Identity: Some General Considerations

Robert Bass
Department of Philosophy
Coastal Carolina University
Conway, SC 29528
rhbass@gmail.com

By "identity", I shall be referring to numerical identity, not to similarity, however great.¹  Numerical identity I shall take to be the identity of particulars, where a particular is simply something which is not a universal and, therefore, not susceptible of multiple exemplification.

There is a type of case in which it seems trivially straightforward to apply a conception of numerical identity -- when only a single particular is being referred to. However, if we are to discuss cases in which there is some question whether the relation of numerical identity holds, we have to assume that there is at least one way,² which may possibly be applied more than once, of picking out the particular or particulars involved. I shall speak of such a way of picking out a particular as individuation and shall speak of what is picked out by an act of individuation as a referent, leaving it an open question whether two or more referents are numerically identical.

I shall refer to something as a criterion of identity if it logically must hold if two referents are to be identical. Criteria for identity may be of two kinds. Some, discussed -- not exhaustively -- immediately below, are formal, applying in any case of identity between referents. Others may depend on facts about the particular in question. For example, it may be that humans are essentially embodied and that, therefore, there can be no disembodied phase in the existence of a human being, while there might be other beings -- angels, say -- for whom embodiment is not necessary to their continued existence. By contrast, I shall term the various fallible means that we use to determine whether two referents are identical, tests.

Identity (=) is often characterized in terms of satisfying the three conditions, symmetry, transitivity, and reflexivity. For symmetry, this means that

(x)(y) (x = y  y = x).

That is, for any pair of referents, x and y, if x is identical to y, then y is identical to x. For transitivity,

(x)(y)(z) ((x = y & y = z)  x = z).

That is, for any triple of referents, x, y, and z, if x is identical to y and y is identical to z, then x is identical to z. For reflexivity,

(x) x = x.

That is, for any referent, x, x is identical to itself.

These are certainly plausible as conditions on what it means to correctly ascribe identity. However, this set of conditions is not, as Wiggins has pointed out, sufficient to distinguish identity from other possible relations. Congruence and consanguinity, for example, also meet these conditions. (Wiggins 1971, p. 5) To fill this gap, he proposes (and I accept -- with reservations to be discussed below) as a further condition for the holding of the identity relation that we should employ Leibniz's Law:

(x)(y) (x = y  (F) (Fx ≡ Fy)).

That is, for any pair of referents, x and y, if x is identical to y, then, for all predicates, F (suitably restricted so as to avoid problems associated with intensional contexts³), x is characterized by F just in case y is also characterized by F. (Wiggins 1971, §§1.2, 1.7)

There is a further condition that has sometimes been proposed as necessary for the relation of identity to hold, the Identity of Indiscernibles:

(x)(y) ((F) (Fx ≡ Fy)  x = y)

That is, for any pair of referents, x and y, if for all predicates, F (again, suitably restricted), x is characterized by F just in case y is also characterized by F, then x is identical to y. Less abstractly, if two referents have all their properties in common, then the acts of individuation which picked out the two referents have picked out the same particular.

I believe that we should reject the Identity of Indiscernibles. For it either is or is not the case that there is something to a particular other than its universal properties. Suppose that there is something to a particular other than its universal properties. Call this, whatever it is, the haecceity of a particular. Then, two particulars might have all their properties in common while differing in haecceity. Suppose on the other hand that there are no haecceities. Then, there is nothing to a particular except its universal properties. This might seem more favorable for the truth of the Identity of Indiscernibles. But universal properties are properties which may be multiply exemplified. Then, however, it appears that any particular characterized by nothing but universal properties could also be multiply exemplified. That is, if it is unique, it is only contingently so, and even if the principle of the Identity of Indiscernibles holds in fact, it does not do so as a logically necessary condition upon the ascription of identity.

As mentioned above, when there is a single referent, there is no difficulty in ascribing identity, for, by reflexivity, all particulars are self-identical. Problems begin to arise only when two or more referents are individuated. One of the sorts of cases in which this may occur is when the acts of individuation occur at different times and the question to be addressed is whether the referents are or are not instances of the same particular at different times.⁴

Consider now some of the questions that may be raised about a particular individuated at different times. First, can such a particular undergo change without the change undermining claims that the two referents are identical? This has a direct bearing on questions about personal identity since, on almost any account, persons are beings who endure and change through time. Initially, it might appear not, for then there would be some characteristic that it had earlier and lacked later or vice versa, in violation of Leibniz's Law. For example, it will be true of the particular, a, that, at time t₁, it is characterized by the property F, while at the later time t₂, a will not be characterized by F. If that's correct, then it would be entailed that no identity relation could hold between referents at different times if the later referent is in any respect different from the earlier.

This conclusion can be avoided, however. It can be handled by building in temporal indices to the predicates ascribed to the referents. For example, it could be said of an apple that it has the property of being green at t₁ and red at t₂. Since, even at t₁, it can be said to have the tenseless property of being red at t₂, and at t₂, to have the tenseless property of being green at t₁, the fact that a change occurs between t₁ and t₂ does not show that there is a violation of Leibniz's Law and therefore does not show that the referent at t₁ cannot be identical to the referent at t₂.

At this point, a deeper issue looms. Are we to take all of the properties of a referent (including tenseless properties) as relevant to testing whether two referents are identical? Suppose that we are. Then, it appears that we are committed to some variety of determinism, at least for any changing particular, different stages in the history of which are assumed to be stages in the history of the same being. Every statement about a change that it undergoes will have to be true tenselessly (or false tenselessly). But if it is true tenselessly of the green apple at t₁ that it is a red apple at t₃, then it is ruled out that something else may happen between t₁ and t₃ -- such as being consumed before it is ripe at t₂.

Consider an apple which is green at t₁, is possibly consumed at t₂, and is possibly red at t₃. Suppose it does survive to ripen by t₃. Thus, it is true that the apple is red at t₃:

(1) Rat₃.

Is that compatible with

(2) ~Rat₃?

Put that simply, I think the answer is yes -- but only because there are ambiguities in what is meant by (2). Among other things, it could mean that at some time earlier than t₃, something might have happened which would prevent the apple from being red at t₃.

Suppose however that we apply temporal indices to the modal operators as well and interpret the operators as referring to physical necessity and possibility. Thus, to say that t₁Bt_n is to say that there are at t₁ physically sufficient conditions for the occurrence of B at t_n -- which is to say that Bt_n can be inferred from some set of facts obtaining at t₁ in conjunction with natural laws. Similarly, to say that t₁ Bt_n is to say that there is no set of facts obtaining at t₁ (conjoined with natural laws) from which ~Bt_n may be inferred. And, of course, to say that ~t₁Bt_n is to say that there is some set of facts obtaining at t₁ (conjoined with natural laws) from which ~Bt_n can be inferred.  

Obviously, given this usage,

(1)   Rat₃

is not compatible with

(3) t₃~Rat₃.

For, trivially, ~~Rat₃ may be inferred from (1). And of course, equally trivially, ~t₃~Rat₃ may be inferred from (1). This does not exactly line up with common uses of terms like physical necessity because an event or state of affairs is ordinarily said to be physically necessary only if there is some prior state of affairs, given which the event or state of affairs in question is necessary. By contrast, on the present usage, every true statement about a state of affairs represents a case of physical necessity while every false statement is a case of physical impossibility. However, this usage does not pre-judge any questions about whether determinism is true or not because it may be the case that for a (true) statement such as Bt_n, there is no time prior to t_n at which it is true. It may be that for all t_m, where m < n, t_mBt_n and t_m~Bt_n.

What happens, though, when we try to describe the properties of the changing apple in these terms? Some of its properties it has tenselessly; they are true of it at any time that it exists. For example, at any time that it exists, it is an apple that grew on a particular branch of a particular apple tree at a particular time. There may also be properties that it does not have throughout its existence, such as being attached to a particular branch, but which can also be said to be properties that it tenselessly possesses if described with temporal indices. How should a property like this be represented? I suggest:

(4) (t_n) Fat_n.

That is, for all times, the apple has the property F at t_n. (We can restrict this to times at or after the beginning of the existence of the apple so as to leave open the possibility that the apple never comes into existence. We might adopt the notational convention that "t_n" refers, when used with respect to a particular which comes into existence at some time, only to those times such that n ≥ 0 and reserve the negative numbers to refer to times prior to the origination of the particular.) To make this complete, it should be noted that temporal indices will in some cases have to be built into the definition of the property F.

Let's return now more directly to the case of the ripening apple. Suppose

(1)   Rat₃ .

Let's abbreviate this by saying that (1) is equivalent to

(5) Qa.

Is this a tenseless property? If it is, then we can say that

(6) (t_n) Qat_n,

from which it follows that

(7) Qat₁.

That is, Rat₃ is true at t₁. But if that is the case, then so is

(8) □t₁Qat₁,

which is equivalent to

(9)~t₁ ~Qat₁.

That is, at t₁, it is not possible that the apple not be red at t₃. More generally, if all of the apple's properties are tenseless or if every time-indexed property has a tenseless correlate from which it may be deduced, then there will be no time during the apple's existence at which any of its properties could have been otherwise than they are. In other words, so far as it applies to the history of that apple, determinism will be true.

Suppose, on the other hand, that determinism is false with respect to the history of the apple. At t₁, say, the apple may be consumed at t₂ or may be red at t₃ (but not both):

(10) t₁ (Cat₂ v Rat₃), and

(11) ~t₁ (Cat₂ & Rat₃).

Suppose that the apple is in fact consumed at t₂:

(12) Cat₂,

from which it follows that

(13) □t₂Cat₂.

If (13) is true, then of course there is a tenseless correlate of sorts, namely,

(14) (t_n) t_n □t₂Cat₂,

but it is not a tenseless correlate from which (12) can be deduced. Thus, if determinism is not true with respect to the history of the apple, there is a property that the apple possesses at t₂ which it does not possess at t₁.

Apparently, this would lead to the conclusion that, for logical reasons, none of the events in the history of a changing particular which is self-identical through changes could have occurred differently. But surely, whether or not determinism is true (even when restricted to being true with respect to the history of changing particulars) is not just a matter of the logic of identity.

It might be suggested that this problem could be avoided by including modal operators within the predicates allowed. For suppose it were said that what is tenselessly true of the green apple at t₁ is that it is possibly consumed at t₂ and possibly red at t₃ but not both. Then, it will be tenselessly true at t₃ that the apple is green at t₁ and possibly-consumed-at-t₂-or-possibly-red-at-t₃-but-not-both. This does not seem to be sufficient, however. Suppose that the apple does survive to ripen by t₃. There is then a further property that it has, namely, being red at t₃, which is not equivalent to anything that could be tenselessly expressed with affirmative possibility operators at some earlier time.

Thus, if we are to allow that referents that differ in properties may be identical without making determinism logically true for all enduring particulars, then there are good reasons for segregating the properties of a particular into at least two classes. We would like to be able to say, for example, that it is neither essential to the existence of some particular apple that it be consumed before it ripens nor essential that it ripen before it is consumed. More generally, some of the properties of the particular will be essential to it and therefore correctly ascribable at any time at which the particular exists, while others will be non-essential and thus may vary without affecting the identity of the particular. Leibniz's Law will then apply only to the properties that are essential to the particular.

That is, if we are going to speak about pairs of referents as identical -- i.e., as being stages in the history of an enduring but changing particular, then we seem committed to some variety of essentialism.

But the essentialism to which we find ourselves committed may be taken in two quite different ways. We may accept the view, part of a substantial philosophical tradition, that essences are metaphysical. That is, for a given particular, there is some fact of the matter that determines which of its features are part of its essence and which are not, said fact being entirely independent of our classifications.⁵ To use a classical example, it is said that, though Socrates could have been Socrates without being snub-nosed -- hence, being snub-nosed is not an essential property of Socrates -- Socrates could not have been or have become a number. I am unsure that something of this sort really states a claim that should be said to be "part of the essence" of a particular. It may be that it is only an instance of the logical truth that something that is a particular can't be identical to something that is not a particular. Nonetheless, I am willing to concede that, in some cases, a claim of this sort may be defensible, so I do not wish to flatly deny that there are any metaphysical essences. However, for almost all the properties of complex and changing particulars that we would be interested in, there does not seem to be any way, other than appeal to intuitions that are notorious for not being convincing to those who do not already share them, to settle questions about whether the presence or absence of a given property is part of the metaphysical essence of that particular. (Could Socrates have been stupid? a woman? a seventeenth-century German?⁶

In addition to problems about divergence of intuitions, there is a further problem for metaphysical essentialist views. In some cases in which it seems reasonably clear what is entailed by an essence, it appears that the essence of a given particular may entail incompatible properties. Quine made this point with the example of a cycling mathematician:

Mathematicians may conceivably be said to be necessarily rational and not necessarily two-legged; and cyclists necessarily two-legged and not necessarily rational. But what of an individual who counts among his eccentricities both mathematics and cycling? Is this concrete individual necessarily rational and contingently two-legged or vice versa? Just insofar as we are talking referentially of the object, with no special bias toward a background grouping of mathematicians as against cyclists or vice versa, there is no semblance of sense in rating some of his attributes as necessary and others as contingent. (Quine 1960, p. 199)

I find this a persuasive objection to metaphysical essentialism, but I do not take it (as Quine does) to show that there are no metaphysical essences, but rather that, at least for the sorts of cases in which we are interested in the non-formal criteria for identity of complex and enduring particulars, the only sorts of clear-cut entailments that we can confidently recognize are relative to our classifications or conceptual schemes. I shall, in any case, assume that such essentialism as we have reason to endorse in order to deal with issues related to the identities of persons over time is an essentialism relative to our classifications.^{7, 8}

References

Parfit, Derek (1984). Reasons and Persons. Oxford: Oxford University Press.

Plantinga, Alvin (1974). The Nature of Necessity.   Oxford: Clarendon Press.

Quine, Willard Van Orman (1960). Word and Object. Cambridge: M.I.T. Press.

Wiggins, David (1971). Identity and Spatio-Temporal Continuity. Oxford: Basil Blackwell.

Comments? I'd love to hear.