Time Travel ; Probability and Impossibility 0198842503, 9780198842507

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Time Travel ; Probability and Impossibility
 0198842503, 9780198842507

Table of contents :
Dedication
Contents
Acknowledgements
List of Figures
The Grandfather Paradox
Introduction
1. MODES OF TIME TRAVEL
1. Modes of Time Travel
2. PARADOXES
2. The Self-Visitation Paradox
3. The Time Discrepancy Paradox
4. The Double Occupancy Problem
5. The Bootstrapping Paradox
6. Changing the Past
7. The Grandfather Paradox
3. PROPOSALS
8. Constrict Theories
9. Inconsistency Theories
10. Incapacity Theory
11. Impossability Theory
4. PROBABILITY AND PLANNING
12. Probability
13. Decision Theory
14. The Tourist Paradox
Bibliography
Index

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OUP CORRECTED AUTOPAGE PROOFS – FINAL, 28/1/2020, SPi

Time Travel

OUP CORRECTED AUTOPAGE PROOFS – FINAL, 28/1/2020, SPi

OUP CORRECTED AUTOPAGE PROOFS – FINAL, 28/1/2020, SPi

Time Travel Probability and Impossibility NIKK EFFINGHAM

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Nikk Effingham 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019957835 ISBN 978–0–19–884250–7 DOI: 10.1093/oso/9780198842507.001.0001 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Dedicated to the memory of Sarah ‘Sally’ Clements

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Contents Acknowledgements List of Figures The Grandfather Paradox

xi xiii xv

Introduction 0.1 What Is Time Travel? 0.2 Structure 0.2.1 Part 1: Modes of Time Travel 0.2.2 Part 2: Paradoxes 0.2.3 Part 3: Proposals 0.2.4 Part 4: Probability and Planning 0.2.5 Symbols

1 1 3 3 3 5 6 7

1. MODES OF TIME TRAVEL 1. Modes of Time Travel 1.1 Discontinuous Time Travel 1.1.1 Prayers, Prophecies, and Providence 1.2 Retrograde Time Travel 1.2.1 Tachyons 1.2.2 Retrocausal Quantum Physics 1.3 Warped Time Travel 1.3.1 Circular Time 1.3.2 Warped Spacetime 1.4 Frozen Time and Chimerical Time Travel

11 11 12 13 14 16 17 17 19 22

2. PARADOXES 2. The Self-Visitation Paradox 2.1 Self-Visitation 2.2 Chorology 2.2.1 Exact Location and Multi-Location 2.2.2 Bloating and Multi-Location 2.2.3 Gainsayers 2.3 Fundamental Conflicts 2.4 The Problem of Multi-Located Intrinsics 2.5 Sider’s Possibility Argument

27 27 29 29 31 34 35 36 37

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viii



3. The Time Discrepancy Paradox 3.1 Temporal Discrepancies 3.2 Ideology and Ontology 3.2.1 Extended Personal Time 3.2.2 A Sketch of External and Personal Temporal Ideology 3.2.3 The Ontology of Personal Time 3.2.4 Reducing Personal/External Time 3.2.5 Reduction and Relativistic Spacetimes

42 42 43 43 43 45 46 48

4. The Double Occupancy Problem 4.1 Philosophical Indulgences 4.2 The Motion Solution 4.3 The Cheshire Cat Problem

51 51 52 54

5. The Bootstrapping Paradox 5.1 Types of Bootstrapping 5.2 The Impossibility of Bootstrapping 5.2.1 Against I 5.2.2 Against N B

59 59 60 61 63

6. Changing the Past 6.1 The Paradox of the Changing Past 6.2 Ludovicianism 6.2.1 The Second Time Around Fallacy 6.2.2 The Open Future 6.2.3 Bilking 6.3 Non-Ludovician Change 6.3.1 Indexed Worlds 6.3.2 Universe-Indexed Worlds 6.3.3 Hypertemporal Worlds 6.3.4 Do Indexed Worlds Allow for Change? 6.3.5 The Doppelgänger Objection 6.3.6 Making Sense of Hypertime 6.3.7 The Metaphysical Possibility of Changing the Past

66 66 67 67 69 71 73 73 74 76 79 82 84 90

7. The Grandfather Paradox 7.1 Variations 7.1.1 Less Interesting Variations 7.1.2 The Explanatory Paradox 7.1.3 The Post-Mortem Indicative Paradox 7.1.4 The Non-Morietur Subjunctive Paradox 7.2 Proposed Solutions 7.2.1 Dead Options 7.2.2 Live Options

91 91 92 93 94 97 98 99 100

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ix

3. PROPOSALS 8. Constrict Theories 8.1 The Conjecture and the Principle 8.2 Laws and Metaphysical Possibility 8.3 Rewriting the Paradox

103 103 104 107

9. Inconsistency Theories 9.1 Dialetheism and the Grandfather Paradox 9.2 The Indiscriminate Inconsistency Problem 9.3 Discriminate Inconsistency Theory

109 109 110 113

10. Incapacity Theory 10.1 David Lewis on the Grandfather Paradox 10.2 The Argument from Impossibility 10.3 The Argument from Analysis 10.3.1 Lewis’s Analysis of ‘Could’ 10.3.2 Countermodality and Counterfactuals 10.3.3 The Standard Analysis of ‘Could’ 10.3.4 Against the Argument from Analysis

116 116 117 119 119 121 124 126

11. Impossability Theory 11.1 Impossible Abilities 11.1.1 Ability Principles and Multiple ‘Can’s 11.1.2 Impossability Theory and the Grandfather Paradox 11.2 The Argument from Countermodal Physical Possibility 11.2.1 From Countermodality to C P P 11.2.2 From C P P to P- P Being False 11.3 Spencer’s Argument 11.4 Beyond the Grandfather Paradox 11.4.1 Analogous Paradoxes 11.4.2 New Lines of Investigation 11.4.3 The Escher Machine Paradox

128 128 128 130 131 132 136 136 137 138 139 142

4. PROBABILITY AND PLANNING 12. Probability 12.1 Time Travel Scenarios 12.1.1 Ludovician Probability vs. Non-Ludovician Probability 12.1.2 Types of Causal Loop 12.2 No Causal Loop Scenarios 12.3 Negative Causal Loops 12.3.1 Credence Shift 12.3.2 A Note on the Principal Principle 12.3.3 A Case Study

147 148 148 148 151 152 152 155 155

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12.4 Positive Causal Loops 12.4.1 Heinlein Values 12.4.2 Eternal Universes and Infinitely Distant Particles 12.4.3 Mixed Causal Loops 12.4.4 Ever More ‘Realistic’ Cases 12.4.5 Killing Pappy 12.5 Non-Ludovician Probability 12.5.1 Bootstrapping at Universe-Indexed Worlds 12.5.2 Probabilities of Facsimile Cases 12.5.3 E 12.5.4 Back to Negative

158 159 160 162 165 168 170 170 171 172 174

13. Decision Theory 13.1 Plague 13.2 Counterfactual Motivations 13.2.1 Backtracking vs. Causal Contexts 13.2.2 Causal Counterfactuals in Time Travel Scenarios 13.2.3 The Rationality of Autoinfanticide 13.3 Causal Motivations 13.3.1 Causation in Plague 13.3.2 Two Types of Causation 13.3.3 Back to Plague 13.4 Chance Motivations 13.5 Other Newcomb Cases 13.5.1 Events in the Past 13.5.2 Other Cases 13.6 Summary 13.7 Appendix: Perfect Predictors 13.7.1 Two-Boxers and Perfect Predictors 13.7.2 Ahmed’s Argument

176 178 179 179 180 183 185 185 186 187 188 190 190 193 195 196 196 197

14. The Tourist Paradox 14.1 The Tourist Paradox 14.2 Objections 14.2.1 Bad Objections 14.2.2 Restricted Time Travel 14.3 Probability, Decisions, and Tourism 14.3.1 The Ultimate Banana Peel 14.3.2 Rationality and the Ultimate Banana Peel 14.4 The Accidental Time Machine 14.4.1 The Dangers of High-Energy Experiments 14.4.2 Policy and Response 14.4.3 The Cosmic Ray Test 14.4.4 The Nielsen-Ninomiya Card Test 14.5 The Laws of Nature

199 199 201 201 204 205 205 207 209 209 211 213 214 217

Bibliography Index

219 241

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Acknowledgements Thanks to the members of the time travel reading group at the University of Leeds, in particular Alex Buckley, George Darby, Jonathan Robson, and Duncan Watson. The issues concerning probability directly stem from that reading group. It took me a decade to figure out what I had to say about that problem; I hope they’re still interested in what I think the answer is. The University of Birmingham’s support was invaluable in giving me research leave early on in the project. I also received an Undergraduate Research Scholarship, which went to Harriet Walters. I thank all of the philosophers at UoB, with special thanks to Kit Fine, Ema Sullivan-Bissett, Scott Sturgeon, and Heather Widdows, and (most importantly) to Nick Jones, Alex Miller Tate, and Alastair Wilson (who have each read drafts and given extensive comments). Jen Elliot helped with diagrams, so thanks to her. And Casey Elliot and Iain Law deserve a special thanks for helping me innovate and achieve almost every day. Academics outside of Birmingham have also helped me with this book. Cian Dorr was very kind and helped a lot with the chapter on decision theory, as did Daniel Nolan and David Braddon-Mitchell. Lee Walters has often schooled me on the nuances of modal logic and counterfactuals. When it came to non-classical logic, invaluable help was provided by Thomas Brouwer, Daniel Nolan, Josh Parsons, and Gareth Young. Malcolm James Price helped with various portions of the book, particularly those concerning probability. The Centre of Time, at the University of Sydney, awarded Alastair Wilson and me a grant to run the ‘Probability and Time Travel’ workshops at the University of Birmingham and the University of Sydney. I’d like to thank the audiences of those workshops; special thanks to Sam Baron, Sara Bernstein, John Bigelow, Ben Blumson, David Braddon-Mitchell, Anthony Eagle, Kristie Miller, Daniel Nolan, and Stephanie Rennick. Portions of this book were presented at: the Slovak Metaphysical Society’s Modal Metaphysics: Issues on the (Im)Possible III conference; two joint sessions (2013 in Exeter and 2015 in Warwick); and research seminars at the Queen’s University Belfast, University of Manchester, University of Sydney, University of Illinois Urbana-Champaign, and University College Dublin. I collectively thank the audience members who are too innumerable to mention. This book was used in a reading group when I was Anderson Visiting Fellow at the University of Sydney in 2016 at which Jane Loo and Naoyuki Kajimoto gave me many excellent comments. Alasdair Richmond also provided many helpful comments. Last but not least, thanks to the kind anonymous referees for OUP whose detailed comments helped considerably improve this manuscript.

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xii



It is never just academics who make things possible. Many friends and acquaintances have contributed via Facebook—the list is too long for me to name individuals, so a collective acknowledgement must suffice (although specific thanks to Robert Barnard, Kate Heartfield, Keith Jones, Jonathan Laidlow, and Ian Sanderson). For pastoral support I’d like to thank Rhiannon Blake, Gavin Byrne, Robert Cryer, Alex Fannon, Sarah Gilbert, Adrian Hunt, Vidya Kumar, Rhia Malik, Luke McGarrity, Kieren McGuffin, Shannon Oates, Alex Orakhelashvili, Ben Smart, Stephen Smith, Anne-Marie Vassiliadis, and Bob (the dog). I’d like to thank Laura Garside and Harry Parker for providing me board and lodgings whilst I was on research leave, and both James and Mark McDonagh for their companionship whilst there. The staff at my local Starbucks have been fabulous; thanks to Hamish, Bethan, Faye, Ella, Emily, Asia, Manuel, Joe, Karl, Russell, and Leah. Finally, last but not least, I’d like to thank Nostalgia Comics and its staff.

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List of Figures 1.1. Möbius battle. Reprinted from xkcd: A webcomic of romance, sarcasm, math, and language under the terms of a Creative Commons Attribution-NonCommercial 2.5 Generic license (CC-BY-NC-2.5) 18 3.1. Minkowski diagram 49 4.1. The failure of the motion solution with extended objects 53 4.2. Fixing the motion solution by progressive parts time travelling 54 4.3. Temporal parts, given the standard understanding 56 4.4. Temporal parts allowing a tangential orientation 57 4.5. Tangential temporal parts 57 6.1. UniverseD indexing 74 6.2. UniverseF indexing 75 6.3. Exterminous hypertemporal indexing 78 6.4. Conterminous hypertemporal indexing 78 6.5. Recombining exterminous hypertime 89 6.6. Recombining conterminous hypertime 90 12.1. Negative 156 12.2. Mixed 163

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The Grandfather Paradox

The bulk of the book discusses a particular version of the Grandfather Paradox (see §7.1.3). For ease of reference I summarize the premises and argument here. T T’ P: Time travel is metaphysically possible. S P: If time travel is metaphysically possible, then it’s metaphysically possible for me to be in the correct circumstances (i.e. circumstances where nothing except my not pulling the trigger would save my grandfather from dying before he met my grandmother). A A: If I were in the correct circumstances, then I would have the ability to pull the trigger of the rifle (in those circumstances). K4: The accessibility relation between worlds is transitive. P-A P: If (in circumstance C) an agent τ can ϕ, then there’s a metaphysically possible world at which τ ϕs in circumstance C. C T: At a world where I shoot Pappy in the correct circumstances, a contradiction is true at that world. N S W: It’s not the case that there’s a world at which a contradiction is true. The Paradoxer’s argument against time travel is: 1. T T’ P is true. [Assumption for reductio] 2. At w₁: I am in the correct circumstances. [1 and S P] 3. At w₁: I can pull the trigger of the rifle (in the correct circumstances). [2 and A A] 4. At w₂: I pull the trigger of the rifle in the correct circumstances. [3, P-A P, and 4] 5. At w₂: A contradiction is true. [4 and C T] 6. There is a world at which a contradiction is true ∧ ¬ there is a world at which a contradiction is true. [5 and N S W] 7. T T’ P is false. [1, 6, and reductio ad absurdum]

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Introduction 0.1 What Is Time Travel? This book is about time travel. What I mean by ‘time travel’ is not the innocuous process of things persisting into the future at the sedate pace of one second per second—to cry ‘I have travelled here from the 1970s!’ when I enter a room is strictly speaking true, but misleading. Even when one’s persistence into the future is expedited (either because they’re playing by the laws of physics, and utilizing Lorentz Contractions, as in Niven’s A World Out of Time [1976] or Pohl’s The World at the End of Time [1990], or simply ‘teleporting’ to the future, as in Irving’s ‘Rip van Winkle’ [1819]), the rest of this book has little to say about that. Rather, I am interested in the more problematic idea of travelling backwards in time, or otherwise interacting with that which has already been. It’s not that travelling forwards in time isn’t a type of travelling in time—of course it is!—but merely that in this book ‘time travel’ is restricted to refer only to the future affecting the past. Indeed, some other ways of the future affecting the past are attenuated, non-literal, or metaphorical and I don’t mean for them to count as ‘time travel’ either. For instance, in 2000 Lance Armstrong won the 2000  Tour de France. When his title was revoked in 2012 we might think that it’s no longer the case that he won in 2000  and that the past was changed [Barlassina and del Prete 2015]. Or if a married man wakes up one morning having slept with someone else, it might yet be within his power to determine whether the event of the previous night had a certain property or not (i.e. the property of being the end of his marriage) [Peijnenburg 2006]. Or if I die and cease to be, we might think that harms which befall me after my death nevertheless manage to affect me right now [Pitcher 1984: 185–6]. Or perhaps remembering the past, and visualizing being there, could be a form of ‘travelling in time’ [Suddendorf and Corballis 2007]. The philosophical import of such cases does not appear to have much in common with the tribulations facing the existence of time machines—for instance, they don’t have problems analogous to the Grandfather Paradox. That said, I’ll stipulate that ‘time travel’ doesn’t include these cases either, at least as I use the term in this book. Further clarifications can be made. I’ll use ‘time travel’ in either a narrow or a broad sense. Used narrowly, I intend cases in which material objects go back in time. Fictional examples are replete: the Doctor from Doctor Who travelling through time in his TARDIS; Dr McCoy stepping back in time in Star Trek’s ‘The City on the Edge of Forever’ [1966]; Marty McFly using the DeLorean in Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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2  Back to the Future [1985] to travel back to the 1950s. In each, a concrete object travels back in time. In 12,000  the Doctor steps into the TARDIS; in 1963 , he then steps out. In 2366 , Dr McCoy steps through a portal into the past; in 1930  he steps out. In 1985  the DeLorean accelerates to 88 mph; in 1955  it crashes into a barn. Material objects have gone into the past, so this counts as time travel narrowly construed. Taken more broadly, ‘time travel’ captures the narrow cases plus cases in which no concrete object travels back in time and yet similar philosophical issues nonetheless arise. Cases include those of ‘mere backwards causation’, where information or causal processes propagate back in time with no accompanying object. For instance, I might deliver an imprecatory prayer asking God to have had an enemy struck dead by now. If God answers, and strikes dead a man last week in virtue of my entreating Him today, then that’d be a case of backwards causation involving no concrete object. It’d be a case of time travel broadly, not narrowly, construed. Similarly, in many stories (e.g. Guthrie’s A Romance of Two Centuries: A Tale of the Year 2025 [1919] or Gibson’s The Peripheral [2014]) there’s a device allowing information, but no concrete object, to travel back in time. Again, that’d be time travel in the broad sense. Broad cases are just as problematic as the narrow cases. If God answered imprecatory prayers, can I now beseech Him to strike my grandfather dead back in 1930 ? If I can phone the past, can I now coax an assassin into doing away with him? These questions are functionally identical to the questions surrounding the Grandfather Paradox (q.v.) and the philosophical issues raised are basically the same. Thus, what I talk about in the narrow cases applies more generally to these broader cases. However, solely for the purpose of example, I generally talk just about the narrow cases. What I won’t do is give an explicit definition of ‘time travel’. Some have attempted such definitions: Lewis [1976: 145–6] defines it in terms of a disconnect between two types of time (‘personal’ and ‘external’ time, which I discuss fully in Chapter 3), whilst others define it in terms of a disconnect between personal time and ‘cosmic time’ (see }3.2.5 for ‘cosmic time’) [Fano and Macchia Forthcoming], or in terms of closed timelike curves [Arntzenius 2006: 602; Charlton and Clarke 1990; Smeenk and Wüthrich 2011: 5, 26]. Smith [2013] discusses the shortcomings of definitions along such lines and I offer no explicit definition in their place. At best, we can only ostensively define ‘time travel’, as I did above. That such an ostensive definition is the only option isn’t a bad thing since no term subject to philosophical investigation has ever received a definition which was at once both interesting and successful. Philosophical industries, both cottage and large-scale, analyse terms like ‘knowledge’, ‘causation’, ‘artwork’, ‘moral goodness’, etc. None have succeeded. Some uninteresting definitions, successful only because they’re defined by cognate notions (e.g. defining mereological overlap using mereological parthood, weak location using exact location, or subset using set membership), are true. Likewise, technical or stipulated terms (e.g. ‘temporal part’, ‘perdurantism’,

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

3

etc.) can be defined. Every other term must live without. This doesn’t threaten the study of epistemology, causation, aesthetics, ethics (etc.). Nor, then, does the lack of an agreed-upon definition of ‘time travel’, ‘time machine’, ‘backwards causation’ (etc.) threaten my project.

0.2 Structure This monograph is both a comprehensive examination of the existing literature as well as a defence and exploration of a solution to the Grandfather Paradox which allow for time travel’s possibility. My conclusion at the end of it is that, whilst time travel is metaphysically possible, your credence of it ever coming about should be effectively zero—that is, for all practical purposes, time travel is impossible. For a book on time travel this might seem a dreary conclusion, but it instead has pressing ramifications. I will argue that, given such an expectation to not be in a time travel scenario, one should expect bad things to befall those who conspire to get into one. The upshot is that we should pay extra special care to ensuring that nothing, no matter how small or microscopic, ever travels in time. This book proceeds in four parts. The first part deals with the preliminary issue of how one might travel in time. The second part discusses six paradoxes of time travel. In particular, it introduces the focus of the third part of the book: the Grandfather Paradox. In that third part I examine various solutions to the Grandfather Paradox, arguing that the best answer is that we can sometimes have the ability to do the impossible. The fourth part explores how, given that conclusion, probability and decision theory function in time travel cases, ending by detailing the dangers of investigating time travel.

0.2.1 Part 1: Modes of Time Travel I start with a preliminary discussion of how one might travel in time. Obviously there are no known actual examples, so Chapter 1 draws those examples from fiction, speculative physics, and historical thought. This chapter is mainly expository, serving as exegesis for later reference.

0.2.2 Part 2: Paradoxes Part 2 of the book focuses on the paradoxes of time travel.¹ Three of the paradoxes are fairly straightforward, being relatively easy—and yet also instructive—to solve. ¹ One extra paradox worth noting, which I don’t deal with in Part 2 of the book, is ‘the Destination Paradox’: Were presentism true, we can’t travel to the past because it doesn’t exist [Abbruzzese 2001:

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4  The first is the Self-Visitation Paradox: If a 6’ tall time traveller returns to a time when they were 4’ tall, then a contradiction (i.e. that they’re both 4’ tall and 6’ tall) is true. The second is the Time Discrepancy Paradox: given time travel’s possibility, it is apparently both true and false that in five minutes’ time a time traveller will be in the Jurassic Period. The third is the Bootstrapping Paradox: if time travel were possible, objects could exist for which there is no explanation. I discuss these in Chapters 2, 3, and 5, respectively. None of these paradoxes are a serious threat to the possibility of time travel. All three, however, offer a chance to introduce concepts which see service in the rest of the book. In the case of the Self-Visitation Paradox I argue that the apparent contradiction of a time traveller being two different ways at the same time is not a contradiction at all—in cases of multi-location (such as time travel), an object can be both 4’ tall and 6’ tall simultaneously. In the case of the Time Discrepancy Paradox, I develop the distinction between personal and external time, arguing that we can reduce temporal relations to causal ones. In the case of the Bootstrapping Paradox, it is revealing to examine the ungrounded fears that one might have about the impossibility of such bootstrapped things. (And, as an example of how these concepts can be useful, in Chapter 4 I rely upon them to solve a problem for a particular type of time travel involving what, in }1.2, I’ll call ‘retrograde time travel’.) Chapter 6 discusses the Paradox of the Changing Past: changing the past is impossible; time travel necessitates changing how the past was; therefore, time travel is impossible. There are two ways out of that Paradox. The first is Ludovicianism (‘Ludovicus’ being the Latin version of ‘Lewis’ and David Lewis being this theory’s most famous, although not first, proponent). Ludovicians deny the second premise: Time travel is possible but time travellers won’t change the past. Instead, every attempt to change the past will be thwarted by some event: a bird swoops down at the wrong moment, causing me to shoot feathers rather than Hitler; a traffic jam stops me from getting to the 1964 grassy knoll in order to

36; Al-Khalili 1999: 184; Anon 2018; Dwyer 1978; Godfrey-Smith 1980; Goff 2010: 68; Lockwood 2005: 128; Pickup 2015: 387–8n1; Richmond 2003: 304–5; Wasserman 2018: 38–49; see also Curtis and Robson 2016: 191–4 and Harrington 2015: 243–4]. I won’t pick that argument up in this book because Keller and Nelson [2001] have already explained in detail where it goes wrong: the presentist believes the future doesn’t exist, yet accepts that we move forward in time at the rate of one second per second; similarly, then, even though the past doesn’t exist, there should be no impediment to travelling to the past. Having nothing to add to their erudite discussion, I am happy to dispense of it in this footnote (although I don’t deny that more could likely be said—see, e.g., Daniels [2012], Hall [2014], Miller [2005], Monton [2003], Pruss [2013: 72], Sengers [2017: 10–12], and Sider [2005]). This isn’t the only point of contact between temporal metaphysics and time travel. Combining time travel with temporal passage has proved problematic (see Savitt [2005] and Dowe [2009] for attempts to avoid the problem). Further, some people have argued that Growing Block Theory (whereby the past and present exist, but not the future) is intimately connected with—perhaps even required for—time travel [Goff 2010; van Inwagen 2010], an idea which is mirrored in fiction (see Hyams’s Timecop [1994], Baxter and Clarke’s The Light of Other Days [2000], and Acres’s Timemaster [1984]). I also neglect discussion of these issues.

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shout ‘Duck!’; I have a heart attack seconds before figuring out the crucial details of the flux capacitor I’d otherwise have used to travel back in time and kill my grandfather; and so on. The second solution is that time travel involves travelling to different universes (or different hypertimes), which allows for the past to be changed. Both replies are tenable. However, I see no reason to think time travel requires extra universes (or what have you) and so spend most of the rest of the book considering worlds at which time travel is Ludovician, at which there aren’t multiple universes, and at which the past can’t be changed. Chapter 7 discusses the paradox which occupies the third part of the book, namely the Grandfather Paradox. If I travelled back in time to 1930 and killed my grandfather—call him ‘Pappy’—before he meets my grandmother, then a contradiction would be true, i.e. that Pappy is alive in 1930 and that Pappy is not alive in 1930.

0.2.3 Part 3: Proposals The Grandfather Paradox is the perennial focus of most discussions of time travel. I do not break with that tradition. Chapters 8–11 discuss various responses to the Paradox. One might think that the Grandfather Paradox is enough to show that time travel is impossible. Call that the ‘Paradoxer’s Solution’. I do not dedicate a chapter to that solution. This is because there is little to say, not that it is intellectually indefensible or philosophically weak. Whilst the Paradoxer doesn’t have a magnum opus defending their position, we should explain the absence of a prolonged defence of the Paradoxer’s position sociologically, rather than philosophically, for time travel’s study proves attractive mainly to those who already believe that it’s possible. This is partially because it attracts sci-fi aficionados with a semi-romantic attachment to its possibility and partially because it’s hard psychologically (and, in today’s results-orientated academic environment, prudentially) to devote large tracts of time to defending the impossibility of something so speculative. It would therefore be remiss to not treat the Paradoxer’s Solution with an appropriately reverential attitude—the Paradoxer’s Solution is what we should opt for were no other response to the Paradox effective. Since it is the default option, the Paradoxer’s Solution needs no chapter of its own. It is incumbent upon the opposition—discussed in Chapters 8–11—to provide a compelling alternative. Chapter 8 discusses whether scientific theories can cast light on the Paradox. I argue that (in general) they cannot, since they truck in what is physically possible whilst it is metaphysical possibility which is relevant to the Grandfather Paradox. Instead, when physicists suggest that, say, the laws of physics ruling out time travel solves the Grandfather Paradox, they’re merely concealing a commitment to the more straightforward Paradoxer’s Solution.

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6  Chapter 9 introduces ‘inconsistency theory’. Inconsistency theorists are unabashed at there being cases where one’s grandfather is both dead and alive. For instance, dialetheists could believe that me murdering Pappy in 1930 just results in a true contradiction which we should not baulk at. Charitably accepting dialetheism for the purpose of argument, in Chapter 9 I argue that, nevertheless, inconsistency theorists can only avoid the Grandfather Paradox by accepting bizarre commitments better left repudiated. I end the third part of the book by examining two theories closely allied to Ludovicianism. The Ludovicians break into two camps. Most of them are ‘incapacity theorists’: in at least some sense, time travellers lose the ability to kill Pappy. In Chapter 10 I argue that contemporary advances in understanding the philosophy of impossibility show that incapacity theory is unmotivated. The other Ludovician camp are the ‘impossability’ theorists: time travellers have the ability to kill Pappy—that is, the ability to do the impossible—even though at no metaphysically possible world will they use that ability. In Chapter 11 I defend this theory, adopting it as the correct solution to the Grandfather Paradox for the remainder of the book.

0.2.4 Part 4: Probability and Planning The final part explores what the world would be like were impossability theory true. Chapter 12 tackles the topic of probability in time travel scenarios, examining the probability of different causal loops. For instance, given Ludovicianism, I will fail to kill my grandfather were I to return to 1930 to murder him. One thing which might stop me is that I have a stroke. Does that mean that the probability of having a stroke increases in time travel cases? Is time travel hazardous to my health? If so, how hazardous is it? Or another question: if time machines were commonplace, should we expect lots of people to be their own mothers and fathers? Or very few? Another: should we expect time travellers from the future to come back and tell us how to make time machines? Or not? Impossability theorists say that physical possibility outstrips metaphysical possibility and that some physically possible situations are metaphysically impossible. On the back of that, it’s natural to further assume that (at least some) metaphysically impossible situations (e.g. me shooting my grandfather dead) have a positive objective chance. That’s not to say that we should expect them to occur for we should never expect the metaphysically impossible to occur. So our credence of a proposition being true in a time travel case is the chance of that proposition being true conditional on no metaphysically impossible situations coming to be. Therefore, metaphysically impossible situations can have a positive objective chance of coming about even though we shouldn’t award a positive credence mirroring that. All of this said, I will argue that our credence of any

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time travel taking place should be effectively zero and that we should not expect time travel to take place. Time travel further poses unique difficulties for decision theory, discussed in Chapter 13. Imagine I arrive in a time machine. In a delirium, my future self tells you that the world is doomed because nuclear war is imminently going to break out. He (me?) then expires. Things look bleak since, given Ludovicianism, time cannot be changed and it seems inevitable that the prophesized nuclear war will take place. But then someone suggests the following plan: torture me until I can’t think straight and merely believe a nuclear war has taken place; poison me with polonium; send me back in time to ‘warn’ everyone about the (imaginary) nuclear war. War would then be avoided (or, at least, would only be as likely to happen as it would be if I hadn’t come back from the future in the first place) and we’d be in a logically consistent, Ludovician-friendly, scenario. So the question is whether it is rational to torture me. I compare such situations to Newcomb cases. In Newcomb cases there are two solutions: one-boxing and two-boxing. One-boxers would torture me and two-boxers would not. I argue that even if you are a two-boxer in normal, non-time travel Newcomb cases, you should nevertheless one-box in time travel cases. Since those who one-box in normal cases should likewise onebox in time travel cases, it follows that, no matter what, we should one-box in time travel cases, i.e. I should be tortured. Chapter 14 ends the book by bringing together the conclusions about probability and decision theory and considering the Tourist Paradox: if time travel is possible, where are the time travellers from the future? Given the conclusion of Chapter 12, human civilization is vastly more likely to be destroyed (by, e.g., an asteroid strike) than it is to successfully travel in time. We should not, then, expect visitors from the future. Further, we should alter our current behaviour for it is rational to avoid conducting experiments which might bring about time travel. This is because, were time travel physically possible, then were an experiment to be on course to bring about time travel—even if that time travel took place on the smallest and most microscopic of quantum scales—we should expect some event (e.g. an asteroid smacking into the Earth) to stop it. Given Chapter 13’s conclusion that we should one-box in time travel cases, it’s rational to avoid conducting such experiments in the hope of avoiding an apocalypse. Armed with the knowledge that conducting such experiments constitutes an existential threat, we should take care to ensure that no experiment is conducted which could bring about an event involving time travel. I end by briefly detailing the practical ramifications of this.

0.2.5 Symbols I use standard symbols for logical notation: ¬, ∧, ∨, and  for negation, conjunction, disjunction, and the material conditional; ∃ and 8 for the existential and

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8  universal quantifier; □! for the ‘would’ counterfactual conditional; r, x, y . . . are variables; Greek symbols like φ and ψ stand for any proposition you care for (I reserve τ as a variable for agents and the lower-case ϕ as a variable for actions). Mainly for reasons of presentation, I sometimes talk explicitly about propositions—usually, this is not for any technical reason regarding logic or metaphysics, but solely to make the text easier to read and digest. I represent propositions using the standard notation of enclosing the proposition in angled brackets, i.e. ‘hφi’.

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PART 1

MODES OF TIME TRAVEL

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1 Modes of Time Travel We start with an examination of ways in which one might travel in time. This primer on time travel is a useful foundation and serves as important background information. Obviously, there are no known extant methods of time travel, so this chapter instead focuses on methods mooted in philosophy, theology, physics, and fiction. I taxonomize the methods of time travel into three categories: discontinuous (discussed in §1.1); retrograde (§1.2); warped (§1.3) [cf. Monton 2009]. There are two other phenomena, frozen time and chimerical time travel, which aren’t really types of time travel, but are very similar—I discuss them in §1.4.

1.1 Discontinuous Time Travel In many fictional accounts, people time travel discontinuously by teleporting from a later time to an earlier one without travelling through the intervening interval. Examples include Rowling’s time turner from the Harry Potter series, Star Trek’s Q’s ability to move one through time with the click of his fingers, and sundry time machines from films such as The Terminator [1985] and Looper [2012]. Discontinuous time travel also includes: stories of someone simply waking up sometime else (e.g. Twain’s A Connecticut Yankee in King Arthur’s Court [1889], Ramis’s Groundhog Day [1993], and Liman’s Edge of Tomorrow [2014]); most narratives describing one’s mind time travelling (e.g. X-Men: Days of Future Past [2014/Claremont et al. 1981], Morneau’s Retroactive [1997], or H.P. Lovecraft’s ‘A Shadow Out of Time’ [1936]); and cases of mere backwards causation involving action at a temporal distance with no concrete thing travelling in time. There might be tricky questions concerning such discontinuous travel. For instance, what happens if one’s destination is already occupied by matter? Given that the traveller arrives instantly, there’d be no time for the matter to be ‘pushed aside’ by the instantaneously arriving time traveller. (Would the target interpenetrate the material at the destination? Is that even possible? [Pitkin 1914: 524–5]). Or consider that discontinuous time travellers must move in both time and space [Bernstein 2015: 162–3; Read 2012 144–5]—for instance, in the comic Strontium Dog [1978–] the titular protagonist has ‘time grenades’ which (discontinuously) send their victims a few seconds through time but leave them in the same place, so they die in the vacuum of interplanetary space with the planet they were on having Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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moved by. Does that even make sense in a relativistic universe, like ours, with no preferred frame of reference? But neither problem has anything to do with time travel per se. These are instead worries ancillary to the possibility of teleportation. If I teleported through space alone, I’d face the same questions. What if I teleported into a wall? And wouldn’t we have the same issue about whether I have to take into account what inertial frame of reference the Earth is in? So it isn’t the ‘time travel’ element of the mode of travel which is causing the problems.

1.1.1 Prayers, Prophecies, and Providence Bigelow [2001: 58] says that time travel stories don’t appear until the 1800s. This isn’t quite true. Not only is there Samuel Madden’s 1733 Memoirs of the Twentieth Century, in which an angel returns from the future, but in the Talmud (Menahot 29b) Moses travels forwards in time before returning to the present (and appears to do so discontinuously). Moreover, given a broader definition of time travel, stories involving divine (or otherwise supernatural) intercession through prophecy, providence, and prayer will qualify. And there are many such pre-1800 stories: the Oracle of Apollo’s prophecy of Laius’s demise at the hands of his son; the Sanskrit Mahabharata that tells of Kansa receiving a prophecy of his death at the hands of a child of Devaki; the witches’ (easily misinterpreted) prophecy of Macbeth’s becoming King; Jonah’s prophecy that the city of Nineveh would receive God’s wrath; and so on. In each, a prophet speaks of a future event. Since there’s no suggestion of ‘augury particles’ mediating the information by zooming back from the future, these are cases of discontinuous time travel, broadly understood. The examples also vary as to what theory of time travel appears to be true [cf. Williams 2018: 218]. Both Oedipus’s Laius and Mahabharata’s Kansa find their prophesized fate unavoidable. Not only do their attempts to escape it lead to its fulfilment, but the prophecy plays a causal role in its own coming about. Such stories dovetail well with the Ludovician model of time travel. However, other prophecies allow for the future to change. For instance, God spares Nineveh in light of Jonah’s actions; in that case, some non-Ludovician model of time travel would be called for. Prayer, as well as prophecy, connects with the philosophy of time travel [cf. Nahin 1999: 274–5]. Whilst few believe that God can change the past (see Hudson [2014] and Lebens and Goldschmidt [2017] for exceptions), it is more popular to believe that backwards causation can play a role in cases where an already realized outcome is unknown to an agent who then prays for the past to have taken a certain course. Imagine I witness a car crash. It is already settled whether my father was involved in that crash, although I don’t know either way. Ignorant of

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the facts, I pray to God that he wasn’t involved. Whilst some think it’s pointless to conduct such prayers (e.g. Geach [1969] and the Talmud Berakoth 60A), others believe that God knows that I will pray in the future and therefore answers my prayer before I give it, thus ensuring that my father is safe [Brown 1985; Timpe 2011]. Given such an understanding, future events (e.g. my praying) would causally affect the past via God’s intercession (and, since nothing mediates God’s desires and temporal events, this would again be a case of discontinuous time travel, broadly construed). For both prayer and prophecy we could give an alternative reading sans time travel. Prophetic agents claiming φ will transpire might be merely ‘Gaborian’, i.e. using their powers to make sure φ comes about rather than things in the future affecting their knowledge of it. Similarly for prayer. For instance, Molinists believe God knows what we will do in the future not because of backwards causation but because God knows what choices we’ll make in every circumstance we might find ourselves in and then he plans accordingly, e.g. knowing already that I’ll later be in a situation in which I’ll pray, God acts now to save my father. For our purposes, I needn’t settle whether or not this is the correct theory of prayer. That there is at least one (reasonable) understanding of prayer/prophecy which involves backwards causation is, by itself, enough to show two things: first, contrary to Bigelow, the possibility of time travel (broadly construed) appears prior to the nineteenth century; second, the philosophy of time travel is relevant to the philosophy of religion (not that this is news—see, e.g., Craig [1988], Effingham [2015b, 2018], Leftow [2004], Potter [2004], and Richmond [2013]).¹

1.2 Retrograde Time Travel The second method of time travel is retrograde time travel whereby a time traveller persists in a reverse direction. Where a normal object moves through space (or remains stationary) whilst persisting through time—i.e. moving through time in a forward direction—a retrograde time traveller moves through space whilst persisting backwards through time (or remains stationary, although we’ll see there are problems if they don’t move—see Chapter 4). Unlike discontinuous time travellers, a retrograde time traveller traces a continuous, unbroken path through spacetime (and, unlike warped time travel, discussed below, does so without moving through any strange regions of spacetime). ¹ Nor is prayer and prophecy the only point of connection between philosophy of religion and the philosophy of time travel. The worry that God’s foreknowledge impinges on free will [McCann 2012] is just a spin on the Grandfather Paradox. In both, we have an ability—e.g. my ability to shoot my grandfather before he met my grandmother or my ability to do as I choose—conflicting with knowledge of the future—e.g. God’s providential knowledge of the future or a time traveller’s knowledge of the future. What is said about the Grandfather Paradox will presumably apply to the theological issue and vice versa.

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The time machines of early fiction travel in a retrograde fashion, the most famous being H.G. Wells’s Traveller’s machine from The Time Machine [1895]. Where Wells’s machine stood stock still whilst it whisked its way through time, the inventions of other early authors tended to move through space: the first story to feature a time-travelling vehicle, Enrique Gaspar’s El Anacronópete, Viaje a China-Metempsícosis [1887], imagines that time itself is the product of the Earth’s spinning, and that a super-fast flying machine travelling counter to that rotation will go back in time; in MacKaye’s The Panchronicon [1904] the time machine travels back by flying tight loops around the North Pole; in Forde’s Time Flies [1944] it shoots off into space in order to make its way back in time. Later authors more inventively toy with how the machine might travel through the temporal, rather than spatial, dimension. In Watson’s ‘The Very Slow Time Machine’ [1978], it travels back in time at the same rate it would otherwise persist forwards; the pilot lives their life backwards, growing ever less insane from the isolation as the years progress. In ‘The Red Queen’s Race’ [1949], Isaac Asimov tests the boundaries of logical possibility with a retrograde machine that travels into the past at the rate of one century per day of external time (q.v.)—if it leaves on Monday, it’ll be in the Dark Ages by the end of the week! Retrograde time travel also features in physics: tachyons and advanced waves are described in §1.2.2–3; further, Tippett and Tsang [2017] describe a shell of exotic matter (q.v.) which works as a retrograde time machine (although the shell achieves this by warping spacetime and so blurs the boundary between the retrograde and warped modes of time travel). Cases where the speed of light is exceeded and time starts going backwards are also cases of retrograde time travel. In fiction we might have in mind Donner’s Superman [1978] (where we watch the Earth revolve in reverse as Superman travels back in time during its finale), Pearl’s The Space Eagle series [1967, 1970] (where a superluminal drive is activated and missiles track back to where they were launched from), or the rocketship from the TV series It’s About Time [1966]. Similar oddities crop up in the physics literature, where warping spacetime might propel things at faster-than-light speeds [Alcubierre 1994; Clark, Hiscock, and Larson 1999; Hiscock 1997] (although, as with the exotic matter shell, this crosses the border over into time travel via a warped spacetime).

1.2.1 Tachyons There are two prime examples of retrograde time travel in physics. The first is (the theoretical existence) of the ‘tachyon’. It’s generally thought that relativity rules out things travelling faster than the speed of light. Since the speed of light is constant in every inertial reference frame, no matter how fast one goes, light always outpaces you by thousands of kilometres per second. Prima facie, then,

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nothing can go faster than the speed of light. But there’s a hole in the dogma, for this shows only that nothing can accelerate past the speed of light. Were something to start off outpacing light, that would be consistent with relativity. Deriving from the Latin for ‘swift’, Feinberg [1967] calls particles which start off, and always remain, travelling faster than light ‘tachyons’. Whilst tachyons appear as far back as Lucretius’s De Rerum Natura [Caldirola and Recami 1980], they’re relevant to time travel because of relativity. To make sense of light’s constant speed, relativity dictates that—according to our perspective—things ‘slow down’ when they are in different inertial frames of reference. Things that travel superluminally ‘slow down’ so much that they go in reverse, i.e. go back in time. Tachyons have other weird features, such as having a weird mass—perhaps they have negative mass (and thus lose energy when they accelerate and decelerate when they gain energy, meaning they can never decelerate below the light barrier), imaginary rest mass [Bilaniuk, Deshpande, and Sudarshan 1962: 720; Ehrlich 2003; Feinberg 1967: 159; see Recami 1986 and Hill and Cox 2012 for objections], or imaginary relativistic mass (q.v.). The first serious examination of tachyons was by Bilaniuk, Deshpande, and Sudarshan [1962] (see also Feinberg [1967], Bilaniuk and Sudarshan [1969], Bilaniuk and Brown et al. [1969], and Terletskii [1966]; Everett and Roman [2013: 62–75] provide an accessible introduction). Whilst not everyone believes tachyons give rise to causality issues [Bers et al. 1971; Fox 1972], it’s not hard to see why we might think that they could. The common example in the literature is of an ‘antitelephone’ which communicates with the past using tachyons [Benford, Book, and Newcomb 1970] (and, playing that time travel role, tachyons appear in a few places in fiction such as Benford’s Timescape [1980], Carpenter’s Prince of Darkness [1987], and Carter’s ‘Synchrony’ in The X-Files [1997]). People have searched for tachyons. Whilst there’s been a possible observation [Clay and Crouch 1974], the result was never replicated and should almost certainly be disregarded [Clay 1988; Bhat et al. 1979; Hazen et al. 1975]. No other search has turned up anything. There have been two main methods of searching for tachyons. One such method is illegitimate. Čerenkov radiation is emitted when particles exceed the speed of light in a given medium. Some have searched for tachyons by looking for the Čerenkov radiation they’d supposedly emit. But no anomalous Čerenkov radiation has been detected [Alväger and Kreisler 1968; Davis, Kreisler, and Alväger 1969]. However, it’s implausible that this method would detect tachyons in the first place. The laws of nature are invariant in all frames of reference. That said, consider a tachyon’s frame of reference. (Even if there are no actual tachyons, it’s still legitimate to talk about how the world looks from their frame of reference—the reference frame is there, even if nothing’s in it.) In that? frame of reference it’s us who are travelling at superluminal velocities. So, by this thinking, we should be emitting Čerenkov radiation. And since we are not

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continually losing energy, we are clearly not emitting such radiation, either in our own frame of reference or in a tachyon’s. The conclusion must be that, since tachyons have odd properties (e.g. negative mass, slow down when approaching gravitational sources etc.), somewhere in the mathematics predicting the Čerenkov emission, a crucial fact about these strange properties has been omitted which, when included, will show that superluminal particles needn’t emit such radiation. Indeed, Mignani and Recami [1973] claim just that. (Considering how we are from a superluminal reference frame also shows that tachyons can have an imaginary relativistic mass—that is, a relativistic mass measured by a complex number—because that is the relativistic mass we have in a superluminal frame of reference; perhaps some tachyons—e.g. those putatively emitted during proton decay (q.v.)—have a real relativistic mass and an imaginary rest mass, but we can at least say that not all tachyons must be like that.) The second method of detection focuses on a possible role tachyons might play in proton decay. If a proton decayed (a big ‘if ’!), it would gain kinetic energy as it is ‘kicked back’ by whatever particle it emits during the decay process. But the proton can’t lose mass (because the number of baryons which exist must be conserved). So, if energy is to be conserved, the proton’s increase in kinetic energy must be balanced by the production of something with negative mass and energy—that is, by a tachyon being emitted during the decay. Thus, even if tachyons were invisible, we’d witness the kick back during proton decay [Everett and Roman 2013: 71–5; see also Everett 1976]. But in bubble chamber experiments, no such decay has yet been found [Baltay et al. 1970; Danburg et al. 1971]. This all said, the case for the possibility of tachyons seems pretty weak (although some recent work has been more positive [Yakovlev et al. 2010]). In Chapter 14 I’ll explain why this conclusion might be premature and why the lack of observations concerning proton decay tells us little about whether tachyons are, or are not, physically possible.

1.2.2 Retrocausal Quantum Physics A second physical theory allowing for retrograde time travel is Wheeler and Feynman’s ‘Absorber Theory’ [1945, 1949; see also Feinberg 1967]. Playing on the invariance of time in solutions to physical equations, if there are waves travelling forward in time (‘retarded waves’), then there are solutions to the equations whereby waves travel back in time (‘advanced waves’). Just as retarded waves move forward in time at a certain rate, advanced waves travel back in time at a given rate—that is, they time travel in a retrograde manner. Just as with tachyons, we might use them in building an antitelephone (as happens in Baxter’s Manifold: Time [1999]). Obviously, whether there are advanced waves is an open question, as is whether they could be used to build an antitelephone; it is worth

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noting that positive answers to the former are correlated with negative answers to the latter. Advanced waves feature elsewhere. Cramer’s transactional interpretation of quantum mechanics [1986, 1988] uses time-travelling waves to solve issues with quantum entanglement and the Einstein-Podolsky-Rosen Paradox. Given that I am only cataloguing types of time travel, and the details of quantum mechanics aren’t relevant, I refer the reader to other sources for more information [Berkovitz 2008; Corry 2015; Costa de Beauregard 1977, 1979; Davidon 1976; Dowe 1996; Kastner 2013; Pegg 2008; Price 1984, 1994; Rietdijk 1978; Schulman 1986; Sutherland 1983].

1.3 Warped Time Travel Sometimes a place, rather than a machine, is responsible for one’s travelling back in time. Because spacetime would then be distorted, call this the ‘warped’ method of time travel. At its simplest, we might imagine a straightforward doorway into the past, e.g. the Guardian of Forever from Star Trek’s ‘The City on the Edge of Forever’ [1967], the vortex from the finale of Raimi’s Evil Dead II [1987], or the time gate from Shirrefs and Thomson’s Tomorrow’s End [1991]. Such gateways would directly connect future to past; passing through them would mean no time spent ‘in transit’. But we could instead imagine an extended region through which one must travel in order to get to the past [cf. Bernstein 2015: 166]. Perhaps this separate region is enormous (maybe even an entire universe in itself!). In Zelazny’s Roadmarks [1979], aliens create a separate realm consisting of a long highway with turn-offs leading to different historical eras; as you move along one direction of the highway, the turn-offs lead to later and later points in time. In Herek’s Bill and Ted’s Excellent Adventure [1989] the time machine travels to an alternative dimension called ‘the circuits of time’; depending upon where in that universe the machine enters and exits, the time machine enters or exits at different points of time in our universe. Alternatively, we might imagine that the separate region is much smaller. For instance, in Winterbottom’s Time Riders [1991] time travellers go from one time to another via a short, misty passageway, whilst the Daleks from Doctor Who use ‘time corridors’ to travel back in time.

1.3.1 Circular Time Before moving to the examples of warped spacetime in physics, consider an example from historical thought: time repeating itself such that all events that have occurred will occur again. This idea appears in both ancient Greek [Sorabji

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1983] and Indian philosophy [Rocher 2004], as well as in Taoism, Hinduism, and the Mayan religion [Earman 1995: 203]. It features in the works of Nietzsche (Gay Science §285, §341, and Thus Spoke Zarathustra), as well as seeing service as a fictional trope—see, for instance, Futurama’s ‘The Late Philip J. Fry’ [2010] in which time travellers to the far future end up returning to the past, or Randall Munroe’s xkcd comic strip presented in Figure 1.1. (And, extending the idea of ‘closed timelike curves’ which I discuss in §1.3.2, there have been arguments from physics to back up this possibility [DeDeo and Gott 2002; Gödel 1949b; Gott and Li-Xin 1998; Ho and Weiler 2013; Nerlich 1994; Päs et al. 2009].) There are two ways to understand this idea of time repeating itself. One way is ‘linear repetition’: the same events occur again, in the same order, but at different (future) times. That is, some events, e₁, e₂ . . . e1,000, repeatedly occur: first at times t₁, t₂ . . . t1,000, then again at times t1,001, t1,002 . . . t2,000, and again at times t2,001, t2,002 . . . t3,000, and so on. Perhaps the events occurring at t1,001, t1,002 . . . t2,000 are mere qualitative duplicates of those occurring at t₁, t₂ . . . t1,000; perhaps they are

Figure 1.1. Möbius battle. Reprinted from xkcd: A webcomic of romance, sarcasm, math, and language under the terms of a Creative Commons Attribution-NonCommercial 2.5 Generic license (CC-BY-NC-2.5) .

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numerically identical and infinitely multi-located throughout the future [cf. Williams 2018: 206–7]. In either case, this wouldn’t be a case of warped time travel. The other, more pertinent, way is ‘circular time’ [Le Poidevin 1991: 123–43]: the events play out at t₁, t₂ . . . t1,000, and, whilst tn is later than tmn-dimensional space [Wasserman 2018: 35; see also Richmond 2018] or in a spacetime with multiple temporal dimensions [Wasserman 2018: 35]. If we’re focusing only on metaphysical possibility, these solutions should be perfectly palatable. Nonetheless, this chapter sketches how to solve the problem without using those ‘exotic’ solutions. Stipulating that such solutions aren’t to be used is probably arbitrary. But I wear these worries on my sleeve and admit that consideration of the Double Occupancy Problem is, as I’ve already said, likely a philosophical indulgence. But because the time travel literature includes a healthy body of work discussing the Double Occupancy Problem, and because discussion of the problem showcases some of the ideas and concepts from Chapters 2 and 3, I include the following solution. Uninterested readers may skip to Chapter 5.

4.2 The Motion Solution Imagine a point particle is moving. When travelling back in time it won’t strike itself, since it has moved [Dowe 2000: 445–6]. Observers would see two particles rushing at one another and then vanishing upon meeting—just as Feynman suggests. But what works for point particles works less well for extended objects. Imagine an extended object composed of (extended) parts p and p* where, at t₁, those parts are exactly located at r₁ and r₂, respectively. It is moving at a rate of one region unit per temporal unit. Imagine that the object—ergo both p and

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!!! t3

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Figure 4.1. The failure of the motion solution with extended objects.

p*—begins retrograde time travelling at t₃. Even in motion it runs into itself because its back end, p, interpenetrates its earlier front end, p* (at r₄ at t₃). See Figure 4.1. We might turn to Wells himself for a solution. In The Time Machine, the Time Traveller says, ‘Don’t you know that every body, solid, liquid, or gaseous, is made up of molecules with empty spaces between them? That leaves plenty of room to slip through a brick wall’. This suggestion might work, but only for objects extended in a particular fashion. Imagine an object is composed of a finite number of point particles which are spread across an extended region of space. Which region does the object exactly occupy? In one sense—a loose sense—it exactly occupies the extended region. In another sense—a strict sense—it exactly occupies a scattered region of zero volume (i.e. where the as are the atoms of such a composite, the scattered fusion of the regions which each a exactly occupies). Say that such an object is ‘non-veritably extended’. It is obviously different from a continuous, non-scattered object which exactly occupies the extended region in the strict sense of exact occupation. Say that the latter such object is ‘veritably extended’.¹ Wells’s explanation works for non-veritably extended objects that don’t have parts which are themselves veritably extended. If the non-veritably extended object is in motion, each point particle part of the retrograde time traveller will have moved and won’t collide with its earlier self. The future versions of its parts—now moving back in time—would compose a composite which only interpenetrates its earlier self in the same way that I ‘interpenetrate’ a ship when ¹ More formally: Region r is extended =df between every part of r there’s at least one path such that no portion of the path crosses something which isn’t a part of r. x is a veritably extended object =df (i) x is an object; (ii) x exactly occupies an extended region R; and (iii) for every decomposition of x into parts, p₁, p₂ . . . (where pn exactly occupies region rn) the fusion of r₁, r₂ . . . is identical to R. x is a non-veritably extended object =df x is an extended object but is not veritably extended.

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Figure 4.2. Fixing the motion solution by progressive parts time travelling.

I’m inside one of its cabins. The composite object only ‘double occupies’ in some loose sense. Such objects can travel back in time. But having already accepted that solving the problem is an indulgence, it’s thereby worth discussing how a veritably extended object might retrograde time travel. The best fix is to have the parts of the object start time travelling at different times, as Le Poidevin [2005] discusses (see also Richmond [2018]). If, at t₃, p* travels in time but p does not (and only starts to do so at t₄), then it will manage to miss itself. See Figure 4.2. However, note that Figure 4.2 is only a sketch of what should be said since p and p* are themselves extended. The more accurate statement (which the figure doesn’t depict) is that, given that space is continuous, a veritably extended object must: (i) have a decomposition into infinitely thin parts, p₁, p₂ . . . ; (ii) time travel over (at least) an interval, T, where T is composed of instants, t₁, t₂ . . . ; and (iii) be such that pn starts its retrograde journey at tn. (Because they time travel back bit by bit, such retrograde time travellers would look somewhat strange—Carroll [2018] includes videos showing what they’d look like.)

4.3 The Cheshire Cat Problem Likening retrograde travellers to Lewis Carroll’s Cheshire Cat, which fades away bit by bit, Le Poidevin [2005] has argued that this motion solution does not work. My understanding of his objection is that there is a two-fold problem. First problem: the abundancy problem. Whilst time travelling, there’s ‘too much’ of the time traveller [Le Poidevin 2005: 346]. See Figure 4.2; at t₃ the time machine appears to be too big, in that it swamps a region twice its size—if it’s normally two metres long, it’d be four metres long at t₃. I don’t see this as

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problematic, for it’s just a variation on the worries already debunked in Chapter 2 about self-visiting time travellers. In the same way that Marty doesn’t end up being 10’ tall if his 6’ tall later self visits his 4’ tall earlier self and has him stand on his shoulders (for, instead, Marty is merely both 6’ tall and 4’ tall), the time machine doesn’t end up being four metres long (for, instead, it is merely two metres long ‘twice over’). Second problem: the existence problem. Imagine I shoot upwards, with my parts time travelling in order from head to toe. At some instant near the end, all that will exist of me will be two sets of toes: one set going forwards in time (and about to reverse temporal direction) and another set already travelling back in time. When I am just toes, do I exist at that time? Assume that I don’t exist at that external time. In that case, my time-travelling self ‘ceases to exist’. The worry, then, is that I’ll trace a broken, discontinuous, path through spacetime [Le Poidevin 2005: 346]. Unlike some others, I have no problem with discontinuous time travel, and am untroubled by numerical identity stretching across spatiotemporal gaps, but I accept that the dialectic demands that retrograde time travel must be a mode of time travel distinct from the discontinuous mode. Therefore, I must exist throughout the process of travelling back in time. But merely having me not exist at that external time doesn’t mean I travel discontinuously. Were the (fusion of the) non-time-travelling toes not identical to Nikk, then the (fusion of the) time-travelling toes wouldn’t be either. Indeed, consider tlast, being the last instant of time at which there’s enough of non-timetravelling Nikk to count as still being Nikk. At tlast the non-time-travelling amount of me will be mirrored by a similar amount of time-travelling Nikk bits; their fusion will therefore also count as being Nikk at tlast. Thus non-time-travelling Nikk (i.e. that extended temporal part of me which doesn’t travel through time) will trace a path through spacetime, up to time tlast, as will time-travelling Nikk (i.e. that extended temporal part of me which does travel through time). Since non-time-travelling Nikk and time-travelling Nikk will touch one another at tlast, it follows that Nikk—being the fusion of non-time-travelling Nikk and timetravelling Nikk—traces a continuous path through spacetime (a path which is followed, at times after tlast, by things—e.g. some toes—which would ordinarily count as being bits of me but which, in this situation, are just random bits). But this fix is ultimately unhelpful. There will only be the ‘same amounts of Nikk’ if, when retrograde time travelling, my bits go back in time at the same rate at which I persist forward in time. If my bits instead go back in time at the rate of a thousand years a second, then there’ll be a lot less of time-travelling Nikk when there’s a large amount of non-time-travelling Nikk, i.e. prima facie there’ll be times when non-time-travelling Nikk exists and time-travelling Nikk does not, threatening my continuous path through spacetime. My solution to the existence problem only shows that there are metaphysically possible cases where the

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existence problem can be overcome; it doesn’t show that all cases of retrograde time travel go unthreatened by the existence problem. Once again, with an eye on philosophical indulgence, I’ll therefore try and avoid the existence problem via a different route. Before getting to that route, it’s perspicuous to consider an alternative way that Le Poidevin cashes out the existence problem: toe fusions aren’t people; when just my toes exist, I therefore can’t be a person; assuming I’m essentially a person, that means I can’t exist at that time. Put another way: given the standard definition of ‘instantaneous temporal part’,² the fusion of all my toes must be my temporal part at that time; since no fusion of just toes is ever intrinsically a person, given P@ I can’t be a person at that external time. If I’m not a person at that time, and am essentially a person, I can’t exist at that time. We can solve this version of the problem by embracing what I said in §2.2.2 about giving up on the standard definition of what a temporal part is (which is also along the same lines as the solution to the Cheshire Cat problem put forward by Carroll, Ellis, and Moore [2017]). Contrary to the standard definition, we should say that temporal parts can be spread across multiple external times—where the temporal parts according to the standard understanding are depicted in Figure 4.3, we should instead treat the temporal parts of a retrograde travelling object as being those depicted in Figure 4.4. The temporal parts of retrograde time travellers will therefore be orientated tangential to the temporal parts of spacetime. Now it is easy to see how I can retrograde time travel whilst tracing a continuous path through spacetime. If my temporal parts are orientated in that fashion, the path I trace through spacetime will look like that in Figure 4.5. It’s clearly t4

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Figure 4.3. Temporal parts, given the standard understanding.

² That is: x is an instantaneous temporal part of y at time t =df (i) x is a part of y; (ii) x exists at, but only at, time t; (iii) x overlaps every part of y at time t.

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Figure 4.4. Temporal parts allowing a tangential orientation.

t

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Figure 4.5. Tangential temporal parts.

continuous. Problem solved! And now we can return to the original question of whether I exist at an external time when just my toes exist. We can deploy the chorological notions from §2.2.2 to solve this. Imagine that I dangle just my toe inside a room. I exist in the room in some sense but not in other senses—in terms of temporally relativized chorological relations, I am weakly located inside the room even though I’m not exactly located at the room (nor exactly located at a sub-region of the room). The same applies when I’m time travelling, although now it will be atemporal chorological relations that are salient. When I’m time travelling and just my toes exist at a time, I am (atemporally) weakly located at that external time but don’t have a temporal part which is exactly located at a subregion of that external time. So I exist at that time in one sense (since I’m

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atemporally weakly located at it) but not another (for no temporal part of me is atemporally exactly located at a part of that time). Just as there’s no problem with there being two senses of existing in a room when it comes to toe dangling, there are no problems with there being two senses of existing at an external time when it comes to retrograde time travel. The existence problem can be solved (and without recourse to extra spatial or temporal dimensions). Interestingly, this explanation raises problems for the tensed theory of time. Imagine the Doctor retrograde time travels and during the journey he gets bored. Because of the tangential orientation of his temporal parts during that period, the personal time at which he’s bored isn’t consociated with any external time (for, recall, personal times are simply temporal parts, and for a personal time to be consociated with an external time is for that temporal part to exactly occupy some part of the external time, which no tangentially orientated temporal part will do). If the personal time at which the Doctor is bored isn’t consociated with an external time, then there’s no external time at which the Doctor is bored (for consociation simply is the relation which holds when what you’re like at a personal time matches what you’re like at its consociated external time). If there’s no external time at which he’s bored, then the Doctor’s boredom is somehow untethered from history. I’m not overly worried by this result (indeed, if relativistic spacetimes were spacetimes with no external time, as discussed in Chapter 3, it’s no more worrying than objects being a certain way at a personal time embedded in a relativistic spacetime). But the tensed theorist should be worried by this result. Tensed theorists invariably believe that consciousness is intimately connected with being present, i.e. that my being conscious of ψ requires it presently being the case that I’m experiencing that ψ is the case. But the temporal part of the Doctor which is bored never has the spotlight of presentness shine down upon it—the spotlight only ever illuminates portions of his bored slices. Presumably, since the spotlight never shines on the tangentially orientated temporal parts of retrograde time travellers, tensed theorists would have to say that they’re never conscious. Tensed theorists are in the unenviable position of having to say that retrograde time travellers become phenomenal zombies for the period of their time travelling. That merely travelling in time might shunt an agent from having qualia to losing them seems like a hard bullet to bite, so it appears to me that this commitment is a strike against the tensed theorist.

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5 The Bootstrapping Paradox Bootstrapped things are things which, at least in part, are responsible for their own existence via time travel. They are a common trope in time travel fiction. The Bootstrapping Paradox is an argument to the effect that these strange entities are so strange that we should think they’re impossible and, because time travel permits them, time travel is therefore likewise impossible. §5.1 provides numerous different examples of bootstrapped entity; §5.2 discusses whether they are a reason to think time travel is impossible. I argue that they are not.

5.1 Types of Bootstrapping When used in the context of time travel, ‘bootstrapping’ draws its meaning from Heinlein’s story [1941] ‘By His Bootstraps’. The protagonist is manipulated into using a time machine by a dictator from the future. At the story’s conclusion, it turns out that the dictator is his future self and that his travelling in time was a necessary condition of his travelling in time. It’s an example of a ‘causal loop’ where an event ec (at least partially) causes an event ee and ee (at least partially) causes ec. It’s called ‘bootstrapping’ because the event is the cause of its own existence; it has pulled itself into existence in the same way that one might metaphorically pull oneself up by one’s bootstraps. Fictional causal loops are replete. In Herek’s Bill and Ted’s Excellent Adventure [1990] the heroes constantly engage in bootstrapping devilry to overcome obstacles delivered by the plot. When they need to open a prison cell with a specific key they don’t have, they promise that they’ll later steal the key and place it in an easy-to-find place. Lo and behold, they find the key behind a nearby rock. In Nolan’s Interstellar [2014] humans from the future save mankind, bootstrapping its survival. Similarly, in The Hitchhiker’s Guide to the Galaxy [Adams 1979] a spaceship from the present affects the past, accidentally bootstrapping into existence all life in the cosmos. Conversely, the destruction of humanity is bootstrapped in Taylor’s Escape from the Planet of the Apes [1971] when the arrival of apes from the future leads to the cataclysmic events precursing the ascendancy of apes over humans. And so on—there are many more examples to be had. Information, as well as causal chains, can be bootstrapped. In Heinlein’s story the protagonist stumbles across a book containing crucial information. After many years, and shortly before he realizes that he’s the dictator, he copies the Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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information from the notebook into a new one. Eventually that new notebook is left for his past self to stumble across. The question arises: where did the information in the notebook come from? Again, further fictional examples are commonplace. In Burgess’s ‘The Muse’ [1968] a historian goes back in time to meet Shakespeare. Discovering the Bard is less than talented, the historian hands him the Shakespearean plays from the future, bootstrapping them into existence. In Nimoy’s Star Trek IV [1986] Scotty gives twentieth-century scientists the information to make ‘transparent aluminium’—information he only knows because he learnt it from their ‘discovery’. And, again, these are just some of the many examples on offer. Entire objects can be bootstrapped into existence. The most famous example is from another of Heinlein’s stories: ‘—All You Zombies—’ [1959]. Heinlein’s protagonist is found as a baby on the doorsteps of an orphanage. Later in life she is seduced by a man, becomes pregnant, and her baby is taken away. She then goes on to have a sex change and he then travels in time, seduces his earlier self, and takes the baby to the doorsteps of the orphanage. The protagonist is their own mother and father! (Spierig and Spierig’s Predestination [2014] is directly based on it; it is also the subject of Analysis Problem No. 18 [Harrison 1979; MacBeath 1982].) Another example common in the time travel literature is Szwarc’s Somewhere in Time [1980] in which a man is given a pocket watch in 1972 by an elderly woman before travelling to 1912 and leaving the pocket watch behind. The woman’s past self then picks it up and waits to give it to him (to then take back to the past etc.). The pocket watch ‘comes from nowhere’ and is a bootstrapped object. Note that whilst it’s natural to think that bootstrapped objects require their underlying matter to likewise be bootstrapped—thus increasing the overall mass energy of the universe for the period of time that the causal loop involving the object takes place—that needn’t be the case [Hanley 2004: 134–5]. Imagine that the parts of Szwarc’s watch are each individually crafted in a factory just like any regular watch part. At time t those parts replace the worn-out parts of an old watch. The watch is then sent back in time. Gradually its parts become worn and, at time t, are replaced with (of course!) the past versions of the parts. The spent watch parts are then discarded just like regular watch parts and thrown in a bin. The watch parts, and their underlying matter, are not bootstrapped; the watch itself, however, is. Whilst I believe that both bootstrapped objects and bootstrapped matter are possible, it is worth distinguishing the two.

5.2 The Impossibility of Bootstrapping The Bootstrapping Paradox has two premises. First: time travel necessarily involves interacting with one’s own past and bringing about a bootstrapped causal chain

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(or creating bootstrapped information, or making a bootstrapped object). Call that claim I. Second: bootstrapped entities are metaphysically impossible. Call that N B. Given T T’ P and I, time travel necessitates bootstrapping but, given N B, that’s impossible. By reductio, T T’ P is false [cf. Hanley 2004: 124; McCall 2010; Wasserman 2018: 154–65]. Just as with the Self-Visitation Paradox and the Time Discrepancy Paradox, I wouldn’t consider it a particularly worrying paradox. Nevertheless, it is illuminating to see where it goes wrong.

5.2.1 Against I I has problems [cf. Monton 2009]. Before turning to good reasons for thinking that it’s false, first debunk one bad reason. We might believe that, just as long as they were very careful, time travellers could avoid interacting with the future. This is a standard trope in science fiction; fictional time travellers often creep around being careful not to influence large-scale geopolitical events. Whilst some think this is a serious response [Hanley 2004: 130; MacBeath 1982: 417], I doubt that it can be anything more than a mere MacGuffin for science fiction authors to rely upon. Consider a time travel case which prima facie involves no interaction. I travel to 1930 to a remote part of the Sahara Desert. I appear for a split second and then quickly leap back into my time machine to return to the present. One might think that briefly stepping out into 1930s Africa would have no causal impact on my future. But one would be wrong—at least if we assume that physical states/interactions have a continuum of possible outcomes [cf. Monton 2009: 63–4; Rea 2015]. I don’t mean to suggest that there’s some ‘butterfly effect’ whereby a tiny change in the past results in radical future changes (a la Ray Bradbury’s ‘A Sound of Thunder’ [1952] or The Simpsons’s ‘Treehouse of Horror V’ [1994]). My appearance in the Sahara presumably won’t lead to Hitler winning World War II. It won’t make a discernible, macroscopic difference to the future. But whilst my brief jaunt makes no easy-to-see macroscopic changes, I do make microscopic changes. Arriving in the past, my mass casts out gravity waves and subtly alters everything within the future light cone of my arrival point. Hitler might not win World War II because I travelled to the 1930s, but it does mean that he’ll lose it with the particles which compose him being in a slightly different arrangement than they’d otherwise have been. The perturbations will continue through to the present. And that means that, because I stepped out of a time machine back in the 30s, I’ll enter the time machine with my particles arranged in a such-and-such way rather than a so-and-so way. Thus, my stepping into the time machine, which causes me to step out into 1930, is causally affected by my

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stepping out in 1930. Thus, treading carefully in the past would only help with I in worlds with substantially different physics than our own. Whilst it is not until §6.3 that I introduce them in-depth, indexed worlds (at which time travel involves travelling to universes or hypertimes) would be another way to deny I. At an indexed world my stepping into the time machine takes place at one universe and my stepping out into the Sahara takes place at another; there will be no causal interaction [Deutsch and Lockwood 1994a; Visser 2003]. However, this works only for indexed worlds, not every world at which time travel is possible—since I will later argue that only some worlds at which time travel takes place are indexed worlds, we can set aside this response. So, finally, we turn to my reasons for thinking that I is false. One reason is that travelling in time might dually involve travelling outside one’s own light cone. The causal loop in the Sahara example arises because I’ve time travelled back to within my own light cone. Had I travelled outside of my past light cone, there’d be no causal loop. Since it’s metaphysically possible that I travel both in time and space, I is false. For instance, in Lovecraft and Price’s [1932] ‘Through the Gate of the Silver Key’ the protagonist, Randolph Carter, projects his mind into an alien on a planet that both lies far in the past and far, far away— indeed, outside his past light cone. Unable to remember the ritual to return his mind to the contemporary period, Carter (in the body of the alien) goes back to Earth to find it again. He embarks on a mammoth journey across the gulf of space, travelling faster than the speed of light. He arrives on Earth a few years after he originally left. So Carter travels in time, outside his light cone, and returns only after his departure point. Carter’s time travelling involves no interaction at all with his own history, even at the level of sub-atomic arrangements. There are other reasons to think that time travel needn’t necessarily involve causally affecting the past. Time travellers might become ‘ethereal’ and unable to affect the past, as in Dickens’s [1843] A Christmas Carol where Scrooge can’t affect anything when travelling with the Ghost of Christmas Past [cf. Read 2012: 141; Wasserman 2018: 9].¹ In other fictions, one’s time travelling mind inhabits one’s earlier body, forced to observe and unable to interact—see Resnais’s Je t’aime, je t’aime and Vonnegut’s Timequake [1997], as well as Long’s [1929] ‘The Hounds of Tindalos’ in which the time traveller similarly experiences the lives of everyone who ever lived. Alternatively, there could be a spacetime where one section goes

¹ This doesn’t work if the ethereal agent has the power to affect the material world, but refrains from using that power. Examples of such agents are the time-travelling ghost from Lowery’s A Ghost Story [2017] (who sometimes affects the past when travelling in time but sometimes chooses not to) or the time travellers in Barjavel’s Le Voyageur Imprudent [1944] who have a suit that can turn them ethereal and back again. Agents who refrain from acting nevertheless end up in a causal loop—what Chapter 12 calls a ‘negative causal loop’—which is just as problematic as one in which the ability is exercised.

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forwards in time and another area goes backwards, but where the two parts can’t freely interact. Such scenarios appear in the philosophical literature [Tooley 1997: 63–8] as well as the physics literature: given some assumptions about entropy and the arrow of time, time can flow backwards in other universes [Carroll, S. 2008] or in localized areas of our own universe (e.g. within black holes [Bousso and Engelhardt 2015]). Were interaction between such regions only one-way (for example, if things can enter the black hole where time goes backwards, but cannot return), then these would be time travel scenarios which could not involve a causal loop. So I is false. Causal loops are not metaphysically necessitated by travelling in time. But this is barely a victory. These cases avoid I but at a cost, for if this were the only way to avoid the impossibility of time travel, then interesting time travel (e.g. travelling to within one’s own pastwards light cone) would be ruled out.

5.2.2 Against N B Fortunately, N B is unmotivated; it is unclear why we should believe it. Whilst I grant that bootstrapped things are weird, I don’t see why we should think they’re impossible. All too often reference is made to the impossibility of bootstrapped things with no discussion of why they’re thought to be impossible (see, e.g., Visser [2003]). Consider some possible motivations and see why each fails. First motivation: Deutsch and Lockwood worry that bootstrapped information breaches the Knowledge Principle ‘that knowledge can come into existence only as a result of problem-solving processes, such as biological evolution or human thought’ [Deutsch and Lockwood 1994a: 70; see also Lockwood 2005: 174–5]. But the Knowledge Principle is false. Consider the possibility of ‘Boltzmann Brains’ which (given quantum physics) can appear from nowhere loaded with fully formed, cogent mental states, including mental states about what the world’s like which, by chance, are true. Imagine a race of Boltzmann Brains come into existence and one materializes from the ether believing everything true about mathematics. That Brain would truly believe how to prove Fermat’s Last Theorem. By itself that might not count as knowledge because we might well accept some anti-luck constraint on knowledge. But the Boltzmann Brains that the proof is shared with would surely come to know it to be true. And that’d be knowledge which ‘came into existence’ as a result of something other than biological evolution or sentient thought. (A less exotic counterexample is Smith’s [1997], who offers the counterexample of someone who, because they mishear something, comes to believe something both profound and true.)

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Second motivation: the problem might be that bootstrapped things are apparently inexplicable [e.g. Al-Khalili 1999: 183]. But this merely puts bootstrapped entities in the same—unproblematic—boat as other things which apparently have no explanation [Fulmer 1980; Ho and Weiler 2013: 2; Levin 1980; Lewis 1976: 149; Meyer 2012; Smith 2005: 392, 2013: §3.2]. Consider Lewis’s example of the decay of a tritium atom. If its decay is ultimately inexplicable, then it can’t be a problem to say that causal loops and bootstrapped objects are inexplicable either. If the decay is instead explicable, then it’ll be explained by a law of nature. And that law is either deterministic or it’s irreducibly stochastic. Either way, bootstrapping can be similarly law governed: were tritium decay irreducibly stochastic, then there’s no problem with the appearance of bootstrapped things being genuinely stochastic as well; were tritium decay instead secretly deterministic (and governed by hidden variables), causal loops could likewise be governed by laws and variables we don’t know about. What’s sauce for the tritium goose is sauce for the bootstrapping gander. Third motivation: Hanley—who himself has no problem with bootstrapped entities—suggests that the problem might be that bootstrapped objects end up having no consistent age [2004: 133]. But it’s possible that there could be ‘timeless worlds’ at which duplicates of things existing at one time slice of our world exist in a timeless state [Effingham and Melia 2007]. Such things would have no age whatsoever. Thus, contrastandard age properties don’t make an entity impossible. Fourth motivation: Hanley [2004: 134] further suggests that there’s a problem with there being bootstrapped artefacts engineered by no-one—artefacts, Hanley suggests, should have intentional creators. But then he points out that some artefacts are engineered by no-one (e.g. a large stone used as a doorstop). Equally, we could also baulk at calling things like Szwarc’s watch an ‘artefact’. Perhaps it’s instead just a very weird naturally occurring thing which is qualitatively similar to many manufactured artefacts (an answer which seems even more plausible if, as per the response to the second motivation, it turned out that the ‘watch’ exists because of some law of nature). Fifth motivation: bootstrapping has problems with the essentiality of origin.² For instance, Szwarc’s watch has no origin and so would lack an essential property other things had. But not only is this no threat to bootstrapping in general (for bootstrapped things like Heinlein’s bootstrapped person will have an origin, i.e. themselves), it’s also not clear why this is a problem. For instance, God has no origin but it’d be a weak argument to move from that to atheism.

² This motivation came up in discussion with Murali Ramachandran who said his students had mooted it as a possibility.

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Sixth motivation: McCall [2010] has an interesting argument based upon issues with aesthetic value.³ He argues that some portion of an artwork’s aesthetic value comes from its having been creatively made; copies of artworks are valuable, but less valuable insofar as they lack that creative input. Imagine an artwork bootstraps itself into existence because an artist copies its future version (so the painting—the object constituted by the canvas—is bootstrapped, not the canvas itself or the matter involved). McCall wonders how that artwork can be valuable given the absence of creativity. (Similarly, if a copy of an artwork is necessarily less valuable than the thing it is copied from, we get a contradiction since the bootstrapped painting will be less valuable than itself.) The solution is that the bootstrapped artwork is less valuable than at first thought. Its sole value comes from its intrinsic, visual qualities; no value is added because of creative input, for no creativity was involved. It’s similar to a case where we discover that Jackson Pollock didn’t intentionally create his famous drip paintings and instead just knocked over a bunch of paint pots one evening whilst drunk. We’d rightly judge Pollock’s paintings to still be aesthetically pleasing and valuable, but less valuable than we’d previously thought because they lacked intentional creative input. (Similarly, we should then judge copies of the drip paintings to be no less valuable merely in virtue of being copies; we might, though, think copies were less valuable because they lacked the correct causal/ historical connection to Pollock.) Seventh motivation: Mellor [1998] argues that, in causal loops, frequencies fail to be appropriately connected to the law of large numbers. This argument has been extensively dealt with elsewhere [Berkovitz 2001; Dowe 2001; Paul 2001; Riggs 1997; Wasserman 2018: 165–70] and I have nothing to add, so pass over it here. In short, N B is without motive. And without a champion to make its case, the Bootstrapping Paradox is no threat to T T’ P. Nevertheless, this discussion has introduced: the idea of causal loops and bootstrapping; the possibility of time travel with no bootstrapping; and the inevitability of causal loops were one to travel back to within one’s own light cone. All of these things will be referred to later in this book.

³ This problem case is discussed further by Caddick Bourne and Bourne [2016, 2017], including a response from McCall [2016]; Miller [2017] also discusses it, taking a substantially different reading but nevertheless arguing that bootstrapping is possible.

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6 Changing the Past 6.1 The Paradox of the Changing Past Some have suggested that time travel necessarily involves changing the past. Combined with the claim that changing the past is impossible, it follows that time travel is impossible [Grey 1999: 57; Harrison 1971: 2–3; Hospers 1997: 121–2; for discussion, see Wasserman 2018: 74–7]. Imagine the following scenario. There is a street in my home city of Liverpool called ‘Bold Street’. I never walked down interwar Bold Street, thus hAt 1930: ¬Nikk is in Bold Streeti is true. Bold Street is said to have been the location of numerous ‘time slips’ where people have ‘slipped’ back in time before returning to the present [Anon 2011]. Imagine that T T’ P is true and that I fall through one of those holes in time, travelling back to 1930. Now hAt 1930: Nikk is in Bold Streeti is true. But that’d mean the following is true: [At 1930: ¬Nikk is in Bold Street] ∧ [At 1930: Nikk is in Bold Street] That’s not quite a contradiction since it’s not of the form φ∧¬φ. But it entails a contradiction given the (eminently plausible): t /¬ C: ▽

t ¬ φ $ ¬ ▽ t φ] [▽

t stands for some specific concatenation of temporal operators, e.g. (Where ▽ ‘At t₁:’, ‘At t₁: At t₂:’, ‘At t₁: At t₁:’ etc.)

Thus time travel involves bringing about contradictions and, by reductio, T T’ P is false. Call this the Paradox of the Changing Past. As far as I know, the only support for this Paradox is historic and the existing literature concerning the debate is thought to have buried it. To understand the solutions we must first draw a distinction between two ways of making a difference to the past: causally affecting the past and changing it [Brier 1973; Fulmer 1980: 151–2; Harrison 1971: 7; Williams 2018: 193, 222–3]. To causally affect the world is to make how it is at one time be a different way from a slightly earlier time. For example: I causally affect the world if I make it the case that the lights are on in a room rather than off because the lights are off at one time and, a short time later, the lights are on because of something I’ve done (e.g. flicking the light

Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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switch). For you to causally affect the past would be for there to be two past moments, t and t+, where t is earlier than t+, and for something you do now to make how the world is at t+ different from how it was at t. If I travel to 1930s Bold Street then something I do now (e.g. falling through a time slip) makes an instant in the past (e.g. an instant in 1930 when I’m in Bold Street) different from a previous instant (e.g. a slightly earlier instant in 1930 when I’m not in Bold Street). Clearly, time travel requires the past to be causally affected by events from the future. Equally clearly, there’s no contradiction in causally affecting the past. My travelling to Bold Street only entails propositions like hAt 1929: ¬Nikk is in Bold Street ∧ At 1930: Nikk is in Bold Streeti and that’s no more contradictory than hAt 10 a.m.: ¬ It is raining ∧ At 11 a.m.: It is rainingi. The other way to make a difference to the past would be to change the past. For instance, if an instant, t, from 1930 went from being one way earlier on in the day to (after using a time machine) being a different way, the past will have changed. Perhaps there are other ways for the past to change (for instance, a time machine making t two contradictory ways by my being both in Bold Street and not in it, or a time machine allowing one to ‘replace’ the past). Sharpening up precisely how to understand what ‘changing the past’ consists in is a task in itself, but the gist is clear: changing the past involves something like what we see in fiction, where we put right what once went wrong. For example: in Kleiser’s Flight of the Navigator [1986] the hero travels to 1986 from 1978, thus apparently vanishing for eight years, before his future self returns to 1978, now making it the case that he never vanished at all; in Back to the Future [1985] Marty McFly changes history so that his parents’ lives are closer to the capitalist ideal; in Curtis’s About Time [2013] the protagonist secretly rewrites history to make up for his romantic shortcomings. To change the past in the above example would be to make it the case that I walk down Bold Street in 1930 even though I knew that I didn’t. Changing the past seems prima facie contradictory since it sounds as if it involves the truth of propositions of the form hAt t: ¬φ ∧ At t: φi (which, given t /¬ C, entail a contradiction). ▽ With these two distinct understandings in place, we can see the two ways in which the Paradox of the Changing Past goes wrong.

6.2 Ludovicianism 6.2.1 The Second Time Around Fallacy The first solution is to say that, where the unproblematic act of causally affecting the past is obviously required for time travel to be possible, the prima facie contradictory act of changing the past is not. To think it’s required is to have become confused. Those who believe time travel is possible don’t believe that there

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is a version of history where I’m not in interwar Liverpool and then, after I fall through the time portal, there’s a version of history according to which I am in interwar Liverpool. Far from it. Instead, there’s only one version of history, where history is such that I emerge from a time slip in 1930s Bold Street and then, years later, fall down that time slip in the twenty-first century. Those moved by the Paradox of the Changing Past have committed the ‘Second Time Around Fallacy’ whereby one fallaciously believes there are two versions of history; in reality, we should think there is only one [Daniels 2012; Dwyer 1975, 1978; Elliott 2019; Fulmer 1980, 1983: 32–3; Harrison 1971: 5–7; Horwich 1987: 116; Lewis 1976; Miller 2008: 189; Putnam 1962; Smith 1997: 365, 2017; McCall 2010: 647; Williams 2018: 193–4, 222–6]. Those who endorse this response, believing that time travel is possible but that changing the past is not, are called ‘Ludovicians’ (being the Latinized version of ‘Lewis’, who popularized this view in his famous 1976 paper). As far as I know, no-one any longer suggests that Ludovicianism is anything but damning to the Paradox of the Changing Past. (Although an extension of the Paradox of the Changing Past—the Grandfather Paradox—is more compelling, and more challenging for the Ludovician to tackle, as the rest of this book demonstrates.) So, at best, the Paradox shows that time cannot be changed. Given Ludovicianism, no matter what a time traveller tries to do in the past, nothing will change—all established facts remain unaltered, all events known to have occurred shall occur, all wrongs remain unrighted. We always end up with a consistent narrative, no matter how unlikely that narrative turns out to be. Amongst philosophers, Ludovicianism is the most popular treatment of time travel [Baron and Colyvan 2016, Forthcoming; Craig 1988; Dwyer 1975, 1978; Eldridge-Smith 2007; Fulmer 1980; Harrison 1971: 6–7; Horwich 1987; Lewis 1976; Newton-Smith 1988: 182; Putnam 1962: 669; Robson and Curtis 2016; Sider 2002; Smith 1997; Thom 1975, inter alia]. It also crops up in physics. Bilaniuk et al. explicitly discuss it [1969], whilst Echeverria, Klinkhammer, and Thorne [1991] argue against Polchinski’s Paradox (see §7.1.1) on the grounds that there are scenarios where time travel occurs without contradiction. That sounds like Ludovicianism to me. (And I’ll argue in §8.1 that those who endorse ‘Novikov’s Principle of Self-Consistency’ are also Ludovicians.) And Ludovician stories—or, at least, prima facie consistent narratives containing unlikely events conspiring to prevent paradoxes—appear throughout fiction. Early examples include Heinlein’s bootstrapping stories from Chapter 5 as well as the likes of Weisinger’s ‘Thompson’s Time Travel Theory’ [1944], Rocklynne’s ‘Time Wants a Skeleton’ [1941], Marker’s La Jetée [1962], and Moorcock’s Behold the Man [1969]. More recent examples sustaining the Ludovician spirit include Laloux and Hernádi’s Les Maîtres du Temps [1982], Gilliam’s 12 Monkeys [1995], Lynch’s Inland Empire [2006], Vigalondo’s Los Cronocrímenes [2007], and Hidalgo’s La Casa del Fin de los Tiempos [2013].

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Ludovicians often shore up their position that the past cannot be changed by saying that this isn’t something peculiar about the past. We can no more change the past than we can change the future [Horwich 1987: 116; Lewis 1976: 150; Putnam 1962: 669; cf. Garrett 2017]. Given the future is closed (i.e. that all propositions about the future have a fixed truth value), then nothing I do can change the future—if, e.g., Dave will die in 2077  because he recklessly flies an airplane, then nothing can change that fact. Note, though, that Dave nevertheless causally affects that situation, since it’s his reckless piloting that causes him to crash. So says the Ludovician: time travel not changing the past is only as weird as the future never changing—that is, not weird at all.

6.2.2 The Open Future §6.3 discusses non-Ludovician theories which allow for the past to be changed. Before getting to those theories, the rest of this section discusses some issues which arise given Ludovicianism’s denial that the past can change. We can start with the future being open. The future is open iff at least some (i.e. the contingent) propositions about the future have an alethic status which changes when the time they are about becomes the present moment. Perhaps these propositions about the future (‘future-orientated propositions’) are false but later some may become true [Markosian 2012]. Perhaps some are true and some are false but the present actions of agents can change their truth value [Geach 1977; Todd 2011]. Perhaps there’s a third truth value (e.g. ‘indeterminate’) which future-orientated propositions have, becoming true or false when the things they’re about become present [Bourne 2006; Łukasiewicz 1968; Slater 2005; Tooley 1997] or, similarly, that they have no truth value until later on [MacFarlane 2003]. Perhaps they’re both true and false and later lose one of their truth values (see Miller [2008]). Whatever the details, were one to sign up to this sort of theory the Ludovician would no longer be able to shore up their claim that changing the past is impossible on the back of changing the future being impossible—if the future can be changed, why not the past? There’s another reason Ludovicianism pairs badly with an open future.¹ For purpose of example, assume that ‘future-orientated propositions’ start off being indeterminate (rather than false, or true and false etc.) before becoming true/false. Imagine that the world consists just of a man in a box. He has a rifle and, high up

¹ Miller [2008] and Slater [2005] both discuss a similar argument to the one I am about to present here. See Wasserman [2018: 49–55] for discussion. Further, Norton [2018] offers a version of Chapter 14’s Tourist Paradox against certain types of open future.

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and out of reach on the wall in front of him, is a target. At t₁₀ a wormhole will open in front of the target. Anything passing through it travels back to t₅. Imagine that, at t₅, a wormhole opens and a bullet shoots out. Time travel has occurred! So, at t₅ the following is true: (1)

At t₅: the wormhole opens and a particle from the future shoots out.

(1) appears to imply that (also at t₅) the following future-orientated proposition is true: (2)

At t₁₀: the man will fire the rifle.

But were (2) true, at t₅ the future wouldn’t be open because some future-orientated proposition about a contingent matter would be true rather than indeterminate. The open-future Ludovician might offer one of two responses: The Inevitability Response: (2) is indeterminate at t₅. Whilst (2) is inevitable (and will turn out to be true), it’s not currently true. The Indeterminacy Response: (1) isn’t true, it’s indeterminate. Whilst it’s determinate that a bullet has left the portal, it’s not determinate that it’s from the future until t₁₀. So (1) can’t be true until t₁₀. (Horacek [2005: 432] advances a claim similar to this.) Neither response works. Consider, first, the open-future theorists who endorse a supervaluational approach to future truths [Barnes and Cameron 2009; Thomason 1970]. Say that world w is a possible future (at t) iff the history of w is identical to actual history up until t. Supervaluationists believe that φ: is true (at t) iff at every possible future (at t) φ is true; is false (at t) iff at every possible future (at t) φ is false; is indeterminate (at t) iff φ is true at some (but not all) possible futures (at t). Next, either (2) is true in all possible futures or it isn’t. To begin with, assume that it is. In that case, given we’ve assumed supervaluationism, (2) is true and not indeterminate and the Inevitability Response won’t work. Further, if (2) is true at a possible future, so is (1); thus, if (2) is true at all possible futures then (given supervaluationism) (1) is true as well i.e. it’s not indeterminate and the Indeterminacy Response won’t work either. So for the future to be open we must drop either supervaluationism or (2) being inevitable (that is, true in every possible future). If supervaluationism is false, then there’s nevertheless still a problem; at least, there is if we assume that the man has free will concerning the firing of the gun. Open-future theorists tend to be incompatibilists, endorsing: I: If it’s physically determined that agent τ ϕs, then τ has no choice about whether they ϕ or not.

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If (at t₅) it’s true in all metaphysically possible futures that the man fires, then it’s metaphysically necessary that, given the facts up until t₅, he will fire the gun. Assuming—as is standard—that all metaphysical necessities are physical necessities, it’s physically determined that the man will fire the gun. Thus (given I) the man has no choice about firing the gun.² The only option left is to deny that (2) is true in every possible future. Assume that only in some possible futures is (2) true; it is false in others. That’s straightforwardly incompatible with the Inevitability Response, so if we opt for this route, we must accept the Indeterminacy Response. Problems nevertheless still arise with free will. Imagine that the laws of nature rule out (1) being true unless (2) ends up being true (for instance, the laws rule out ex nihilo appearances of bullets and mandate that the gun is never going to misfire). A bullet could then only appear at t₅ without (2) being true were a miracle to occur, i.e. a miracle whereby a bullet appears ex nihilo, or which causes the gun to miraculously misfire. So in a possible future at which a bullet appears at t₅ and (2) is false, a miracle occurred. But that means that, whilst the bullet appearing and (2) being false is metaphysically contingent and happens in some possible futures, it doesn’t happen in any physically possible future. So, given that a bullet appears at t₅, (1) and (2) are physically necessary. And if it’s physically necessary that the man fires the gun, then (given I) the bullet appearing means that the man loses his choice to shoot the gun. In conclusion, open futurists are unlikely to be attracted by Ludovicianism.³ If (like myself) you’re not a fan of the open-future theory, this is a mark against it; if you are a fan of the open-future theory, this is a mark against Ludovicianism (and, probably, the possibility of time travel in general).

6.2.3 Bilking The second issue Ludovicianism has is with ‘bilking’ [Black 1956]. Imagine that in 2018  a woman appears, claims to be my granddaughter from 2070 , and tries to kill me. Fending her off, she vanishes in a weird machine. Call that event e₂₀₁₈. I come to believe that in 2070  my granddaughter will step into a time machine to come and kill me. Call that event e₂₀₇₀. I try to prevent e₂₀₇₀. If I succeed, e₂₀₇₀ won’t occur. If it doesn’t occur, then, since the past can’t change, ² In Chapter 11 I’ll argue that not every metaphysical necessity is a physical necessity. But I imagine that the incompatibilist/open-future theorist would nonetheless endorse the following replacement principle, leaving them in just as problematic a position as before: I*: If it’s metaphysically determined that agent τ ϕs, then τ has no choice about whether they ϕ or not. ³ Indeed, I think the only open-future-friendly models of time travel will likely involve conterminous hypertemporal indexing (q.v.).

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time travel won’t have happened; instead e₂₀₁₈ must’ve been caused by something else, e.g. an elaborate set-up or a murderous assassin appearing ex nihilo. If I fail to prevent e₂₀₇₀, my attempts to prevent e₂₀₇₀ will be strangely thwarted. My attempts at suicide all misfire; a vasectomy turns into an appendectomy by mistake; all assassins who hunt for my granddaughter when she’s finally born mysteriously slip on banana peels when they try to fire their guns. The crucial claim the bilker makes about this is that, in cases where I fail to prevent e₂₀₇₀, it’s unreasonable to believe e₂₀₇₀ caused e₂₀₁₈; far more reasonable, they say, is that e₂₀₁₈ caused e₂₀₇₀ and we’re in a case of weird forward causation but with no time travel. Thus, succeed or fail, no time travel takes place. Since e₂₀₁₈ and e₂₀₇₀ are just placeholders, all time travel is impossible. Perhaps the problem is merely epistemological rather than metaphysical, at best showing that rational agents are always unjustified in believing in backwards time travel rather than it being metaphysically impossible [Ben-Yami 2007; Wasserman 2018: 152]. But we can set aside that issue since the crucial claim is unpersuasive anyhow. Bilking scenarios from the wider literature tend to focus on simple cases: Black [1956] focuses on a man predicting coin tosses in advance of the coin being flipped and Dummett [1964] imagines a chieftain dancing after a successful hunt in order to ensure its success. If a man predicts ‘heads’ and nothing stops the coin then coming up heads (e.g. everyone who refuses to flip the coin has an involuntary arm spasm, every magnet deployed to stop it flipping spontaneously demagnetizes etc.), it’s reasonable to think the man’s a con artist or that he possesses some demonic wizardry. Similarly, if nothing can stop the chieftain dancing (e.g. all bonds are broken, all bullets fired at him bounce off of his preternatural iron-like hide etc.), we might suspect that successful hunts cause chieftains to dance. But whilst time travel tales may be weirder than the forward causal stories in the case of coin flips and tap-dancing kgosi, they are far less weird in more complex cases like that of my granddaughter. In the forward causal story, according to which no time travel occurs, there are too many unanswered questions: why did she appear ex nihilo in 2018? Why does she look exactly like my granddaughter in 2070? Why do they both have the same mental states of hate and rage? Where does my granddaughter vanish to (in 2070) when she steps into the alleged time machine if it’s not into the past? Assuming our best physics justifies thinking time travel is possible, why ever believe the weird ‘forward causal’ story instead? Or another example. Imagine I throw all my enemies through a time portal to the Cretaceous Period. It’d be deliberately intransigent to suggest that the appearance of people in the past isn’t caused by me shoving them through the gate and that, instead, I pushed them through the gate because they died 75 million years ago. And as we unearth multiple skeletons of immense age, each evincing fatal wounds from velociraptors and each with dental records matching those of my nemeses, it seems absurd for me to say that it’s not my fault they died. Bilking wouldn’t

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stand up in a court of law when I was brought to trial for murder! Whilst there are some scenarios where a forward causal story is a better explanation, there are lots where it is not. ‘Bilking’ arguments are by the by.

6.3 Non-Ludovician Change Ludovicianism is one response to the Paradox of the Changing Past. But it’s not the only one. The rest of this chapter discusses ‘indexed worlds’ at which time travellers can change the past.

6.3.1 Indexed Worlds Propositions are often ‘indexed’ to elements. For instance, if it’s raining, then it’s not raining simpliciter, it’s raining now; hIt is raining in Liverpooli is a ‘metaphysically incomplete’ proposition, a ‘metaphysically complete’ proposition would specify when it was raining in Liverpool, e.g. hAt 12:04 on 3 May 2016 in Liverpool it is rainingi. Roughly speaking, an ‘indexed world’ is: (i) A world at which propositions are indexed to an element we tend not to think they’d normally be indexed to. Complete propositions are of the form hAt ζ: At t: φi where ζ is that extra element; (ii) Time travellers leave an element when they go into the past and arrive at a new element. The elements are ordered such that time travellers cannot return to the past of the original element (for instance, if, at ζ₁, I time travel from 2018 to 1930, I arrive at 1930 in ζ₂ and can never return to ζ₁). This chapter discusses two types of indexed world: one where propositions are complete only when indexed to universes and those where completeness requires propositions to be indexed to hypertimes. This discussion isn’t exhaustive of what indexed worlds there might be: Lockwood’s worlds of ‘space-timeactuality’ (see §6.3.5) are a substantial variation on universe-indexed worlds; Bernstein’s [2017] theory of the ‘movable objective present’ is a substantial variation on the hypertemporal theory; Lebens and Goldschmidt [2017] similarly introduce a variant of the hypertemporal theory whereby we can do without hypertimes; and so on. Some indexed worlds might even index to regular times rather than universes or hypertimes. In that case, in 2015 hAt 2015: At 1930: ¬Nikk is in Bold Streeti is true, but come 2018 (when I’ve already used my time machine), the past has changed and hAt 2018: At 1930: Nikk is in Bold Streeti is now true. Theories of time travel along these lines have been

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discussed by Goff [2010], Meiland [1974],⁴ and Wasserman [2018: 99–106]. For the sake of space, I won’t discuss these variations and will limit my discussion just to the indexed worlds discussed below.

6.3.2 Universe-Indexed Worlds At universe-indexed worlds there are multiple universes. Time travellers leave one universe and arrive at another which is qualitatively identical to the universe they left up until the moment of their arrival. Once in that new universe, the time traveller can causally affect its history however they want. Universe indexing appears both in fiction and physics. In fiction, it is often a mere MacGuffin wrapped in technobabble (e.g. Clee’s Branch Point [1996], Mehta’s I’ll Follow You Down [2013], and Gentry’s Synchronicity [2016]). At other times it is presented in a fashion more authentically representing the theory offered in this chapter (e.g. Baxter’s The Time Ships [1995], Carruth’s Primer [2004], and Scott’s Déjà vu [2006]). In the physics literature it appears in the form of Deutsch and Lockwood’s theory, explained below [Deutsch and Lockwood 1994a; Deutsch 1997: 289–320; Lockwood 2005: 322–30], either as the proposed answer to the Grandfather Paradox [Al-Khalili 1999: 185–91; Greene 2004: 455–8; Gribbin 1992: 202] or as, at the least, a plausible, live option [Bronnikov and Rubin r6

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Figure 6.1. UniverseD indexing. ⁴ One may read the title of Meiland’s 1974 paper—‘A two-dimensional passage model for time travel’—as being about hypertemporal worlds (see Loss [2015: 3]). But, at some junctures, Meiland indicates otherwise. When indexing past truths to a present time, he never hints that the indexed element is some new ‘hypertime’ rather than a regular time. Moreover, a key part of his argument requires only that the (regular, non-hypertemporal) past changes over time: ‘Briefly put, my position is that the past itself is a continuant. Being a continuant, the past exists at different times and therefore can be different at one time from what it is at another time’ [1974: 160; his emphasis].

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Figure 6.2. UniverseF indexing.

2013: 161–2; Davies 1995: 251]—although some are suspicious that the theory isn’t truly grounded in quantum physics [Dunlap 2016]. There are two ways of understanding universe-indexed worlds, differing over what counts as a universe: universes are maximal spatiotemporally connected regions disconnected from one another (‘universesD’); or universes (and their contents) fission, forming a branching structure whereby a universe is one complete branch (‘universesF’). Figure 6.1 depicts a universeD-indexed world; Figure 6.2 depicts a universeF-indexed world. Deutsch and Lockwood argue for a universeD-indexed world. They believe that time travel is best explained using quantum computation which—when interpreted using the Everett interpretation—gives us time travel within a universeD-indexed world [Deutsch 1991: 3206]. (Abbruzzese [2001] also has something like this in mind when he discusses universe-indexed worlds.) In universeF-indexed worlds, universesF overlap with one another, branching off at the point of a time traveller’s arrival. You might be familiar with universeFindexed worlds from Doc Brown’s explanation of time travel in Zemeckis’s Back to the Future II [1989]. In the physics and philosophical literature, though, it receives little explicit endorsement [Gardner 1987: 7–8] (although those who are not themselves universe indexers often think universe indexers have in mind universeF indexing [Carroll, J. 2016: 184–6; Lewis 1976: 152; Richmond 2003; Toomey 2007: 253–4; Williams 2018: 225]). Nonetheless, it’s not hard to see why one might latch onto it. Deutsch focuses on universeD-indexed worlds because he treats the ‘many worlds’ of Everett’s many-world interpretation as being many universesD [Deutsch 1985]. If we thought they were instead many universesF [Saunders and Wallace 2008: 299], a world of time travel would instead be a universeF-indexed world.

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6.3.3 Hypertemporal Worlds The other commonly discussed type of indexing uses ‘hypertimes’ instead of universes [Goddu 2003, 2011; Hanley 2009: 339; Hudson and Wasserman 2010; van Inwagen 2010]. Just as with universe-indexed worlds, there are two types of hypertemporally indexed worlds (which is no coincidence because hypertemporal indexing will turn out to be a spin on universe indexing). The first type of hypertemporally indexed world is ‘conterminously indexed’. Imagine, as Goddu [2003] does, that a second hypertemporal series exists alongside the regular temporal series we’re familiar with. The situation is analogous to watching a film on an old-style VCR whilst a clock hangs on the wall. We have two temporal series (time according to the VCR and time according to the clock). Normally, as the clock advances so, too, does the VCR. However, when the tape is rewound the series come apart for the clock advances and the VCR readout rolls back. Time and hypertime are meant to be similar: usually, when an instant elapses, a hyperinstant also elapses, but a time machine can cause time to move back even though hypertime advances. For example, using Ts for hypertimes and ts for regular times, imagine time and hypertime are in sync until T₁₀₀/t₁₀₀ (so at T₁ it’s t₁, at T₂ it’s t₂ etc.). At T₁₀₀ (which is also t₁₀₀) I use a time machine to return to t₃₀ and time ‘rolls back’—at T₁₀₁ it’s now t₃₀ again (and at T₁₀₂ it’s t₃₁ etc.). This is a ‘conterminous’ version of hypertime because times are conterminous with the hypertimes, and as the temporal series changes (either by advancing or ‘rolling back’), the hypertemporal series changes too (although always by advancing).⁵

⁵ I believe Loss [2015] also argues for conterminous indexed worlds, for he has two ‘types’ of time (even though they’re ‘types’ of a single temporal dimension). Van Inwagen [2010] also seems to have them in mind. It’s also the only way I can make sense of Cramer’s [1986] ‘pseudotime’, which plays a role in his transactional interpretation of quantum mechanics (mentioned in §1.2.3). Cramer says that when an emitter emits a particle, the absorbers it might end up striking are first sent ‘offer waves’. These move forwards in time and strike the absorbers, which then respond with ‘confirmation waves’. The confirmation waves, which have a strength proportional to the probability of the particle being absorbed by that absorber rather than another, travel back in time and strike the emitter. Then the emitter, based upon the strength of the confirmation waves, emits the particle towards one of the absorbers. The process of the offer/confirmation waves moving between emitter/absorber takes place in ‘pseudotime’ which, Cramer says, is ‘presented [as] a time sequence external to the process’ [1986: 661n14]. Whilst Cramer warns us not to take his pseudotemporal explanation literally, one would ultimately need some literal explanation of what is meant to be going on [Berkovitz 2008: 727; Maudlin 2002: 198], so it’s worth tarrying over what a literal understanding of pseudotime would amount to. I believe it’s best borne out by a world of conterminous hypertime for different parts of hyperhistory end up being eradicated. If hypertime and time are in sync until T₁₀₀ and I time travel to t₃₀, then the events of T₃₀–T₁₀₀ won’t be part of history from T₁₀₁ onwards—they are an interval which will never have been. Such intervals sound like Cramer’s pseudotime: when the time travelling confirmation waves roll back history by travelling backwards through time, the historical record of the offer/ confirmation waves travelling between emitter/absorber are wiped away and end up in a temporal sequence outside of the new historical one (a ‘pseudotemporal’ sequence). I do not suppose that this brief discussion comes close to telling us whether or not taking pseudotime literally helps or hinders the transactional interpretation of quantum mechanics—my point is merely that conterminous hypertime can put some meat on the bones of Cramer’s rough and ready pseudotemporal talk.

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As it stands, this is a depressing form of time travel since time travel ‘kills’ everyone you leave behind by cutting short their futures [cf. van Inwagen 2010: 18n23]. Moreover, there could be cases where time stops moving forwards because it keeps looping. For instance, in Postiglione’s Tiempo Muerto [2016] the lead character, Franco, tries to change time to stop his girlfriend, Julia, from having died; he succeeds, but then dies himself in the past; Julia (now alive) then tries to change time to stop Franco from dying; she succeeds, but then she dies in the past; and so on. If time kept rolling back every time one travelled in time, a repeating loop like this would mean regular time never moved past a certain point (even though hypertime would keep advancing). We could even imagine CosmoTerrorists building a time machine which went back in time a split second only to activate itself to, again, go back in time a split second—hypertime would move forever forward, but only to replay the same, arbitrarily short, interval. How frightening! Conterminous hypertime is only one way of understanding hypertime. A world of exterminous hypertime also has two temporal series but, unlike with conterminous hypertime, time can pass without hypertime passing. (Hudson and Wasserman [2010], Hudson [2014], and Tognazzini [2016] explicitly discuss worlds of exterminous hypertime; Bernstein [2015] discusses something similar to it.) Riffing on van Inwagen’s example [2010: 15–19], imagine God watching the entirety of four-dimensional spacetime in front of her—it stands to her as a film reel stands to the director, with each scene laid out in front of her such that she can see them all at the same time. Imagine God then reached down and started to toy with spacetime. First, she reaches down and squishes my grandfather, ‘Pappy’, in 1930. Instantly, the future is changed from her perspective such that I am never born. Changing her mind, God then sculpts a new Pappy and puts him back into 1930. The future is changed once again, back to how it was. The best way to understand such a tale is to imagine that, in addition to the temporal series of the spacetime God is looking at, there’s a second, separate, temporal series (i.e. the hypertemporal series) which God is sitting in. At T₁ God sees our universe as it is. At T₂ she reaches down and squishes Pappy in 1930. At T₂, hAt 1930: Pappy is alivei is now false and the rest of post1930 history is also different such that, at no regular time at T₂, am I born. Because God decides she quite likes me, she puts Pappy back as he was at T₃. Thus every (regular) time at T₃ is how it was at T₁. No contradictions are true, for the apparently contradictory state of affairs of Pappy being dead and alive obtain at different hypertimes. If that story makes sense—and I think it does—time travel situations can be explained in the same way. Time travellers leave one hypertime and travel to the next. There they can change the past. For instance, I can travel in time, leaving t₂₀₁₉ at T₁. I’d then travel to t₁₉₃₀ at the next hyperinstant, T₂, where, at T₂, I kill

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Pappy (and subsequently change history at T₂ such that the times from t₁₉₃₀ onwards are all different from what they were at T₁).⁶ The ontology of a hypertemporal-indexed world is suitably similar to that of a universe-indexed world. Consider exterminous hypertime first. At a world of exterminous hypertemporal indexing, an entire regular temporal series exists at each hyperinstant. Each regular temporal series is composed of hyperplanes of points and then the fusion of those hyperplanes is related to another fusion of such hyperplanes by relations of hypertemporality. The ontology is the same as that of a world of universeD indexing. See Figure 6.3. And similar reasoning applies to a world of conterminous hypertime and of multiple universesF. See Figure 6.4; the ⁶ One quibble concerns the talk of moving to the ‘next’ hypertemporal instant. That needs hypertime to have a discrete, rather than a continuous, structure. If a temporal series necessarily had to be continuous, exterminous indexers would then have to say that a time traveller moved forward in hypertime by a fixed unit, skipping the intervening hyperinstants. That would raise awkward questions concerning the arbitrariness of the fixed interval. With an eye on moving on to other concerns, set aside such worries.

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left-hand side depicts a world of conterminous hypertime whilst the right-hand side is the same diagram but with some reorganization. Clearly the right-hand side is the same as the world of universeF indexing—the only difference is the addition of hypertemporal relations. So the ontological commitments of universe and hypertemporal indexing are the same; the types of worlds differ only over what relations hold between the things at those worlds.

6.3.4 Do Indexed Worlds Allow for Change? At (at least some) indexed worlds, the past can be changed by travelling in time. The opening gambit to show this imagines a universe-indexed world. In it, I travel to the past and arrive in 1930s Bold Street. There are now two universes, U₁ and U₂. At U₁ I am not in 1930s Bold Street; in U₂ I am. So goes the gambit: the past has been changed. But some complain that this is not changing the past at all [Baron 2017; Lewis 1976: 152; Smith 1997: 365–6]. If the past has been changed, with respect to what proposition has the past changed? The proposition h[At U₁: At 1930: ¬ Nikk is in Bold Street]∧[At U₂: At 1930: Nikk is in Bold Street]i is forever true. That never changes. In the past I’ve always been in 1930s Bold Street in U₂ and in no sense am I ever in 1930s Bold Street in U₁; that never changes, so no change has happened. Were this worry founded, then even were indexed worlds metaphysically possible, then they wouldn’t ground the possibility of non-Ludovician change in time travel cases. I think this objection is unfounded (Goddu [2011] and Law [2019] offer arguments along similar lines to that which follows). Set aside time travel, universe-indexed worlds, and hypertimes for just a moment. Simply consider the variation of an object’s properties at both different regions of space (e.g. a metal cone which is wide at one end and narrow at another) and different times (e.g. the cone being shiny at one instant and rusty at a later instant). Conservatives about change will say that only the temporal variation qualifies as a type of change, i.e. only the change from shiny to rusty qualifies as change. We might not, though, be a conservative. Liberals would say that the variation of the cone from being wide to narrow is also a type of change; temporal change is just a particular type of change, rather than a metaphysically distinct brand of phenomenon. Liberals should say that the past has changed at the universe-indexed world. The past is lamentably Nikkless in one place (i.e. U₁) but the act of time travel causes it to be joyously Nikkful in another (U₂). When asked which proposition has changed, the answer is simple: The metaphysically incomplete proposition

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hAt 1930: Nikk is in Bold Streeti has changed. And it’s no problem for metaphysically incomplete propositions to be the subject of the change. For instance, if it goes from not raining in Liverpool at t to raining in Liverpool at t+, then the subject of the change is the metaphysically incomplete proposition hIn Liverpool, it’s rainingi. Indeed, this shows that it was always an odd thing to think that a metaphysically complete proposition like h[At U₁: At 1930: ¬ Nikk is in Bold Street]∧[At U₂: At 1930: Nikk is in Bold Street]i was ever going to change, for when we consider just non-indexed worlds it’s not as if the metaphysically complete proposition h[¬At t: In Liverpool, it’s raining]∧[At t+: In Liverpool, it’s raining]i could ever change either. Nevertheless, the world manages to change with respect to Liverpool’s dreariness and does so because the subject of that change is a metaphysically incomplete proposition. No surprise, then, that changing time in time travel cases involves changing the truth of a metaphysically incomplete proposition. Of course, one might instead be a conservative about change. In the same way that one might think that changing from being wide to narrow is no type of change, the spatial variation of Nikklessness in U₁ to Nikkfulness in U₂ is no type of change either. That’s a fair enough complaint. But conservatives about change can then turn to hypertemporal worlds. In a hypertemporally indexed world there are two hypertimes, T  and T +, where the former is hyperearlier than the latter. At T, in 1930 I’m not in Bold Street; at T +, in 1930 I am in Bold Street. The conservative can now say more or less the same as the liberal would say of the universe-indexed world: the change occurs with respect to the metaphysically incomplete proposition hIn 1930: Nikk is in Bold Streeti; the metaphysically complete proposition h[At T : At 1930: ¬ Nikk is in Bold Street]∧[At T +: At 1930: Nikk is in Bold Street]i may never change, but that’s only as worrisome as the metaphysically complete proposition h[¬At t: In Liverpool, it’s raining]∧[At t+: In Liverpool, it’s raining]i never changing at non-indexed worlds, i.e. not worrisome at all. The key difference is that the variation of the truth of the metaphysically incomplete proposition is along a (hyper)temporal dimension, rather than a spatial dimension, and so qualifies as change even for the conservative. In conclusion, just as time allows the world to change from being one way to another, hypertime (and, for the liberal, the multiverse) allows the past to change from being one way to another. And that these indexed worlds allow for that seems substantially far enough away from what the Ludovician had in mind that this counts as non-Ludovician change. What it won’t allow for is for the past to be ‘scrubbed away’. If you want to go back and put right what once went wrong, you can’t make it the case that the wrong is no longer any part of reality’s record—it’ll still be the case that at some earlier hypertime, or at some other universe, the wrong takes place. But that’s no threat to saying that the past has changed. In a regular non-indexed world, when

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I put my hand on the stove and it hurts, I am still satisfied by the change in the world I get when I remove my hand from the scorching hot hob. Whilst the past is indelibly scarred by my pain, it is enough that I am no longer in pain. Similarly, whilst you can never make it the case that hyperearlier some past wrong never took place, it is enough that hyperpresently, and at all hypertimes from hereon in, the past wrong never took place. Lebens and Goldschmidt [2017: 5–8] argue that this stronger type of change, whereby time travel allows for the past to be scrubbed away, is in fact possible. I believe they are incorrect. Assume hAt t₁: φi is true. At a hypertemporal world, that fact can change; hAt T₁: at t₁: φi is true but we act to make hAt T₂: at t₁: φi false. Call that change ‘C1’. Given a third temporal dimension (composed of hyperhyperinstants, T*₁, T*₂ . . . ), we can change whether hAt T₁: at t₁: φi is true. In that case, hAt T*₁: at T₁: at t₁: φi would be true but hAt T*₂: at T₁: at t₁: φi would be false. Call that change ‘C2’. We can go again, for instance ‘C3’ would change hAt T*₁: at T₁: at t₁: φi from being true to false in a fourth temporal dimension. Lebens and Goldschmidt imagine a world with an infinite number of higher-order temporal dimensions. At that world, we conduct an infinite number of changes (i.e. a supertask of changes!) such that Cn has been carried out for any value n. Once the supertask has been completed, hAt t₁: φi wouldn’t be true at any order of hypertime. However, this supertask isn’t possible. Represent a temporal operator n of any order by ‘At T l’ where n is the ‘order’ of the dimension and l is some point along that dimension (thus ‘At t₁’ and ‘At T¹₁’ are the same operator, as are ‘At T₁’ and ‘At T²₁’, as well as ‘At T*₁’ and ‘At T³₁’). Were Lebens and Goldschmidt correct then, after the supertask has been conducted, hAt t₁: φi won’t be true at any concatenation of temporal operators, i.e. no proposition of the form hAt T mi: . . . at T²j: at T¹₁: φi will be true at a supertasked world. I admit the possibility of worlds with an infinite number of temporal dimensions, as well as worlds at which no proposition of the form hAt T mi: . . . at T²j: at T¹₁: φi is true. But such worlds qualify neither as worlds where hAt t₁ φi has changed and been ‘scrubbed away’, nor as worlds at which the appropriate supertask has been conducted. ‘Changed’ and ‘carried out’ are past-tensed verbs; for our time machine to have changed hAt t₁ φi to being false—i.e. for the supertask to have been carried out—requires there to be some time, of some order, at which hAt t₁ φi is true, i.e. for there to be some true proposition of the form hAt T mi: . . . at T²j: at T¹₁: φi. But ex hypothesi the supertask means that no such proposition is true. Thus we should conclude that there is no world at which the supertask has been conducted; there’s no way to distinguish a world at which hAt t₁ φi is simply false (i.e. it’s never, in any sense, been true) from a world where the time travelling supertask has been conducted and hAt t₁ φi has been changed to being false. Since there is no such (logically possible) world, the supertask is apparently impossible. Thus Lebens and Goldschmidt haven’t demonstrated that the past can be ‘scrubbed away’ in the requisite fashion.

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6.3.5 The Doppelgänger Objection A different worry is that one can’t change the past because the contents of the indexed element that one arrives at aren’t the contents of the past that you left. For instance, in the case of universe indexing, the worry is that if I go back to Bold Street, I arrive at a universe which looks like 1930, and get to walk down something which looks like Bold Street, but in actuality it isn’t. The ‘past’ I arrive at is no more my past than an intricate, well-staged mock-up of the past qualifies as being the past; for all intents and purposes, I may as well have stayed at home and simply created a simulacrum of Bold Street instead [cf. Smith 1997: 366]. In that case I’ve not changed the past with respect to the (incomplete) proposition hAt 1930: Nikk is in Bold Streeti because Bold Street exists only at the first universe and doesn’t have me in it during 1930. This objection has been levelled against both universe indexing [Abbruzzese 2001: 37; Arntzenius and Maudlin 2013; Dainton 2010: 126–7; Hewett 1994; Richmond 2003: 303–4, 2008: 44] and hypertemporal indexing [Forbes 2010; Smith 1997: 365–6]. This objection doesn’t work. At (at least some) indexed worlds, the contents of the destination element can ‘count as’ the contents of the origin element. With an eye on remaining neutral over what it takes for one thing to count as another—for it need not be that numerical identity holds between the different Bold Streets—I’ll say that the times and their contents need to be ‘appropriately connected’ in order to appropriately count as one another. So imagine I go back in time to kill Pappy in 1930; if non-Ludovician change is possible, we should be able to change the past so that he goes from being alive in 1930 to being dead in 1930. The challenge can then be restated thus: why is Pappy from one element (‘PappyO’) appropriately connected to his alleged doppelgänger in another element (‘PappyA’)? Some answers are better than others. Deutsch and Lockwood [1994b] say that qualitatively identical things are appropriately connected. But PappyA isn’t qualitatively identical to PappyO—in 1930 he’s wondering why someone’s pointing a rifle at him whilst PappyO isn’t. Presumably the suggested fix would be that things are appropriately connected iff they were previously qualitatively identical. But that can’t be right either. Philipp, Landgrave of Hesse, was Prince Wolfgang of Hesse’s twin. Assume, as isn’t entirely unreasonable, that at one point in their mother’s womb (say, when they were one-celled organisms) they were—solely by chance—qualitatively identical. Philipp and Wolfgang are not thereby appropriately connected for the rest of their lives. If they were, we would hold Wolfgang responsible for Philipp’s participation in the Nazi programme of forced euthanasia. Because we don’t stop for one minute to consider holding someone responsible for the crimes of their identical twin, intuitively their being qualitatively identical at one time doesn’t ground their being appropriately connected thereafter. Similarly for PappyA and PappyO.

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This response, though, takes us to the ballpark of the correct solution: PappyA/ PappyO might be exactly like Philipp and Wolfgang in another respect, i.e. they’re ‘fission counterparts’ of one another. Consider a fission case like that of Philipp and Wolfgang fissioning from the same zygote, an amoeba splitting in two, or the Ship of Theseus. Call the initial object (e.g. the zygote, original amoeba, or ship before we start doing funny things with replacing the planks) the ‘parent object’. The ‘fission counterparts’ are the objects we end up with after the fissioning event (e.g. Philipp and Wolfgang, the two amoebas, or both the ship made of reassembled planks and that which is made via continually replacing planks). Intuitively, fission counterparts were both once the parent object, i.e. they are appropriately connected to the parent object. (For example: if a criminal commits a crime and then fissions, you should punish both of his fission counterparts.) Equally, intuitively the fission counterparts aren’t one another, i.e. aren’t appropriately connected to each other. (For example: if a person splits and one fission counterpart murders someone, you aren’t justified in punishing the other fission counterpart.) Fission counterparts are numerically distinct, yet appropriately connected, things. If PappyA and PappyO are like fission counterparts, the problem is solved. And that’s a natural solution for both the universeF-indexed world and—since they have the same ontology—a world of conterminous indexing. So at least some indexed worlds don’t face a problem with appropriate connection. It’s less clear how well this would work at other indexed worlds, i.e. at universeD-indexed worlds, or worlds of exterminous hypertime. Perhaps we’d be able to say a similar thing about universeD-indexed worlds. Pappy could be an object composed of temporal parts from different universesD. Or, in the same way that enduring entities are multi-located at multiple spacetime regions, Pappy might be a multi-located entity exactly located at different spatiotemporal regions at different universesD. Both options are far from unproblematic, though. A natural understanding of universeD-indexed worlds would have it that Pappy isn’t causally intraconnected between the different universesD—that is, his temporal parts (given the former perdurantist theory) or his states at different universesD (given the latter endurantist theory) aren’t causally connected. It’s then hard to see how he can be a unified person across universesD without the appropriate causal connections [cf. Effingham 2015b: 32–6]. The response looks more plausible at worlds of exterminous hypertime. Goddu [2011: esp. 17n8], for instance, says that Pappy is an object composed of temporal parts from the different hypertimes. If we were to then persist across exterminous hypertime just as we persist across time, there’ll be (possibly immanent) causal connections between Pappy’s temporal parts at the different universesD, solving the issue that is a problem for universeD-indexed worlds. But now there’ll be causal overdetermination: how I am at time tn at hypertime T₂ will be caused not only by my pre-tn-temporal parts from T₂ but also by my hyperearlier tn-temporal part from T₁. However, as an anonymous referee pointed out, overdetermination

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is at best a reason to think we don’t actually live in an exterminously hypertemporally indexed world, not that such worlds are metaphysically impossible. A final alternative is to ditch the universes of universe-indexed worlds entirely. Lockwood appears to do that in his 2005. Where he agrees with Deutsch that quantum mechanics correctly explains how time can be changed, he opts for a many-minds interpretation of quantum mechanics rather than a many-worlds interpretation [Lockwood 1996]. Now, rather than there being a multiverse, there is a single manifold of ‘space-time-actuality’ across which a mind manages to have distinct, conflicting perspectives [Lockwood 2005: 316]. Given this theory, there won’t be a doppelgänger objection—at least, not for anything which qualifies as a mind. In any case, whatever the demerits some indexed worlds have in coping with doppelgängers, we can conclude, at the very least, that some indexed worlds don’t have a doppelgänger problem (i.e. universeF-indexed worlds and worlds of conterminous hypertime).

6.3.6 Making Sense of Hypertime A special problem faces hypertemporal indexers, namely what on earth hypertemporal relations are meant to be! We might be able to coin the words ‘hyperearlier’ and ‘hyperlater’, and then talk with gay abandon of hypertimes, but it’s not implausible to think that this is all just a meaningless word salad for which we have no idea of to what it is meant to correspond. Especially given the key role hypertemporal relations play in allowing the past to change for conservatives about change, we can only be happy if we properly understand hypertemporal ideology. I canvass two solutions: Loss’s, which I think doesn’t work, and a recombinatorial solution, which I think does. Loss [2015] argues that even though time is one-dimensional, there are two temporal series. When activated, a time machine moves forward in one temporal series and back in the other. Whilst he seems not to accept hypertime—indeed he specifically contrasts his view with hypertemporal views—insofar as he believes in two temporal series which often advance in sync with one another, he qualifies as a conterminous hypertemporal indexer (at least, as I’m using the term). Loss starts by imagining a world at which the regular temporal series advances forwards inexorably. In 2018 I activate a putative time machine. Upon activation, the world returns to looking exactly as it did in 1930 with the exception of the machine staying how it is. This is effectively the form of ‘time travel’ I discussed back in §1.4; the worry introduced there was that whilst the world looks like it’s 1930 all over again, that isn’t really the case—all that has happened is that the world around me has been morphed. But Loss says time travel has happened. There is a second temporal series, he says, and whilst I’ve merely moved forwards

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in one of them (and it’s still 2018), I’ve moved backwards in the other (and it’s now 1930 all over again). So, in at least one sense, I’ve travelled in time. Loss has two reasons for believing that the described world contains a second temporal series. The first reason is that whilst, in external time, I’ve travelled forwards to 2018, Loss believes that in the personal time of the universe, I’ve gone backwards to its 1930 personal time. To travel in time is to travel back in the universe’s personal time. But given how I understand external time, this is impossible. In Chapter 3 I argued that external time just is the universe’s personal time, so I can’t allow for Loss’s distinction between external time and the universe’s personal time—for me, they are both the same. Loss has another argument for believing that there is a second temporal series. Imagine a world identical to our own up until 1990 at which point, for ten years, the world consists of nothing but spheres bumping into one another. Call these the ‘sphere years’. After the sphere years, i.e. in the year 2000, the universe becomes a qualitative duplicate of the actual world from 1991 onwards. Loss says that were we to become aware that this was the world we lived in, we’d develop two ways of talking about history. One way would recognize that the sphere years took place. When talking in that context we’d say that the events of 1990 (e.g. the first Starbucks opening) took place eleven years before the events we otherwise would’ve thought took place in 1991 (e.g. the Gulf War). But it’d be cumbersome to constantly recognize the sphere years. We’d quickly develop a second way of talking whereby we’d ignore the sphere years and would instead say that the Gulf War took place only a year after Starbucks opening. So we’d have two perfectly acceptable contexts, according to which different temporal intervals separate different events. Call the first context—the context in which we recognize the sphere years—the Recognition Context. Call the second context—the context in which we conveniently ignore the sphere years—the Ignored Context. Loss thinks that in the case of time travel, we can similarly have multiple temporal series. Fundamentally, there’s only one temporal series, but there are two different ways/ contexts in which we can talk about it. In one context, we recognize the years between 1930 and 2018 where I grew up, lived, and made a time machine; in that context—the Recognition Context—I have travelled forward in time rather than back to 1930 (and in that context I’ve not travelled in time; I’ve merely morphed the world around me). In another context, we ignore those years. In that context— the Ignored Context—we exclude from consideration the years between 1930 and 2018. In the Ignored Context, says Loss, I’ve travelled in time, for in the same way an event occurring just after the Sphere Years is, in the Ignored Context, only seconds after 1990 rather than a decade, I am now only seconds later than 1929 rather than eighty-eight years. But Loss is wrong to think that this counts as time travel, for in neither context does anything travel in time. Clearly nothing travels in time in the Recognition Context. In that context I’ve only travelled forward in time and morphed the

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world around me—it is explicit that, in this context, no time travel has taken place. But in the Ignored Context, I also fail to travel in time. To see why, first consider the Sphere Years scenario again. It makes no sense, if we’re in the Ignored Context, to ask how long an interval there was between the last sphere bumping and the Gulf War. It makes no sense because we are ignoring the sphere bumpings. Similarly, in the Ignored Context we are ignoring the ‘previous history’ where I grew up, lived, and made a time machine in 2018. And if we are ignoring that previous history, then we can’t say that I travelled from there; to say that I travelled from a place would be to acknowledge it, rather than ignore it. So in this context, nothing has come from the future. In this context, nothing has travelled in time. Compare: imagine you were in a context where quantification was restricted so as to exclude the USA; you will be promptly stymied if, in that context, I ask you where Barack Obama was born. For instance, if I say, ‘Forget about Hawaii— ignore it, and focus just on where we are. That said, where does Obama come from?’, my first sentence might implore you to trump other contextual demands and not answer ‘Hawaii’, but it’s hard to see how that can happen given the question itself. The question ends up being absurd. Similarly, in the Ignored Context we can’t acknowledge that time travel has taken place. And since I don’t travel in time in either the Recognition Context or the Ignored Context, there’s no context in which the world Loss describes counts as a world at which time travel has taken place. That said, we can set aside Loss’s theory as a possible explanation of hypertime (and of time travel in general). Nevertheless, I believe that we can make sense of hypertime. We can understand hypertime as being an extra dimension of time, standing to the regular temporal series as forward/backward stands to left/right or up/down. Just as a flatlander might be initially sceptical of up/down, but then have it explained to them that there’s an extra spatial dimension, we should not be overly suspicious of the additional (hyper)temporal dimension. To understand hypertime in this way, we need to show two things: (i) that there being two temporal dimensions is possible; (ii) that at least some bitemporal worlds are the way the world would be according to either or both of the exterminous or conterminous indexer.

6.3.6.1 Bitemporality Is Possible Start with bitemporality’s metaphysical possibility.⁷ I will advance a ‘recombinatorial argument’ for the possibility of bitemporal worlds. It’s popular to accept a

⁷ There’s already an extant argument for two-dimensional time by Thomson [1965] and Macbeath [1993] (it’s been extensively discussed [King 2004; Richmond 2000; Nusenoff 1977; Wilkerson 1973, 1979]). We can’t rely on that here for there are (justifiable!) worries that their argument already relies upon time travel being possible [Nusenoff 1976: 341; Oppy 2004].

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principle of recombination whereby something is possible if it can be ‘recombined’ out of entities (or their duplicates) plus any pattern of fundamental relationships. Defining an acceptable principle of recombination can be tricky and competing versions are available [Darby and Watson 2010; Efird and Stoneham 2008; Nolan 1996; Wasserman et al. 2004]. For argument’s sake I’ll capture it with the following two principles: R: Necessarily, for any series of (possibly identical) objects, x₁, x₂ . . . , and any series of (possibly identical) perfectly natural relations, R₁, R₂ . . . , where A(Rm) is the adicity of Rm there exists a metaphysically possible world at which R₁(x₁ . . . xA(R1)), R₂(xA(R1)+1 . . . xA(R1)+A(R2)), R₃(xA(R1)+A(R2)+1 . . . xA(R1)+A(R2)+A(R3)), and so on. D: Necessarily, for any series of objects, x₁, x₂ . . . , and any series of (possibly identical) cardinals, n₁, n₂ . . . , there is a metaphysically possible world at which n₁ duplicates of x₁ exist, n₂ duplicates of x₂ exist etc. Given these principles, we can guarantee the possibility of worlds with varying numbers of dimensions. As a precursor, consider worlds with extra spatial dimensions. Imagine there were but two points standing in the quantitative relation ‘__ is spatially separated from __ by one metre’. Imagine that relation is perfectly natural. Such a world would be one-dimensional. Given D, there’s a world of three points. Given R, those three points can all stand in that same quantitative relation to one another. Such a world would be two-dimensional. Given D, there’s a world of four points. Given R, those four points can stand in that quantitative relation to one another. Such a world would be three-dimensional. And so on. Thus, there can be a world with any number of spatial dimensions. In a similar fashion, we can guarantee a world with any number of temporal dimensions. If some quantitative temporal relation—e.g. ‘__ is later than __ by one second’—was perfectly natural, then an n-dimensional time would, for any n, be possible. Conclusion: as long as the quantitative temporal relation is perfectly natural, R and D entail the metaphysical possibility of bitemporality. Perhaps the quantitative temporal relation isn’t perfectly natural because temporal relations aren’t perfectly natural.⁸ For instance, maybe the fundamental ⁸ Alternatively, perhaps some temporal relations are natural, but they are not quantitative temporal relations. Imagine quantitative relations are derivative of perfectly natural non-quantitative temporal relations. For instance, comparativists say that quantitative relations are derivative of comparative relations [Dasgupta 2013], and a temporal relation like: In the pair the first member is later than the second member to twice the extent that the first member of is later than its second member. is the perfectly natural relation—the relation ‘__ is later than __ by one second’ is derivative and imperfectly natural. This isn’t a problem because comparative relations can also be recombined to get a

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relation is spatiotemporal rather than temporal. I believe it’s likely that similar recombinatorial reasoning nevertheless bears out the possibility of worlds which have multiple temporal dimensions. Whilst I might be wrong about this, this isn’t too worrying for it’s enough that temporal relations could be perfectly natural even if they actually aren’t. At a world at which temporal relations are natural, bitemporality would be metaphysically possible and thus (given a modal logic at least as strong as K4) it’s actually metaphysically possible. And it is possible that some (quantitative or comparative) temporal relation is natural. It’s possible because Newtonian physics, whilst actually false, could have been true—that is, it’s true at some world. And since the obvious interpretation of Newtonian physics takes some temporal relation to be perfectly natural, it follows that there’s at least one world at which the relation is perfectly natural. Thus bitemporal worlds are metaphysically possible.

6.3.6.2 Some Bitemporal Worlds Are Hypertemporal Even given bitemporality’s possibility, it doesn’t follow that hypertime is possible, for there can be bitemporal worlds which aren’t hypertemporal worlds. For instance, physicists have thought multiple temporal dimensions might account for certain physical phenomena [Bars 2006] but they’re not thereby imagining that the world is similar to that imagined by the hypertemporal indexer. Bitemporality isn’t sufficient for hypertemporality. So the second step is showing that recombination further guarantees the possibility of hypertemporal worlds. To do this, play on the crucial difference between time and space: its directionality. Things are temporally distant from one another in either a futurewards direction or pastwards direction [Earman 1974; Horwich 1987; Maudlin 2007: 103–42; Price 1996]. That said, we can construct an exterminous hypertemporal world. Start with a world with just one temporal dimension. The times at that world—i.e. the hyperplanes—are temporally separated; see Figure 6.5(a). Given D, there’s another world consisting of two such spacetimes (i.e. a world of two island universes). See Figure 6.5(b). Given R, there’s a world where the first spacetime is earlier than the second—that is, where fusions of hyperplanes stand in the same temporal relations to one another as the hyperplanes composing them stand in to one another. That world is depicted in

world of two-dimensional time. Imagine there are some instants: t₁, t₂, t₃ . . . In a world of onedimensional time there’s just such a comparative temporal relation, R, such that R for all i and j. Given D, there’s a world with extra instants t1*, t2*, t3* . . . in addition to t₁, t₂, t₃ . . . .Given R, there’s a world at which t1*, t2*, t3* . . . also form a one-dimensional temporal series (i.e. R, for all i and j). So at that world there are two, separate, onedimensional times. Given R, there’s another world at which, say, R for all values of i and j, i.e. there’s a world where every instant in the second temporal dimension is temporally separated from a corresponding instant in the first temporal dimension to a degree comparable to the temporal separation between certain instants within the first dimension. And that is what it would be for there to be a bitemporal world.

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(b)

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Temporal relations/ Direction of time

Hypertemporal relations/ Direction of hypertime

Figure 6.5. Recombining exterminous hypertime.

Figure 6.5(c) whereby the regular temporal relations are unshaded arrows and the hypertemporal relations are shaded arrows. Those temporal relations are the same, but we can distinguish temporal and hypertemporal separation by the direction they run in: temporal separation goes in one direction; hypertemporal separation goes in another. Directionality of time also plays a key role in capturing the possibility of conterminous hypertime. In worlds of one-dimensional time, every time is a hyperplane and each hyperplane is removed from the others in only one direction; see Figure 6.6(a). In a bitemporal world the hyperplanes can be temporally separated in two directions; see Figure 6.6(b). Regular temporal separation is separation along one direction, whilst hypertemporal separation is separation along the orthogonal direction. And, given R, we can arrive at a world corresponding to that envisaged by the conterminous indexer. Consider Figure 6.6(c). At that world, there are four times/hyperplanes. The first two are such that one is both later and hyperlater than the other. The third is hyperlater but is not later than the second—it is instead earlier than the second. Time has ‘rolled back’ at such a world. Time then continues; the fourth time is both later and hyperlater than the third. At such worlds hypertime inexorably moves on even though time sometimes rolls back, i.e. it’s a world of conterminous hypertime.

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(b)

(c) Direction of time

Direction of hypertime

Figure 6.6. Recombining conterminous hypertime.

6.3.7 The Metaphysical Possibility of Changing the Past So we have it: the past can be changed. That does not mean that the past can change at every world. For instance, at the least—assuming there’s no necessarily existing Godlike agent that can move things through time—there’ll be some worlds at which no agent has the power to time travel. More saliently, it is consistent to believe in worlds of both non-Ludovician time travel (i.e. indexed worlds) and Ludovician time travel (i.e. where the past can be causally affected, but not changed). Van Inwagen [2010: 5], himself a proponent of the possibility of hypertemporal worlds, takes just that view. Indeed, I have noted the possibility of changing the past mainly for the sake of completeness—I am far more interested in time travel at Ludovician worlds than I am in time travel at indexed worlds, for I see little reason to think our world would be an indexed world even were time travel physically possible (of course, a physicist like Deutsch will likely disagree!). From Chapter 7 onwards the vast majority of the book focuses on what would be the case were Ludovicianism true and we were in a non-indexed world at which time travel was possible. We start in Chapter 7 with the most famous of all paradoxes: the Grandfather Paradox.

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7 The Grandfather Paradox The Grandfather Paradox is a quasi-successor to the Paradox of the Changing Past. Where the Paradox of the Changing Past argues, less than persuasively, that time travel would necessitate a paradoxical state of affairs (and is therefore impossible), the Grandfather Paradox focuses more persuasively on the idea that time travel could bring about such states of affairs. Solutions to the Grandfather Paradox occupy our attention in Part 3 of this book and this chapter is dedicated to laying it out in detail. §7.1 explains exactly what the paradox is, breaking it down into easily digestible premises. §7.2 then briefly discusses the options for escaping it, ready for the fuller discussion in Part 3.

7.1 Variations The heart of the Grandfather Paradox is this: imagine I travel back in time to 1930 with a plan to shoot my grandfather—call him ‘Pappy’—stone-cold dead.¹ Who knows why I’d do such a thing [cf. Smith, N. 2005]: perhaps it is a fancy way to kill myself or perhaps Pappy was unkind to me when I was younger and I’m seeking revenge. Whatever the details, you are to imagine that I am stood, pointing a gun ¹ Issues girding the Grandfather Paradox originally appear in one of the first fictional tales of time travel: Enrique Gaspar’s El Anacronópete, Viaje a China-Metempsícosis [1887]. Two characters— Benjamin and Don Sindulfo—discuss time travel: Benjamin wonders whether, by using a time machine, the Moors could be stopped from taking over Spain in the eighth century. Don Sindulfo argues that it’s impossible. Gaspar’s characters make clear the issues at stake in the Grandfather Paradox: can one change the past? If not, what stays our hand? During the late 1920s these worries become showcased in the form of ‘autoinfanticide’ whereby a time traveller kills their grandfather before he met their grandmother. The first mention I know of comes from a preface by Hugo Gernsback, accompanying Kirkham’s short story ‘The Time Oscillator’, in the December 1929 issue of Science Wonder Stories (although, Nahin [1999: 254] notes, Gernsback appeared to be referencing an already extant idea). Stories playing on the idea quickly followed (e.g. Schachner’s ‘Ancestral Voices’ [1933], in which a time traveller kills a Hun warrior, and Palmer’s ‘The Time Tragedy’ [1934], in which a time traveller kills his own grandfather). But those stories were quite unlike Gaspar’s novel and Gernsback’s exordium for they failed to deal with the metaphysical issues involved in the Grandfather Paradox in just the same way that stories in which things move about generally fail to tackle Zeno’s Paradox. Whilst their narratives featured grandfathers and ancestors being killed by time travellers, those deaths were mere plot devices; no heed was given to the philosophical upshot of such an act. This changes in the 1940s with René Barjavel’s Le Voyageur imprudent [1944]. In Barjavel’s novel the protagonist travels in time and accidentally kills his great-great-grandfather. He then fades from history. At the end of the book, Barjavel captures the relevant philosophical worry: ‘He killed his ancestor. Therefore he does not exist. Therefore he did not kill his ancestor. Therefore he exists. Therefore he killed his ancestor. Therefore he does not exist. Stop, this is madness!’ [1944: 183].

Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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at Pappy, ready to shoot him. Something is allegedly contradictory about this scenario. Different variations of the paradox differ over what is contradictory. Those variations are the focus of this section.

7.1.1 Less Interesting Variations Firstly, note that some variations of the paradox are less interesting, philosophically speaking. For instance, I might kill someone other than my grandfather (e.g. my grandmother or a distant ancestor); I might accidentally, rather than intentionally, kill Pappy; I might skip the ancestor murder entirely and just kill myself as a child. Or the gruesome elements might be eliminated: I might prevent the Great Depression by taking the correct single malt back in time to catch Winston Churchill in just the right mood, staving off the introduction of the 1925 British Gold Standard Act; I might get a black astronaut to be the first man on the moon by saving JFK and having him install Ed Dwight on Apollo 11; or I could simply shake hands with my future self and then, when I get to my future self’s point in personal time, not shake hands with myself. Any change will do. All of these variations are fine variations, but they don’t add anything significantly different to the paradox for our purposes here. Some variations do add something significantly different. Some worry that the problem with the Grandfather Paradox involves free will [Al-Khalili 1999: 178–9; Feinberg 1967: 1092; Hawking 1998: 166–7; Nahin 1999: 293; Price 1994]. We could avoid this issue by focusing on a variation involving no free-willed agents [cf. Earman 1972: 231; Hanley 2004: 146; Hawthorne 2000: 628; Ismael 2003: 305; Roache 2009: 606; Wasserman 2018: 142]. The earliest discussions of the Grandfather Paradox within physics have exactly that focus. Polchinski, in a letter responding to Morris, Thorne, and Yurtsever’s article ‘Wormholes, Time Machines, and the Weak Energy Condition’ [1988], considers a billiard ball-based variant of the Grandfather Paradox [Echeverria et al. 1991: 1079–81; Friedman et al. 1990: 1925]. Imagine a billiard ball heading towards a wormhole at time t₁. Unimpeded, it will enter the wormhole at t₃ and emerge at t₁. Imagine further that it exits at an angle such that it will strike its earlier self at t₂, preventing the billiard ball from entering the wormhole. No free-willed agents appear in Polchinski’s variation of the paradox, but we are nevertheless on course for a contradiction given the possibility of time travel. Since we could focus on billiard balls rather than grandfathers and guns, issues of free will aren’t relevant to the Grandfather Paradox.² Nonetheless, for the sake of example I’ll keep with the traditional case of murdering Pappy. ² Although that’s not to say that time travel in general isn’t relevant to free will—see King [1999] and Rea [2015].

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7.1.2 The Explanatory Paradox Something is problematic about the scenario where I’m about to kill Pappy. Somewhere, a contradiction must lurk. But what is the contradiction meant to be? The following three sub-sections discuss the options. Books on popular science most often cast the paradox as being a problem about explanation. For instance, Carroll writes: [T]he standard scenario is to imagine going back into the past and killing your grandfather before he met your grandmother, ultimately preventing your own birth. The paradox itself is then obvious: If your grandparents never met, how did you come into existence in order to go back and kill one of them? [Carroll, S. 2010: 100; my emphasis]

Carroll draws our attention to a ‘post-mortem’ world at which I have killed Pappy. Carroll’s worry is that there are explanatory problems at such a world—given Pappy is killed by me in 1930, there’s no longer any explanation for that chain of events, for I no longer exist in the future. Other popular science books contain renditions along similar lines [Clegg 2011: 244; Davies 1995: 249; Deutsch 1997: 293; Everett and Roman 2013: 130; Gott 2001: 11; Greene 2004: 249–50; Gribbin 1992: 199; Mallett 2007: 249–51; Toomey 2007: 145]. The explanatory paradox isn’t a very good paradox. It is easily solved. One solution is to say that it is explicable. Pappy’s death is explained by me shooting him; that, in turn, is explained by me getting in a time machine with murderous intent; in turn, that’s explained by some perceived slight Pappy inflicted upon me in the 1980s; and so on and so forth. My killing Pappy also means these things are false, but that just means they’re true and false, i.e. the propositions are still true! And if they are true, then they can play the necessary role in explanation. Of course, it’s also true that I never shot him, never got into the time machine, and that Pappy was dead and so never slighted me (etc.); in general, this would require contradictions to be true. But if contradictions being true is your real worry, then you should solely worry about contradictions rather than worrying about explanation—that is, you should endorse the post-mortem indicative paradox discussed below. A second solution would be to admit that there might just be some inexplicable things. As discussed back in Chapter 5, that some things are inexplicable is less worrying than you might at first imagine, for lots of people believe that, say, the start of the universe or the decay of a tritium atom are inexplicable. With a choice of two solutions available it is better to simply set aside the explanatory paradox.

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7.1.3 The Post-Mortem Indicative Paradox If you read physics journals, rather than books on popular science, the Grandfather Paradox is couched differently. The paradox is still presented as focusing on a post-mortem world but the worry is that at that world a contradiction is true, rather than there being problems about explanation. For instance, Friedman and Higuchi [2006: 114] worry that, at such worlds, there is no consistent evolution of the world state; other physicists follow suit, although more often in the guise of Polchinski’s Paradox [Lossev and Novikov 1992; Thorne 1994: 508–11; Visser 1995: 212–13]. Some philosophers also have this version of the paradox in mind [Craig 1988: 143–4; Earman 1972; Fulmer 1980: 153; Goff 2010: 67; Harrington 2015: 246–7; Malament 1984: 98; Pezet 2017: 333]. I take this version to be canonical. Call someone who thinks time travel is proven impossible by the Grandfather Paradox a ‘Paradoxer’. Were time machines possible, says the Paradoxer, I could travel to 1930 before Pappy met my grandmother. There I could be in circumstances where: I’m pointing a rifle at Pappy; I am a crack shot; Pappy is definitely my grandfather; Pappy won’t rise from the dead three days after dying; no local miracle will prevent me firing the rifle if I choose to do so; and so on. Call these the ‘correct circumstances’; every factor preventing a contradiction— other than my choosing not to pull the trigger—is ruled out by being in the correct circumstances. In those circumstances, says the Paradoxer, I could nonetheless pull the trigger. That means that there’s a world at which I do pull the trigger in the correct circumstances. But that’s impossible! Thus time machines are impossible. To clarify their thinking, compare the problem to the ‘Escher Machine Paradox’. An Escher Machine is a robot combined with a mechanical carving tool. It carves blocks of wood into miniature staircases and has a ‘Penrose setting’ which, when activated, means the machine carves the wood into a Penrose staircase (i.e. a staircase which makes four ninety-degree turns as it ascends and yet connect backs with itself to form a continuous loop). Imagine I claimed to have an Escher Machine in the boot of my car. Clearly I am wrong for, if I did, I’d have a machine that could bring about metaphysical impossibilities. The Paradoxer has similar thinking in mind, but for time machines. Since discussing the post-mortem indicative paradox is the focus of much of this book, I will break it down in detail. Reuse T T’ P as an assumption for reductio. The possibility of me being in the correct circumstances is then guaranteed by: S P: If time travel is metaphysically possible, then it’s metaphysically possible for me to be in the correct circumstances.³

³ The conditional in S P is a material conditional.

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The next step plays on the idea that, were I in the correct circumstances, I could shoot Pappy for I have ‘what it takes’. The English ‘could’ confusingly admits of multiple (similarly related) disambiguations [cf. Fara 2010: 67], e.g. the de dicto sense whereby ‘φ could be true’ just means φ is true at an epistemically possible world, or the actuality entailment sense whereby ‘Agent τ could ϕ’ expresses the proposition that, at some previous time, τ ϕ’d. Here, I’ll treat ‘could’ as the subjunctive of ‘can’, where ‘can’ is being read as attributing an ability to an agent: agent τ can ϕ is to say that τ has the ability to ϕ; τ could ϕ (in circumstance C) is to say that, were C the case, then it would be the case that τ would have the ability to ϕ. To illustrate: when discussing whether to hire a car, if I say, ‘I could drive the car’, I’m saying that, were a car hired, I would have the ability to drive it. Another example: whilst watching a high-octane action movie, the hero drives a car through a fighter jet and I whisper under my breath, ‘I could do that’; I am (falsely and humorously) saying that, had I been in that situation, I would have had the ability to duplicate that feat. So when the Paradoxer says that were I in the correct circumstances I could pull the trigger, they mean: A A: If I were in the correct circumstances, then I would have the ability to pull the trigger of the rifle (in those circumstances).⁴

The above premises use modal terms (‘possible’, ‘could’, ‘would’). As is standard, they should be translated into possible world talk. That in mind, we can add two principles dealing with possible worlds. First: accessibility between worlds is (at the least) transitive, i.e. the correct modal logic is at least as strong as K4 [Garson 2013: 220]: K4:

The accessibility relation between worlds is transitive.

Second: P-A P: If (in circumstance C) an agent τ has the ability to ϕ, then there’s a metaphysically possible world at which τ ϕs in circumstance C. [Spencer 2017: 465] With these premises in place, it is a good time to pause and sketch the first stage of the Paradoxer’s argument. Assume T T’ P is true. Given S P there’s a world, w₁, at which I am in the correct circumstances. Modus ponens holds of the counterfactual conditional and so, given A A, at w₁ I can pull the trigger. Given

⁴ The conditional in A A is obviously a subjunctive counterfactual conditional.

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P-A P, at w₁ there’s a world, w₂, at which I succeed in pulling the trigger in the correct circumstances. Given K4, we can trim that down to: There’s a world, w₂, at which I pull the trigger in the correct circumstances. This lemma in place, return to the Paradoxer’s substantive premises. By definition, in the correct circumstances Pappy is alive in 1930; similarly, since the trigger is pulled, Pappy must be dead in 1930 because he’s in those circumstances. Therefore, at w₂, hAt 1930: Pappy is alive ∧At 1930: Pappy is deadi is true. That’s not of the form φ∧¬φ and so is not a contradiction (since being dead isn’t the negation of being alive). A contradiction would follow only if we believed: □ [At 1930: Pappy is dead → At 1930: ¬ Pappy is alive] But I deny just this principle! Given that gainsayers can be true (see §2.2.3), Pappy can be dead in 1930 without him not being alive. Imagine that in 2018 I kidnap my ninety-year-old grandfather and time travel back to 1930s Peru, where I shoot him dead. In that case, Pappy would be dead and alive in 1930 but it’d be false to say that he’s not alive in 1930 (for he is alive, given that he’s good and well in Liverpool). A gainsayer is true but a contradiction is not. However, gainsayers require the incompatible properties in question to be had by distinct temporal parts of an entity (i.e. by different temporal parts of Pappy). It is inconsistent for one and the same instantaneous temporal part of an entity to have incompatible properties. In the case of the Grandfather Paradox, and unlike the situation of kidnapping a nonagenarian and murdering him in 1930s Peru, at w₂ it is the selfsame instantaneous temporal part of Pappy which is dead and alive. So there is a contradiction true at w₂. We should endorse: C T: At a world where I shoot Pappy in the correct circumstances, a contradiction is true at that world. But contradictions aren’t possible: N S W: It’s not the case that there’s a world at which a contradiction is true. So there both is and isn’t a world at which a contradictory proposition is true. That is itself a contradiction. By reductio, T T’ P is false. Here’s a truncated summary of the argument: 1. T T’ P is true. [Assumption for reductio] 2. At w₁: I am in the correct circumstances. [1 and S P]

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3. At w₁: I can pull the trigger of the rifle (in the correct circumstances). [2 and A A] 4. At w₂: I pull the trigger of the rifle in the correct circumstances. [3, PA P, and K4] 5. At w₂: A contradiction is true. [4 and C T] 6. There is a world at which a contradiction is true ∧ ¬ there is a world at which a contradiction is true. [5 and N S W] 7. T T’ P is false. [1, 6, and reductio ad absurdum] (An additional summary of both the premises and the truncated argument is reproduced at the start of this book for ease of reference.) The focus of the paradox is world w₂. Since it’s a world at which I have shot Pappy dead in the correct circumstances, it’s a ‘post-mortem’ world. And the contradiction true at w₂ is a conjunction of propositions in the indicative mood (see step 6 of the above). Hence I call this the ‘post-mortem indicative paradox’.

7.1.4 The Non-Morietur Subjunctive Paradox Interestingly, the philosophical community tends to focus on a different variation of the Grandfather Paradox. Rather than focusing on a post-mortem world at which Pappy has died, the focus is on a ‘non-morietur’ world at which Pappy doesn’t die because I decide not to pull the trigger. At that world, even though I have chosen not to kill Pappy, the worry is that I nevertheless have the ability to kill him. But because killing Pappy in the correct circumstances is impossible, I also don’t have that ability. Thus a contradiction is true at the non-morietur world: I both do and do not have the ability to kill Pappy. Since that’s a conjunction of propositions in the subjunctive mood, this is the ‘non-morietur subjunctive paradox’. Lewis’s 1976 paper talks in these terms and others have either followed suit [Arntzenius 2006: 605, 611; Dainton 2010: 127; Grey 1999: 64; Rennick 2015; Vihvelin 1996: 315; Vranas 2009: 520; Wasserman 2018: 72–3] or were already talking in those terms [Gorovitz 1964: 366–7]. It’s also not hard to see the nonmorietur subjunctive version as the more natural response to the Ludovician once you’ve heard their solution to the Paradox of the Changing Past: the Ludovician solution is that I could go back in time and causally affect the past, but that I cannot change the past; the non-morietur subjunctive paradox tells a tale where nothing in the past changes (and so Ludovicianism is not a solution) but where we nevertheless apparently have a contradiction. At the end of the day, I see little difference between the post-mortem indicative version and the non-morietur subjunctive version. It has almost the same premises,

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although it doesn’t need K4. From the definition of ‘correct circumstances’ and C T, were the trigger pulled in the correct circumstances, a contradiction would be true—thus, given N S W, there is no world at which I pull the trigger in those circumstances. Given P-A P, if I can pull the trigger (in the correct circumstances), then there is a world at which I pull the trigger (in the correct circumstances); thus, by modus tollens, I can’t pull the trigger in the correct circumstances. Given T T’ P and S P, there’s a non-morietur world, ẅ, where I am in the correct circumstances. And, given A A, at ẅ I can pull the trigger. Having already established that I can’t pull the trigger at any world, at ẅ I both can and cannot pull the trigger, i.e. a contradiction is true at ẅ and so there can’t be a possible world such as ẅ. So we again arrive at a contradiction (namely that ẅ exists but does not exist), albeit by a slightly different route. Nothing appears to hang on this different presentation of the paradox and I don’t think there are any interesting differences between the post-mortem indicative paradox and the nonmorietur subjunctive paradox. And few would suggest otherwise.⁵ I believe philosophers generally focus on this version because Lewis [1976] did and Lewis only focuses on the non-morietur version because he is doing something subtly different than dealing with the Grandfather Paradox. Lewis tasks himself with discovering whether a consistent time travel story can be told. Many time travel stories—including the narrative of the post-mortem indicative paradox—are clearly inconsistent. So Lewis never pauses to consider such narratives. Instead, he focuses on stories which are consistent at first blush (e.g. the narrative of the non-morietur paradox) and investigates whether they contain a hidden contradiction (e.g. the subjunctive contradiction). So there is no great mystery as to why philosophers follow Lewis’s set-up. That said, I’ll keep my focus on the postmortem indicative paradox.

7.2 Proposed Solutions There’s only one way out of a valid paradox: Deny a premise. I end this chapter by sketching some options, the more interesting of which are followed up on in Part 3 of this book.

⁵ The only explicit dissenter I know of is Harrington [2015: 248] who argues that weak constrict theory (see Chapter 8) is true and that this resolves the post-mortem paradox but not the non-morietur version. But not only do I think weak constrict theory is false—see Chapter 8—I am also unclear why it’d solve one paradox but not the other.

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7.2.1 Dead Options Denying either of K4 or S P is unlikely to go anywhere. K4 is unlikely to be the problem because the suitably similar non-morietur subjunctive paradox can do without it. And S P is unlikely to be the culprit because it’s just a placeholder. Recall it: S P: If time travel is metaphysically possible, then it’s metaphysically possible for me to be in the correct circumstances. To deny S P is to deny that it’s possible that I be in the correct circumstances. But ‘the correct circumstances’ are just a placeholder, as is grandfather murder. If we denied S P, then we could simply redux the Grandfather Paradox, replacing having the ability to kill Pappy with the ability to get into the correct circumstances. After all, what would stop me? Would my car break down on the way to the spot that’d allow me to be in the correct circumstances? That’s little different than my rifle jamming every time I go to pull the trigger! So if we deny S P, then we end up where we began, in that some other premise has to be denied anyhow. Another, more workable, option which I nonetheless won’t follow up is to rope in indexed worlds from Chapter 6. At an indexed world, time travellers can go back in time and kill their grandfathers without fear of contradiction, for their ancestor would be dead at one universe (or hypertime) and alive at another. At an indexed world C T is false. But whatever one thinks about indexed worlds and changing the past, it is unlikely they can help with the Grandfather Paradox (indeed, the only philosopher I know to explicitly suggest they can is Goddu [2003: 31], and even then it is only a passing suggestion). It no more helps to say that at an indexed world C T is false than it does to say that the metaphysical possibility that everything is immobile helps with Zeno’s Paradox, or the possibility that everyone could be mute and illiterate (and thus that it’s possible that there are no liar sentences) bears on the Liar Paradox, or that the world might not include statues helps solve the statue/lump paradox. Only if liar sentences were metaphysically impossible would that bear on the Liar Paradox; only if mereological nihilism were necessarily true would we have a solution to the statue/lump paradox; only if Parmenides were correct and all motion necessarily an illusion would we avoid Zeno’s Paradox. Similarly, only if it were metaphysically necessary that time travel required indexed worlds would we have a solution to the Grandfather Paradox—if indexed worlds are merely metaphysically possible, we need to keep looking for a further response. Those who discuss indexed worlds tend to be discussing them solely to show that the past can be changed, a fact requiring only the metaphysical contingency of indexed

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worlds—indeed, van Inwagen [2010: 5] says exactly this, explicitly allowing for Ludovician worlds. One might decide to go ‘all in’ on indexing, claiming that indexed worlds are metaphysically necessary (at least, if time travel occurs). I won’t discuss that option in this book because the only reason to consider it would be if Ludovician time travel was problematic. Since I ultimately defend the tenability of Ludovicianism, I’ll set it aside. So, except where otherwise noted, I’ll assume that the world isn’t indexed, not because I believe it’s definitely true—for all I know Deutsch and Lockwood are correct and time travel does involve travelling to other universes!—but because that assumption seems to me to be the more interesting one to make and the one that is most charitable to the Paradoxer.

7.2.2 Live Options Turn to more workable options. First up is the Paradoxer who draws the lesson that T T’ P is false. Prima facie the Paradoxer has a strong case. We should default to their solution unless we have a good alternative response. Paradoxing therefore doesn’t need a chapter dedicated to it—if the other responses to the Grandfather Paradox aren’t good responses, it’s presumably true. There’s little need for an extensive discussion of its merits. That leaves three premises which can be denied. We might deny N S W, i.e. accept that contradictions are metaphysically possible. Such dialetheic responses to the Grandfather Paradox have been underdiscussed—I remedy this in Chapter 9 when I discuss ‘inconsistency theory’. I argue that inconsistency theory ends up in the same boat as indexing, i.e. interesting only if Ludovicianism were problematic (which it isn’t). The two remaining positions correspond to already extant positions taken by self-avowed Ludovicians. The incapacity theorist denies A A. They say that in the correct circumstances I am simply unable to shoot Pappy; as I take a bead on him back in the 1930s I’m rendered unable to pull the trigger even though trigger pullings are something I can normally do. Meanwhile, the impossability theorist denies P- P. They say it’s metaphysically possible for agents to have the ability to do the impossible—agents can do metaphysically impossible things even though (because we know we live in a metaphysically possible world) they won’t do metaphysically impossible things. I discuss the shortcomings of incapacity theory in Chapter 10 and defend impossability theory in Chapter 11. Before all of that, though, Chapter 8 discusses something that might be conspicuously absent: what does contemporary science have to say about the Grandfather Paradox? That chapter argues that the answer is ‘Very little’.

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PART 3

PROPOSALS

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8 Constrict Theories Before moving to the philosophical proposals discussed in Chapters 9–11, one might wonder what the ‘scientific’ responses are to the Grandfather Paradox. This chapter argues that they’re either (i) ever so slightly confused misstatements of either the Paradoxer’s Solution or of one of the Ludovician solutions (presumably incapacity theory) or (ii) straightforwardly wrong-headed.

8.1 The Conjecture and the Principle Some physicists believe that some law of nature, or collection of laws, prevents time travel. This has been dubbed the ‘Chronology Protection Conjecture’— henceforth simply ‘the Conjecture’ [Carroll, S. et al. 1994; Dyson 2004; Grant 1993; Hawking and Ellis 1973; Hawking 1992]. The proposed protective laws vary: perhaps traversable wormholes can’t exist because exotic matter is impossible to manufacture; perhaps wormholes are impossible to form in the first place; perhaps tachyons are physically impossible [Skinner 1969: 189]. Whatever the case, if the Conjecture were true, then time travel would be physically impossible. Other physicists think that time travel might be possible, but suggest that there’d then be a law of nature preventing paradoxes from coming about. This is called ‘Novikov’s Principle of Self-Consistency’—henceforth ‘the Principle’. (Some physicists endorse the same idea without using this name, for instance Pegg [2008] and Price [1994: 317].) It is also a device often deployed in fiction. In Niffenegger’s The Time Traveller’s Wife [2003] the protagonist becomes inexplicably paralysed when put in circumstances where he can prevent previously witnessed events; in Moorcock’s ‘Pale Roses’, once you travel in time you can only return back to your own period for, at best, a short visit ‘owing to the properties of Time itself’; and all sorts of fictional laws have been mooted, each of which is similar to the Principle (e.g. ‘the Blinovitch Limitation Effect’ from Doctor Who or the ‘the fifth law of Causal Determinism’ in Happy Accidents [2000]). How the Principle would stay my hand when I’m aiming at Pappy is a good question. Some talk as if the Principle is a fundamental law of nature [Friedman et al. 1990: 1915], whilst others think it’s entailed by more fundamental laws of nature (e.g. the principle of least action [Carlini et al. 1995; Carlini and Novikov 1996], general relativity [Richmond 2004: 177], or attempts to quantize gravity Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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[Visser 1995: 256]). Whatever the case, the law(s) would be somewhat peculiar. If I am aiming at Pappy whilst Malcolm, my (non-time-travelling) qualitative duplicate, is aiming at another man, the Principle has it that I am physically constrained to not pull the trigger whilst Malcolm is not. Since Malcolm is my duplicate, the only difference between us are our extrinsic properties/relations, thus the Principle must be sensitive to such things (e.g. sensitive to what path an object has previously traced through spacetime). This sensitivity is precisely its peculiarity for we don’t expect laws to (e.g.) affect two qualitatively identical electrons differently just because only one of them earlier passed through Detroit. This peculiarity has put some people off the Principle [Everett 2004: 124023.1; Everett and Roman 2013: 145; King 1999; Toomey 2007: 259], although it’s not totally implausible since some laws, e.g. the principle of least action, are already peculiar in just this respect (although such laws fall under suspicion precisely because they bespeak of some peculiar, unwholesome character [Stöltzner 2003; Yourgrau and Mandelstam 1968]). Indeed, the Principle is weirder still. As Horacek argues, were there a Principle, it’d constrain time travellers not just with regards to attempted murder but any activity; time travellers would be nomologically constrained to only do what they actually did, such that the Principle ‘would need to hold a time traveler in a very tight grip indeed, and in a manner that is quite mysterious’ [Horacek 2005: 431]. Further, since a time traveller can no more acquire the abstracted property of being such that Pappy has died than kill Pappy themselves, upon arriving in the past seemingly everyone else in the universe will suddenly be bound by the laws of nature to not kill Pappy either (and, similarly, everyone will be bound to make the universe only how it actually turns out to be). That the mere arrival of a time traveller affects the abilities of everyone in the universe seems intellectually repugnant. So, the Principle is prima facie bizarre. However, for the rest of this section I’ll put aside this worry and imagine that the Principle is not as peculiar as some worry; it’ll nevertheless turn out to be less than helpful in escaping the Grandfather Paradox.

8.2 Laws and Metaphysical Possibility That the Principle or the Conjecture might be laws of nature is an open question; I’m no physicist and have nothing to add to what the laws of physics actually are. But mounting a response to the Grandfather Paradox on the back of such laws is a hard sell. Presumably the idea would be that the Conjecture being true means that time travel is metaphysically impossible (i.e. T T’ P is false). Similarly, the Principle being true is presumably meant to indicate that no time traveller has the ability to bring about a paradox (i.e. A A is false). Call the former a ‘strong constrict theory’ (because all time travel is ruled

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out) and the latter a ‘weak constrict theory’ (because only some time travel is ruled out). Some people have suggested that some constrict theory is true. Friedman et al. [1990: 1925] introduce the Grandfather Paradox and then clearly imply that the Principle allows us to avoid it. And the idea that the laws of nature bear on the Grandfather Paradox is also common in popular science books where the Conjecture or the Principle (or both) are proposed as a response to it [Clegg 2011: 246–7; Gott 2001: 16–20; Mallett 2006: 251; Toomey 2007: 259–60]. Philosophers also get in on the act: Earman [1995: 175–9], Harrington [2015: 248], Sorabji [1986: 224], and Stevenson [2005] all think some version of constrict theory is true. Empirical investigation may tell us a lot about the philosophy of time travel. For instance, if we travelled in time, we’d know that T T’ P was definitely true. But except for such cases, empirical investigation won’t shed any light on the Grandfather Paradox because the Paradox concerns whether or not time travel is metaphysically possible [cf. Goddu 2003: 20; Mancuso 2014: 212]. Focus on the Conjecture for now (although substantially the same concerns apply to the Principle). Were the Conjecture true, that’d only settle that time travel was physically impossible [cf. Horacek 2005: 430], and that’s not what we need. Were the Conjecture true—if, for instance, exotic matter was physically impossible— then the Grandfather Paradox would still have bite because we could still speculate about how the world might, or might not, have been different. Even if exotic matter doesn’t actually exist, and is physically incapable of existing, could it have existed? That is: is it metaphysically possible for it to have existed and the world otherwise remain the same, thus allowing for time travel? Cognately: even if some law of physics prevents exotic matter from being made, and exotic matter is physically necessary to hold open wormholes, is there nevertheless a metaphysically possible world where some miracle happens, the laws of nature are suspended, and the exotic matter is manufactured (or, alternatively, the wormhole miraculously remains open when a time traveller passes through)? There’s nothing wrong with thinking that time travel is the result of a miracle; whilst some people think miracles are metaphysically impossible, that’s not everyone. And the idea that miracles might bring about time travel is already extant in both philosophy (for God might toy with time; see §1.2.1) as well as fiction (God time travels in Gilliam’s Time Bandits [1981], angels in Kripke’s Supernatural [2008], and the Devil can send people through time in Beerbohm’s ‘Enoch Soames: A Memory of the Eighteen-Nineties’ [1916]). To settle whether the Conjecture is true tells us nothing about time travel’s metaphysical impossibility. One objection would be that T T’ P is false because the Conjecture is true by metaphysical necessity. But why believe such a thing? Law necessitarians, who believe that the actual laws are the laws at all metaphysically possible worlds, would be justified in believing this (at least if they further added

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that miracles were also metaphysically impossible). But I don’t believe law necessitarianism is true. Absent law necessitarianism, I don’t see any argument for believing that the Conjecture is metaphysically necessary. Alternatively, the strong constrict theorist might just build it into their position in the first place, saying that strong constrict theory just is the claim that it’s metaphysically necessary that there’s a law of nature ruling out all time travel. But now the difference between the Paradoxer’s Solution and strong constrict theory is eroded. The strong constrict theorist would have to say that, as a matter of stipulated, brute fact, certain combinations of laws (i.e. those permitting time travel) are metaphysically impossible. But since no natural laws explain why metaphysical possibility is constrained thus and so rather than so and thus, the only explanation for that impossibility would be some purely philosophical explanation. The strong constrict theorist therefore thinks that purely philosophical speculation means we can conclude that it’s metaphysically necessary that there’s a law of nature ruling out time travel. But the Paradoxer argues that purely philosophical speculation means we can conclude that time travel is impossible—they say exactly the same thing as the strong constrict theorist but without taking a diversion through the laws of nature. If purely philosophical concerns dictate that there’s no world lacking such a law, why shy away from thinking that purely philosophical concerns more simply dictate there being no world at which time travel takes place? Why bother with the diversion? So either strong constrict theory fails to be relevant to the Grandfather Paradox or there’s no substantive difference between it and the Paradoxer’s Solution. The strong constrict theorist might present themselves as a minor twist on the Paradoxer’s Solution, saying that the ban on time travel being mandated by physical laws, not mere metaphysical fact, is important. But it isn’t; it’s superfluous. The motive for saying it’s important would presumably be because one endorsed: C:

If φ is metaphysically necessary, then φ is a law of nature.

But C has clear counterexamples. It’s not a law of physics that 2+2=4 or that murder is immoral. Further, popular metaphysical views of laws bear out C being false. Consider the Ramsey-Lewis theory [Ramsey 1978; Lewis 1973a, 1983]: a proposition is a law of nature iff it features in the best system of laws (where the system is best iff it has the best balance of simplicity and strength). Necessary propositions won’t feature in the best system because neglecting to include them will make the system simpler with no reduction in strength. So, given the Ramsey-Lewis view, were the Conjecture necessarily true, it wouldn’t be a law of nature. Or consider a rival theory, the Dretske-Tooley-Armstrong theory [Armstrong 1983; Dretske 1977; Tooley 1977], whereby propositions of the form hEvery F is a Gi are laws iff the property Fness stands in the natural necessitation

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relation to the property Gness. The natural necessitation relation holds contingently; thus, even if every F is a G, Fness may fail to naturally necessitate Gness and a metaphysically necessary proposition needn’t be a law of nature. In short: the best that can be said about strong constrict theory is that it’s a minor variation on the Paradoxer’s Solution rather than an interesting response in its own right [cf Dowe 2007: 729–30]. And similar thinking applies to weak constrict theory and thinking that the Principle is true by metaphysical necessity—that theory would at best be only a minor variation on incapacity theory. So I set aside constrict theories. Physicists who think the Conjecture solves the Grandfather Paradox should realize that they’re just Paradoxers. And physicists attracted to the Principle as a solution should realize that they’re just Ludovicians. Indeed, it’s already debated in the literature whether true propositions which explain the lack of paradoxes should further count as laws [Dowe 2007: 729; Kutach 2003] and some physicists certainly already talk about the Principle without implying that it’s a law of nature [Dolanský and Krtouš 2010; Mikheeva and Novikov 1993]. I suggest that all Principle-supporters follow suit and become loud and proud Ludovicians.

8.3 Rewriting the Paradox The constrict theorist might argue that I’ve unfairly loaded things in my favour by setting up the Grandfather Paradox to be specifically about metaphysical possibility. A constrict theorist might say that the Paradox needs to be cashed out without mentioning metaphysical possibility. So goes the thought, we should instead assume for reductio that time travel is physically possible and try and reach a contradiction on the back of that. Were that so, then constrict theories, physics, the Principle/Conjecture etc. would all be eminently relevant to the Grandfather Paradox. But this would be wrong-headed. We must focus on the broader modality of metaphysical possibility rather than physical possibility because the Grandfather Paradox is a thought experiment and thought experiments—even those conducted by physicists—focus on that broader modality. Consider Galileo’s thought experiment from De Motu (also from his Due Nuove Scienze) disproving a cornerstone of Aristotelian physics. Galileo assumes, for reductio: M A A: tion with their mass.

Ceteris paribus objects fall faster in propor-

Imagine there are two lead balls: one of mass n (‘Balln’); one of mass 2n (‘Ball2n’). Given M A A, Balln falls at speed m whilst Ball2n falls at 2m. Imagine a cord is then attached between them: Balln now falls faster, pulled

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down by Ball2n’s faster descent; in turn, Ball2n’s velocity will be retarded by the slower-moving Balln. Together they will fall at speed m* where mCh(φj¬you ϕ) should you ϕ in order to bring about φ. In Boxes this tells against one-boxing: at the point at which you make your decision, Derren has (or has not) filled Box B; the objective chance of it having money in it is therefore either zero or one and there’s now nothing you can do to raise that chance; one-boxing is irrational. Two-boxing, on the other hand, raises the objective chance (from zero to one!) of you leaving £1k richer than if you took only one box. So you should two-box. But this reasoning must be tweaked in order to deliver the correct reasoning in Autoinfanticidal. In Autoinfanticidal it’s physically possible that any one of the following three propositions ends up being true: N G:

You don’t gamble and don’t become a millionaire.

P G: I G:

You gamble. I fail to kill Pappy. You lose everything. You gamble. I kill Pappy. You become a millionaire.

The order of values is obvious: I G > N G > P G

because they feature a remarkable and unlikely event. So the counterfactual is false in context CN, i.e. your belief state wouldn’t counterfactuallyN depend upon the fluctuation.

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Given the Chance Motive, you should gamble if doing so raises the objective chance of I G without significantly increasing the objective chance of P G. I G is metaphysically impossible, but that doesn’t mean that your actions can’t raise its objective chance. Indeed, were you to gamble, then the objective chance of I G would sky rocket, with only a tiny increase in the chance of P G. Two-boxers who focus solely on the objective chance function must recommend gambling in Autoinfanticidal. Since we’re agreed that you should not gamble in Autoinfanticidal, something is therefore wrong with the Chance Motive when it focuses on objective chance. Two-boxers moved by the Chance Motive start by saying that one-boxers go wrong because they focus on the wrong probability function. I say the same of the traditional two-boxer: instead of focusing on the objective chance function, they should focus on the ‘adjusted chance function’. The adjusted chance of something happening is simply its objective chance conditional on metaphysical impossibilities not coming about. That is, where ‘ChA (φ)’ represents the adjusted chance of φ (and where, recall, ⏈ stands for the proposition that no metaphysical impossibilities come about): ChA (φ) = Ch(φj⏈) Only when physical and metaphysical possibility come apart do the functions deliver different outputs. That is, in every case other than a time travel case (or a case involving Reapers etc.) ChA(φ)=Ch(φ). So in Boxes you should still two-box. But in cases like Autoinfanticidal the adjusted chance and objective chance diverge. Gambling increases the objective chance of I G without significantly increasing the objective chance of P G, but it doesn’t increase the adjusted chance of I G at all. Since I G is metaphysically impossible, ChA (I Gjyou gamble)=0 and can never shift from that value. Moreover, gambling significantly increases the adjusted chance of P G since ChA (P Gjyou gamble)=1. By focusing on adjusted chance, the recommendation is to not gamble in Autoinfanticidal, which is the right answer. In Plague, if we focus on adjusted chance, then the recommendation is to trick Enemy. Even though it doesn’t affect the objective chance of Enemy contracting the virus, tricking Enemy raises the adjusted chance of that happening. Indeed, because the chance of a metaphysically impossible case of time travel is so high (i.e. 1), if Enemy is tricked, then the adjusted chance of them contracting a virus similarly raises to near one. If you don’t trick them, then both the objective chance and adjusted chance of that happening are the same, i.e. miniscule. So those motivated by the Chance Motive—once it has been tweaked to be about adjusted chance and not objective chance—should trick Enemy in Plague.

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13.5 Other Newcomb Cases §§13.1–13.4 have made the case for two-boxers to trick Enemy in Plague—that is, to take the option analogous to one-boxing when in a time travel Newcomb scenario. There are other such time travel Newcomb cases which differ from Plague in ways which raise interesting questions. Plague is characterized by two features. Firstly, the ‘subject’ of the Newcomb reasoning—in Plague’s case, the event of Enemy being wiped out by a virus—takes place in the future. Secondly, in Plague you are not already involved in a causal loop and, because your credence of time travel occurring should be close to zero, you should not suspect that you will become so involved. We can imagine Newcomb cases which vary both features.

13.5.1 Events in the Past Imagine a case which involves no causal loop but in which the ‘subject’ of the Newcomb reasoning is a past event: Asteroid: Exactly the same as Plague except that, rather than Enemy’s being wiped out being settled by some indeterministic future event, Enemy can only be wiped out by an event depending wholly and solely upon a past event which has already occurred or not. In this case, that event is an asteroid being on a collision course with Enemy. If the asteroid is already on course, Enemy will die, come what may; if it is not on course, Enemy will not die. You have no way of knowing whether the asteroid is on course or not and no earthly power can influence its current trajectory. We might think that because it’s a fixed fact already whether the asteroid is on the correct course or not, this makes a difference whether to trick Enemy. But I doubt that any convinced two-boxer who has come over to my side regarding tricking Enemy in Plague would then continue arguing that one should ‘two-box’ in Asteroid. It seems intuitive to me that once the two-boxer accepts the strangeness of ‘one-boxing’ in Plague, it’s hard to see what would make them resist doing the same in Asteroid. To put some meat on the bones of that intuition, consider a pair of cases which, like Autoinfanticidal/Homicidal, make clear what is rational to do in certain cases. First, consider the non-time travel scenario: Door: I cut myself badly, falling through a door into the next room. I’m injured and can’t come back though the door. It is unlikely that I will survive without help. You are a skilled first aider who can easily save my life, although

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providing help will ruin your clothes by getting blood over them. You value my survival far more highly than you do your clothes. In Door it’s clearly rational to save me at the cost of your clothes (and irrational to let me bleed to death). Next, consider a similar time travel case: Timegate: The same as Door. However, the door is a (two-way) portal back to the 1930s. We both know this. In every respect other than which time period they are located in, our environs are qualitatively identical to those in Door. (To quell fears that you should have zero credence of time travelling back into one’s own past light cone, imagine that the past destination is outside of that cone.) It strikes me as obvious that in Timegate you should still go through the portal, bandage me up, and then both of us can return to the present day. I find it intuitively intolerable that there being a time portal present makes any difference (for instance, imagine there were two doors and you can’t quite remember which was the time portal—does it make sense for you to ignore my hollers and cries whilst you check which it was?). That pair of cases in place, go back to considering the various motivations for being a two-boxer. Consider a two-boxer moved by the Counterfactual Motive. In Timegate, such a two-boxer must say that my bleeding to death in the past appropriately counterfactually depends upon the future action of you stepping through the time portal (or not). In Timegate it’s tricky to evaluate that counterfactual because in causal contexts worlds are close when they have a past as close to the actual world’s past as possible, so it’s hard to know what to say about counterfactuals concerning my bleeding to death. But that the closeness of worlds is (partially) determined by their past histories is only a crude statement of the more mature position actually taken in the literature. Those who have set about crafting a more sophisticated understanding of counterfactuals have often taken care to preserve the possibility of exactly these sorts of counterfactual dependencies of past events on future events [Lewis 1973b: 566, 1979a: 464]. And given that the more sophisticated accounts allow that the (already fixed) fact about my not bleeding to death can counterfactually depend upon some future act (even in the causal context), then in Asteroid the asteroid being on such-and-such a course can counterfactuallyL depend on you tricking Enemy. So you should trick Enemy! Next, consider a two-boxer moved by the Causal Motive. Such a two-boxer will admit that, given time travel is possible, you can causally affect the past. In Timegate, we just have the straightforward case of you staunching my blood loss even though that effect happens earlier than its cause of you stepping through the time portal. And if the (already fixed) fact about my not bleeding to death can be caused by some future act, then in Asteroid the asteroid being on such-and-such a course can be within your (counterfactualL) causal control in the future in exactly

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the same way that in Plague the appearance of the virus was under your (counterfactualL) causal control. Again, in Asteroid you should trick Enemy. It gets more complicated when we consider the two-boxer motivated by the Chance Motive. Either the asteroid is coming or it is not. So the objective chance of Enemy’s destruction is already zero or one. Here’s the problem: if the objective chance is already one, then so is the adjusted chance and no act you take will increase it; if the objective chance is already zero, then so is the adjusted chance and no act you take will increase it; so either way, it’s impossible to increase the adjusted chance of Enemy’s dying and, thus, it’s irrational to trick Enemy in Asteroid. But the same thinking applies to Timegate! Either I bled to death or not. So the objective chance—and therefore also the adjusted chance—of me having bled to death is already zero or one. Nothing you do can raise it. Thus, if you should only act to bring about desirable outcomes if your action can raise the adjusted chance of bringing about that outcome, in Timegate you should not bother to try and save me. And that’s not right, for clearly I should be saved in Timegate. What we need is another tweak to be made to the details of the Chance Motive which bears out saving me in Timegate—we’ll then see that the tweak also bears out tricking Enemy in Asteroid. The problem would be easily fixed if past events could have objective chances other than zero or one. Eagle has argued that this can be the case for certain events, e.g. events involved in time travel cases [2011: 291, 2014: 145; see also Cusbert 2018]. So, Eagle believes there is an ‘Eagle function’, ε, which outputs the same results we would normally expect as the objective chance function except for past events involved in time travel cases, where it outputs a value other than the zero or one which we normally think the objective chance function would output. Whilst Eagle goes on to argue that ε is the objective chance function, we needn’t trouble ourselves with that issue. Even if Ch(me dying in the past) equals either zero or one, ε(me dying in the past) does not. Further, ε(me dying in the pastjyou come to help me)>ε(me dying in the pastjyou stay in the future and do nothing). What’s important is not whether ε is the chance function or not, but whether or not ε is the probability function relevant to rational decision making. If, in Timegate, we think ε is the function relevant to decision making, we get the correct answer that you should come and save me. And since focusing on the probability function normally thought of as the objective function (i.e. the function which says past events always have probability zero or one) delivers the wrong result in Timegate, it follows that two-boxers moved by the Chance Motive should initially focus on ε. I say ‘initially’ because, in light of Autoinfanticidal, the probability function we end up focusing on is the ‘adjusted Eagle function’, call it ‘εA’, which is such that: εA (φ) = ε(φj⏈)

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Focusing on εA delivers the correct results both of saving me in Timegate and not gambling in Autoinfanticidal. Similarly, the adjusted Eagle probability of bringing about a virus rises in Plague if you trick Enemy, so you should trick Enemy in Plague. Finally, the adjusted Eagle probability of an asteroid being on a collision course with Enemy rises in Asteroid if you trick Enemy, so you should trick Enemy in Asteroid. Tweak in place, the problem is solved in favour of the two-boxer ‘oneboxing’ in time travel cases like Asteroid.

13.5.2 Other Cases Plague does not involve being in a causal loop. But we can consider cases in which you are in one. Whilst unlikely that you’ll ever be caught up in a causal loop, it’s nevertheless a metaphysical possibility, so such cases are worth considering. Once in a causal loop you will be in one of three positions: knowing everything about what action you’ll take; not knowing what action you’ll take, but otherwise having complete knowledge of the outcome; not knowing what action you’ll take and yet having partial knowledge of the outcome. The first type of case is less interesting. To deliberate is to deliberate over what action one will take, so if you know what action you’ll take, there’s no room for genuine deliberation [Baker 2016; Fernandes 2020; Fulmer 1980: 155] (although there is probably room to ask whether the action taken was nevertheless rational or irrational). That said, I won’t consider those sorts of cases further. The second type of case has already been discussed in the literature: Pauper: Knight will fight in tomorrow’s battle. If Knight buys new armour, his objective chance of surviving will be 0.99, otherwise it’ll be 0.5. Buying new armour will make him destitute and he’ll have to live on as a pauper. Prizing his life more highly than his riches, Knight is about to buy the armour. But then Knight uses a crystal ball and comes to know for certain that he’ll survive the battle unscathed, though not whether he bought the armour or not. [Price Unpublished] Pauper is a type of Newcomb case. Traditional two-boxer reasoning will support purchasing the armour anyhow since it, e.g., improves his objective chance of survival [Bales 2016: 1499–1500]. Traditional one-boxer reasoning favours not buying the armour since, if you’re certain you’ll survive the battle, that guarantees the best possible outcome. As with Plague, Asteroid etc. I recommend that the two-boxer take the ‘one-boxing’ option: Knight should save his money—indeed, knowing that his safety is assured, he may as well make a name for himself and fight naked.

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The third type of case has also been discussed in unpublished work by Nolan, Braddon-Mitchell, and Dorr. Here’s a sample scenario: War: A time travelling DeLorean appears. Delirious, and suffering from radiation poisoning, I stumble out of it. ‘North Korea’s leader is mad’, I say, ‘He has decided that if an indeterministic lottery held next week has a particular result, then he will start a nuclear war. Unfortunately, it has that result and the ensuing conflict will kill us all!’ My future self then dies. It looks like the world is going to end next week. But it occurs to you that you could kidnap me, torture me until I’m no longer thinking straight and can be made to believe anything, poison me with polonium, make me think a nuclear war has taken place—even though none has—and bundle me into a DeLorean to go back in time to ‘warn’ you of what was going to happen. War is a Newcomb case. Torturing me is the ‘one-box’ option (since seemingly that won’t biff-cause an indeterministic lottery to go one way rather than another, nor reduce the objective chance of nuclear war etc.). Not torturing me is the ‘twobox’ option (for the nuclear war is either on its way or it’s not). Similar scenarios aren’t hard to find. Nolan imagines that you time travel to the past to a rather unusual archaeological dig where whatever you unearth you get to keep. You are particularly keen on getting hold of a particular statue. Checking the aged remains of tomorrow’s (probably, but not certainly, veridical) newspaper brought back from the future, you discover that a person with one thumb will discover the statue. You value the statue more than your thumb; you see everyone else appears to have their thumbs; you have a pair of hedge clippers with you. Do you then cut off your thumb? Braddon-Mitchell imagines that you’ve just received bad news from your doctor. Because of some unhealthy behaviour of yours, you have contracted cancer. You have a time machine, although you understand that time cannot be changed and you cannot prevent yourself engaging in the unhealthy behaviour. But it then occurs to you that you can travel to the year 5000  and get a ‘neuralyzer’ which allows you to effortlessly rewrite anyone’s memories. With the neuralyzer in hand you can travel back in time, convince yourself to not engage in the unhealthy behaviour, then travel to minutes before the present and change your memories so that you think you’ve engaged in the unhealthy behaviour, and think you’ve just had a portentous meeting with your doctor about cancer, when none of that really happened—in that case, you presumably don’t have cancer. Whilst that’s a metaphysical possibility, is it rational to try and bring it about? Clearly, it’s a Newcomb case. More cases abound, I’m sure. And in each, I recommend that no matter how many boxes you take in Boxes, you should take the option analogous to ‘oneboxing’ in these time travel cases. I should be tortured! You should cut off your

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thumb! (The neuralyzer case is different because you’ve not, as yet, travelled in time and thus I recommend not trying to do it lest you give yourself a heart attack; the case is nevertheless instructive.) The reasoning is the same as that of Asteroid/ Plague: nuclear war counterfactuallyL depends upon (and is prevented by) my being tortured (and you should have a high credence that torturing me will lower the adjusted Eagle chance of such a war); unearthing the statue counterfactuallyL depends upon (and is caused by) you cutting off your thumb (which likewise increases its adjusted Eagle chance); not having cancer counterfactuallyL depends upon (and is caused by) your escapades through history with the neuralyzer; and so on.

13.6 Summary Time travel gives rise to interesting Newcomb cases. If someone would one-box in a non-time travel case like Boxes, they should clearly take the option analogous to ‘one-box’ in the time travel cases as well. If someone would two-box in Boxes, I have argued at length that they should nevertheless take the option analogous to ‘one-box’ when in these weird time travel cases. I’ve shown this for three motives for two-boxing. There are other motives for two-boxing, but I hope that this chapter provides the roadmap of what to say for all such motives. For instance, whilst above I considered the Counterfactual Motive for two-boxing, one might instead believe that we should be motivated by indicative conditionals and that this also bears out two-boxing [DeRose 2010]. Assuming that indicative conditionals receive a possible worlds analysis [Davis 1979; Lycan 2001; Nolan 2003; Stalnaker 1975; Weatherson 2001], I’d imagine that the strange and funky things which go on in time travel cases (with possible worlds, impossible worlds, evaluating closeness in different contexts, and all the sorts of things already discussed in §13.2) will again bear out that the correct indicative conditional motivating our actions tells in favour of one-boxing in time travel cases even when they tell in favour of two-boxing in Boxes. Or consider Lewis’s motive for two-boxing in terms of dependency hypotheses [Lewis 1981a]. A time travel case would be a case where there’d be but a single dependency hypothesis and Lewis is already clear that in those cases we should do what the one-boxer would do [Lewis 1981a: 11; see also Bales 2016: 1501–3]. So we should two-box in Boxes but one-box in time travel cases. For both of these two-boxer motivations this is only a sketch of the tack to take, not a full-bodied defence. Nonetheless, it keys us into the tactics which would hopefully work to tailor alternative motivations to lead to the same conclusion. Chapter 14 describes the practical upshots of Chapters 12 and 13 on how we act—right here, right now—in the world we live in. Plague shows that there can be practical effects on our actions merely because of the threat of someone turning on

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a time machine. As Chapter 14 discusses, it turns out that similar thinking means that even though no time machines exist, we should bear it in mind when making current decisions.

13.7 Appendix: Perfect Predictors 13.7.1 Two-Boxers and Perfect Predictors Boxes is a standard Newcomb case. But the stereotypical example involves a perfect predictor: Oracle: You enter a room containing two boxes, A and B. You can take either or both boxes. However, five minutes ago, Sybil—a perfect predictor—predicted how many boxes you’ll take. Sybil filled them accordingly, filling Box B with £1m if she predicted ‘one box’ and nothing if she predicted ‘two boxes’. Either way, she filled Box A with £1,000. You know Sybil did this, cannot look inside the boxes etc. Which box(es) do you take? I’ve studiously avoided discussion of perfect predictor cases like Oracle. This is partially because they seem to some to be so weird and implausible [cf. Bar-Hillel and Margalit 1972; Schlesinger 1974]. Absent a crystal ball, or other time travel mechanism for prediction, it’s hard to understand how Sybil could have such powers (and if Sybil used such time travel mechanisms, then Oracle wouldn’t really be a Newcomb case and everyone should clearly take just one box [BenMenahem 1986: 199; Mackie 1977: 218]). But perhaps Sybil is, e.g., a Laplacian Superscientist living in a deterministic universe [Schmidt 1998] or some such. That said, I accept cases like Oracle are metaphysically possible. Here’s the snag: this chapter’s argument would have it that we should one-box in Oracle. For instance, imagine you’re moved by the Counterfactual Motive. Consider: B*: a millionaire.

Were an agent to select Box B in Oracle, they would become

Unlike B in Boxes, in Oracle B* is true even in the causal context. That is, whilst in Boxes the Counterfactual Motive drives one to select two boxes, in Oracle it drives one to select one box. To see why, apply (my modified version of) the Stalnaker-Lewis analysis to B*. Where O stands for the proposition that you select one box and M is the proposition that you become a millionaire, B* translates as: Some O∧M-world is closer to the actual world than any O∧¬M-world.

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We must hold fixed everything we are certain about in Oracle, e.g. hold fixed that Sybil is a perfect predictor. And the only worlds where Sybil is a perfect predictor (and the rules of the Newcomb game are what they are etc.) and O∧¬M is true are metaphysically impossible worlds. Therefore, when we evaluate the counterfactual in context CL, those worlds must be far from actuality. Thus, some O∧M-world is closer to the actual world than any O∧¬M-world. Hence B* is true and in Oracle those moved by the Counterfactual Motive should one-box. Similar thoughts apply to the other motives. For instance, selecting one box counterfactuallyL causes you to become a millionaire, so those moved by the Causal Motive should one-box. So we have it: unlike Boxes, I recommend that when we have a perfect predictor we should one-box. I’m fine with this and side with those who believe precisely the same, i.e. believe that you should one-box with a perfect predictor but two-box with an imperfect one (for a list of such people, see Ahmed [2014: 172n16–17]).

13.7.2 Ahmed’s Argument However, Ahmed [2014] argues against exactly this position, saying that we cannot have disparate treatments of Boxes and Oracle. Assume, he says, that an agent should one-box in Oracle (because the predictor is perfect) but two-box in Boxes (because the predictor is imperfect). Then consider: Chas & Dave: Just as with Boxes, you are picking boxes which have been filled (or not) by a predictor. The only difference is that there are two predictors: Chas, a perfect predictor, and Dave, who always gets the wrong answer. A predictor was randomly selected; there’s chance n that Chas was selected and 1n that Dave was selected. You don’t know who the predictor was. How many boxes do you pick? How many boxes you select clearly depends on the value of n.⁴ If the predictor is likely to be Chas—even if that chance isn’t one—we should one-box. If it’s likely that it’ll instead be Dave, we should two-box.

⁴ As Ahmed explains, Chas & Dave is directly analogous to the following scenario: Lottery: You can either have (i) a lottery ticket which pays out £1m with probability n; or (ii) a lottery ticket which pays out £1m with probability 1n plus £1k. In Lottery, were n high you should take the lottery ticket and were it low you should take the cash in hand plus the worse lottery ticket. Clearly option (i) in Lottery corresponds to selecting one box in Chas & Dave and option (ii) corresponds to selecting two boxes. Equally clear, then, is that the value of n makes a difference as to whether one should one-box or two-box in Chas & Dave [Ahmed 2014: 174].

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But, says Ahmed, randomly selecting between Chas (who always gets it right) and Dave (who always gets it wrong), with chance n of getting Chas, is no different from receiving a prediction from an imperfect predictor who has temporal parts that get it right and temporal parts that get it wrong, where we have chance n of speaking to the temporal parts which get it right. And if those situations are no different, then—assuming that we should treat Oracle and Boxes disparately and believe that the value of n makes a difference to Chas & Dave—the value of n must also make a difference to Boxes, i.e. one-boxing is correct, not two-boxing. But this isn’t right. There is a difference between Chas & Dave and the imperfect predictor of Boxes. We one-box with Chas/Sybil because—in the presence of a perfect predictor—our choice counterfactuallyL causes (or is the counterfactualL dependent of etc.) the perfect predictor’s choice. And this is never the case for Derren. Even if the temporal part of Derren which predicted my actions gets it right, there’s no likelihood whatsoever of my eventual choice counterfactuallyL causing that temporal part to get it right. We only get counterfactualL causation in weird scenarios involving time travel or perfect predictors (or Reapers etc.). Derren’s temporal part happening to get it right is quite unlike Chas’s or Sybil’s perfect disposition to get it right each and every time; in Boxes there’s no counterfactualL causation at all. So Boxes and Chas & Dave aren’t analogous after all. We treat situations differently only when we start to suspect that certain counterfactualsL are true. Consider: B’ C: Box B containing £1m counterfactuallyL depends upon my selecting one box. In Oracle we are certain that B’ C is true, i.e. Crϗ(B’ C)=1. In Chas & Dave it’s true only if Chas were selected to predict, i.e. Crϗ(B’ C)=n. But in Boxes there’s no probability of B’ C being true, i.e. Crϗ(B’ C)=0. Even if Derren predicts correctly, it’s not the case that his prediction counterfactuallyL depends on my choice. In each case, one’s fever for one-boxing tracks the likelihood of there being weird counterfactualL connections. Since there are such connections in Oracle, might be such connections in Chas & Dave, and (by hypothesis) definitely aren’t any in Boxes, my disparate approach to Oracle and Boxes can be justified.

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14 The Tourist Paradox The Tourist Paradox is an inductive argument for the physical impossibility of time travel based upon the lack of time-travelling tourists from the future (§14.1). There are various extant objections (discussed in §14.2) but I believe that the best answer is that there are no tourists from the future because, no matter what the laws of nature might allow—and no matter what the objective chance of time travel might be—our credence of there being time travellers should have been effectively zero all along. Alongside this line of thinking, I further argue that we should regulate our current activities where they might have a non-negligible chance of bringing about time travel events (§14.3). The sorts of high-energy experiments we currently conduct qualify as a potential area of concern; thus we should consider adding additional levels of regulation to such experiments (§14.4). The chapter ends with a discussion of naturally occurring time travellers, such as tachyons. I argue that their apparent absence doesn’t falsify theories predicting their existence (§14.5).

14.1 The Tourist Paradox Thus far, this book has focused on the metaphysical possibility of time travel. I now turn to an argument against its physical possibility: were time travel physically possible, we’d likely see time travellers from the future; their absence is therefore evidence that time travel isn’t physically possible [Al-Khalili 1999: 161; Fulmer 1980: 155–6; Grey 1999: 56; Hawking 1992: 610; Horwich 1987; Reinganum 1986; Wasserman 2018: 22n64]. This is the ‘Tourist Paradox’. Examine the argument in more detail. Assume, for reductio: F: Time travel is physically possible and is feasible for a sufficiently advanced society to achieve. An ‘advanced society’ is one with technological and scientific understanding millennia in advance of our own. Our global society is comparatively young, having only discovered electricity 200 years ago; an advanced society would be as far removed from ours as our own is removed from protohistorical societies which had only just begun to master clothing, agriculture, and masonry. Such a society need not even be biological, perhaps being composed of artificially Time Travel: Probability and Impossibility. Nikk Effingham, Oxford University Press (2020). © Nikk Effingham. DOI: 10.1093/oso/9780198842507.001.0001

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intelligent machines created by organic forebears. Note also that F is stronger than merely saying that time travel is physically possible. F could be false because time travel was physically possible but only during conditions present at the Big Bang, or only with resources no society can reasonably be imagined to muster. Having clarified F, return to the argument. Assume both: D: Advanced societies almost certainly discover everything within their purview. A:

Mankind will eventually become an advanced society.

Given F and D, an advanced society would (almost certainly) know how to travel in time. Given A, mankind will come to know how to travel in time. It isn’t hard to imagine that, if mankind discovered time travel, they would travel back in time to the present day. Perhaps literal tourists will visit the past, a common fictional trope (see, inter alia, Silverberg’s [1969] Up the Line, Kilworth’s ‘Let’s Go to Golgotha!’ [1975], and Polák’s Zítra vstanu a opařím se čajem [1977]). Perhaps people visit the past for other reasons. It has already been noted that it may play a role in paleontology [Allen 2020]—see also the quantum palaeontologists of Silberling’s Land of the Lost [2009]. Indeed, fiction suggests other reasons: see the geologists studying past environments in Robson’s Gods, Monsters, and the Lucky Peach [2018]; the time travelling thief from Star Trek: The Next Generation’s [1991] ‘A Matter of Time’; or the filmmakers looking to save a few bucks on film sets in Harrison’s The Technicolor Time Machine [1967]. Or perhaps mankind’s interaction with the past is more low-key, involving merely sub-atomic time travel (for instance, at the end of Benford’s [1980] Timescape stray tachyons from communications between future starships are overheard). Whatever the mode of interaction—whether it’s literal tourism or mere subatomic intrusions into the past—I’ll collectively call such interactions ‘sightseeing’. The next premise is: S:

If mankind could travel in time, it’d almost certainly sightsee.

Given the above premises, mankind is almost certainly going to try and sightsee the present. But this is problematic because: N-: There are no sightseers (i.e. nothing which has come from the future is currently interacting with our present time). By induction, one of these premises is probably false. The proponent of the Tourist Paradox argues that the most likely premise at fault is F—time travel is physically impossible (or effectively impossible because it’s not feasible).

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14.2 Objections Who knows whether F is true or not? I’m no physicist and I can’t make a case for time travel’s physical possibility—if you’ve decided to read this book in the hope of answering that question, then you’ve made a poor book-buying decision. But, regardless of whether F is true or false, this chapter argues that the Tourist Paradox is not a good reason to deny it. I offer my own argument for this in §14.3. However, because I don’t know of any in-depth philosophical discussion of the Tourist Paradox, §14.2 discusses the prospects of some of the extant objections before moving to my own.

14.2.1 Bad Objections Some objections are less than convincing. Par for the course is to moot a possibility which, were it actually true, explains why we don’t see any time travellers even though F is true. But all too often it seems the mere possibility of such a thing is where the argument ends. That’s not enough. To provide a compelling response to the Tourist Paradox the argument must be that the possibility is more likely to be true than it is likely that F is false—given that the possibility of time travel is exotic, it’s not unreasonable to think F is fairly likely false and so that’d be quite a task indeed. All too often the objections to the Tourist Paradox fall foul of this. For instance, one might take a dim view of mankind’s future and deny A. Were mankind to die out, no time machines would be invented and no sightseeing would occur. The most pessimistic might believe that the end is just around the corner and that, in the next 200 years or so, a catastrophe—nuclear war, antibiotic resistant bacteria, errant asteroids, supervolcanic eruptions, artificial intelligences running riot, etc.—will result in our extinction. But whilst it’s possible that some catastrophe occurs, it’s not likely. For instance, an informal survey at the Global Catastrophic Risk Conference [Sandberg and Bostrom 2008] ranked it at roughly 20 per cent by 2100 — and I daresay that their estimations of the dangers of nanotech and artificial intelligence (which account for half of that risk) are somewhat over-egged. It’s hard to see why one wouldn’t think that it was more likely that the laws of nature simply didn’t allow for time travel (i.e. that it was more likely that F was false). A more optimistic alternative changes the time scale. Thinking back to §1, even were time travel physically possible, it might well involve creations on a planetary scale (e.g. rotating black holes, stellar-scale warpings of spacetime etc.). Whilst mankind might last the next few centuries, the more optimistic pessimist says that it won’t last the many millennia it needs to in order to get to the appropriate stage

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of technological mastery. Even then, it’s hard to make the case that this is probable. If a civilization manages to go from occupying one planet to many—that is, goes from being monoglobal to being polyglobal—it becomes hard to see what can threaten it. Monoglobal civilizations may be threatened by nuclear war, asteroid strikes, volcanic activity etc., but not polyglobal civilizations. Whilst some events can destroy polyglobal civilizations (e.g. a gamma ray burster sweeping clean entire sections of the galaxy), it’s a lot less likely that such a polyglobal civilization becomes extinct than a monoglobal one. And as long as it trundles on, it should eventually reach the required level of technology (even if, during the period that it takes to get there, there are troughs where advancement stalls or even retreats). Just as long as one thinks mankind will manage to travel to other planets within the relatively tight time frame of the next few centuries—a far more achievable aim than engineering stellar-scale time machines!—it will be likely that mankind will survive into the far future. (For more on mankind’s eventual fate, see Baxter [2001] and Hanlon [2008].) Alternatively, we might doubt S. Perhaps there’s no reason for people from the future to return to the past. Perhaps they know everything they feel they need to about the past and see no point in returning [Al-Khalili 1999: 191; Grey 1999: 56]. Or perhaps the creation of time machines is a resource-heavy activity and that, no matter how interesting our television shows are or how tasty our fusion cuisine is, our future descendants are unlikely to want to squander their precious assets visiting the twenty-first century [Deutsch and Lockwood 1994a: 72]. I don’t deny that this is possible. But I doubt that we should say that this is probable. Societies throughout history have been interested in history, exploration, and experimentation. Whilst it’s admittedly possible that they might lose a zest for these things, it’s hard to see why one would think that, faced with the Tourist Paradox, we should come to believe it’s more likely that this is the case than that F (which is exceedingly speculative!) is false [Casati and Varzi 2001: 581; see also Goddu 2007 for a list of reasons to travel through time]. Nor should we forget that ‘tourists’ need not be actual tourists. Even were our descendants to have little motive to return to the present day on the grounds of mere intrigue, they have every reason to travel back in time for other purposes. No matter what happens, our descendants will need the bare essentials for life, such as somewhere to live and energy to live off. And no matter how long mankind might live, they’ll know that, inevitably, we’ll be left eking out an existence using whatever meagre energy can be harnessed. But if they can travel in time then they can go back and utilize the energy of the hundreds of billions of stars which have already burnt out back in the past. That wasted energy need not be lost! It’s hard to see why any civilization would not take that chance. And if mankind travels back to the dawn of time, the galaxy should be teeming with our descendants. Evidently, this is not the case. So it’s hard to believe that S is more likely to be false than F.

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Finally, we might deny N-. Perhaps the tourists are here but are making efforts to hide [Arntzenius 2006: 611]. But this sounds implausible given the scenario just discussed. If mankind’s descendants have returned from the future to farm the universe for energy, and have been at it for billions of years, then they should be scattered across every star system in the heavens. But when we look out, we see no sign of any other intelligent life [Davies 2010; Webb 2002]. Just as this is evidence that there are no alien races, the absence of any sign of our descendants is compelling evidence that there are no time travellers from the future. In any case, why would they hide? Whilst it’s a common trope in science fiction that time travellers should make efforts not to reveal themselves lest they ‘contaminate the timeline’—see Anderson’s Millennium [1989] as well as innumerable episodes of Doctor Who and Star Trek—that’s merely a plot device, not a serious philosophical position. Given Ludovicianism, the timeline cannot be ‘contaminated’, for time can never change. So if there are time travellers nearby, we should wonder why they aren’t open about their presence. What monsters they would be to not have cured cancer, solved world hunger, and taken other low-cost, highvalue actions. For a convinced Ludovician, there’s nothing to fear by giving the natives of twenty-first-century Earth vast medical advancements for, upon your return to the future, you won’t find that anything’s changed. Again, then, the probability that time travellers are with us but hiding seems a lot lower than that of F being false. An alternative reason for denying N-O is that the world might be nonLudovician, i.e. an indexed world. Al-Khalili [1999: 161] suggests this, arguing that N-O might be false, for if we lived in a universe-indexed world, then, whilst there could be time travellers interacting with the present moment, we would not see them since they’d be interacting with the present moment in another universe. (Similar thoughts would apply were the world hypertemporally indexed.) In §12.5 I laid down the groundwork for understanding probability at indexed worlds. That understanding bodes badly for this response to the Tourist Paradox. There are two relevant factors: the chance of us travelling to the past at some future time and the chance that the arrival of time travellers in the past scuppers us travelling to the past at that future time.¹ Since the introduction of universe indexing is meant to make for a unique response to the Tourist Paradox, I can safely assume, for the purpose of argument, that the first factor is high. If the second factor is low, it follows that it’s very likely that there are scads of universes, ¹ By limiting myself to two factors, I caricature the situation. I ignore the possibility that time travellers might merely lower the chance of us travelling in time, rather than preventing it entirely. I ignore the possibility that their arrival might raise that chance. I ignore the chance of time travellers arriving in the past and then travelling again, deeper into the past. I incorrectly assume that the chance of these two factors is static at every one of the universe indexer’s universes. However, caricature that it is, I take the fundamental lesson of the main text to nevertheless be apposite.

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of which only one would be a universe in which no time travellers appear. Given that it’s equiprobable which universe we’re in, it would then—contrary to AlKhalili—be very likely that we’d witness time travellers from the future. The higher the second factor is, the less likely we’ll be in such a universe, although only if the second factor was such that the arrival of time travellers almost certainly prevented more time travel (and why would we think that?) is it not most likely that we’re in a time travel universe (and even if time travellers arriving almost certainly prevented any more time travel taking place, it’d still be roughly 50/50 whether we saw time travellers). In any case, this assumes that people in the original universe (at which no time travel is witnessed) manage to travel back in time just the once. Given the conceit of the Tourist Paradox, we should assume otherwise and believe that there’s a high chance of lots of people trying to travel back to the past many times over. To each attempt corresponds a different universe.² Thus, even were the second factor high, there’d still be many, many more universes at which time travellers arrive in the past than the sole universe at which this doesn’t happen, i.e. it’d be overwhelmingly likely that people in our situation would witness time travellers arriving from the future. Thus, the introduction of indexed worlds does nothing to independently resolve the Tourist Paradox.

14.2.2 Restricted Time Travel A better objection to the Tourist Paradox is that time travel might be restricted: time machines can’t travel to times before they were constructed. That time travel might be restricted in this way is a feature of the more ‘plausible’ time machines discussed back in Chapter 1, e.g. wormholes, Krasnikov tubes etc. And if it is restricted, S would be false because time machines couldn’t be used to travel in time to a point before they were constructed. We shouldn’t expect to see any time travellers because no time machines have yet been built; the lack of time travellers only demonstrates that as yet no-one has discovered time travel [Arntzenius 2006: 611; Deutsch and Lockwood 1994a: 72; Lockwood 2005: 153–4; Richmond 2001: 307]. We can get a lot of mileage out of this answer. But it’s not without issue. For instance, if wormholes, Krasnikov tubes etc. can be manufactured, it’s not outlandish that they could occur naturally [Gott 1991; Ori 2005: 4; Päs et al. 2009] or, alternatively, that some alien race has already created such a warping which can be used by ourselves at some point in the future. In both cases, even though the

² The multiple time travellers can’t all go to the same universe, for if time travellers at universeindexed worlds try and travel to different past times, they end up in different universes [Effingham 2012b].

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human race hasn’t yet figured out how to make such time machines, we could nevertheless be in a position to encounter time travellers from the future who have utilized such things. Further, whilst this objection leaves open that F is true, a closely related principle is nevertheless still (probably) false: F*: Time machines which can travel to points before their creation are both physically possible and feasible for a sufficiently advanced society to achieve. To cave on F* is to admit that Tourist Paradox-style reasoning rules out the feasibility of manipulating tachyons, controlling advanced waves, constructing Alcubierre warp drives etc., for if such things were feasible, we’d see time travellers arriving in their warp vessels or we’d read Morse code messages amongst the experimental data of mass-sensitive physics experiments caused by tachyons sent from high school children in the forty-fifth century. So whilst the solution would solve the Tourist Paradox and rescue F, it’d only be by opening up a similar paradox that demonstrated that F* is probably false. If you worry, like I do, that philosophical reasoning like the Tourist Paradox can’t prove even that more limited conclusion, you should be suspicious of the ‘restricted time travel’ response to the Tourist Paradox and be open to alternative solutions. And such an alternative is available, at least assuming that we live in a Ludovician world. And this alternative is such that we don’t need to worry about whether this ‘restricted time travel’ response is a good one or not since the Ludovician is obliged to accept that alternative regardless, and it explains the lack of time travellers quite apart from whether time travel is restricted or not. It is to this answer that we now turn.

14.3 Probability, Decisions, and Tourism 14.3.1 The Ultimate Banana Peel Having briefly skimmed existing responses, this section advances a new objection grounded in the work of Chapters 12–13: assuming we’re in a Ludovician world, our credence of being in a time travel situation—even a time travel situation where time travellers are remaining incognito—should be effectively zero; thus the most likely explanation of why there are no time travellers is that we are unlikely to ever travel in time—perhaps unlikely to even discover how to travel in time (as Deutsch suggests [Lockwood 2005: 169]). We should therefore be effectively certain that, were F true, one or the other of A, D, or S must be false.

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(At least, that’s true if we live in a Ludovician world. If the actual world were indexed, this thinking wouldn’t apply and a different response would be called for. I suspect it’d be more reasonable to think that the absence of time travellers did demonstrate that F was false—or, at the very least, that F* was false. The conclusion of the Tourist Paradox should be disjunctive: either the world is Ludovician, yet we should expect no time travellers, or F/ F* is false.) If our credence of time travel coming about is effectively zero, then time travel is impossible for all practical purposes. And you might think that there’s therefore little difference between my position and thinking that F is false—after all, if your credence that someone ever travels in time is effectively zero, how different is that from thinking that time travel is physically (or, at least, practically) impossible? It turns out that there’s a huge difference between living in a world where time travel is physically impossible (or otherwise practically unachievable because of some physical constraint) and living in a world where it’s physically possible. To see why, imagine that we lived in a world where it was absurdly easy to create a time machine—that is, the laws of nature were such that there was a very high objective chance of people travelling in time. For instance, imagine that combining easily available household ingredients creates a potion which whisks the drinker back into the past. You would not want to live in such a world for exactly the same reason that someone who was certain that they were about to try and kill their grandfather (using a time machine that cannot malfunction) should start fearing for their health (which was the conclusion of §12.4.5). Given that it’s effectively certain that no-one travels back in time, we’re certain that no-one drinks the potion and would have a very high credence that the ingredients are never combined in the right order. Everyone who intentionally attempts to do so will fall foul of some incident: some might change their mind; the more bullish might get off lightly and mix the ingredients wrong or accidentally spill the contents everywhere; the less fortunate will suffer some calamity, such as a heart attack. Indeed, some will be struck dead not because they tried to make the potion but because at some point in the future they would otherwise have tried. For instance, someone who otherwise would’ve become a curious chemist might die in childbirth. And the same thinking applies to those who might accidentally make the potion. Imagine that the correct combination of ingredients was unknown; in that case, cooks who experimented with recipes would have a high objective chance of making the potion by accident and we should expect such cooks to regularly fall foul of some calamity at a vastly increased rate compared to noncooks. Extending this thinking to its natural conclusion, these facts would pose an existential threat to everyone. Whilst numerous people might suffer a string of unlikely accidents in order to prevent them from consuming the potion, the objective chance of that is exceedingly low. But a single event which prevents

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potion consumption and has a low objective chance—but nowhere near as low as the string of unlikely events—is easily imaginable. For instance, rather than a string of heart attacks and potential cooks dying in childbirth, everyone could be wiped out by nuclear war or an asteroid strike. If everyone was dead, no potion could be made. The chance of such a catastrophe might be quite low, but not as low as tens (or hundreds!) of thousands of people each suffering a string of unlikely events conspiring to stop them swigging the potion. Recalling I C from Chapter 12, i.e. that our credence in ψ should equal the objective chance of ψ coming about given that metaphysical impossibilities do not occur, then those living at such a world should have a high credence that a global catastrophe is about to occur—Armageddon is the ‘ultimate banana peel’ which trips up every potential time traveller by ensuring that they don’t exist in the first place. The moral of this fictional potion story carries over into the more plausible cases where the objective chance of us travelling in time is zero given our current technological understanding, but will be substantially higher were mankind to become a polyglobal galactic civilization that can create stellar-scale spatiotemporal warpings. If D and F are true, there’ll be a high objective chance of mankind’s descendants discovering how to do this, even though that discovery lies far in the future. And were S true, there’d be a high objective chance of them creating one to use. We know that some event, or string of events, will prevent them. Imagine an example typifying that string of events: the scientists of 120,000  repeatedly lose the relevant paperwork; each scientist who would come up with the crucial information dies by accident before they make their discovery; the construction of the time machine is plagued by faults; and so on, throughout the millennia. Then imagine an example typifying the single, globally catastrophic, event: a nuclear war wipes out mankind before we become polyglobal and we’re prevented from making a time machine in the 120nd millennium in virtue of us having died out back in the third. The objective chance of the former, millennia-long, chain of unlikely events is—by many orders of magnitude—lower than that of the contemporary nuclear war. And we know that, in such cases, we should then adjust our credences accordingly: were F, A, D, and S true, we should have a high credence of a nuclear war (or some other catastrophe) wiping us out in the near future. In other words, we should believe that the ultimate banana peel is on its way.

14.3.2 Rationality and the Ultimate Banana Peel The truth of F is outside our control. And it’s desirable that A and D are true. But it’s within our power, as a species, to determine whether S is true or not. If we inculcated within everyone an

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understanding that time travel was an actively bad thing to pursue, S would no longer be true. And if we did this then we’d avoid whatever catastrophe would otherwise be expected to befall us, for if we believed mankind would intentionally avoid time travel, we’d no longer have a high credence that some catastrophe will wipe us out prior to becoming a polyglobal society. We can avoid the ultimate banana peel. One objection would be that it’s irrational to do this. Either the catastrophe is on its way or not: if nuclear war is going to happen, it’s going to happen or not; either an asteroid is or isn’t on its way to destroy civilization; either the volcanic super-eruption is or isn’t going to occur etc. But I argued in Chapter 13 that it is rational to do things like this. Even though there is no ‘biff’ causal connection between our intention to not travel in time and a catastrophe wiping us out (or not), it’s nevertheless rational to avoid time travel, and make concerted efforts to ensure that S is false, because we should ‘one-box’ in time travel cases. A more subtle objection is that we end up in a situation analogous to the Toxin Paradox. In the Toxin Paradox an agent is offered money if they honestly promise that a year later they will willingly consume a deadly toxin. They don’t get the money for consuming the toxin; they get it for merely promising to do so. No rational agent can sign up to this deal, for they know that, when later comes around (and they already have the money), they will not consume the poison. Thus, they cannot honestly promise to take it in the future. One might think mankind is in the same position. Even if it makes sense for us to currently form an intention to not build a time machine in the 120nd millennium, when we get to that stage (and have become a polyglobal civilization that knows for certain that no catastrophe took place back in the third millennium), there’d be no such compunction. Thus mankind will start their time travel experiments and we know that any current efforts to make S false will ultimately fail—we should, therefore, expect a global catastrophe to be on its way (at least, were F true). But the cases are dissimilar. In the Toxin Paradox there is no continuing incentive to intend to take the toxin. Our descendants, however, will have a continuing incentive to not travel in time. Whilst, as a polyglobal species, they needn’t worry about species-wide existential threats, they’ll know that anyone who tries to travel in time will find their project beset by problems—quite possibly problems terminal to those involved (e.g. scientists being struck by heart attacks or an explosion wiping out its participants). They’ll know that all such projects will result in failure. Who, in their right mind, would want to be involved in a project doomed to fail and which raises the probability of death for all involved? No-one will want to build a time machine, even in the polyglobal future. (See also Casati and Varzi [2001: 581–2], who discuss a substantially similar claim.) So I have a revised response to the Tourist Paradox. If the reasoning of this book is correct, it is very likely that S is false because no well-informed

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agent will ever try to travel in time even if, in theory, they know how. Thus there’s no reason to think F is false on the back of the lack of time travellers.

14.4 The Accidental Time Machine 14.4.1 The Dangers of High-Energy Experiments The practicalities of making S false might look easy. Above, I painted a picture of a future mankind, far removed from the present day, having to go to great lengths to travel in time. I’ve imagined that our descendants would have to engage in grand schemes on vast scales, creating stellar-sized spacewarps, steering the movements of cosmic strings, creating wormholes in interplanetary space, and so on. These things are nothing like the magic time-travelling potion; these are not the sorts of things that one does by accident. Nor do we need to imagine special measures required to prevent such time machines—it doesn’t take a dictator to prevent people stellar engineering! But time travel needn’t take place on that scale and the existential threats arise no matter what scale of time travel event takes place, no matter how small or microscopic. Consider contemporary high-energy experiments (‘HEXs’). In a HEX, particles are smashed against one another at close to the speed of light. Currently, the highest energies are generated at CERN’s Large Hadron Collider (LHC). In December 2015 it achieved energies of 13 TeV; as of writing, it has not yet been powered up to its maximum 14 TeV. It has already been suggested that such experiments might bring about sub-atomic cases of time travel: Ho and Weiler [2013] and Nielsen and Ninomiya [2008, 2009] have suggested that the LHC might allow particles to travel back through time; Aref’eva and Volovich [2008] have suggested that the LHC could result in traversable wormholes at the microscopic scale. Time travel events due to such HEXs might happen accidentally, in a way that the time machines made by stellar engineering could not. If HEXs had a non-negligible objective chance of bringing about time travel events when run at (or above) a certain energy level, HEXs would pose an existential threat to humanity. Even trying to run the HEX would make a catastrophe more likely to wipe out mankind. The reasoning is just the same as that of §14.3.1. We are certain that time travel won’t occur. Assume that a HEX we are planning to run would have a high objective chance of bringing about a microscopic time travel event. For the purpose of argument, imagine that if the LHC experiments are conducted at 14 TeV then it will most definitely bring about such an event. It follows that the HEX will be prevented either by some catastrophe or some string of unlikely events continually besetting the attempt to run the HEX (which would have to prevent it from being conducted not just now, but for the rest of eternity). The objective chance of something like the latter is far lower

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than the objective chance of the catastrophe. As placeholders, imagine that only a nuclear war occurring next year (objective chance: one in a thousand) or a string of unlikely events (lost paperwork, funding stops, enough magnets blowing up in the LHC to make people quit; objective chance: one in a million) could prevent the HEX. Since we know for certain that the HEX won’t run, we should be almost certain that nuclear war will break out. We can lay that out more formally using some toy figures. Say that: t₁ is the time at which the agent is considering whether to conduct a HEX at t₂; T T is the proposition that we successfully conduct HEXs and microscopic time travel events occur; W is the proposition that nuclear war breaks out next year and everyone dies; S is the proposition that a string of unlikely events prevent mankind from conducting HEXs at that energy level; ⏈ stands for no metaphysical impossibilities coming about; and ⊥ stands for metaphysical impossibilities coming about. Using the toy values from above, the objective chances at t₁ would be: Ch(T T∧⊥)=0.998999 Ch(W)= 0.001 Ch(T T∧⏈)0 Ch(S)= 0.000001 Recall Chapter 12’s principle for determining credences: I C: the case.

CrΓ(φn) is the value Ch(φnj⏈) would take were Γ

Given that T T, W, and S are mutually exclusive outcomes, it follows that a well-informed rational agent’s credence, prior to conducting the HEX, should be Crϗ(T T) = Ch(T T∧⊥j⏈) + Ch(T T∧⏈j⏈)  0 Crϗ(W) = Ch(Wj⏈)  0.999 Crϗ(S) = Ch(Sj⏈)  0.001 Thus, we should have an exceedingly high credence that nuclear war is about to take place. Nielsen and Ninomiya [2008, 2009] argue for the same sort of conclusion. They also point out that unlikely events have befallen our efforts to conduct HEXs. The Texas Superconducting Super Collider was beset by problems and cancelled in 1993. The LHC met a string of issues: funding issues; an explosion between magnets; a piece of baguette overheating a magnet; a contributor being arrested

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on terrorism charges [Harrell 2009]. That’s (worrying!) evidence that a future HEX run by the LHC would bring about time travel and that these accidents have occurred in order to delay the HEX just long enough for some singular event to make sure it never takes place. That singular event might be a catastrophe (e.g. a nuclear war), but it need not be. It could, instead, be the scientific community putting in place mechanisms whereby we shy away from conducting such HEXs, thereby intentionally preventing events with a positive objective chance of bringing about time travel.

14.4.2 Policy and Response If this is right, then the urgency of acting on these worries about time travel isn’t to be solely left to our descendants many millennia hence. Instead, right here and right now, we are obligated to implement certain policies in light of these fears. But the policy that should be implemented should not be one which assumes, as I did in §14.4.1, that the LHC running at 14 TeV would definitely bring about a microscopic time travel event. A policy based on such a claim would be obvious fearmongering, for no-one has demonstrated that such a thing would definitely happen. Indeed, whilst the above-cited papers suggest such microscopic events are epistemically possible, those speculations are far from solid and doubtlessly few would fix a high credence in the 14 TeV experiment bringing about microscopic time travel. The key point, however, is that the resulting dangers to humanity, even if the risk is quite low, must be treated seriously. I am no expert on those risks. I have no idea to what extent we should take seriously the above physics papers on HEXs bringing about time travel. But what I can say is that if the risk is judged to be anything but most minimal, the experiment should not be conducted. One might ask what it takes for a risk to count as ‘most minimal’. I don’t know the answer to that question either. But that’s not an objection to my position, for this question of minimality is a question we can ask of any existential risk. It’s obvious to take pains not to do things which have, say, a 10 per cent probability of wiping out humanity (and we routinely take such pains by, say, deploying political leverage to avoid global thermonuclear war); it’s less obvious what to do when the probabilities are far less likely and/or harder to judge. An example of an existential threat which is far less likely is an asteroid strike. It poses an existential risk even though it is highly unlikely. An example of an existential threat which is hard to judge the probability of is the possibility of LHC experiments creating ‘strangelets’ that would bring about a cascading process converting the entire planet into ‘strange matter’, destroying us all. Strangelets are very speculative, and the exact probability of their being created in the LHC is hard to judge. (Similarly, there have been speculative worries about the creation of other existential threatening things in the LHC, e.g. black holes,

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magnetic monopoles, and a vacuum decay-causing event.) Nevertheless, for both low-level existential risks and existential risks which are hard to judge the probability of, we do not presume to ignore those risks given that the potential downside—i.e. the destruction of the human race!—is of such great disutility [Bostrom 2002, 2013]. And indeed we don’t ignore those risks—we take action! In the case of the threat of asteroid strikes, a serious amount of money is invested in things like NASA’s Near Earth Object Programme and the EU’s NEOShield project, with an eye on preventing impact events. In the case of the worry that the LHC might bring about the destruction of the world, it was considered as a possibility and a report written discounting the risk [Blaizot et al. 2003] (indeed, legal cases were mounted on those grounds [Adams 2009]). And other HEXs have been subjected to similar risk analysis, and—again—such risks have been discussed and discounted [Bostrom and Tegmark 2005; Jaffe et al. 2000]. So whilst I’ve no idea exactly what probability should be judged ‘safe’, or what to do when estimating those probabilities is difficult, this just puts my worries in the same boat that many other existential risks are in. Since we act (in some fashion) in light of those risks, we should act (in some fashion) in light of these risks and, before conducting any HEX, ensure that the chance of a microscopic time travel event arising from such a HEX is negligible. So I am not histrionically suggesting that we ban all HEXs. I merely suggest that we police them with one more existential risk in mind than we currently worry about. What I do suggest are banned are experiments explicitly intended to bring about time travel, no matter how low we may think their objective chance of succeeding. No such experiment should be approved. And, before we think that such a ban would be gratuitous, note that some people have tried such things—for instance, Ronald Mallett is trying to do just that [Storr 2016]. Banning the conduct of experiments explicitly intended to send something back in time is straightforwardly rational: either (i) the experiment will fail because time travel is physically impossible or it’s possible but this particular experiment doesn’t raise the chance of it happening (in which case, it’s a pointless experiment to conduct); or (ii) the experiment might bring about time travel, and does raise the objective chance of time travel happening (in which case, it poses a serious existential risk and should not be conducted). Either way, one should never conduct such experiments. One objection might be that extra policy or regulation concerning these matters is unjustified because, since time travel is impossible and stories about it happening are mere fictions, there is no need to worry about those risks any more than we should worry that a physics experiment might accidentally bring Voldemort back into the world. In response, I offer this entire book as a refutation of thinking that it is clear or obvious that time travel is impossible. Further, this chapter itself gives us reason to think that (at least some) physicists who doubt that time travel is possible have over-egged their suspicions, since some will have concluded that time travel is physically impossible on the grounds that there are no tourists from

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the future. Having now corrected that reasoning, some physicists may wish to look again at their sceptical stance towards time travel’s physical possibility.

14.4.3 The Cosmic Ray Test Worries about the existential risks of HEXs have often been mollified by consideration of cosmic rays. Some cosmic rays are of exceedingly high energy and will produce collision energies in excess of those produced in the LHC. For instance, the Oh-My-God Particle, detected in 1991, was an ultra-high-energy cosmic ray that would have had a collision energy roughly sixty times that of collisions in the LHC. Where worries are raised that a HEX will produce strangelets (or black holes, or . . . ), since the energies produced in particle accelerators are less than the collision energies routinely produced in nature, the HEX is presumably safe to conduct (for if such energies were an existential risk, we’d all have died long ago). This ‘cosmic ray test’ for existential threats has been explicitly relied upon when it comes to time travel events in HEXs. Luminaries such as Brian Cox [Highfield 2008] and James Gillies [Leake 2009] both cite the cosmic ray test when they rubbish worries about the LHC bringing about time travel. They are wrong to do so, for the cosmic ray test does not apply to the specific worries developed in this chapter. Imagine that we know that the objective chance of a time travel event occurring in such highly energetic cosmic ray collisions was high, but less than one. There’d nevertheless be a chance that no cosmic ray collision has ever brought about a time travel event. Even if it was minute, given how probability works in time travel cases—and given that we should be effectively certain that time travel has never happened—we should now be effectively certain that this minute chance has been actualized, i.e. we should be effectively certain that all such collisions have failed to result in a time travel event, even though the chance of that is very low. And none of this has any bearing on the probability of global catastrophes. If the human race died off, then the number of these ultra-highenergy cosmic ray collisions would remain unchanged. Since an apocalypse wouldn’t facilitate lowering the number of such naturally occurring collisions, we should not expect to be wiped out by nuclear war or asteroid strikes, for whilst it might be shockingly unlikely that cosmic rays result in no time travel events, that becomes neither more nor less shocking were the human race annihilated. Nevertheless, generating artificial high-energy collisions would still pose an existential threat because, clearly, the human race’s annihilation does facilitate stopping those sorts of events. Whilst there’d be an unusual proliferation of naturally occurring high-energy collisions, each—against the odds!—failing to produce a time travel event, that makes no difference to the chance of the HEX high-energy collisions doing likewise. Compare: full in the knowledge that

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smoking has a high chance of giving you cancer, just because those around you have smoked forty a day and remained cancer free doesn’t mean that you should think yourself safe to smoke; being surrounded by unusual events of a low chance doesn’t license you to engage in the risky behaviour yourself. Similarly, even given that (naturally produced) cosmic rays always beat the odds to produce no time travel event, that doesn’t mean catastrophes won’t happen were we to attempt to activate the LHC. Thus, unlike with many other existential risks connected with HEXs, the cosmic ray test is not a good indication that HEXs are safe to run.

14.4.4 The Nielsen-Ninomiya Card Test I’ve already noted that Nielsen and Ninomiya [2008, 2009] argue for something substantially similar to my proposal. They recognize that time travel events must be thwarted and that the knock-on effect of this is the raising of the posterior probability of prima facie unlikely existential threats. They further argue that activating the LHC at the higher energy levels might be a bad idea. Interestingly, they offer up a proposal for settling whether the LHC should be activated at such energy levels. This sub-section discusses a simplified version of that proposal; I argue that, unfortunately, it won’t work. Take a deck of one million cards (which we could always simulate on a computer). One card is the ‘Stop Card’; the rest are blank. An agreement is reached by the international community such that, when we intend to conduct a HEX with a higher energy than that generated previously, a single card is drawn at random. If, against all the odds, it is the Stop Card, then no experiments will ever again be performed at that energy level or higher. The chance of pulling a Stop Card is one in a million. The chance of an existentially threatening catastrophe is (e.g.) one in a billion. If the relevant HEX we’re thinking about performing would have a high objective chance of bringing about a time travel event, then (since pulling the Stop Card prevents all future HEXs posing a threat) we should have a high credence that the Stop Card will be pulled (since the card pull is a thousand times more likely than the catastrophe). And if the relevant HEX doesn’t have such an objective chance of bringing about a time travel event, then we’re almost certainly not going to pull it by accident, and so will pull a blank card and run the HEX. In effect, the card test safely ‘corrals’ the lowprobability single event which stops dangerous HEXs into being the result of a simple card test rather than being a nuclear war wiping out humanity (or an asteroid smacking into the Earth etc.). There are two problems with this test. First Problem: the probability of pulling the Stop Card has to be a lot higher than the objective chance of a catastrophe wiping out humanity. But the chance of

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a catastrophe wiping out humanity is far from miniscule. In the above example, I imagined that the chance of an existentially threatening event was one in a billion. That’s overly optimistic by many orders of magnitude. Imagine instead that the chance of the existential threat was a more reasonable one in 100,000. For the drawing of the Stop Card to be a thousand times more likely than the catastrophe—that is, for the pulling of the Stop Card to be almost certain were the HEX to have a chance of bringing about time travel—there’d have to be only one hundred cards. Whilst that’d make the chance of pulling the Stop Card almost certain in cases where the HEX was liable to bring about time travel, it’d make the scientific community a lot less disposed to agree to never conduct a HEX again if the Stop Card is pulled, for, with a deck of one hundred cards, there’d be a 1 per cent chance that the Stop Card is pulled by fluke rather than because anything funny is going on concerning probability and existential disaster. A one in a hundred chance is unlikely, but doubtlessly not unlikely enough to convince the international scientific community (which would have to include all future scientists as well) to never again run a HEX at that energy level. Thus I doubt that we can develop a test which will almost certainly generate the Stop Card were time travel a risk, whilst simultaneously almost certainly not generate a Stop Card when there is no risk. Second Problem: imagine the HEX would bring about a time travel event. A Stop Card is drawn and we don’t conduct the HEX—indeed, we never again conduct a HEX at that energy level or higher. In that case we can say that our planned efforts to conduct a HEX caused the card draw (in the ‘counterfactualL’ sense of cause, rather than the ‘biff’ sense; see §13.3.3). Further, the Stop Card would be evidence that a disaster would occur should the HEX be conducted. But now imagine that a Stop Card is drawn and that we try and conduct the HEX anyhow. In that case, the drawn card would not be evidence that the HEX was going to cause a catastrophe. This is because the HEX either is or is not prone to bring about time travel. If it isn’t prone, then, clearly, the unlikely card draw was mere luck and not the result of any weird counterfactualL causation. On the other hand, if the HEX is likely to bring about a time travel event, then some catastrophe will occur to prevent the HEX. Whilst that catastrophe would be (counterfactuallyL) caused to occur by our efforts to conduct the HEX, the Stop Card being drawn would not have been counterfactuallyL caused by our efforts in that situation. Our proposed intentions (counterfactuallyL) cause things to happen (e.g. drawing a Stop Card) only when the effect they cause prevents the HEX being conducted (i.e. the Stop Card draw is only counterfactuallyL caused when it is the case that the card draw manages to successfully prevent the HEX and all such future HEXs). Since it was the catastrophe and not the card draw which prevents us from conducting the HEX, it is the former and not the latter which is caused by our efforts. In that case, the card draw would also be a mere

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fluke. So, should we draw the Stop Card and go on to conduct the test, whatever happens, it’ll turn out that the Stop Card wasn’t evidence of anything; drawing the Stop Card is only evidence in cases where we don’t go on to conduct the test. Our actions after the card draw can make it the case that the Stop Card draw is not evidence of a looming disaster. Since running the HEX is valuable to us, we would prefer to run the test. So, if it were up to us, we’d prefer the drawing of the Stop Card to not count as evidence. Since that’s within our power to bring about, even once the Stop Card has been drawn we may as well act to make it the case that there’s no longer any evidence justifying not running the HEX, i.e. we may as well run the HEX and ignore the results of the card draw. One might object that these sorts of actions cannot ‘make’ something evidence or not. Intuitively, drawing the Stop Card either is evidence or it is not. Similarly, you either should take it as justification for not running the test or you shouldn’t, and these things are fixed regardless of your later actions. Normally, that’d be sensible reasoning. But it’s only sensible reasoning because this is the reasoning of the two-boxer. And whilst that reasoning is sound in non-time travel cases (like Boxes from Chapter 13), it’s unsound in time travel cases. In a time travel case it is rational to ‘one-box’ and act in order to change the evidence. Imagine there is videotape evidence of me murdering my best friend. I manage to abscond and travel back in time to a point prior to the murder. There, I convince my friend to fake the murder on videotape, before getting him to hide whilst the police investigate and arrest my earlier self, only emerging once earlier-me has absconded in a time machine. In such a case, I’d have changed the videotape from being evidence of a murder to being evidence of a weird practical joke. Given the argument from Chapter 13, it’s rational for me to do this sort of thing. Similarly, then, it’s rational for us to act to make the Stop Card being drawn to no longer count as evidence. Nor can Nielsen and Ninomiya object by denying the arguments advanced in Chapter 13 and saying that we should two-box in time travel cases. If one should two-box in time travel cases, then Nielsen and Ninomiya’s original rationale for conducting the card test is scuppered since two-boxers will reason that the catastrophe stopping the HEX is either on its way or not and the results of the card test make no difference to that. Thus, we may as well run the experiment no matter what the results of the card test. Since only one-boxer reasoning leads us to conduct the card test in the first place, we are correct to use that one-boxer reasoning to prevent the card draw from being evidence of a disaster. The Nielsen-Ninomiya card test is an interesting puzzle concerning decision theory in time travel cases. It is much under-discussed (indeed, the only other discussion appears to be Eckhardt [2013]). Nevertheless, I don’t think we should run it, or anything like it, to determine whether to conduct a HEX or not.

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14.5 The Laws of Nature We are nearly at the end of both this chapter and this book. Before finishing, consider one last issue. I’ve argued that we shouldn’t expect time-travelling tourists, nor should we expect signals from the future. Similarly, we should not expect to see naturally occurring time travel phenomena like wormholes—even if they are physically possible, we shouldn’t expect any to actually exist. Or another example: some suspected that the universe might rotate in just the right way for there to be closed timelike curves. Now it’s obvious that this should never have been expected. Or another example: some suspect that, somewhere in the universe, there could be cosmic strings rotating in just the right way to warp spacetime and create a galactic highway to the past. Again, it’s obvious now that there are no such things. Or another example: as explained in §1.2.2, some suspected that tachyons are produced when protons decay. We have looked for evidence of this and found none. But now it’s clear that this doesn’t mean that the laws of nature don’t allow protons to decay, nor do they ban the existence of tachyons; given what’s been said, even if such proton decay were physically possible, we simply shouldn’t expect to observe it. Given this, our observations might be misleading us as to the true laws of physics. Imagine you are weighing up two theories of nature, T₁ and T₂. They vary only over the following propositions: D: Protons have a relatively high objective chance of decaying. When they decay, they emit tachyons. P: To bring about some desirable state of affairs, one should ϕ rather than ¬ϕ. (The desirable state of affairs could be anything. For instance, it could be correctly estimating the age of the universe, or positively determining the presence of Everettian many worlds, or curing cancer, or . . . ). T₁ entails D and P. T₂ entails ¬D and ¬P. If we are divided only over which of T₁ and T₂ is true, then we might try and figure out whether or not to ϕ by testing to see whether protons decay and produce tachyons: if they do, P is true and we should ϕ; if not, P is false and we shouldn’t ϕ. But given everything that has been said, no matter what the laws of nature are, we won’t see any tachyons produced. Thus, no matter what the laws of nature are, we’ll come to believe that T₂ is true and won’t ϕ. But this might be a bad idea! Since D can be true, and there can be a high objective chance of tachyons being produced (even though we never see any), we might well be best advised to ϕ even in light of the absence of tachyons. In short: if a theory predicts that there’ll be time travel events (and there are such theories!), attempts to falsify such a theory by checking for such events (and there have been such attempts!) are pointless.

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Moreover, imagine that we eventually come to believe P. We might start to suspect that a theory, T₃, is true (according to which ¬D and P). But it’s not hard to imagine that T₃ is a gerrymandered mess of a theory and that trying to make ¬D consistent with P leaves us with a theory scoring low on various theoretical virtues concerning parsimony, unity of explanation etc. We’d then waste a lot of time and resources trying to develop a theory avoiding that mess. Given what I’ve said, there is an alternative option open to us: even in the absence of decaying protons and tachyons whizzing about, we can nevertheless be justified in believing that T₁ is true. Thus, what’s been said in Chapters 10–13 feeds directly into evaluating which scientific theory is true. For all we know, the correct scientific theory is one such that there’s a huge objective chance that protons decay (and for the world to be abuzz with tachyons) even though, against the odds, there are none. Similarly, it might be one predicting the existence of traversable wormholes, or a world replete with timelike curves etc. It is consistent to believe such a scientific theory even in the absence of the time travel phenomena they predict as being commonplace. I finish with brief mention of a single exception. I’ve argued that we should not expect there to be any time-travelling entities or events. This is the case except when it is physically necessary that a time travel event occurs. If time travel is physically necessitated, then we should expect such time travel events. (Compare: if you’re certain that you’re already in a causal loop—e.g. War from Chapter 13— then you should be certain that some time travel event will occur.) So in cases where the objective chance of some time travel event occurring is 1, our credence that time travel will occur is likewise 1. This is important because some theories demand just that. The transactional interpretation of quantum mechanics, whereby quantum effects are explained by backwards causation, is one such example (see §1.2.3). Recall: when particles are emitted and strike an absorber, the absorber sends a time-travelling advanced wave back to the emitter which affects what the particles in the past do. If the transactional interpretation were the correct interpretation, then—by physical necessity—these waves must be emitted and received. Were we to believe it to be physically necessary, our credence of being in a world with time-travelling advanced waves (all forming logically consistent causal loops) would be equal to 1. Thus, whilst I think the armchair philosophy of this book can rule out that we’ll ever see tachyons or wormholes, it doesn’t rule out that no time travel is taking place around us, and there is hope yet for theories like the transactional theory of quantum mechanics.

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Index ability see ‘could’, ‘incapacity theory’, and ‘impossability theory’ absorber theory see ‘quantum physics; retrocausal physics’ admissible information 148–49 adultery 33–34 advanced wave see ‘quantum physics; retrocausal physics’ Ahmed, Arif 197–98 Alcubierre warp drive 14, 205 Banana Peel Mechanism 169, 205–07 Bernstein, Sara see ‘Movable Objective Present’ bilking 71–73 Bill and Ted 60, 149–50, 158–59, 168, 172 bitemporality 86–89 black hole 19, 63, 201, 211, 213 Boltzmann brain 63 bootstrapping 4, 60–65 non-Ludovician bootstrapping 170–75 probability of 149–50, 158–70 Braddon-Mitchell, David 193–95 butterfly effect 61 causal loops 43, 59–60, 65, 148–51; see also ‘bootstrapping’ mixed 162–69 negative 152–58 positive 158–62 time travel without causal loops 151–52 causation as a motivation in decisions 177 backwards 2 causal theory of time 47–48 counterfactual and biff 186–88 immanent 39–41, 46–50, 83 transeunt 39 change see ‘paradox; of the Changing Past’ conservative/liberal understandings 79 Cheshire Cat problem 54–58 chimerical time travel 22–24 chorology 30–35 exact location 30 multi-location 4, 19, 29, 34–41, 83 Chronology Protection Conjecture 103; see also ‘paradoxer’ and ‘constrict theory’

circular time 17–19, 48 classical mechanics 161–62 conditional see ‘counterfactual’ and ‘indicative conditionals’ confounding factors 164–68 constrict theory 103–08 defining weak and strong 104–05 weak 98n5, 111, 141 context Ignored/Recognition contexts 85–86 introduction of CL/ CN 122–23 recherché 117 correct circumstances, definition of 94 cosmic rays 213–14 cosmic time 2, 50 could 94–95 canmetaphysical/canphysical 129 Lewisean analysis 119, 130n1; see also ‘incapacity Theory’ standard analysis 124–25 counterfactual backtracking vs. causal 179–80 scepticism 181–82 as a motivation in decisions 176 Stalnaker-Lewis analysis 120, 191 countermodality 118, 119–25, 131–37, 152 amended Stalnaker-Lewis analysis 121 obverse counterpossibles 120, 122–23 decision theory 7, 176–98 ‘Why Ain’cha rich’ 184n1 dependency hypothesis 195 defining time travel 1–2 destination paradox see ‘paradox; destination paradox’ dialetheism 109; see also ‘gainsayer’ and ‘inconsistency theory’ discontinuous time travel 11–13, 55 divine time travel 105; see also ‘prayers’ and ‘prophecy’ Doctor Who 1–2, 17, 42–46, 58, 103, 112, 203 doppelgänger objection see ‘non-Ludovicianism; doppelgänger Objection’ double-occupancy problem 51–58

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endurantism 35, 36–37 Escher machine paradox see ‘paradox; Escher machine paradox’ essentiality of origin 64 eternal recurrence see ‘circular time’ exotic matter 14, 21, 103, 105 extension 53–54 external time 2, 4, 14, 22, 42–50, 55–58, 85 Feynmann’s theory of antimatter 51–52 fictional truth 112–13 fission and fission counterparts 74, 83–84 free will 13, 70–71, 92, 128–29, 137 frozen time 22–24 fundamentality 35n4, 149 gainsayer 34–35, 37, 96 Galileo Galilei 107–108 ghost 40, 41, 62 Goldschmidt, Tyron 12, 73, 81 Grandfather Paradox 5–6, 13n1, 22, 68, 74, 91–100; see also ‘constrict theory’, ‘incapacity theory’, ‘inconsistency theory’, and ‘impossability theory’. explanatory paradox 93 non-morietur subjunctive paradox 97–98 post-mortem indicative paradox 93–97, 126–27 growing block theory see ‘paradox; destination paradox’ Heinlein values (Њ) 159–66 Heinlein, Robert 59–60, 68, 150 hypertime see ‘non-Ludovicianism; hypertemporal indexing’ incompatibilism 70–71; see also ‘free will’ inconsistency theory 5–6, 100, 109–15, 132, 141 indexing see ‘non-Ludovicianism’ indicative conditionals 179, 195 intrinsic properties (and multi-location) 36–37 knowledge principle 63 Krasnikov tube 21, 47, 204 Large Hadron Collider 209–11 laws of nature 216–18 Dretske-Tooley-Armstrong 106–07, 133 logical laws 121–22 Necessitarianism 105–06 Ramsey-Lewis view 106 Least Action, Principle of 103–04 Lebens, Sam 12, 73, 81 Lewis, David see ‘Ludovicianism’ and ‘incapacity Theory’

location see ‘chorology’ Lockwood’s theory of spacetime-actuality 73 logic K4 modal logic 88, 95, 98–99 paraconsistent logic (RM₃) 134–36 Loss, Roberto 74n4, 76n5, 84–86 Ludovicianism 4–5, 12, 67–73, 97, 99–100, 107, 203, 205; see also ‘incapacity theory’ and ‘impossability theory’ and probability 147–170 Mallett, Ronald 212 Meiland, Jack W. 74n4 Mellor, Hugh 65, 128 mental time travel 1, 11, 62, 84 metaphysical possibility 108 Movable Objective Present (MOP) 73 multiple dimensions of time 52; see also ‘bitemporality’ and ‘hypertime’ multiple location see ‘chorology; multi-location’ multiverse see ‘non-Ludovicianism; universe indexing’ Newcomb cases 176, 196–98; see also ‘decision theory’ Nielsen, Holger and Ninomiya, Masao 210, 214–16 Nolan, Daniel 193–95 non-Ludovicianism 12, 73–90 account of change 73–81 and bootstrapping 62 and the Grandfather Paradox 99–100 and the tourist paradox 203–04 definition of conterminous/exterminous hypertime 76–77 definition of universeD/universeF 74–89 doppelgänger objection 82–84 hypertemporal indexing 76–81, 84–90 probability and non-Ludovicianism 148–49, 170–75 universe indexing 74–75, 78–84, 99, 100, 113–15, 203–04 Norton’s dome 161 Novikov’s Principle of Self-Consistency 68, 103–08, 129, 141 open future 69–71 Paradox of the Changing Past see ‘Paradox; of the Changing Past’ Paradox Bernadette’s paradox 140, 142 Contact paradox 140–41 Destination paradox 3n1

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 Escher machine paradox 94, 118, 142–43 Fitch’s paradox (‘knowability paradox’) 139 Grandfather paradox see ‘Grandfather paradox’ Liar paradox 99, 109 of Divine Impeccability 138–39 of Omnipotence 138–39 of the Changing Past 4–5, 66–90, 91, 97 of the Stone see ‘Paradox of Omnipotence’ Polchinski’s paradox 68, 92, 94 Tourist paradox 7, 69n1, 199–218. Toxin paradox 208 Zeno’s paradox 91n1, 99 Paradoxer 5, 94, 95, 100, 117–19, 123, 129, 131; see also ‘Chronology Protection Conjecture’ Penrose staircase see ‘Escher machine paradox’ perdurantism 28–31, 33, 35, 36, 37–41 personal time 2, 22, 35, 37, 42–50, 58, 85 Polchinski, Joseph see ‘Paradox; Polcinski’s Paradox; prayer 12–13 presentism 37; see also ‘Paradox; Destination Paradox’ Principal Principle 154n2, 155, 170, 175 probability 6–7, 147–75, 188–93, 205–07, 211–16 adjusted chance function 189 and subjunctive counterfactuals 181–82 credence (‘CrΓ(φ)’) 148 Eagle function 191–93 equiprobability 172–74 objective chance (‘Ch(φ)’) 148 of doomsday 201 the chance function and decision theory 177 prophecy 12–13, 23 propositions 8 amputated 114 future orientated propositions 69 metaphysical completeness 73 proton decay 16, 217 quantum physics Everett Interpretation 74–75, 173–74, 217 many-minds interpretation 84; see also ‘Lockwood’s theory of space-time-actuality’ quantum gravity 104 retrocausal physics 16–17, 76n5, 218 rapid rotation 19–20 Reaper 140–42, 155, 185, 189, 198; see also ‘Paradox; Bernadette’s paradox’

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Recombination, Principle of 87, 133 relativity, theory of 12, 20, 48–50, 104 retarded wave see ‘quantum physics; retrocausal physics’ retrograde time travel 13–17; see also ‘double occupancy problem’ Second Time Around Fallacy 68 Self-Consistency, Novikov’s Principle of see ‘Novikov’s Principle of SelfConsistency’ self-visitation 4, 27–41, 45, 61 Ship of Theseus see ‘fission and fission counterparts’ Somewhere in Time 60, 150, 170. speed of light see ‘superluminal’ Spencer, Jack 137 superluminal 14, 21, 39n3; see also ‘tachyon’ tachyon 14–16, 21, 51, 104, 200, 205, 216–18 Talmud 12–13 teleportation 1, 11–12 temporal Parts 2, 27, 30–31, 33–41, 44–49, 55–58, 83, 96, 197–8 definition 33–34 temporal relations before and after 44n1 directionality 88–90 quantitative and non-quantitative 87 reduction of external temporal relations 46–50 tensed theory of time 50, 58; see also ‘open future’, ‘presentism’, and ‘growing block theory’ timeless worlds 23, 64; see also ‘chimerical time travel’ Tourist paradox see ‘paradox; Tourist paradox’ Transactional interpretation of quantum mechanics see ‘quantum physics; retrocausal physics’ tritium atom decay 63–64, 93 tyrannosaur rex 158 universalism, mereological 28–29 Vihvelin, Kadri 117 warped time travel 17–21 wormhole 20–21, 69–70, 92, 103, 105, 165n5, 204, 209, 217–18