Abstract
I propose formulating identity criteria as generic statements of ground, thereby avoiding objections that have been made to the more usual formulations.
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Notes
There exists in the literature another, loosely related, line of criticism to the effect that identity does not require a definition or analysis—either in general or in application to a particular kind of object. As Williamson (2013, p. 145) puts it, ‘there is nothing clearer or more basic than identity in terms of which it could be defined. To require a definition or explanation of identity specifically for F’s would be to compound the mistake; ‘=’ is not ambiguous in its application to different kinds of objects’. In responding to this line of criticism, we might distinguish between those definitions or explanations which are meant to provide an aid to the understanding and those which are meant to provide some kind of metaphysical underpinning for the application of the notion; and, in the latter case, the present critique of these criticisms will still apply.
This use of ‘generic’ should not be identified with its use in linguistics, though there are some affinities.
The factive/nonfactive distinction for ground is discussed in Fine (2012, pp. 48–49).
The notion of null ground is discussed in Fine (2012).
Williamson (2013, p. 118) recognizes this difficulty and appeals to the spatio-temporal regions occupied by a person in place of stages. But our identity theorist may not even believe that persons exist in space.
The beginnings of such a theory are to be found in Fine (1985).
I have dropped the variable subscripts in the formulations (S) and (P) above. However, this requires that we have some means of distinguishing between variables that play a generic role and those that play a purely quantificational role.
I find it interesting, in this connection, that Burgess (2012, p. 97) appeals to what appear to be arbitrary objects in attempting to explain how ‘there is a genuine and neglected puzzle about what makes arbitrary A and B identical or distinct’.
Dorr in unpublished work has appealed to a distinctive form of variable binding—or, alternatively, to the λ-notation—in formulating real definitions and Martin Glazier, again in unpublished work, has appealed to a distinctive form of variable binding in formulating metaphysical laws.
We should, of course, distinguish [λxy. Set(y) ∧ Set (y): x = y], a relation only defined on sets, from [λxy: Set(y) ∧ Set (y) ∧ x = y], a relation defined on all objects whatever but one whose correct application is limited to sets. [λxy.Set(y) ∧ Set (y): ∀z(z ∈ x ≡ z ∈ y)] is plausibly regarded as a ground for the first of these relations though not for the second and it may even be thought that [λxy: Set(y) ∧ Set (y) ∧ ∀z(z ∈ x ≡ z ∈ y)] is not a ground for [λxy: Set(y) ∧ Set (y) ∧ x = y], since a ground for [λxy: Set(y) ∧ Set (y) ∧ x = y] must go through its conjuncts. If this is right, then restricted lambda-terms will be required to give proper expression to identity claims, with the restriction playing the same role as the ‘conditions’ under the formulation in terms of arbitrary objects.
Indeed, the definition of an A-model given on pp. 24–25 of Fine (1985) already captures the general idea behind a causal model, although the application to causal modeling is not something I had in mind in giving the definition. Oddly enough, the semantics for quantified relevance logic developed in Fine (1988) also appeals to arbitrary objects.
Lowe (1998, Section. 2.7 and (1991)) also attempts to reduce indirect to direct criteria. My reduction is different in that it distinguishes the conditions under which one thing grounds another from the grounds themselves and for this reason, I believe it is not subject to the criticisms that Williamson [91] has leveled against Lowe’s account.
References
Burgess, A. (2012). A puzzle about identity. Thought, 1, 90–99.
Fine, K. (1985). Reasoning with arbitrary objects. Oxford: Blackwell.
Fine, K. (1988). Semantics for quantified relevance logic. Journal of Philosophical Logic, 17, 27–59.
Fine, K. (2012). Guide to ground. In B. Schnieder & F. Correia (Eds.), Metaphysical grounding. Cambridge: Cambridge University Press, 8-2; reprinted online in ‘Philosophers Annual’ for 2012 (eds. P. Grim, C. Armstrong, P. Shirreff, N-H Stear).
Frege G., (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner [J. L. Austin (The foundations of arithmetic: A logico-mathematical enquiry into the concept of number, 2nd revised ed. 1974). Oxford: Blackwell].
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Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell.
Lowe, E. J. (1991). One level versus two level identity criteria. Analysis, 51, 192–194.
Lowe, E. J. (1998). The possibility of metaphysics: substance, identity and time. Oxford: Oxford University Press.
Pearl, J. (2009). Causality (2nd ed.). Cambridge: Cambridge University Press.
Salmon, N. (1987). The fact that x = y. Philosophia, 17(4), 126–132; reprinted as chapter 8 in Salmon (2005).
Salmon, N. (2005). Metaphysics, mathematics, and meaning. Oxford: Clarendon Press.
Williamson, T. (1986). Criteria of identity and the axiom of choice. Journal of Philosophy, 83(7), 380–394.
Williamson, T. (1990/2013). Identity and discrimination. Oxford: Basil Blackwell. reissued and revised in ‘Identity and Discrimination’ [2013].
Williamson, T. (1991). Fregean directions. Analysis, 51, 194–195.
Acknowledgments
I should like to thank Ted Sider, Fatema Amijee and Martin Glazier for their very helpful written comments and members of the audiences at Austin, Birmingham, CUNY, Oberlin, Oxford and Oslo for many helpful oral comments.