α, and f(γ) = α. The task accordingly reduces to that of filling in the dots in the schema

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The desired M-relative falsifications of f(α) = α, f(β) = α and f(γ) = α (to fill in the respective vertical dots) are as follows:

We refrain here from inserting these into the previous schema, since the sideways spread would be too wide for the page to contain. The reader will appreciate that these insertions would be completely straightforward, if one had paper wide enough. Our exposition of the rules for verifying and/or falsifying sentences (or saturated formulae) is not yet complete. We have not yet seen how to verify a conditional claim (one of the form ϕ → ψ), nor how to falsify a disjunctive claim (one of the form ϕ ∨ ψ). In order to see these rules in action, we turn to the fifth truth and the fifth falsehood in our list. The fi fth truth is the claim ∀x(f(f(x)) = x → ¬P(x)). As we inspect the diagram M, we realize that this universal generalization is true for want of a counterinstance. No individual in the domain is such that it is identical to its own f f-image, yet has the property P. Each of α, β, and γ fails to be such a counterinstance, and in interesting ways. First, α is not its own f f-image. (It is, however, in the extension of P.) Secondly, β, likewise, is not its own f f-image; but, also, β is not in the extension of P. Indeed, it is easier to tell that the latter is the case, than that the former is the case. So, as far as β is concerned, its failure to be a counterinstance to the claim ∀x(f(f(x)) = x → ¬P(x)) is more easily shown by showing ¬P(β) than by deriving ⊥ from f(f(β)) = β. Thirdly, γ fails to be a counterinstance to the claim ∀x(f(f(x)) = x → ¬P(x)) for a similar reason: P(γ) is false. Our verification of the claim ∀x(f(f(x)) = x → ¬P(x)) of course proceeds by verifying each of its three instances f(f(α)) = α → ¬P(α) f(f(β)) = β → ¬P(β) f(f(γ)) = γ → ¬P(γ)

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and we are proposing to do this, instance by instance, as follows:

The missing falsification is as follows:

The missing verification in the middle we have seen before, and the rightmost one is similar to it. They are as follows:

Once again, it is left to the reader to insert these last three bits of detailed working into the appropriate places (indicated by the vertical dots) in the preceding verification-schema. Some comments are now in order on the new rules that have just found application within the last example. First, we have seen that there are two ways to verify a conditional ϕ → ψ: one can falsify its antecedent ϕ, or one can verify its consequent ψ. Both these methods found application in the verification above of the claim ∀x(f(f(x)) = x → ¬P(x)). Its α-instance was verified by falsifying the antecedent; while the β- and γ-instances were verified by verifying their consequents. The general form of the rule for verifying a conditional is accordingly

where the missing dots on the left indicate the presence of a verification of the consequent ψ, while those on the right indicate the presence of a falsification of the antecedent ϕ. Our manipulations of identity statements in the last example were also novel. The step

234 Neil Tennant

is a special case of the rule

where u1 is a saturated D-term and α, α1 are individuals in the domain

A moment’s refl ection reveals that this is the canonical way to establish a conclusion of the form f(u 1) = α. How does one work out that f(u 1) is indeed the object α? One fi rst fi nds the object α1 denoted by the contained (saturated) term u 1. (See the second premise of the rule just stated.) Then one fi nds the object (call it α) denoted by the (saturated) term f(α1). (See the fi rst premise.) That object α is thereby shown to be (the denotation of the term) f(u 1). (See the conclusion.) So this rule for identity simply teases out what is involved in computing function-values stage-by-stage. The rule just stated can be generalized to n-place functions as follows:

where f is a primitive n-place function sign, u1, . . . , un are saturated D-terms and α, α1 , . . . , αn are individuals in the domain

The computation of the values of the (saturated) terms u1, . . . , un proceeds in parallel. We turn now to our fi nal example, which is the fi fth falsehood: P(f(α)) ∨ P(f(β)) In order to falsify a disjunction ϕ ∨ ψ, one needs to falsify each disjunct

Applied to the case at hand, we obtain the falsification schema

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The missing falsifications to be inserted here are

and

The fi nal steps of the last two falsifications were applications, for the case n = 1, of the identity rule

where A is a primitive n-place predicate, t1, . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain

In order to motivate the formulation of a few more rules governing the construction of verifications and falsifications, let us consider a slight embellishment of our earlier model M. Let M′ be the following model, identical to M on the vocabulary of M, but with a new, two-place predicate Lxy (to be read as ‘x is to the left of y’). The extension of L is represented by dashed arrows in the diagram for M′.

Note that all the rules of the model M are rules of the model M′. We extend our earlier list of sentences whose truth-values are to be determined by means of the rules framed here.

236

Neil Tennant True-in-M′

False-in-M′

(1)

∃xP(x)

∀xP(x)

(2)

∃xf(x) = x

∀x¬f(x) = x

(3)

∃x(P(x) ∧ ¬f(x) = x)

∀x(¬P(x) → f(x) = x)

(4)

∀x(¬P(x) ∨ ¬f(x) = x)

∀x∃yf(y) = x

(5)

∀x(f(f(x)) = x → ¬P(x))

P(f(α)) ∨ P(f(β))

(6)

f(f(f(α))) = f(f(α))

f(f(f(α))) = α

(7)

L(f(α), f(β))

∃xP(x) ∧∀x(f(f(x)) = x)

The seventh sentence on the left, L(f(α), f(β)), is true in M′. Its M′-relative verification is:

The fi nal step here is an instance, for the case n = 2, of the following identity rule:

where A is a primitive n-place predicate, t1, . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

The careful reader will have noticed, from the three substitution rules already stated for identity, that there is a pattern to them that would be completed by having the following rule:

where f is a primitive n-place function sign, u1, . . . , un are saturated D-terms, and α, α1 , . . . , αn are individuals in the domain.

An example in which this rule fi nds application is the falsification of the sixth sentence on the right in the list above: f(f(f(α))) = α The rule is applied in the fi nal step of the following falsification, whose rightmost premise is the claim being falsified:

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There is only one more ‘model-relative’ rule (or axiom) that needs to be specified in order to have an adequate set of rules for the verification and falsification of claims in a fi rst-order language with identity. It is the rule of refl exivity of identity, stated only for individuals α in the domain of the model in question: where α is an individual in the domain

This rule enables one to derive the identity claim t = u (where t and u are saturated D-terms of any complexity) when each of those terms has been verified as denoting some same individual α:

Note that this inference is an application of our earlier substitutivity rule for n-place atomic predicates A—identity being a two-place atomic predicate that can take the place of A in the statement of that rule. In our model M, for example, the claim f(f(f(α)))=f(f(α)) is true. Its verification, whose fi nal step is of the form just displayed, is

Consider a conjunction that is false in M, such as ∃xP(x) ∧ ∀x(f(f(x)) = x). (This is the seventh sentence on the right in the list above. It is the righthand conjunct that is false.) A falsification of the whole conjunction in such a case must proceed by falsifying the culprit conjunct. The general form for falsification of a conjunction ϕ ∧ ψ via its right conjunct ψ is as follows:

Of course there must also be a way of falsifying a conjunction when it is only its left conjunct that is false:

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Even when both conjuncts are false, the most economical route to the falsification of the conjunction is to focus on the conjunct with the easiest falsification. Hence there is no need to add to, or alter, the form of these last two rules for the falsification of conjunctions. We are now in a position to take stock by listing all the rules to which we have had recourse in verifying or falsifying our chosen example sentences above. It is worth stressing that these are rules for determination of the truth-value of an arbitrary sentence, using as synthetic ‘fi rst principles’ the (inferentially coded) basic information in the model in question. The analytic ‘fi rst principles’ are the rules themselves. They characterize the meanings of logical expressions in terms of their roles in determining sentences as true, or as false, in any model. Rules for Verification and Falsification of Primitive Saturated Formulae

where A is a primitive n-place predicate, t1, . . . , tn are saturated D-terms, and α1 , . . . , αn are individuals in the domain.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

where A is a primitive n-place predicate, t1, . . . , tn are saturated D-terms, and α1, . . . , αn are individuals in the domain.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

where f is a primitive n-place function sign, u1, . . . , un are saturated D-terms, and α, α1 , . . . , αn are individuals in the domain.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

where f is a primitive n-place function sign, u1, . . . , un are saturated D-terms, and α, α1 , . . . , αn are individuals in the domain.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ where α is an individual in the domain

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

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239

Rules for Verification and Falsification of Saturated Formulae with a Connective Dominant

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Rules for Verification and Falsification of Saturated Formulae with a Quantifier Dominant

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

240 Neil Tennant We have noted above how quickly one can incur sideways spread in writing down a detailed verification or falsification. This feature militates against the actual construction of these otherwise very illuminating and detailed constructions for sentences (and saturated formulae) relative to a given model M. As soon as one has three or more individuals in the domain of M, along with nested quantifiers (especially when they occasion the use of the two rules that require investigation of all instances of a quantified claim), the blow-up, in the form of sideways spread, is prohibitive. But the resulting construction is only ever as deep as the longest branch within the analysis tree of the sentence (or saturated formula) being evaluated. Moreover, in cases where the domain is infinite, some of these verifications and falsifications will contain steps (for the verification of a universal, or the falsification of an existential) that require infinitely many premises (in the form of instances of the quantified claims in question). In such cases the constructions cannot be written down. Instead, they exist only as infi nitary mathematical objects: labeled trees (where the labels are at least fi nite!) that can have infinite branching, albeit only with branches of finite length. Ultimately, the present ‘inferentialist’ approach to formal semantics via the verifications and falsifications illustrated above requires no more powerful mathematical machinery than is needed in order to vouchsafe the existence of these (rather modest) kinds of infi nitary object.

2

2.1

GENERAL CONSEQUENCES OF THE RULES OF EVALUATION

Identity

Our rules of evaluation, which are framed in the metalanguage, allow for reflexivity of identity only in the form ⎯⎯ α=α

where α is an individual in the domain

This is not the same as saying that the rule ⎯ t=t

where t is a term of the object-language

is valid. But its validity is not hard to establish, using constructive reasoning in the metalanguage, for models in which all names in the object language denote, and all function signs represent total functions. Every such model M for any language containing the extra-logical expressions involved in the term t has in its domain some individual α for which there is a verification V, say, of the claim t = α. Applying the fi rst rule for verification of primitive saturated formulae, one obtains the verification

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Thus it follows from our rules of evaluation in the metalanguage that reflexivity of identity holds generally for terms t of the object-language. A similar account can be given of the validity of the rule of substitutivity of identicals for terms t, u in the object-language: where

Reflexivity of identity and substitutivity of identicals therefore become available as general rules of deduction. They preserve truth from their premises to their conclusions. (In the case of reflexivity, of course, the set of premises is empty. Its conclusion is true in every model of any language containing the extra-logical expressions in t.)

2.2

Non-Contradiction

It is clear that the basic axioms and rules in the atomic diagram of a model are coherent, in the sense that for no primitive saturated formula does the model contain both the axiom allowing one to infer it from no assumptions, and the rule allowing one to infer ⊥ from it. This point generalizes. Lemma 1. Let M be a model for the extra-logical vocabulary in an Msaturated formula ϕ. Then there cannot be both an M-relative verification of ϕ and an M-relative falsification of ϕ. Proof. By induction on the complexity of ϕ. The basis step is obvious. Inductive hypothesis: Assume that the result holds for all M-saturated subformulae of ϕ. Inductive step: By cases, according as ϕ is of the form (i) ¬ψ, (ii) ψ1 ∧ ψ2 , (iii) ψ1 ∨ ψ2 , (iv) ψ1 → ψ2 , (v) ∀xψ, or (vi) ∃xψ. In what follows, λ and μ (with or without numerical subscripts) will be parts of the atomic diagram of M, and V and F (with or without numerical subscripts) will be M-relative verifications and falsifications. We shall also assume that the individuals in the domain of M are α1, . . . , αn , . . . Case (i). Any M-relative verification of ¬ϕ would take the form

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Neil Tennant

and any M-relative falsification of ψ would take the following form

But then ψ would have both an M-relative verification and an M-relative falsification, contrary to the inductive hypothesis. Case (ii). Any M-relative verification of ψ1 ∧ ψ2 would take the form

and any M-relative falsification of ψ1 ∧ ψ2 would take one of the following two forms:

Either way, we would have both an M-relative verification and an M-relative falsification of one of the conjuncts, contrary to the inductive hypothesis. Case (iii) and Case (iv) are similar. Case (v). Recall that the individuals in the domain of M are α1, . . . , αn , . . . So any M-relative verification of ∀xψ(x) would take the form

and any M-relative falsification of ∀xψ(x) would take the form

The instance ψ(αi) would therefore have both an M-relative verification and an M-relative falsification, contrary to the inductive hypothesis. Case (vi) is similar.

QED

Inferential Semantics for First-Order Logic 3

243

SPECIAL FEATURES OF THE RULES OF EVALUATION

The foregoing rules of evaluation permit the construction of proof-like objects (verifications and falsifications; or, evaluation proofs and evaluation disproofs). They do, however, have some special limiting characteristics.

3.1

The Undischarged Assumptions

First, the ‘undischarged assumptions’ of a verification are always either (saturated) primitive formulae, or rules equivalent to negations thereof. When one constructs a verification using the primitive (positive or negative) information in a model (its atomic diagram Λ), there is no complexity in the undischarged assumptions involved (apart from the negation signs in negative literals). The same holds for a falsification, except that the (saturated) formula being falsified may itself be complex. But it will be the only complex formula among the undischarged assumptions of the falsification. So: apart from the complex formula being falsified (when the construction in question is a falsification), construction by means of our rules allows us to ‘reason away from’ at best primitive formulae (and negations thereof). We can emphasize this point by adding mention, within the statement of our rules, of the primitive information upon which the evaluation rests. We shall use λ, λ1, λ2 as variables ranging over subsets of the atomic diagram Λ. (We use the Greek letter lambda to suggest ‘literals’.) We shall illustrate the point by reference to the rules for the connectives and the quantifiers. ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

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Neil Tennant

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

3.2

Conclusions

The second limiting characteristic is that the rules for falsifi cation have ⊥ as their main conclusions, and as conclusions of their subordinate ‘disproofs’. So: the only way to ‘reason away from’ a complex formula (by means of our rules of evaluation) is to reason towards absurdity.

3.3

Domain-Dependence of Quantifier Rules

Thirdly, the rules for verification of universals and for falsifi cation of existentials call for as many subordinate proofs as there are individuals in the domain (one subordinate proof for each individual). And this involves infi nite sideways branching when the domain of the model is infinite. (Remember, our verifi cations and falsifications are model-relative. They are not like deductions in general. The job of a deduction—which is always fi nitary—is to preserve M-relative truth from its premises to its conclusion, for all models M.)

3.4

The Contrast with Rules of Deduction in General

General rules of deduction allow one in general to reason away from (fi nite) sets of sentences of any complexity to sentences of any complexity. Of course, primitive sentences (and negations thereof) can stand as

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245

assumptions of deductions; and absurdity can stand as a conclusion (in which case the deduction is called a reductio ad absurdum, or refutation, of its set of undischarged assumptions). But deduction in general involves reasoning from a (fi nite) set of complex sentences as assumptions, to a complex conclusion. We are now in pursuit of rules of inference governing such reasoning, rules in accordance with which more general proofs can be constructed—more general, that is, than our model-relative evaluation proofs (i.e. verifi cations) and evaluation disproofs (i.e. falsifi cations). When deducing a sentence ϕ from a set Δ of sentences, we are no longer working with the atomic diagram of a particular model. Rather, the sentences in the set Δ (the premises of our sought proof) might all be true ‘simultaneously’ in many different models. The task is to show that in any such model, the sentence ϕ will be true too. That is the job of proof in general. When one has a proof of ϕ whose undischarged assumptions form the set Δ, one must be able to say: any model that verifies every member of Δ verifies ϕ. This means that we cannot use the present rule for verification of universals when it comes to deductive reasoning towards a universal (trying to establish it as a conclusion); nor can we use the present rule for falsification of existentials when it comes to deductive reasoning away from an existential (trying to use it as a premise). For, both these rules call for a specific number of subordinate deductions, one for each individual in the domain of a specific model (relative to which truth-value determination takes place according to the evaluation rule in question). Deductive reasoning, however, is undertaken without any specific model in mind. What is important is only preservation of truth-value from premises to conclusion—so that every model for the premises is a model for the conclusion.

4 FROM RULES OF EVALUATION TO RULES FOR DEDUCTION IN GENERAL Our task now is to fi nd suitable generalizations or analogues of our rules for verification and falsification that can serve as rules governing deduction towards, and deduction away from, complex sentences. As we survey our rules of verification and falsification, certain of these analogues are immediate. A box subscript on a discharge stroke indicates that the assumption in question must have been used, and therefore be eligible to be discharged. (This was obvious in the case of verifications and falsifications, but now needs to be emphasized, since we are moving towards a statement of rules of inference in general.) We shall take the connectives in turn, as we morph the rules for verification and falsification of sentences with a given connective dominant into

246 Neil Tennant more general rules of deduction. These are introduction rules (which tell one how to introduce a dominant occurrence of that connective into the conclusion of an inference), and elimination rules (which tell one how to eliminate a dominant occurrence of the connective from the major premise of an inference).

4.1

Negation

The sought generalizations of the rules for negation are straightforward. 4.1.1

Introduction

The rule for verifying a negation becomes the negation introduction rule upon allowing for more general sets Δ of side-assumptions in the subordinate reductio:

Note that the conclusion ¬ϕ depends only on the assumptions in Δ; the assumption ϕ (for reductio ad aburdum) is discharged by applying the rule. Moreover, as indicated by the box subscript on the discharge stroke, the assumption ϕ must have been used, and be undischarged within the subordinate reductio, in order that the rule be applicable. 4.1.2

Elimination

The rule for falsifying a negation becomes the negation elimination rule upon allowing for more general sets Δ of assumptions in the subordinate proof of the minor premise:

The conclusion rests both on ¬ϕ and on the assumptions in Δ. Note also that we are not allowing the major premise ¬ϕ to stand, itself, as the conclusion of any proof-work above it. Rather, ¬ϕ stands proud as an undischarged assumption. (It could, however, be discharged by subsequent applications of rules of inference, as the proof-work proceeded further in a downward direction).

Inferential Semantics for First-Order Logic

4.2 4.2.1

247

Conjunction Introduction

The morphing of the rule for model-relative verification of conjunctions into the introduction rule (for inferring conjunctions from arbitrary sets of premises) is straightforward:

Note that we allow for Δ1 to be distinct from Δ2 . The conclusion ϕ ∧ ψ depends on their union. 4.2.2

Elimination

Now consider the rule for model-relative falsification of a conjunction:

We want the elimination rule to allow for the derivation of general conclusions θ in place of ⊥:

and we may as well economize by allowing for simultaneous discharge of the dischargeable assumptions. At the same time we require that at least one such assumption should have been used, and therefore be eligible to be discharged—this requirement being indicated by a box affi xed to the inference strokes:

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Neil Tennant

The conclusion θ depends only on ϕ ∧ ψ and the assumptions in Δ. The major premise ϕ ∧ ψ stands proud.

4.3 4.3.1

Disjunction Introduction

As with conjunction, the introduction rule for disjunction is a straightforward generalization of the rule of model-relative verification:

4.3.2

Elimination

The rule for reasoning away from a disjunctive premise ϕ ∨ ψ needs likewise to be generalized so as to permit the deduction of a sentence θ in general, rather than just ⊥. But to this end it would suffice to deduce θ from but one of the cases ϕ and ψ. If the other case closes off with ⊥, then we know the truth does not lie there; hence, lies with the case that leads to θ. So permissible deductive moves would be:

Naturally also if θ is deducible from each case-assumption, then θ should be deducible overall:

And a special case of θ in this last rule is of course ⊥ itself, as with the rule of falsification with which we began. We can sum up the possibilities just canvassed as follows:

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The conclusion (θ or ⊥) depends on ϕ ∨ ψ, the members of Δ1 and the members of Δ2 . The sets Δ1 and Δ2 could be distinct. The major premise ϕ ∨ ψ stands proud. The rule of ∨-Elimination is also known as proof by cases. The two subproofs indicated are called the case-proofs. The rule of thumb is: if either one of the case-proofs ends with ⊥, one may bring down the conclusion of the other case-proof as the overall conclusion.

4.4 4.4.1

The Conditional Introduction

A refutation of ϕ modulo Δ:

guarantees that any model of Δ falsifies ϕ. 2 Thus, by the fi rst half of the model-relative verification rule for the conditional, any model of Δ verifies ϕ → ψ. So the fi rst half of our model-relative verification rule generalizes into the fi rst half of the sought introduction rule as follows:

The second half of our model-relative verification rule needs, however, to be generalized more carefully. We need to allow for the distinct possibility that one might not be in a position to deduce the consequent, given the under-specifi c information Δ at hand (as opposed to the highly specifi c information about a model, which will tell one whether the consequent holds). Since we are now allowing deductions from arbitrary sets of complex sentences, one can imagine a situation in which one has a deduction of the consequent ψ from the antecedent ϕ along with other assumptions forming a set Δ, say:

Remember that deductions are to be truth-preserving. So, every model of Δ that verifi es ϕ verifi es ψ. Therefore one can say of every model M

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Neil Tennant

of Δ: if M verifi es ϕ, then M verifi es ψ —whence, M verifi es ϕ → ψ. That justifi es the following second half of the introduction rule for the conditional:

The use of the diamond here indicates that the subordinate proof need not have used ϕ as an assumption. But if it did, then that assumption will be discharged by application of the rule. In a case where ϕ is not used as an assumption, the justification of ϕ → ψ is immediate by the verification rule: for every model of ϕ would, ex hypothesi, verify ψ, hence also (by the verification rule) verify ϕ → ψ. 4.4.2

Elimination

The elimination rule for the conditional is obtained by straightforward morphing of the rule for model-relative falsification of conditionals:

The conclusion θ depends on ϕ → ψ, all the members of Δ1 and all the members of Δ2 . The sets Δ1 and Δ2 can be distinct. The major premise ϕ → ψ stands proud. The proof of ϕ from Δ1 is called the minor proof; that of θ from Δ2 , ψ is called the major proof (for →-Elimination).

4.5 4.5.1

The Universal Quantifier Introduction

Recall the model-relative rule for verifying a universal claim (without loss of generality here we shall assume that the domain is fi nite): where α1, . . . , αn are all the individuals in the domain

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In the presence of the indicated verifications, one would have a proof of ψ(a) from the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:3

The idea behind the introduction rule for ∀ is that any ‘parametric’ proof to the effect a is in the domain

should justify one in drawing the conclusion ∀xψ(x): _______________(i) a is in the domain

Standard logic is the logic of a ‘logically perfect’ language. The perfection assumption is that every well-formed singular term denotes. In free logic, one gives up this assumption. One allows for ‘empty’ names, such as ‘Pegasus’. One allows also for partial functions, that is, functions that are ‘not everywhere defined’, such as division (which is not defi ned when the divisor is 0). And one needs a free logic if one aims to accommodate the defi nite description operator as a primitive variable-binding term-forming operator. This is because any term of the form ιx(ϕx ∧ ¬ϕx) fails to denote. The same need arises for the set abstraction operator. For, as Russell’s Paradox shows, the term {x | ¬x ∈ x} fails to denote. In free logic, the assumption that a is in the domain is expressed by the formal sentence ∃x x=a, often abbreviated as ∃!a. Thus the rule of ∀-Introduction in free logic is

where ∃!a is the only assumption containing a on which ψ(a) depends

252 Neil Tennant In standard logic, however, the assumption that a is in the domain does not need to fi nd expression. One can limit oneself to proofs of ψ(a) from assumptions making no mention of a:

where ψ depends on no assumptions containing a

4.5.2

Elimination

The elimination rule for ∀ can be stated as a rule allowing for multiple discharge (of assumption-instances of the predicate involved). This is on grounds analogous to those on which we allowed the ∧-Elimination rule simultaneously to discharge all assumption-occurrences of either of its conjuncts in the subordinate proof. The rule of ∀-Elimination is accordingly as follows:

In free logic one would invoke extra premises to the effect that one’s chosen terms t1, . . . , tn do indeed denote:

4.6 4.6.1

The Existential Quantifier Introduction

The introduction rule for ∃ closely resembles the model-relative rule for verifying existentials. In the latter, it sufficed to verify a single instance:

When deducing conclusions from arbitrary sets of sentences, the available instances are no longer saturated formulae using particular individuals α from a domain. Rather, one might prove, for some term t, that ψ(t) holds

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(conditionally on whatever assumptions are being used). Then—on the background assumption that said term t denotes an individual—one would be able to conclude ∃xψ(x):

In free logic one would invoke an extra premise to the effect that one’s chosen term does indeed denote:

4.6.2

Elimination

Recall the model-relative rule for falsifying an existential claim (in the fi nite case): where α1 , . . . , αn are all the individuals in the domain

Similar considerations apply here as applied in our formulation of the rule of ∀-Introduction above. In the presence of the indicated falsifications, one would have a disproof of ψ(a), using the assumption a = α1 ∨ . . . ∨ a = αn , to the effect that a is a member of the domain:4

Accordingly, in free logic the rule of ∃-Elimination would be where the parameter a does not occur in any assumption, other than ∃!a and ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

while in standard logic it would be

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Neil Tennant where the parameter a does not occur in any assumption, other than ψ(a), on which the upper occurrence of θ depends; and does not occur in either ∃xψ(x) or θ

4.7

Summary of Introduction and Elimination Rules

We collect together the introduction and elimination rules for standard, unfree logic. Where sets of assumptions (other than those being discharged) are permitted within subproofs, we indicate this by means of Δ (with or without a numerical subscript). In the rule ∧-E, the box indicates that at least one of the conjuncts ϕ, ψ must appear as an (undischarged) assumption in the subordinate proof. Likewise, in the rule ∀-E, the box at the level of the discharge strokes indicates that at least one undischarged assumption of the form ψ(t) must appear in the subordinate proof. Other boxes attached to discharge strokes indicate that an assumption of the form in question must appear undischarged in the subordinate proof. The diamond in the second half of →, however, indicates that no such assumption need appear; if it does, however, it is discharged by the application of the rule. Introduction

Elimination

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Inferential Semantics for First-Order Logic

255

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ where a does not occur in any member of Δ

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ where a does not occur in ∃xψ(x), θ or any member of Δ

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ where

Important note: Major premises for eliminations stand proud. They are not drawn as conclusions of any proof-work above them.

5

5.1

CONCLUSION

Summary

The foregoing rules are those of core logic. 5 The aim of this study has been to reveal the natural way in which core logic emerges from reflections on how one establishes sentences as true or as false under interpretations, and how one can generalize those movements in thought so as to deal with complex sentences in general, as starting points and as endpoints of trains of reasoning. Elsewhere, I have argued that core logic is the correct logic according to an anti-realist account of meaning (Tennant 1987 and 1997); that it suffices for constructive mathematics (Tennant 1994); that it suffices for hypothetico-deductive testing of theories (Tennant 1985); that it enables efficient automated proof-search (Tennant 1982); and that it is the minimal canon invulnerable to revision, every

256

Neil Tennant

part of which is indispensable for the process of rational belief-revision (Tennant unpublished typescript b). On this occasion, however, the aim has been to describe a natural conceptual route to core logic, beginning with one’s rudimentary grasp of verification- and falsification-conditions, construed in a suitably inferentialist fashion.

5.2

Further Developments

Model-relative verifications and falsifications, as introduced here, are rigorously defi nable as well-understood mathematical objects: they are trees of fi nite depth, whose nodes are labeled by saturated formulae. When the domain of the model is infi nite, the sideways branchings corresponding to verifications of universals and falsifications of existentials will be infi nite. But all branches will be of fi nite length. In a reasonably weak metamathematical theory, one can prove the following. Theorem 1 For any model M, and any saturated formula ϕ, ϕ is true in M (in Tarski’s sense) ⇔ there is an M-relative verification of ϕ ; and ϕ is false in M (in Tarski’s sense) ⇔ there is an M-relative falsification of ϕ The theory of model-relative verifications and falsifications is a structuretheory for truth-makers and falsity-makers. The current philosophical literature on truth-makers appears to be bereft of such a structure-theory. In future work I intend to demonstrate some advantages in conceiving of philosophers’ ‘truth-makers’ as these (appropriately structured) model-relative verifications. Other results, to be presented in detail elsewhere, are the following (Tennant unpublished typescript a). Theorem 2 Given any two proofs in core logic, where the conclusion of the first proof is a premise of the second proof (call it the ‘cut-sentence’), one can effectively find a core proof of the second proof’s conclusion, or of absurdity, from premises of the given proofs other than the cut sentence. Theorem 3 Proofs in core logic provide an ‘effective’ means of transforming (for any model M), M-relative verifications of their undischarged assumptions into M-relative verifications of their conclusions. 6

NOTES 1. This is true not only of Tarski’s original treatment (1956), which invoked infi nite sequences of individuals correlated with object-linguistic variables, but also of the treatment (in Tennant 1978) of Tarski’s approach that appeals,

Inferential Semantics for First-Order Logic

2. 3. 4. 5. 6.

257

more modestly, to fi nitary assignments of individuals to the free variables in a formula. This innocuous-seeming claim requires, and admits of, proof. In the infinitary case, the disjunctive major premise would have to be infinitary. That would pose no problems in principle, however, since the disjunction in question is being invoked only by way of motivation of the main idea. See Note 3. Formerly called IR, or intuitionistic relevant logic, as in Tennant (1982, 1987, and 1997). I would like to thank Jonathan Lear and Alex Oliver for their many helpful editorial comments.

REFERENCES Tarski, A. (1956) ‘The concept of truth in formalized languages’ in his Logic, Semantics, Metamathematics, J.H. Woodger (ed.), Oxford: Clarendon Press, pp. 152–278. Tennant, N. (1978) Natural Logic, Edinburgh: Edinburgh University Press. . (1982) Autologic, Edinburgh: Edinburgh University Press. . (1985) ‘Minimal logic is adequate for Popperian science’, British Journal for Philosophy of Science 36: 325–9. . (1987) Anti-Realism and Logic: Truth as Eternal, Oxford: Clarendon Press. . (1994) ‘Intuitionistic mathematics does not need ex falso quodlibet’ in Topoi 13: 127–33. . (1997) The Taming of The True, Oxford: Oxford University Press. . (unpublished typescript a) ‘Core logic’, paper presented to the Reunion Conference of the Fellowship of the Pittsburgh Center for Philosophy of Science, Ohio University, Athens, OH, June 2008. . (unpublished typescript b) Rational Belief Revision.

Bibliography of Works by Timothy Smiley

1959 1960 1961 1962 1963 1967 1970

1971 1973 1976 1978

1979 1980 1981 1982 1983

‘Entailment and deducibility’, Proceedings of the Aristotelian Society 59: 233–54. ‘Sense without denotation’, Analysis 20: 125–35. ‘Propositional functions’, Aristotelian Society Supplementary Volume 34: 34–46. ‘On Łukasiewicz’s Ł-modal system’, Notre Dame Journal of Formal Logic 2: 149–53. ‘Syllogism and quantification’, Journal of Symbolic Logic 27: 58–72. ‘The independence of connectives’, Journal of Symbolic Logic 27: 426–36. ‘The logical basis of ethics’, Acta Philosophica Fennica 16: 237–46. ‘Relative necessity’, Journal of Symbolic Logic 28: 113–34. ‘Mr. Strawson on the traditional logic’, Mind 76: 118–20. Review of Nicholas Rescher, Many-valued Logic, British Journal for the Philosophy of Science 21 (1970): 405–6. Review of G. H. von Wright, Logical Studies, Journal of Symbolic Logic 35: 460–2. Review of G. H. von Wright, ‘A note on entailment’, Journal of Symbolic Logic 35: 462. ‘Deducibility and many-valuedness’ (with David Shoesmith), Journal of Symbolic Logic 36: 610–22. ‘What is a syllogism?’, Journal of Philosophical Logic 2: 136–54. ‘Does many-valued logic have any use?’ in S. Körner (ed.) Philosophy of Logic, Oxford: Blackwell, pp. 74–88. ‘Foundations of mathematics and mathematical logic’ in F.P. Ramsey, Foundations, D. H. Mellor (ed.), London: Routledge, pp. 7–10. Multiple-Conclusion Logic (with David Shoesmith), Cambridge: Cambridge University Press. ‘A theorem on directed graphs, applicable to logic’ (with David Shoesmith), Journal of Graph Theory 3: 401–6. Review of A. N. Prior, Papers in Logic and Ethics, Journal of Symbolic Logic 45: 180–83. ‘Frege and Russell’, Epistemologia 4: 53–57. ‘The theory of descriptions’, Proceedings of the British Academy 67: 321–37. ‘The schematic fallacy’, Proceedings of the Aristotelian Society 83: 1–17. ‘Hunter on conditionals’, Proceedings of the Aristotelian Society 84: 169–77. Logic Colloquium ’80 (co-editor with D. van Dalen and D. Lascar), Amsterdam: North-Holland.

260 Bibliography of Works by Timothy Smiley 1985 1988 1993 1994 1995 1996 1998

2000 2001

2002 2004

2005 2006

2008 2009

‘The foundations of econometrics: comment’, Econometric Reviews 4 (1985): 93–99. ‘Frege’s “series of natural numbers”’, Mind 97: 583–4. ‘Can contradictions be true?’, Aristotelian Society Supplementary Volume 68: 17–33. ‘Aristotle’s completeness proof’, Ancient Philosophy 14: 25–38. ‘William Calvert Kneale’, Proceedings of the British Academy 87: 385–97. Philosophical Dialogues (editor), Oxford: Oxford University Press. ‘A tale of two tortoises’, Mind 104: 725–36. ‘Rejection’, Analysis 56: 1–9. ‘Consequence, Conceptions of’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 2, London: Routledge, pp. 599–603. ‘Multiple-conclusion logic’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, vol. 6, London: Routledge, pp. 602–4. Philosophical Logic (editor), Oxford: Oxford University Press. ‘Predicate calculus’ in E. Craig (ed.) Routledge Encyclopedia of Philosophy, http://www.rep.routledge.com/article/YO37. Mathematics and Necessity (editor), Oxford: Oxford University Press. ‘Abstraction by recarving’ (with Michael Potter), Proceedings of the Aristotelian Society 101: 327–38. ‘Strategies for a logic of plurals’ (with Alex Oliver), Philosophical Quarterly 51: 289–306. ‘Recarving content: Hale’s final proposal’ (with Michael Potter), Proceedings of the Aristotelian Society 102: 351–4. ‘Kneale, William Calvert’, in Oxford Dictionary of National Biography 31: 872–3. ‘Multigrade predicates’ (with Alex Oliver), Mind 113: 609–81. ‘Popper and the poker’, Philosophy at Cambridge, no.1: 7. Studies in the Philosophy of Logic and Knowledge (edited with T. R. Baldwin), Oxford: Oxford University Press. ‘The theory of descriptions’ in T. R. Baldwin and T.J. Smiley (eds) Studies in the Philosophy of Logic and Knowledge, Oxford: Oxford University Press, pp. 131–61. ‘Plural descriptions and many-valued functions’ (with Alex Oliver), Mind 114: 1039–68. ‘A modest logic of plurals’ (with Alex Oliver), Journal of Philosophical Logic 35: 317–48. ‘What are sets and what are they for?’ (with Alex Oliver), Philosophical Perspectives 20: 123–55. ‘Is plural denotation collective?’ (with Alex Oliver), Analysis 68: 22–34. ‘Sharvy’s theory of descriptions: a paradigm subverted’ (with Alex Oliver), Analysis 69: 412–21. ‘Plural logic’ (with Alex Oliver) in E. Craig (ed.) Routledge Encyclopedia of Philosophy, http://www.rep.routledge.com.

Contributors

Kwame Anthony Appiah teaches philosophy at Princeton University. Among his books are Assertion and Conditionals (Cambridge University Press 1985), The Ethics of Identity (Princeton University Press 2005), and Cosmopolitanism: Ethics in a World of Strangers (Penguin 2007). Thomas Baldwin is a Professor of Philosophy at the University of York. He is the author of G. E. Moore (Routledge 1990) and Contemporary Philosophy (Oxford University Press 2001), and editor of The Cambridge History of Philosophy 1870 –1945 (Cambridge University Press 2003). Since 2005 he has been the editor of Mind. James Cargile is a Professor in the Corcoran Department of Philosophy of the University of Virginia. He teaches a variety of courses related to elementary logic and has published work on related topics. James Doyle is Senior Lecturer in Philosophy at the University of Bristol and Visitor at the Institute of Advanced Study, Princeton. He works mainly in ancient philosophy and philosophy of mind. Ian Hacking is Professeur honoraire (emeritus) at the Collège de France (Chaire de philosophie et histoire des concepts scientifiques) and University Professor Emeritus, University of Toronto. The author of many books, his main project at present is to fi nish his work on styles of scientific thinking, alluded to in the essay printed in this volume. Lloyd Humberstone is a Reader in Philosophy in the Department of Philosophy and Bioethics, Monash University. His main interests are in logic and its applications to philosophical issues. Rosanna Keefe is a Senior Lecturer at the University of Sheffi eld. She is the author of Theories of Vagueness (Cambridge University Press 2000).

262

Contributors

Jonathan Lear is the John U. Nef Distinguished Service Professor at the Committee on Social Thought and the Department of Philosophy at the University of Chicago. His most recent book is Radical Hope: Ethics in the Face of Cultural Devastation (Harvard University Press 2006). Alex Oliver is Reader in Philosophy at Cambridge University and a Fellow of Gonville and Caius College. His main interests are in metaphysics, logic, and philosophy of mathematics. His publications include several papers on plural logic written with Timothy Smiley. Michael Potter is Reader in the Philosophy of Mathematics at Cambridge University and a Fellow of Fitzwilliam College. His books include Reason’s Nearest Kin (Oxford University Press 2000), Set Theory and its Philosophy (Oxford University Press 2004), and Wittgenstein’s Notes on Logic (Oxford University Press 2009). He is the co-editor of Mathematical Knowledge (Oxford University Press 2007) as well as of the forthcoming Cambridge Companion to Frege. Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne, and Arché Professorial Fellow at the University of St. Andrews. His books include In Contradiction (2nd edn, Oxford University Press 2006), Beyond the Limits of Thought (2nd edn, Oxford University Press 2002), Towards Non-Being (Oxford University Press 2005), Doubt Truth to be a Liar (Oxford University Press 2006), and Introduction to Non-Classical Logic (2nd edn, Cambridge University Press 2008). Neil Tennant is Humanities Distinguished Professor in Philosophy at The Ohio State University. His books include Natural Logic (Edinburgh University Press 1978), Anti-Realism and Logic (Oxford University Press 1987), and The Taming of The True (Oxford University Press 1997). His interests lie in logic, philosophy of science, and philosophy of mathematics, and he is currently writing on rational belief revision.

Index

A Abel, Niels Henrik, 102 abstract algebraic logic (AAL), 120 abstractionism abstraction principles, 186, 188–89, 190–91, 192, 194, 203 circularity, 188, 190–91, 201–2, 202–3 classes, 188–91, 195–203 epistemology, 187–88 impredicativity, 187–87, 191 Julius Caesar problem, 187, 194 stability, 193–94 See also conceptions abstraction principles, examples (Access), 196, 202 (Countable), 196–97 (Finite), 196–97 (New V), 196 Newer V, 190 Newer V*, 191 parity, 192 (Set), 196 (V), 188, 192, 193 (V′), 188 accident, fallacy of, 206 Ackermann, Wilhelm, 207 admissible rules. See under rules Alexander of Aphrodisias, 175–76 analyticity, 55–56, 66n8, 136 analytic-synthetic distinction, 4–5. See also holism Anderson, Alan Ross, 112–13, 118, 122n10, 124n21, 124n27 Anscombe, Elizabeth, 80n2, 147 anti-realism mathematical (see under mathematics) semantic, 2–3, 255 See also pragmatism

Apollonius, 98 a posteriori knowledge, 8, 13. See also empirical truth; necessary truth Appel, Kenneth, 86 appetite. See Plato: tripartite soul Appiah, Kwame Anthony, 12, 16, 16n4, 17n13, 155 a priori knowledge, 8, 90, 137–38, 140–41 Archimedes, 95, 96, 98, 100 aretē, 80n2 arguments, elements of, 54. See also propositions; sentences Aristotle Barbara, 20, 23–24, 34, 169 conversion, 24, 171 existential import, 24, 45n3 figure, 173, 175 forms, 169, 173 identity, 206, 220n2 modality, 177 mood, 173 obversion, 24 plural definition of a deduction, 169 restricted quantification, 24 rhetoric, 94 schematic letters 169–70, 176 square of opposition, 45n3 surface structure, 21 plural variables, 170–71 armchair philosophy, 1–2, 7, 8, 11, 13–16 aspiration, 146–54, 155, 158. See also irony, Kierkegaardian; pretense Austin, J.L., 80n4 automated proof-search, 255 axiomatic method, 186–87 axiomatization 113–16 differentiated vs. undifferentiated, 114

264

Index

finite, 116 schematically finite, 116

B Bacon, Francis, 98–99, 100, 101, 104n6 Baker, Gordon, 85 Bambrough, Renford, 84 Barcan, Ruth, 122n4 Barnes, Jonathan, 175 Barwise, Jon, 46n5 Basic Law V. See abstraction principles, examples: (V) Bedeutung. See nominee Belnap, Nuel, 107, 112–13, 118, 122n10, 124n21, 124n27 Benson, Hugh, 68 van Benthem, Johan, 123n18 Bernays, Paul, 88 Blanchette, Patricia, 88 Blok, Willem, 120, 122n7 Bolzano, Bernard, 176–78, 180, 183 Boole, George, 34–35, 46n4 Boolos, George, 177, 189, 192, 201 Bose, S.N., 83 Bostock, David, 178 Boyer, Carl B., 97 Bradley, F.H., 165 Brady, Ross T., 123n17 Brown, Gary I., 104n6 Buridan, Jean, 176, 177, 182

C Callicles, 69, 74–75, 78, 79, 80n2 Cantor’s paradox, 65 Cargile, James, 181, 182 Carnap, Rudolf, 5 Carolino, Pedro, 8 Carroll, Lewis, 5–6 Cayley, Arthur, 103 Chaerephon, 75 Changeux, Jean-Pierre, 87–88, 104n3 Choice, Axiom of, 64 Christendom, 147–49, 162n5 Church, Alonzo, 48–50, 52, 57, 61, 62–63, 166, 167 Clark, Peter, 203n3 classes abstractionist theory, 188–91, 195–203 definite conception (Def), 198–201 iterative conception, 189–91, 194, 197, 198, 199, 200, 201, 202 limitation of size conception, 195–98, 199–202

maximality principle, 197 vs. sets, 188 well-founded, 189–90 See also abstractionism; abstraction principles, examples; limitation of size principles; Russell’s paradox; sets Climacus, Johannes, 145, 148. See also Kierkegaard, Søren Coleridge, Samuel Taylor, 104n7 composition, 213–15 computations, 10–12 Comte, Auguste, 99 conceptions (of mathematical objects), 191–5 concepts analysis, 1–2, 5, 7, 14–16 conceptual knowledge, 3–4, 14 conceptual truth, 1, 8–9, 11–12 See also paradox of analysis concepts, Fregean class-forming, 188–89 indefinitely extensible, 198–201 second-level, 32, 40 See also classes; properties conceptual schemes, 6 conceptual truth 1, 8–9, 11–12 conditional proof, 131–35, 137. See also Deduction Theorem Connes, Alain, 87–88, 90 conservativeness, 202 content recarving, 187. See also abstractionism convention, linguistic, 8 Cook, Roy, 190–91 Cooper, Robin, 46n5 Copeland, Jack, x Copi, Irving, 174 core logic, 255–56 conditional, 249–50, 255 conjunction, 247–48, 254 disjunction, 248–49, 254 existential quantifier, 252–54, 255 generalizing M-relative rules of evaluation, 244–45, 255 identity, 240–41, 255 introduction vs elimination rules, 246 negation, 246, 254 summary of rules, 254–55 universal quantifier, 250–52, 255 See also M-relative rules of evaluation cosmopolitanism, inner, 156

Index counterpart theory, 216–17, 220n7 Crelle, A.L., 102 criterion of identity, 194. See also abstractionism Crombie, Alistair, 93 Crow Nation, The, 159–61 crystallization, 92–97 Cut, 256 Czelakowski, Janusz, 122n2

D Da Fonseca, José, 8 D’Alembert, Jean Le Rond, 99–100 Darwin, Charles, 158 Davidson, Donald action sentences, 181–82 assertion, 72 conceptual schemes, 6 interpretation, 81n8 logical form, 19–21, 182–83 principle of charity, 7, 12, 17n10 quantified sentences, 19–21, 23 theory of meaning, 7–10 deducibility (Bolzano), 176–77 deducibility (proof-theoretic), 108 deduction, rules for. See under core logic deductions as trees, 122n6 Deduction Theorem, 108–11, 121 definite descriptions, ix, x, 19, 51, 172, 210–11 definitely operator, 133–35, 137–40 degenerating research programs, 219 degree theory (vagueness), 218 Denecessitation, 118 derivable rules. See under rules determiners, 22 Detlefsen, Mic, 85, 89–90, 102 dialetheism, 219 Diogenes, Laertius, 91 Dodds, E.R., 80n6 Doyle, James, 81n8 dualism, 213 Duhem, Pierre, 9 Dummett, Michael, 3, 16n5, 89, 193

E Eddington, Arthur, 103 Edgington, Dorothy, 17n13 ego, Freudian, 158–59 Einstein, Albert, 83 elenchus. See Socratic elenchus

265

empirical truth, 8, 15. See also a posteriori knowledge; necessary truth Enderton, Herbert B., 174 entailment, 56, 64–65, 112. See also logical consequence; strict implication; validity epistemicism (vagueness), 140, 143n9, 217 epistemic necessity, 212 Etchemendy, John, 179 Euclid, 97, 205–6 Eudoxus, 91 evaluation, model-relative rules. See M-relative rules of evaluation Evans, Gareth, 180 ex falso quodlibet (EFQ), 25, 56–57, 64 externalism, 11

F Fagin, Ronald, 123nn17–18 family resemblances, 84–85 Fara, Delia Graff, 127, 133, 134, 135–41, 142n3, 142–43n7, 143n8 Fermat, Pierre de, 83 final vocabulary, 154–55, 161 Fine, Kit, 107, 133, 217, 220n4 fission, 215–56. See also fusion Fitch, Frederic B., 124n24 Fitting, Melvin, 123n18 Fodor, Jerry, 94 Font, Josep Maria, 124n21, 124n26 Forbes, Graeme, 178 formal fallacy, 172 formalism, 187 formalization, 180–83. See also under Smiley, Timothy formal consequence. See under logical consequence formal validity. See under validity forms. See logical form forms of life, 146–47 formulae closing an open formula, 224 open vs. closed, 224 saturated D-formula, 225 saturated identity formula, 224 saturating an open formula, 225 Forrest, Peter, 207 Four-color theorem, 86–87 four-dimensionalism, 214, 215–16 free logic, x, 251

266

Index

Frege, Gottlob Aristotle’s logic, 45n3 consistency, 187 empty terms, x and Hilbert, 187 identity, 207, 211 incomplete definitions, 32–34 mathematics, philosophy of, 102 propositional attitude contexts, 212–13 propositional calculus, 108 quantification, 32–34, 36, 188 reason, 195 rejection, xi and Russell, 45n4 sentences, priority of, 16n5 thoughts, composition of, 60 See also abstractionism; nominee; sense Freud, Sigmund, 158 functionalism, 10 fusion, 217. See also fission

G Galileo, 89, 93, 97, 100, 101 Gentzen, Gerhard, 123n16 Gergonne, Joseph, 102 Gettier, Edmund, 14 Gibbard, Allan, 213 Gödelian platonist, 192. See also mathematics: realism vs. anti-realism Gödel, Kurt, 36 grammatical form. See under logical form Grice, Paul, 5, 80n5 groups, finite, 88

H Hacker, Peter, 85 Hacking, Ian, 16n1, 54, 98, 104n1, 104n4 Haken, Wolfgang, 86 Hale, Bob, 190–95 Halpern, Joseph Y., 123nn17–18 Hamilton, William, 102 Hardy, G.H., 103 Herbrand, Jacques, 36, 108 Hilbert, David, 187, 207 historicism, 90–91, 93, 103–4 holism, 5, 7, 9. See also Davidson, Donald: theory of meaning; Quine, W.V.O: meaning; revisablitiy Homer, 79n1 Horsten, Leon, 85, 86, 89 Hughes, G.E., 123n20 humanity, 146–48, 152

Humberstone, Lloyd, 107, 120, 122n3, 123n17, 124n22, 124n25, 124nn27–28 Husserl, Edmund, 100 hypothetico-deductive testing, 255

I id, Freudian, 158, 163n11 ideals, 149, 153–54, 158–59, 160, 161. See also irony, Kierkegaardian; pretense ideas, 176–77 identification, 156 identity contingency, 216–17 vs. indiscriminability, 219–20 necessity, 13, 210 See also counterpart theory; identity of indiscernibles; Law of Identity; LL; Substitutivity of Identicals; transitivity of identity identity-claims, 147, 152, 155–56, 162n4. See also irony, Kierkegaardian; pretense identity of indiscernibles (II), 207 indefinite extensibility, 198–201 indiscriminability. See under identity inferential semantics, 223. See also M-relative axioms and rules of inference; M-relative rules of evaluation infinite domains, 240, 244 inquiries, individuation of, 71 interpretations. See models intersubstitutability salva congruitate, 179, 208–9 intuitionistic relevant logic (IR). See core logic irony, Kierkegaardian earnestness, 145, 146, 148–49 ironic culture, 159–61 ironic questions, 148, 150–51, 153, 156 ironic soul, 157–59 subjectivity, 154, 155, 156 therapy, 158–59 See also aspiration; Christendom; ideals; identity-claims; irony, Rortian; irony, Socratic; pretense; transcendence irony, Rortian, 154–55, 158. See also irony, Kierkegaardian irony, Socratic, 145, 146, 150–52, 156, 160, 162n7. See also irony, Kierkegaardian; Socratic elenchus

Index J Jackson, Brendan, 180 James, William, 16n6 Jané, Ignacio, 203n1 Jansana, Roman, 124n21, 124n26 Jeffrey, Richard, 178 Jespersen, Otto, 179 Julius Caesar problem. See under abstractionism

K Kant, Immanuel analytic truth, 66n8 mathematics, philosophy of, 89–90 mathematics, pure, 101–2, 104n9 mathematics, ur-history of, 90–92 mental faculties, 87 natural science, 93, 97, 104n10 Kaplan, David, 178–79 Keefe, Rosanna, 133, 134, 136, 142nn5–6, 143n7 Keenan, Edward, 41, 42, 44 Kelvin, Lord (William Thomson), 103 Kempe, Alfred, 87 Kemp Smith, Norman, 104n10 Kielkopf, Charles F., 123n11 Kierkegaard, Søren, 144–55, 161n1, 162n5, 162n8. See also irony, Kierkegaardian Kleene, Stephen Cole, 121n2, 122n6, 122n11 Klein, Jacob, 97, 98 Kneale, Martha, 220n2 Kneale, William, 220n2 Knobe, Joshua, 15 know-how. See propositional knowledge Koch, John A., 86 Koyré, Alexandre, 100 Kracht, Marcus, 117, 120 Kraut, Richard, 71 Krieger, Martin H., 83 Kripke, Saul, 12–13, 50–51, 122n11, 124n23, 211 Kuhn, Thomas, 92

L language of thought, 213 Lagrange, Joseph Louis, 102 Lakatos, Imre, 210, 211, 219 Laplace, Pierre-Simon de, 95, 102 Larmor, Joseph, 103 Latour, Bruno, 95–97 Law of Excluded Middle, 134 Law of Identity, 205

267

Law of Non-Contradiction, 208 Lear, Jonathan, 80n7, 81n8, 163n10, 163n12 Legendre, Adrien-Marie, 102 Leibniz, Gottfried, 207 Lemmon, E.J., 119, 124n23, 165, 177 Lewis, David, 8, 216, 220n7 liberalism, 156 limitation of size principles (All), 195, 197–98, 201 (Lim), 195, 197–98, 200, 201 (Ord), 199–202 (Sep), 197–98 (Size), 195–98 See also under classes Lindström, Per, 41 linguistic representation, 10–11 Littlewood, J.E., 104n2 LL, 207. See also Substitutivity of Identicals Lloyd, Geoffrey, 94–95 logical consequence, deducibility (Bolzano), 176–77 deducibility (proof-theoretic), 108 formal vs. material, 176, 182–83 inferential vs. probatory, 118 in normal modal logic, 119–20 vs. logical truth 31 sets of premises vs. premises, 168, 170–71 See also Deduction Theorem; entailment; relative validity; strict implication; validity logical form Aristotelian vs. Platonic, 21 basic schematic form, 167–68, 171–72, 175 deep form, 181 impure forms, 172–74 indeterminacy, 127, 129 linguistic vs. platonic, 61–62 main connective form, 171–72 properties, 166–67, 181 purely invalid forms, 172–74 purely valid forms, 172–74 restricted forms, 52, 168 substitutional notion, 48–53, 167 unique vs. manifold notions, 171–72, 173 validity in virtue of (see under validity) vs. grammatical form 19–21, 48–53, 60–61, 178–80, 208–9 vs. matter, 175–78 See also formalization logically perfect languages, 179, 251

268

Index

logical truth, 31, 176–77 logic, Aristotle’s. See under Aristotle logic, Boolean, 35, 46n4 Lockean model of reference, 194–95 Łoś, Jerzy, 115, 122n7, 123n16 Łukasiewicz, Jan, 169 Lull, Raymond, 99

M Machina, Kenton F., 134, 218 Maddy, Penelope, 193 Makinson, David, 142n3 Massey, Gerald J., 179 material consequence. See under logical consequence material constitution. See composition material implication, 53, 131–33. See also strict implication material validity. See under validity Mates, Benson, 53, 54–55, 61 mathematical truth, 90, 194–95 mathematics constructive, 86, 255 crystallization, 92–97 kidnapped by Plato, 95–97 mathematical logic, 35–36 mathematical truth, 90, 194–95 models, 83–84 motley, 84–85 philosophical vs. practical, 97–98 platonic vs. neurobiologists’ attitudes, 87–88, 104 probability, 99–100 pure, mixed and applied, 82–83, 98–103 realism vs. anti-realism, 87, 88–89, 104, 192–95 Science Studies perspective, 89, 103–4 theoretical physics, 83 See also under Kant, Immanuel Maxwell, James Clerk, 103 McCawley, James, 23 McKay, Thomas J., 45n2 meaning and use, 3 meaning, theories of. See anti-realism: semantic; Davidson, Donald: theory of meaning; pragmatism; realism: semantic; verificationism; Wittgenstein: meaning mental content. See externalism; mental representation; r-content mental representation, 10–11 metalanguage, 132–33, 194, 224–25

Mill, John Stuart, 90 Mitchell, O.H., 34–35 modality, iterated, 54–55 modal logic, x, 109, 110, 117–120. See also Denecessitation; Necessitation models, 223–24, 225–26 modernity, 155–56 Modus Morons, 208 Modus Ponens, 108–10, 117–19, 208 Moh, Shaw-Kei, 110 monster-barring, 210 Moore, Adrian, 88 Mostowski, Andrzej, 41 M-relative axioms and rules of inference domains, 224, 226–27 functions, 226 function-values, axioms of, 226 names, 225–26 predicates, 226, 235 substitutes for models, 224 M-relative rules of evaluation conclusions, 244 conditionals, 231, 232–33, 239 conjunctions, 229, 237–38, 239 disjunctions, 229, 234, 239 domain-dependence, 244 evaluation disproof (falsification), 227, 256 evaluation proof (verification), 227, 256 existentials, 227, 228, 231, 239 negations, 229, 230–31, 239 identity, 228, 233–34, 235, 236–37, 238 infinite sideways branching, 240, 244 limiting characteristics, 243–44 non-contradiction, 241–42 summary of rules, 238–39 undischarged assumptions, 243–44 universals, 227, 229, 230, 239 See also core logic multiple-conclusion logic, x–xi multisets, 123n12

N naturalism, 53 necessary existents, 198 necessary truth, 12–13, 54–56, 90, 131, 211 Necessitation, 110, 112–13, 117–19, 122n5, 216. See also rules: proof vs. inference necessity operator, 109, 112. See also modal logic; modality, iterated

Index Nehamas, Alexander, 161n1 neo-Fregeanism. See abstractionism Netz, Reviel, 91, 93–97, 100, 104n5 Newton, Isaac, 100–3 NF, 64, 200 nominalism, 55, 58 nominee (Fregean reference), 212–13. See also sense non-rigid designators. See rigid designators number, concept of, 65

O objects, trans-world, 215 Ockham, William of, 206, 210 Oliver, Alex, x, xi, 17n9, 167, 169, 179, 208 oracle, 150 ordinals, notion of, 199–202

P paradox of analysis, 1, 2–4, 11–12 Parkinson, G.H.R., 220n3 Peano, Giuseppe, 45–6n4 Peirce, Charles Sanders, 34–35, 46n4 penumbral consequence. See under supervaluationism penumbral truths. See under supervaluationism Pericles, 150 perspectives, external vs. internal, 193–94 Philo of Megara, 142n4 Pigozzi, Don, 120, 122n7 Plato, 51, 79, 88, 90, 100, 101, 102 Forms, 21 kidnapper of mathematics, 95–98 slaveboy demonstration (Meno), 82–83, 94–95 tripartite soul, 157–58, 163n9, 163n11 See also Socrates plurals, xi, 168, 170–71 Pogorzelski, Witold A., 122n7 Polus, 69, 73, 74, 75, 77, 78, 79 Potter, Michael, 67n17, 186, 189–90, 192, 194–95 pragmatism, 2–3, 5, 16–17n6, 193–94 precisifications. See under supervaluationism pretense, 146–54, 155–56, 158. See also irony, Kierkegaardian Priest, Graham, 107, 210, 220n5 principle of charity. See under Davidson, Donald probability, 99–100

269

proof computer-assisted, 86 discovery by Ancient Greeks, 92–95 rhetoric, 96 See also entailment; mathematics; rules: proof vs. inference proper names, 52, 179. See also rigid designators properties Cambridge properties, 210 class-forming, 188–89 time-relative, 209–10 See also classes; concepts, Fregean propositional attitudes, 212–13 propositional knowledge, 3–4, 6, 9, 14 propositions, 54–57, 64–65, 176–77 identity conditions, 59 logical properties, 59 platonic forms, 62–63 Protagoras, 73 Prucnal, Tadeusz, 122n7 psychoanalysis, 158–9 public norms, 5–6 purely referential occurrences, 209, 212. See also Substitutivity of Identicals Putnam, Hilary, 5, 11, 13 Pyke, Steve, ix

Q quantifier phrases, 179–80 quantifiers, binary, 32, 40 conservativity, 42 domain-independence, 44 restricted quantifiers, relation to, 41–44 unrestricted quantifiers, relation to, 44–45 quantifiers, generalized, 41 quantifiers, restricted binary quantifiers, relation to, 41–44 contraposition, 32–34 existential import, 24–25, 37 explication of Aristotle’s syllogistic, 24 Fregean conception, 32 model-theoretic benefits, 31 model theoretic equivalence of R and U, 29–30 model theory for R, 28–30 multiple quantifiers, 33–34 numerical quantifiers, 39–40 proof-theoretic benefits, 23–24, 31, 34 proof-theoretic equivalence of R and U, 26–28 proof theory for R, 23 soundness and completeness of R, 30

270 Index quantifiers, sortal, 22 quantifiers, unrestricted binary quantifiers, relation to, 44–45 contraposition, 34 mathematical logic, 35–36 model-theoretic equivalence of U and R, 29–30 model theory for U, 28–30 multiple quantifiers, 33–34 proof-theoretic benefits, 33–34 proof-theoretic equivalence of U and R, 26–28 proof theory for U, 23 standard logical theory, 22 Quine, W.V.O. analytic truth, 56 conceptual truth, 8–10 grammatical operations, 180 logical truth, 177 meaning 4–6, 17n7 quasi-quotation, 166 necessity, 210–11 NF, 64, 200 See also analyticity; holism

R R. See under quantifiers, restricted Ramsey, Frank, 2, 15, 16nn2–3 Ramsey sentences, 15 Rayleigh, Lord (John William Strutt) 103 r-content, 10–11 realism mathematical (see under mathematics) semantic, 2–3 See also pragmatism reason. See Plato: tripartite soul reducibility, axiom of, 39 reductio ad absurdum, 134 reductionism, 4 reference (Fregean). See nominee reference, Lockean model of, 194–95 reference-shift, 209, 212 refutation by logical analogy, 172 regressive method, 193 relative validity analysis of everyday arguments, 128–30, 132–33 relative to axioms and rules, ix, 127–33, 141–42, 142nn1–2 relative to intended models, 136–42 See also rules: suppressed rules vs. premises relevance logic, 130. See also entailment replacement, axiom of, 201

revisability, 9, 255–56 rhetoric, 96–7, 150 rigid designators, 210, 214 Robinson, Richard, 80n3 Rorty, Richard 154–55, 158, 161, 162n8 Rosch, Eleanor, 84 rules admissible vs. derivable, 113–15, 122n11 differentiated vs. undifferentiated, 114 horizontal vs. vertical, 120–21 locally vs. globally truth-preserving, 116–19, 120 locally vs. globally validity-preserving, 116–19 proof vs. inference, 110–21 sequential, 115 skeletons, 115, 120 soundness and completeness, 117–18 substitution invariance, 115, 123n16 suppressed rules vs. premises, 128–35, 142n2 See also core logic; Deduction Theorem; M-relative axioms and rules of inference; M-relative rules of evaluation Russell, Bertrand (Def) to (Ord), 199 descriptions, theory of, ix, x, 19, 51, 172 identity, 207 influences, 45–46n4 logical forms, 165 mathematics, 90, 94, 101, 102–3 quantification, 36–39, 42 regressive method, 193 self-reproductive process, 198 theory of types, 36, 38–9 Russell’s paradox, 189, 193, 197, 202–3, 251. See also classes; concepts, Fregean; sets

S Sadleir, Lady Mary, 103 Sainsbury, Mark, 180 Schaffer, Simon, 95–96 schema basic, 167 expressive limitations, 167–68 vs. form, 166–67 mention vs. use, 166 plural, 169 See also logical form; substitution instance

Index schematic letters, 166. See also under schema Schröder, Ernst, 46n4 Schurz, Gerhard, 111, 121, 122n6, 123n17, 124n21 Scott, Dana, 107, 120, 124n27 Scott’s rule, 120 second-order logic, 186 self-reproductive processes, 198 sense (Fregean), 212–13. See also nominee; propositions sentences, 54–56, 57–60 sentential connectives, 177–78 separation, axiom of, 197, 201 sets, x, 64, 168–69, 188, 190, 201–2. See also classes Shapin, Steven, 95–96 Shapiro, Stewart, 193, 198–99 Ship of Theseus. See sorites paradox Shoesmith, David, x, 121n1 SI. See Substitutivity of Identicals singularism, 170–71 Sinn. See sense SI validity. See strict implication Skolem, Thoralf, 36 Smiley, Timothy abstractionism (neo-Fregeanism), 67n17, 186 Aristotle’s logic, x, 165, 223 connectives, 107 descriptions, ix, x, 19 entailment, 66n9 formalization, 127, 144, 183 free logic, x lectures at Cambridge, ix, 19, 223 lists, xi logical consequence, ix, xi, 121n1, 223 logical form, 168, 173, 175 multiple-conclusion logic, x possible worlds semantics, x plurals, xi rejection, xi, 69, 107 relative necessity, 107 relative validity, ix, 127–33, 141–42, 142nn1–2, 142n4 rules of proof vs. rules of inference, 107–11, 114, 116, 117, 118, 120, 121, 121n1, 122nn4–5, 124n21 schematic calculi, 113, 122n9 sets, x, 169 sortal quantification, 45n1 substitution and replacement, 107

271

Smith, Robert, 104–5n11 Smith, Robin, 171 Smith’s prize, 103, 104–5n11 Smullyan, Arthur, 173–74 social sciences, 89, 152 Socrates aretē, use of, 80n2 knowledge, analysis of, 14 slaveboy demonstration (Meno), 82–83, 94–95 See also irony, Socratic; Socratic elenchus Socratic elenchus assertion, 71–72 examples, 79–80n1 going back on premises, 72–76 individuation of inquiries, 71 interlocutor-relativity, 77–79 people to the test, 72–76 premises, 69 proof, 76–78 propositions to the test, 70 reader’s perspective, 79 say what you believe, 72–74, 76 thesis, 69 Vlastos’s criticism, 69, 76–77, 79n8 See also irony, Socratic sorites paradox, 137–40, 142–43n7, 217–18 spirit. See Plato: tripartite soul Stavi, Jonathan, 41, 42 Stevenson, Leslie, 45n1 stipulation, 187 Stokes, George,103 Strawson, Peter, 5, 66n2, 172 strict implication, 53–57, 65, 130, 170, 176. See also entailment; logical consequence; validity structural validity. See under validity styles of scientific thinking, 92–93 substitution instance, 49, 166–67, 168, 175 Substitutivity of Identicals (SI) counterexamples, 208–18 degenerating research program, 218–19 denials, 219 fundamental, 207–8 history, 205–7 Sullivan, Peter, 189–90, 192, 194–95 superego, Freudian, 158–59

272

Index

supervaluationism classical logic, 133–34 vs. epistemicism, 140, 143n9 penumbral consequence, 136–39 penumbral truths, 136 precisifications, 133, 137 relative validity, 136–42 semantics, 121, 133–34 validity, 133–34 See also conditional proof, definitely operator suppressed premises. See rules: suppressed rules vs. premises Suszko, Roman, 115, 122n7, 123n16 syllogisms. See Aristotle

T Tait, P.G., 87, 103 Tarski, Alfred, 7, 58, 108, 256n1 tautology, 144, 148, 157 Taylor, Charles, 162n3 Tennant, Neil, 255–56, 256n1, 257n5 term-forming operators, 186–87, 188. See also abstractionism terms closing an open term, 224 open vs. closed, 224 saturated D-term, 225 saturating an open term, 225 Thales, 91 Thomason, Richmond, 121 Thomson J.J., 103 TI. See transitivity of identity Torricelli, Evangelista, 93, 97 totality, notions of, 198–9 transcendence, 152–54, 161. See also irony, Kierkegaardian transitivity of identity (TI), 215, 217 Tredennick, Hugh, 170 tree of knowledge, 98–100 truth theory of, 7–9 truth-conditions, 2–3, 216 truth-makers, 223, 256 T-sentences, 7 See also conceptual truth, empirical truth, logical truth, mathematical truth, necessary truth types, theory of, 36, 38–39

U U. See under quantifiers, unrestricted Uniform Substitution, 113, 117–19, 123n16

universally satisfiable, 176–77 Uzquiano, Gabriel, 203n1

V vagueness. See definitely operator; degree theory; epistemicism; sorites paradox; supervaluationism validity connectives, role in, 178 evaluative notion, 56–57, 62–64 formal vs. material, 176, 182–83 in virtue of form, 48–53, 57–58, 62–64, 174–75, 176 logical vs. structural, 180 recurrence, role in, 178 weak vs. strong dependence on form, 172–74 See also entailment; logical consequence; relative validity; strict implication; supervaluationism: validity van Inwagen, Peter, 217 Vardi, Moshe Y., 123nn17–18 variables, metalinguistic, 166 variables, plural, 169, 170–71 variables, syntactic, 166 verification and falsification, modelrelative rules. See M-relative rules of evaluation verificationism, 2–3 Vlastos, Gregory, 68–79, 80nn1–2, 81n8, 151, 161n1

W web of belief, 5–6 Weir, Alan, 203n2 Westerståhl, Dag, 41, 42, 44 Whitehead, Alfred North, 46n4, 102–3, 207 Wiggins, David, 40–41 Wiles, Andrew, 83 Williams, Bernard, 80n2 Williamson, Timothy, 1, 134, 140, 142n5, 217 Wittgenstein, Ludwig mathematics, 84–85 meaning, 3, 5, 15, 17n8 Wojtylak, Piotr, 122n7 Wright, Crispin, 190–95, 198–99, 208

Z ZF, 196, 199 ZFC, 199