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3.5 Priest's Dialetheism n Young, Gareth (2015) Revenge: dialetheism and its expressive limitations. PhD thesis. http://theses.gla.ac.uk/6415/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Revenge: Dialetheism and its Expressive Limitations Gareth Young MA, Mlitt Submitted in fulfilment of the requirements for the degree of Ph.D. Philosophy School of Humanities University of Glasgow September 2014 2 3 Abstract This thesis is about dialetheism and the problem of revenge. More broadly, it is about truth and what the logical paradoxes tell us about the logical behaviour of truth. One of the driving forces behind the contemporary study of truth and paradox is the problem of revenge: that many, perhaps all, available theories of truth, give rise to further paradoxes, invoking central notions of those theories, which demonstrate that the theory cannot express those notions. This sort of expressive limitation, especially if it involves the very notion invoked to diagnose what goes wrong in paradoxical sentences, would normally be thought a decisive point against a given theory of the paradoxes, were it not for the fact that the problem is so pervasive that every currently available theory has, at some point, been argued to suffer from it. Dialetheism, the view that some contradictions are true, has often been thought to be the only view which has a reasonable chance of avoiding the problem. Indeed, the surge of interest in the view since the first publication of Priest’s In Contradiction, in 1987, defending dialetheism, is probably due in large part to the seeming immunity to the revenge problem that Priest’s view possesses. Its virtue, in respect of revenge, is that its ability to accept, without collapse into incoherence, contradictions, allows it to accept any further revenge paradoxes as merely giving more sound arguments for dialetheia (true contradictions). This thesis argues that this appearance of revenge-immunity is mistaken. Dialetheism, too, has its revenge problems. The seeming virtue of dialetheism, that it can accept the contradictions generated by revenge paradoxes without incoherence, also has its drawbacks. This is because dialetheists are not only able, but compelled to accept the contradictions arising from the semantic paradoxes. This means that contradictions can arise in certain areas where they are undesirable. In particular, there are notions which seem to require consistency in order to be expressible. If we can demonstrate, using revenge paradoxes, that, on dialetheism, predicates putatively representing these notions would have to behave inconsistently, then we can demonstrate that dialetheists cannot express the notions. 4 There are many ways one might wish to carve up the different varieties of dialetheism available. I have separated the view into two broad kinds: metatheoretically paraconsistent dialetheism, on the one hand, and metatheoretically consistent dialetheism, on the other. This distinction decides to which variety of revenge problem the version of dialetheism in question is subject. I take each in turn, and argue that they are each subject to expressive limitations brought about by revenge paradox. 5 Revenge: Dialetheism and its Expressive Limitations Contents Abstract .............................................................................................................................................. 3 Acknowledgements ............................................................................................................................ 7 Declaration of Originality Form – Research Degrees ........................................................................ 9 Chapter 1: Introduction .................................................................................................................... 11 Chapter 2: What is Revenge? ........................................................................................................... 13 2.0 Introduction ................................................................................................................................ 13 2.1 The Phenomenon of Revenge ..................................................................................................... 15 2.2 Too Easy Revenge ....................................................................................................................... 18 2.3 Recipes for Revenge .................................................................................................................... 24 2.4 Varieties of Revenge ................................................................................................................... 40 2.5 Chapter Conclusion ..................................................................................................................... 40 Chapter 3: Metatheoretically Paraconsistent Dialetheism ............................................................ 42 3.0 Introduction ................................................................................................................................ 42 3.1 Priest’s Metatheoretically Paraconsistent Dialetheism .............................................................. 43 3.2 The Semantic Paradoxes ............................................................................................................. 44 3.2.1 Paracompletism ....................................................................................................................... 47 3.2.2 Rejecting the T-Scheme ........................................................................................................... 50 3.2.3 Hierarchical Approaches .......................................................................................................... 54 3.3 Set-Theoretic Paradoxes ............................................................................................................. 58 3.3.1 The Inconsistency of Naïve Set Theory .................................................................................... 58 3.3.2 Priest Against Set-Theoretic Orthodoxy .................................................................................. 62 3.3.2.1 Lack of Motivation for the Cumulative Hierarchy ................................................................. 62 3.3.2.2 Category Theory .................................................................................................................... 66 3.3.2.3 Logic ...................................................................................................................................... 69 3.3 Paradoxes of Inclosure ................................................................................................................ 70 3.4 Summary of Priest’s Case for Dialetheism .................................................................................. 73 3.5 Priest’s Dialetheism ..................................................................................................................... 76 3.5.1 The Teleological Account of Truth ........................................................................................... 79 3.5.1.1 Falsity .................................................................................................................................... 81 3.5.1.2 Untruth.................................................................................................................................. 83 3.5.2 Formal Semantics for Priest’s Dialetheism .............................................................................. 86 3.5.2.1 Extensional Connectives ....................................................................................................... 86 6 3.5.2.2 Priest’s Conditional ............................................................................................................... 88 3.5.3 Paraconsistent Set Theory ....................................................................................................... 94 3.5.4 Paraconsistent Metatheory ..................................................................................................... 99 3.6 Chapter Conclusion ................................................................................................................... 103 Chapter 4: Getting Revenge on Metatheoretically Paraconsistent Dialetheism ......................... 104 4.0 Introduction .............................................................................................................................. 104 4.1 Revenge, Just False and Non-Dialetheia ................................................................................... 106 4.2 Inconsistent Validity and Revenge ............................................................................................ 117 4.2.1 Inconsistent Validity with Transparent Truth ........................................................................ 119 4.2.2 Invalidity with a Non-Contraposable T-scheme ...................................................................
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