Existential Dependency as Identity-Dependence

I spoke a moment ago of the relation of identity-dependence as being the sort of relation we seek. What I propose, accordingly, is this:

(D1**) x depends for its existence upon y = df Necessarily, the identity of x depends on the identity of y.

To say that the identity of x depends on the identity of y —or, more briefly, that x depends for its identity upon y—is to say that which thing of its kind y is fixes (or at least helps to fix) which thing of its kind x is. (By ‘fixes’ in this context I mean metaphysically determines—so that, strictly speaking, the word ‘necessarily’ in (D1**)'s definiens is redundant, since identity-dependence is never a contingent relation.) A fully perspicuous formal definition of identity-dependence is not easy to frame (for reasons which we shall come to in a moment), but the relation can be made sufficiently clear for most of our present purposes by means of examples. For instance, the identity of a set is fixed by the identities of its members and the identity of an assassination is (at least partially) fixed by the identity of the person assassinated. These relationships of identity-dependence are direct consequences of the identity criteria governing the kinds of which the items thus related are instances.¹⁸⁷Thus the identity-dependence of a set upon its members is a direct consequence of the Axiom of Extensionality, which functions as a criterion of identity for sets. Notice, here, that I allow that x may be said to depend for its identity upon y even in cases in which the identity of y alone does not suffice to fix the identity of x: thus a set with two or more members depends for its identity upon each of them, although its identity is only completely fixed by the identities of all of them.

Now, although I have not yet offered a formal definition of identity-dependence,

187 For more on identity-criteria, see Chapter 2 above and also my ‘What is a Criterion of Identity?’

it seems to me that a consequence of any such definition should be the following theorem:

(T4) If the identity of x depends on the identity of y, then, necessarily, there is a function F such that x is necessarily identical with the F of y.

For example: because the identity of a marriage depends on the identities of the two people being married, if x i s a marriage and y and z are the two people in question, (T4) is satisfied in respect of x and y in virtue of the fact that x is necessarily identical with the marriage of y with z—so that in this case the required function is the ‘marriage with z’

function from persons to events. (I ignore the complications created by the fact that, under some legal systems, the same two persons may be married to one another more than once.) Note that the purpose of the second occurrence of the word ‘necessarily’ in (T4) is to ensure that F is a function which picks out x as the referent of the expression ‘the F of y’ in every possible world in which that expression refers to anything at all: thus the event that was actually the marriage of y with z could not have been anything other than the marriage of y with z. (By contrast, the ‘first marriage’

function, for example, does not meet this requirement, because the event that was actually the first marriage of y—say, the marriage of y with z—might not have been the first marriage of y, since y could have married someone else before marrying z.) It seems clear, however, that wherever the identity of an item x depends on the identity of another item y, the criterion of identity for items of x's kind will supply the requisite function F.

Evidently, it would not do simply to replace the conditional connective in (T4) with a biconditional connective and thence attempt to turn it into a definition of identity-dependence, unless at the same time one could impose some suitable restriction on the kind of function involved. For—to give a simple example—if we let y be x (that is, the unit set of x), then we see that x is necessarily identical with the sole member of x: and yet, intuitively, the identity of x depends on the identity of x rather than vice versa. So the ‘sole member’ function would obviously have to be excluded. How to exclude all and only the intuitively inappropriate functions is, however, a problem of some magnitude which remains to be resolved. Perhaps the most promising strategy would be to exclude the sole member function on the grounds that, whereas it is part of the ‘essence’ of x that it contains x as its sole member, it is not part of the ‘essence’ of x that it is the sole member of x (because whereas one cannot understand what x is without understanding that it contains x as its sole member, one certainly can understand what x is without understanding that it is the sole member of x). And then we could generalize this reasoning so as to

exclude any function F which is not such that it is part of the essence of x that it is the F of y, giving us as our desired definition of identity-dependence the following:

(D3) The identity of x depends on the identity of y =dfNecessarily, there is a function F such that it is part of the essence of x that x is the F of y.

We can exemplify (D3) by letting x be z and y be z, in which case we have, as is intuitively correct, that the identity of z depends on the identity of z because, necessarily, there is a function—namely, the ‘unit set’ function—such that it is part of the essence of z that it is the unit set of z.

Obviously, (D3) entails (T4), since it is necessarily the case that if F is a function such that it is part of the essence of x that x is the F of y, then x is necessarily identical with the F of y —although, as the example of x and its unit set demonstrated, the converse is not necessarily the case. That is, it is not necessarily the case that if F is a function such that x is necessarily identical with the F of y, then it is part of the essence of x that x is the F of y: thus, x is necessarily identical with the sole member of x, but (plausibly) it is not part of the essence of x that x is the sole member of x.¹⁸⁸ But, of course, for the foregoing strategy to work and thus for (D3) to be fully vindicated, a perspicuous account of the notion of ‘essence’ is required—and that is a large task which I shall not try to undertake here.

Before proceeding, there is one minor worry that I should try to forestall, namely, that it cannot be correct to define existential dependency simply as identity-dependence—as in (D1**)—because existence and identity are different concepts. My answer is twofold. First, existence and identity are in fact intimately related, via the principle that for an object x to exist is just for there to be something that is identical with x. Second, we shall see in a moment that identity-dependence actually entails the kind of existential dependency defined by (D1): and since the latter relation is certainly an existential one, it surely follows that identity-dependence qualifies as such a relation too. (Later, indeed, I shall suggest a reconciling move, whereby we may call these two relations ‘strong’ and ‘weak’ existential dependency respectively.)

In the light of preceding considerations, we need now to establish two consequences of (D1**). First:

188 Cf. Kit Fine, ‘Essence and Modality’, in James E. Tomberlin, ed., Philosophical Perspectives, 8: Logic and Language (Atascadero, Calif.: Ridgeview, 1994), 4–5. See also Kit Fine,

‘Ontological Dependence’, Proceedings of the Aristotelian Society, 95 (1994–5), 269–90, for more on the notion of essence and an account of ontological dependence which is in some ways quite similar to my own.

(T5) If the identity of x depends upon the identity of y, then, necessarily, x exists only if y exists.

And second:

(T6) If x is not identical with y and the identity of x depends on the identity of y, then the identity of y does not depend upon the identity of x.

These theorems are, I think, relatively easy to prove. As to (T5), clearly, since for x to exist is for there to be something identical with x (which presupposes that x at least has self-identity), x cannot exist unless everything upon which x's identity depends also exists. Thus an assassination cannot exist unless the person assassinated exists; and a set cannot exist unless its members exist. (We can demonstrate (T5) more rigorously with the aid of (T4): for if, necessarily, there is a function F such that x is necessarily identical with the F of y, then since the F of y cannot exist unless y exists, x cannot exist unless y exists.) And as to (T6), this follows from the requirement of non-circularity which is a condition on the adequacy of any criterion of identity. (This non-circularity requirement is clearly related to the explanatory status of an adequate criterion of identity, so that we have by no means abandoned our earlier thoughts on the role of explanation in an account of existential dependency, but have merely fixed more precisely on what kind of explanation is relevant to such a relationship.) For example, given that unit sets are not to be identified with their members, we cannot say both that the identity of a unit set depends upon the identity of its member and that the identity of that member depends upon the identity of that unit set, for this would engender a vicious circle which would deprive both unit sets and their members of well-defined identity-conditions. But note that this still allows us to say that there may be some privileged kinds of items which possess determinate identity-conditions even though though they are not governed by any adequate (because non-circular) criterion of identity.¹⁸⁹ Such items would depend for their identity solely upon themselves (and so, if particulars, would qualify as substances according to (T7) below).

Of course, it might be urged, with some plausibility, that every object x trivially depends for its identity upon itself—and if that is so, then, according to (D1**), every object x likewise depends for its existence upon itself (which was also a consequence, it may be recalled, of our original definition, (D1)). And, certainly, (D3) has this implication, because for any object x, it is necessarily the case that there is a function—namely, the identity function—such that it is part of the essence of x that x is the

189 See further Chapter 2 above and Chapter 7 below.

object identical with x. On the other hand, however, there is something to be said for amending (D3) by making an exception of the identity function, because, for example, it seems odd to say that the unit set of x depends for its identity upon itself in addition to depending for its identity upon x: surely—it might be urged—the identity of x is all that the identity of that set depends on. But, be this as it may, we must in any case be careful to distinguish between the claim that an object depends for its identity upon itself and the claim that an object depends for its identity solely upon itself: even if the former claim is trivially true of all objects, the latter claim is not. (This is important because we do not want our definitions to imply that everything, trivially, is a substance.)