How to Live Without Identity—And Why

¹One would then have to stipulate the totality and right-uniqueness of ⇛ _f .

²It may be instructive to consider an example of the suggested rewriting strategy. Take the formula R(g(x, h(c)), h(h(h(y)))), where ‘g’ is a binary function symbol, ‘h’ a unary function symbol, and ‘R’ a binary relation symbol. We'd rewrite this formula into

Clearly the original notation is more concise and better suited for practical purposes. But since we're asking whether objectual identity is eliminable from logic in principle, such practical considerations carry no weight.

³‘Sign’ is to be read quite generally here so as to include not just names, but also variables.

⁴Readers interested in exegetical issues such as Wittgenstein's own motivation for his convention should consult chapter V, §6 of Fogelin [1976], as well as [White [1977/78], which helpfully sets out Wittgenstein's views against a Fregean background. For technical details see my [2004] and [2008].

⁵In languages containing constants, which we consider below, the referents of constants occurring within the scope of the quantifier binding ‘x’ will also be excluded from the range of ‘x’.

⁶Anyone who has ever taught an introductory logic course knows that many beginning students must be drilled into accepting that ‘∀x∀y’ does not mean ‘for any two objects x and y’, that is, that ‘x’ and ‘y’ may be instantiated to the same value. The viability of W-logic shows that this pedagogical manoeuvre is merely the enforcement of a convention, rather than the correction of a conceptual mistake.

⁷Recall that the range of a bound variable also excludes the referents of all constants occurring in the scope of the quantifier binding that variable.

⁸The triple bar symbol is, of course, a tip o' the hat to Frege's [1879] introduction of identity in the Begriffsschrift. It's crucial to insist that the triple bar can be flanked only by individual constants, never by variables, to avoid the use-mention trouble diagnosed in Frege by, among others, White [1977/78: 178], Heck [2003: 87], and Mendelsohn [2005: 60–1].

⁹Thus McGinn [2000: 8] errs when he writes, ‘If we say “for some x, x is F and x is G”, we are making tacit appeal to the idea of identity in using “x” twice here: it has to be the same object that is both F and G for this formula to come out true.’ For an extended discussion see Humberstone and Townsend [1994].

¹⁰Since the translation easily extends to second- and higher-order logics, there's no reason to think that W-logic might not be able to accommodate any part of mathematics.

¹¹Actually, (1w) is adequate only under the assumption that the domain contains at least one individual besides Alfred, which is certainly a conversational implicature of any use of (1). In a one-element domain, (1w) is vacuously W-true, regardless of Alfred's smartness. A formalization that is adequate even in one-element domains would be ∀x (¬Sa & Sx) & ¬Sa.

¹²That is, we can analyse ‘a is the F’ as Fa & ¬∃x∃y (Fx & Fy) and ‘The F is the G’ as ∃x (Fx & Gx) & ¬∃x∃y (Fx & Fy) & ¬∃x∃y (Gx & Gy).

¹³I don't wish to suggest that this principle has any historical claim to being associated with Wittgenstein. Its designation is only intended to signal that it plays a role in the defence of a broadly Wittgensteinian claim, not that he himself would have endorsed it.

¹⁴To wit, Blackwell's Companion to Metaphysics (cf. [Mulligan 1995]) and the Routledge Encyclopedia of Philosophy (cf. Bacon, Detlefsen and McCarty [1999: 7]). Both neglect to include the modal element and define the arity of a relation simply as the maximal number of objects it relates. In a world of strict narcissists, this would make the amatory relation unary. WAP gives the correct result because it would still be possible for the amatory relation to hold between two individuals, even if it does not in fact do so.

¹⁵Of course the fact that Kermit the Frog and the Incredible Hulk are both green doesn't mean that greenness is binary; it just means that greenness can be multiply instantiated. Similarly the fact that Antony loves Cleopatra and Disraeli loves Mary Anne doesn't mean that the amatory relation is quaternary, it just means that we can conjoin binary amatory facts. The case of Ilsa's loving Rick and Victor (simultaneously) is entirely analogous, the only difference being that the conjoined amatory facts share a relatum. Similar considerations apply to betweenness.

¹⁶There are other moves the friend of identity could make in order to ground a well-formed–ill-formed distinction. One might take recourse to the view that relations contain ‘positions’ or ‘slots’ that serve as receptacles for their relata, and that the arity of a relation is its number of slots. But at least in its most intuitive form, where relational facts are quasi-spatial complexes consisting of the relation with its positions occupied by the relata, such a view can hardly be reconciled with the binarity of objectual identity: Suppose identity has the two slots α and β. Insert any object x into α. Surely x itself cannot now be plugged into β, since it already occupies α. Thus identity facts are impossible. Worse, the same consideration shows that reflexive relational facts are impossible. While there may be ways to side-step these problems (e.g. by allowing two positions to collapse into one in case of reflexive facts, or by filling the positions not with the actual relata but by copies or clones of them), the measures to which one is driven seem so desperate that rejection of a binary objectual identity relation appears much more reasonable.

¹⁷This is subtler than it looks: on the now standard Kuratowski definition of the ordered pair, ‘<x, y>’ stands for the set {{x}, {x, y}}, and hence ‘<x, x>’ stands for {{x}}. Now {<x, x> : x ∈ D}, that is, {{{x}} : x ∈ D}, looks like a binary relation when regarded as a subset of D × D, but like a property when considered as a subset of {{{x}}: x ∈ D}. This shows that the assignment of -arities to sets is relative to a background domain. Another case in point is the empty set, which is trivially, for any n, a set consisting solely of n-tuples. Thus without specification of a background domain, the empty set, considered as a relation, has every arity.

¹⁸Set theory takes, as it were, a contingent notational feature of our natural and symbolic languages, namely the use of order to indicate in which way a relation applies to its relata, and misleadingly turns it into an essential feature of the relation.

¹⁹That there is a distinction between <x, x> and <x> is, incidentally, not obvious, even within set theory. As noted, <x, x> is typically taken to be the set {{x}}. There's no generally accepted set-theoretic definition of the 1-tuple <x>, presumably because the notion is not particularly useful for mathematical purposes. However, given that the only characteristic required of a definition of ordered n-tuples is that the identity of <x ₁, … , x_n > and <y ₁, … , y_n > imply the identity of each x_i with the corresponding y_i , it's perfectly in order to define <x> as {{x}}, for from the identity of {{x}} with {{y}} the identity of x and y certainly follows. But then <x, x> just is <x>, and the arity of the set-theoretic identity relation becomes ambiguous between one and two.

²⁰In fact, intersection is not even well-defined on relations. See Williamson [1985: 258–60].

²¹As noted by an anonymous referee, on the standard construal of the first-order quantifiers, second-order logic allows for a definition of identity as indiscernibility: one can introduce a binary predicate ‘I’ via the Leibniz principle ∀x∀y (I(x, y) ↔ ∀P (P(x) ↔ P(y))), and the semantics of second-order logic then ensures that the interpretation of ‘I’ is objectual identity. However, from our perspective, this involves the same legerdemain as the principle (Ax =) considered above, for the Leibniz principle translates into W-logic as the conjunction of ∀x (I(x, x) ↔ ∀P (P(x) ↔ P(x))) and ∀x∀y (I(x, y) ↔ ∀P (P(x) ↔ P(y))). The former reduces to ∀x I(x, x), and the latter (under the assumption, which we must make anyway, that no two things can share all their properties) to ∀x∀y ¬I(x, y). Thus the Leibniz principle is equivalent to (Ax =) in W-logic, and our earlier comments apply. Similarly for the notion of being one, to be introduced anon.

²²That the contingent sentence (3) can be logically equivalent to the necessary sentence (4) should not be too surprising, given the role rigidity plays in the argument. See, e.g., [Zalta 1988].

²³Our quadrisection of the notion of identity may be compared with Pardey's [1994: 134–5] analysis of identity statements into seven categories. His categories (II) and (V) essentially correspond to the cases of identity statements in which at least one definite description flanks the equality sign. Category (III) consists of statements of self-identity, categories (IV) and (VI) of co-reference claims, and category (VII) of statements concerning the evaluation of functions. This leaves Pardey's category (I), for which he provides the example ‘Scott and the author of Waverley are identical (these supposedly two persons are in reality only one person)’. The parenthetical paraphrase suggests that he is thinking of our notion of ‘being one’.

²⁴Thanks to three anonymous referees for this journal, to Aldo Antonelli, Bill Demopoulos, Kit Fine, Robert May, Richard Mendelsohn, Ulrich Pardey, Helge Rückert, Ansgar Seide, Bartosz Wieckowski and especially Lloyd Humberstone and Tobias Rosefeldt for helpful comments. I am also grateful to audiences at the universities of Bristol, Glasgow, Konstanz, Münster and Nancy, as well as the participants in my Metaphysics of Relations seminar.