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The Strong Theory of Relative Identity Daniel Molto PhD University of York Philosophy September 2014 2 0.1 Abstract This dissertation considers a theory of numerical identity, first presented by P.T. Geach (1962). I label this theory `the strong theory of relative identity'. I suggest that the strong theory of relative identity involves three theses, which I name `GT', `RI', and `SRI'. I argue that each of these theses is logically independent. I consider arguments for and against each of these theses in turn. I conclude that none of the arguments for GT, RI, or SRI are conclusive. However, I also argue that the arguments against GT, RI and SRI are unsuccessful. I argue, further, that the strong theory of relative identity, and GT in particular, is incompatible with classical semantics and classical first-order logic with identity. I consider alternative non-classical logical systems and semantics which might be compatible with the strong theory of relative identity. Finally, I consider the philosophical applications of the strong theory of relative identity. I focus on one area, specifically philosophical theology, and I argue, with respect to the logical problem of the Trinity, that either GT is true or orthodoxy is false. 3 CONTENTS 0.1 Abstract . 3 0.2 Acknowledgements . 5 0.3 Declaration . 6 0.4 Introduction . 7 0.4.1 Chapter 1: The Strong Theory of Relative Identity . 8 0.4.2 Chapter 2: Geach's Argument Against Absolute Identity 11 0.4.3 Chapter 3: RI . 12 0.4.4 Chapter 4: SRI . 12 0.4.5 Chapter 5: Objections to Relative Identity . 13 0.4.6 Chapter 6: The Logic and Semantics of Relative Identity 13 0.4.7 Chapter 7: Applications of Relative Identity . 14 1. The Strong Theory of Relative Identity :::::::::::::::: 15 1.1 Theories of Identity . 16 1.1.1 The Theory of Absolute Identity . 16 1.1.2 Apparent Counter Examples to Orthodoxy . 20 1.1.3 An Alternative Theory . 24 1.1.4 Theses . 25 1.1.5 Sortal Terms . 27 1.1.6 Criteria of Identity . 30 1.2 Relationship Between Theses . 34 1.2.1 Independence of SRI, RI, and GT . 34 1.2.2 How to Evaluate Theories of Identity . 37 1.2.3 Absolute Identity and Arguments by Example . 40 1.2.4 Conclusion . 42 2. Geach's Argument Against Absolute Identity ::::::::::::: 43 2.1 Geach's Argument . 44 2.1.1 Geach's Thesis . 44 2.1.2 Criteria . 47 2.1.3 The Deductive Structure of Geach's Argument against Absolute Identity . 49 2.1.4 Argument 1 . 51 2.1.5 P1 . 52 2.1.6 P2 . 56 2.1.7 P3 . 56 2.1.8 Example . 57 2.1.9 Argument 2 . 58 2.1.10 P2.1 . 58 2.1.11 P2.2 . 59 2.1.12 Quine's Proposal . 64 2.1.13 Geach's Objections to Quine . 66 2.1.14 Surmen . 68 2.1.15 Responding to the Objections . 69 2.2 A Charitably Re-constructed Geachean Argument Against the Existence of Absolute Identity Relations . 76 2.2.1 Argument 3 . 82 2.2.2 P3.1 . 84 2.2.3 P3.2 . 84 2.2.4 P3.3 . 85 2.2.5 P3.4 . 86 2.2.6 P3.5 . 88 2.2.7 Responding to Grelling's Paradox . 91 2.2.8 Conclusion . 96 3. RI :::::::::::::::::::::::::::::::::::: 97 3.1 Arguments for RI . 98 3.1.1 Two Geachean Arguments, River and Waters/Men and Heralds . 99 5 3.1.2 River and Waters . 100 3.1.3 Argument 6 . 103 3.1.4 Possible Alternatives: Wiggins and Lowe . 106 3.1.5 Men and Heralds . 107 3.1.6 `Is' of Instantiation . 109 3.1.7 Griffin’s Argument . 110 3.1.8 Zemach's Argument . 112 3.2 Arguments Against RI . 115 3.2.1 Lowe's objection . 115 3.2.2 Wiggins' Conceptual Objection . 116 4. SRI :::::::::::::::::::::::::::::::::::: 119 4.1 An Argument for SRI . 119 4.1.1 Two `Parallel' Theses . 120 4.1.2 Frege's Relativity Argument . 120 4.1.3 The Thesis . 121 4.1.4 The First Argument . 122 4.1.5 SRI and CT: The Differences . 123 4.1.6 Finding the Principle of Equivalence . 124 4.1.7 Sacchi and Carrara's Objection . 126 4.1.8 Le Poidevin's Objection to SRI . 130 5. Objections to Relative Identity :::::::::::::::::::: 132 5.0.9 A Preliminary Objection . 133 5.0.10 Relative Identity and Singular Reference . 137 5.0.11 Quantification . 140 5.0.12 An Objection: Cain . 142 5.0.13 Further Objections: Quine and Dummett . 144 5.0.14 Conclusion . 148 6. The Logic and Semantics of Relative Identity :::::::::::: 150 6.0.15 Geachean Quantification and the Syllogisms . 150 6.1 Semantics . 153 6.1.1 Prospects for a Semantics for GT . 154 6 6.1.2 Prospects for a Semantics for GT, RI, and SRI . 160 6.2 Logical Systems for Strong Theories of Relative Identity . 163 6.2.1 Stevenson and Van Inwagen's Logics . 166 6.2.2 Routley and Griffin’s Logics . 168 6.2.3 Garbacz's C1 . 170 6.2.4 Conclusion . 176 7. Applications of Relative Identity ::::::::::::::::::: 177 7.0.5 The Doctrine of the Trinity . 179 7.0.6 Social and Latin Trinitarianism . 180 7.0.7 Latin Trinitarianism and the Logical Problem of the Trinity . 184 7.0.8 Time Travel and the Trinity . 186 7.0.9 The Relative Identity Solution to the Logical Problem of the Trinity . 188 7.0.10 Weak Theories of Relative Identity . 191 7.0.11 Strong Relative Identity . 193 7.0.12 Conclusion . 193 Appendix A: List of possible rules of inference for logics of relative iden- tity, with counterexamples 202 List of Abbreviations 205 List of References 208 7 0.2 Acknowledgements Of the many people to whom I am in debt for their help, I must thank, first and foremost, my supervisor and friend Dr. David Efird for his invaluable guidance and unending patience. This work would not have been started and would certainly never have been finished without his tireless assistance. My thanks are due, also, to Professor Tom Stoneham for several crucial sug- gestions and much useful advice on my work. I owe another great debt to my examiners, Professors Tom Baldwin and Ian Rumfitt, for many insight- ful comments on this dissertation. I am further indebted to the rest of the Department of Philosophy at the University of York, to the staff for their expertise, to my fellow postgraduates for much lively discussion, and to the undergraduates for the opportunity to experience the pleasure of teaching philosophy. Finally, I would like to thank my parents and siblings for their support throughout the years, and to the friends I have made in the post- graduate community at York, in particular Dr. Dominic Spengler and Dr. Daniel Gustafsson who have helped me from the outset of this project. 8 0.3 Declaration I hereby declare that this submission is entirely my own work except where due acknowledgement is given. This work has not previously been presented for an award at this, or any other, University. 9 0.4 Introduction Natural languages such as English contain a multiplicity of constructions expressing apparent relations of identity. Some of these relations seem, from the sentences in which they are expressed, to be dyadic. Others seem to be triadic. Apparent dyadic relations of identity in natural language are often found in clauses involving the expressions `...is identical with...', `...is the same as...', `...is equal to...', or simply `... is ...'. Apparent triadic relations of identity are normally expressed in English with the construction `...is the same ... as ...'. This may be completed by a count noun or a mass term. To complicate matters, the expression `...is the same ... as ...' seems to function differently depending on which noun fills the empty place. It seems, prima facie, as if a different relation is involved in the sentence `Clark Kent is the same man as Superman' from that involved in the sentence, `Your eyes are the same colour as the sea'. The latter difference is often characterized by distinguishing between relations of `qualitative identity', such as `... is the same colour as...', and relations of `numerical identity', such as `... is the same man as ...'. The number and nature of relations of identity will be the subject matter of this dissertation. According to one influential view of identity, the appearance of multi- plicity is deceptive, and there is really only one genuine relation of identity. However, according to an alternative view, made famous by the late Peter Geach (1962, 1967, 1980), there is not one relation of identity, but many. Moreover, the supposed single `genuine' relation of identity is, in fact, inco- herent. This dissertation is concerned with the second of these views. More particularly, I will consider a theory which I will call `the strong theory of relative identity'. I will identify this theory with the conjunction of three theses. In what follows, I will define these three theses and consider the ar- guments that have been presented both for and against them. I will conclude that none of the arguments either for or against any of the theses is entirely compelling. The following discussion will be split into seven chapters, the content of which I will briefly outline here. 10 0.4.1 Chapter 1: The Strong Theory of Relative Identity In Chapter 1, I will set the stage by distinguishing between two sets of theo- ries about the logic of identity.
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