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The non-transitivity of the contingent and occasional identity relations

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Abstract

This paper establishes that the occasional identity relation and the contingent identity relation are both non-transitive and as such are not properly classified as identity relations. This is achieved by appealing to cases where multiple fissions and fusions occur simultaneously. These cases show that the contingent and occasional identity relations do not even satisfy the time-indexed and world-indexed versions of the transitivity requirement and hence are non-transitive relations.

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Notes

  1. The challenges are related in that a defender of occasional identity must also defend contingent identity, though not vice versa.

  2. The reason why the fissions and fusions have to be simultaneous is explained in the Appendix.

  3. Some defenders of contingent identity are happy to concede that the contingent identity relation does not satisfy an inter-world transitivity requirement (cf. Nozick 2001, p. 135). However, the subsequent argument will show that it does not even satisfy an intra-world transitivity requirement, which appears to be a highly problematic consequence of the contingent identity thesis.

  4. A similar account can be given for the temporal case, namely sortal-relative occasional identity (cf. Gupta 1980, pp. 22–25, 45 and 85). Such an account consists of an acceptance of a sortal-relative trans-temporal identity relation combined with an affirmation of the possibility of occasional identities. The same considerations apply in that this strategy cannot solve transitivity worries when occasional identities result where the same sortal is at issue.

  5. It is important to note that sortal-relative contingent identity is a rather radical view. Since this view is committed to making the trans-world identity relation a sortal-relative relation, it can be considered as the modal analogue of Geach’s relative identity thesis. Moreover, the predicational-shifts required by a defender of sortal-relative contingent identity lead to a number of problems that have been identified by Fine. These problems derive from the fact that defenders of Abelardian predicates have to argue that many linguistic contexts are opaque even though they appear to be transparent and that ordinary devices for making contexts transparent fail (cf. Fine 2003; also cf. Koslicki 2008, chap 3). Furthermore, as Gray points out, while Gibbard and Noonan might well have provided us with a functioning semantics for contingent identity, they have not given us a plausible account of the metaphysics of contingent identity (cf. Gray 2001, pp. 30–41).

  6. For a discussion of cases involving simultaneous fissions and fusions in the context of counterpart-theoretic accounts of persistence and modality cf. Bader, R. M. ‘Contingent identity, counterpart theory and the failure of transitivity’ (manuscript).

  7. As an anonymous referee has pointed out, the identities at t: A = C as well as at t: C = E would be lost, if it should be possible for Gallois to find a way of establishing at t′: C = B as well as at t′: C = D. This, however, cannot be done by appealing to the I@T principle and Gallois does not recognise any other principle for establishing temporary identities.

  8. It might be suggested that C is not I@T-related to anything, that the fusion of two fissions somehow creates an entirely new object. However, this would be unjustified in the sense that there would be nothing that is identical to C at t′ that could be used for ruling out candidates for identity-at-t that are related to C by means of identity-sustaining relations. Moreover, if C would not be related by identity-at-t relations to B and D in either direction, then we would get a scenario in which \(\mathop{\hbox{B}=\,_{@t}}\limits^{\longleftrightarrow}\hbox{A}\) and \(\mathop{\hbox{D}=\,_{@t}}\limits^{\longleftrightarrow}\hbox{E}.\) This symmetric identity-at-t relation between A and B, as well as between D and E, would undermine the claim of “identity-sustaining relations failing to ensure identity in branching cases” (Gallois 1998, p. 94). We would get these symmetric relations because even though B would split into A and C, C would not be identical to B at t and, consequently, everything that would be identical to B at t would be identical to A at t′.

References

  • Fine, K. (2003). The non-identity of a material thing and its matter. Mind, 112, 195–234.

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  • Gallois, A. (1998). Occasions of identity: The metaphysics of persistence, change, and sameness. Oxford: Oxford University Press.

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  • Gibbard, A. (1975). Contingent identity. Journal of Philosophical Logic, 4, 187–221.

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  • Gupta, A. (1980). The logic of common nouns: An investigation in quantified modal logic. New Haven: Yale University Press.

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  • Noonan, H. W. (1991). Indeterminate identity, contingent identity and Abelardian predicates. The Philosophical Quarterly, 41, 183–193.

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Acknowledgment

For helpful comments I would like to thank Daniel Nolan, André Gallois, Katherine Hawley and Ira Kiourti, as well as audiences at St Andrews and Boulder.

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Correspondence to Ralf M. Bader.

Appendix: The non-symmetry of the identity-at-t relation

Appendix: The non-symmetry of the identity-at-t relation

It might be asked why the fissions and fusions must occur simultaneously. In order to see why this has to be so, it is best to consider a case analogous to Case 1, but where the fissions and fusions are not simultaneous, and then explain why such a case fails.

Case A1:

At t, B and D both undergo fission, B splitting into A and x, while D fissions to form y and E. At t′, we have a fusion of x and y that forms C. Exactly like in Case 1 it seems that we now have the following temporary identities. A and C, as well as C and E, are identical at t, but A and E are not identical at t, thereby undermining the transitivity of identity. A and E are only identical at t to things that are identical to each other at t′. It is only true that at t: at t′: A = E, but this is not equivalent to at t: A = E since the temporal qualifiers are not redundant. Thus, it seems that Case A1 constitutes a situation where at t: C = A and at t: C = E, but A is not identical to E at t.

Applying the time-indexed version of the transitivity relation to the case at hand, we get:

$$ (\hbox{at t:}\; \hbox{A}=\hbox{C} \;\&\; \hbox{at t:}\; \hbox{C}=\hbox{E}) \to\hbox{at t:}\; \hbox{A}=\hbox{E} $$

However, since it is not the case that A = E at t, we can see that transitivity does not hold. Hence, the occasional identity relation does not even satisfy the temporally qualified transitivity relation.

Even though Case A1 appears to be identical in the relevant respects to Case 1, we have to note that it fails because at t: \(\neg(\hbox{A}=\hbox{C})\) and at t: \(\neg(\hbox{C}=\hbox{E}).\) This is because the identity-at-t relation only holds between x and y “if there are a pair of times t and t′ such that everything identical with x at t is identical with y at t′” (Gallois 1998, p. 116).

$$ (\hbox{I@T}) \square(\hbox{x})(\hbox{y})\left[\hbox{x}=\,_{@t}\hbox{y} \leftrightarrow(\exists \hbox{t})(\exists \hbox{t}')(\hbox{z})(\hbox{at t:}\; \hbox{z}=\hbox{x} \to\hbox{ at t}^{\prime}\hbox{:}\; \hbox{z}=\hbox{y})\right] $$

Now, B and C are not linked by the identity-at-t relation because there is a z, namely y, to which C is identical at t′, while B is not identical to this z at t. Since at t: \(\neg(\hbox{B}=\hbox{C})\) and since at t: A = B, it follows that at t: \(\neg(\hbox{A}=\hbox{C}).\) The same holds for the at t: C = E relation that is required for Case A1 to succeed. In this case, there is a z, namely x, to which C is identical at t′, while D is not identical to this z at t. Since at t: \(\neg\)(D = C) and since at t: D = E, it follows once again that at t: \(\neg(\hbox{C}=\hbox{E}).\)

Since A and C, as well as C and E, are connected by a chain of identity-at-t statements, it might seem that we can get the required at t: A = C and at t: C = E. When looking at Case A1, we see that \(\hbox{A}=\,_{@t}\hbox{B},\) \(\hbox{B}=\,_{@t}\hbox{x},\) \(\hbox{x}=\,_{@t}\hbox{C},\) \(\hbox{C}=\,_{@t}\hbox{y},\) \(\hbox{y}=\,_{@t}\hbox{D},\) and \(\hbox{D}=\,_{@t}\hbox{E}.\) However, we cannot get at t: A = C and at t: C = E given that the identity-at-t relation put forward by Gallois is non-symmetrical (cf. Gallois 1998, p. 116). Accordingly, in Case A1 we have \(\hbox{B}=\,_{@t}\hbox{x}\) and \(\hbox{x}=\,_{@t}\hbox{C},\) but \(\neg(\hbox{B}=\,_{@t}\hbox{C}),\) without a violation of transitivity. Since x = B at t and since x = C at t′ and since the identity-at-t relation is not symmetrical, we do not get at t: C = B.

When representing the ‘direction’ of temporal identity by means of vector arrows, we can see that there is no intransitivity since we only get the following identity-at-t statements:

$$ \mathop{\hbox{A}=\,_{@t}}\limits^{\longrightarrow}\hbox{B}, \;\mathop{\hbox{B}=\,_{@t}}\limits^{\longleftarrow}\hbox{x},\; \mathop{\hbox{x}=\,_{@t}}\limits^{\longrightarrow}\hbox{C}, \;\mathop{\hbox{C}=\,_{@t}}\limits^{\longleftarrow}\hbox{y}, \;\mathop{\hbox{y}=\,_{@t}}\limits^{\longrightarrow}\hbox{D}\; \hbox{and}\; \mathop{\hbox{D}=\,_{@t}}\limits^{\longleftarrow}\hbox{E}. $$

Case A2:

We can salvage the idea underlying Case A1 by slightly modifying the example. In Case A1, we can rule out \(\hbox{C}=\,_{@t}\hbox{D}\) by means of the \(\hbox{C}=\,_{@t}\hbox{x}\) relation. At t′: C = x and at t: \(\neg\)(D = x). Accordingly, \(\hbox{C}=\,_{@t}\hbox{D}\) does not satisfy I@T and is hence ruled out. The same applies to \(\hbox{C}=\,_{@t}\hbox{B}\) since at t′: C = y but at t: \(\neg\)(B = y). However, this problem can be fixed by appealing to cases where fissions and fusions occur simultaneously. These cases are such that at one instant we have B and D and at the next instant we have A, C and E. There are no intermediary objects, such as x and y, that are identical to C at t′ that could be used to rule out \(\hbox{C}=\,_{@t}\hbox{B}\) and \(\hbox{C}=\,_{@t}\hbox{D}.\) Accordingly, we get the following identity-at-t relations:

$$ \mathop{\hbox{A}=\,_{@t}}\limits^{\longrightarrow}\hbox{B}, \;\mathop{\hbox{B}=\,_{@t}}\limits^{\longleftarrow}\hbox{C},\; \mathop{\hbox{C}=\,_{@t}}\limits^{\longrightarrow}\hbox{D}\;\hbox{and}\; \mathop{\hbox{D}=\,_{@t}}\limits^{\longleftarrow}\hbox{E}. $$

From this it follows that at t: A = C and at t: C = E, but at t: \(\neg\)(A = E).

In a normal case of fusion, we can use the identity-at-t relation going in one direction from B to C, i.e. \(\mathop{\hbox{B}=\,_{@t}}\limits^{\longrightarrow}\hbox{C},\) in order to establish the identity B = C at t′. This occasional identity can then be used to rule out the identity-at-t relation from C to D going in the other direction, i.e. \(\mathop{\hbox{D}=\,_{@t}}\limits^{\longleftarrow}\hbox{C},\) since at t: \(\neg\)(D = B) which is required by I@T. However, in Case A2 we have no identity-at-t relation going to C that could be used to rule out identity-at-t relations going away from C. This is because the fission cases ensure that no such relations obtain, since the occasional identity at t: A = B rules out \(\mathop{\hbox{B}=\,_{@t}}\limits^{\longrightarrow}\hbox{C}\) since this identity-at-t relation would require C to be identical to A at t′. The same holds for the relation between D and C, insofar as the occasional identity at t: E = D rules out \(\mathop{\hbox{D}=\,_{@t}}\limits^{\longrightarrow}\hbox{C}\) since this identity-at-t relation would require C to be identical to E at t′.Footnote 7 This is what is special about simultaneous fissions and fusions, namely that both the branches connecting C to B and C to D are such that there are no identity-at-t relations ‘going to’ C that could be used for ruling out identity-at-t relations ‘going away’ from C. The identity-sustaining relation between C and B is sufficient for the identity-at-t relation \(\mathop{\hbox{B}=\,_{@t}}\limits^{\longleftarrow}\hbox{C}\) to hold in the absence of there being something at t′ to which C is identical and to which B is not identical at t. At t′ C is only identical to C and hence any object standing in an identity-sustaining relation to C will be related to C by I@T. Since both B and D stand in identity-sustaining relations to C, they are both identical-at-t to C, i.e. \(\mathop{\hbox{B}=\,_{@t}}\limits^{\longleftarrow}\hbox{C}\) and \(\mathop{\hbox{D}=\,_{@t}}\limits^{\longleftarrow}\hbox{C}.\) Footnote 8

Thus, we can conclude that Case A1 fails due to the non-symmetry of the identity-at-t relation. However, we can save the argument by appealing to cases involving simultaneous fissions and fusions. In Case A2 it is the case that at t: A = C and at t: C = E, but at t: \(\neg\)(A = E) which implies that the occasional identity relation is non-transitive.

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Bader, R.M. The non-transitivity of the contingent and occasional identity relations. Philos Stud 157, 141–152 (2012). https://doi.org/10.1007/s11098-010-9623-z

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