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Kripke on the necessity of identity.

1. Kripke's Formalization

In his (1971) essay "Identity and Necessity," Saul A. Kripke considers an argument for the necessity of identity statements. Kripke writes:

An argument like the following can be given against the possibility of contingent identity statements: First, the law of the substitutivity of identity says that, for any objects x and y, if x is identical to y, then if x has a certain property F, so does y:

(1) (x)(y)[(x = y) [contains] (Fx [contains] Fy)]

On the other hand, every object surely is necessarily self-identical:

(2) (x) [] (x = x)


(3) (x)(y)(x = y) [contains] [[](x = x) [contains] [](x = y)]

is a substitution instance of (1), the substitutivity law. From (2) and (3), we can conclude that, for every x and y, if x equals y, then, it is necessary that x equals y:

(4) (x)(y)((x = y) [contains] [](x = y))

This is because the clause [](x = x) of the conditional drops out because it is known to be true. (1)

The continuing interest of the argument Kripke adapts from precursor versions appearing in W.V.O. Quine, Ruth Marcus, and David Wiggins, against the logical contingency of identity relations, consists in its efforts to prove the necessity of identity, logical, metaphysical, or in another unspecified sense, by appeal to the Leibnizian principle of the indiscernibility of identicals. If a = b, then a and b are supposed to share all of their properties in common. The necessity of identity seems assured, therefore, if only we consider, among the identifying properties universally belonging to both a and b, the property of being necessarily identical to a.

The inference is a variation of an argument that is sometimes raised as an objection to or rationale for restricting quantification over all properties in a Leibnizian principle of the indiscernibility of identicals. The modal objection to Leibniz is previously explored by Ruth Marcus in "The Identity of Individuals in a Strict Functional Calculus of Second Order" (1947), A.N. Prior in Formal Logic (1962), and David Wiggins in "Identity Statements" (1965), as background to his Identity and Spatio-Temporal Continuity (1967).2 The core idea of these arguments is fundamentally the same, although Kripke's formulation is inferentially more elegant, more compact, making several new and interesting elements more explicit than his predecessors. We shall abbreviate the Quine-Marcus-Prior-Wiggins-inspired argument that Kripke formalizes as Q-W, and Kripke's "Identity and Necessity" formalization of some selective composite version of the Q-W inference extracted from the passage already quoted from the essay as K(Q-W).

Kripke's K(Q-W) formalization of a version of the Q-W argument depends explicitly on two caveat conditions: (a) []Fa means [][there exists]xx = a [right arrow] Fa, rather than [][there exists]xx = a [conjunction] Fa; (b) quantification in modal contexts and de re modalities, contrary at least to Quine's conclusions reached by means of another application of the same Q-W (or better, Q) argument, are sensibly interpretable. Quine rejects the equivalent of K(Q-W) conclusion (4) as an inadmissible consequence of quantified modal logic, which he rejects outright on the basis of its supporting (4). Kripke, in arguing that all true identity relations are necessary, accordingly uses a version of an inference he attributes to Quine as an argument that is not only different from, but in this respect diametrically opposed to, Quine's.

Kripke's K(Q-W) conclusion is presented as a general result concerning the modal necessity of identity. However, the argument works only if it is presupposed that any x is identical to any y only if "x" and "y" are bound variables whose substituends are all and only rigidly designative terms. This is not how the quantifiers are generally understood. If I write [for all]xFx, asserting that everything has property F, then I mean to say not only that rigidly designated entities like Jane Jones have property F, but also definitely described individuals, such as the woman who lives next door. This restricts Kripke's argument to the formalized equivalent of proper names, appearing in the logical notation as what Kripke regards as rigidly designative object constants. It is this aspect of Kripke's variation of the Q-W argument in connection with Kripke's early understanding of the metaphysics of modal relations pendant on a particular application of Leibnizian identity, in light of its intuitive attractions, that is considered in more detail in the following critical reflections. (3)

2. Universality in the Indiscernibility of Identicals

Kripke introduces the key property of being necessarily identical to an object in a somewhat surprising way. He first observes:
   Since x, by definition of identity, is the only object identical
   with x, "(y)(y = x [contains] Fy)" seems to me to be little more
   than a garrulous way of saying 'Fx', and thus (x)(y)(y = x
   [contains] Fx) says the same as (x)Fx no matter what 'F' is - in
   particular, even if 'F' stands for the property of necessary
   identity with x. So if x has this property (of necessary identity
   with x), trivially everything identical with x has it, as (4)
   asserts. But from statement (4) one may apparently be able to
   deduce various particular statements of identity must be necessary
   and this is then supposed to be a very paradoxical consequence. (4)

Kripke maintains that the reduction of identity conditional predications of the form [for all]x,y[y = x [right arrow] Fx], equivalently but logically more complexly represented as all objects having property F, entitles us to include in applications of the indiscernibility of identicals as substituends for "F", the formalizable name of any property whatsoever. It is this assumption of the higher-order universality of Leibniz's principle, quantifying indiscriminately over any and all properties, by which Kripke believes he is entitled to advance as a particular instance the modally loaded property of being necessarily identical to a specifically, ultimately rigidly, designated individual object. The necessity of an identity statement in Kripke's philosophy is further categorized as potentially a necessary a posteriori proposition in every instance, just as Kripke argues in the particular case of such an identity assertion as that The Evening Star = The Morning Star or Hesperus = Phosphorus. The identity statement is necessary, because it is expressed by means of presumably rigid designators, disallowing transworld referential variance. The identity is additionally a posteriori, despite its necessity, because it represents a discovery that is not disclosed merely to reflection on the sense-meanings of the relevant terms. (5)

Kripke's K(Q-W) formalization of a Q-W argument for the necessity of all identity relations depends essentially on the property of being necessarily identical to the object of a rigidly designative object constant or bound object variable in authorized applications of Leibniz's principle. If we take away the admissibility of the property of being necessarily identical to a specifically designated object in permissible applications of the indiscernibility of identicals, then K(Q-W) is logically incapable of supporting the inference of K(Q-W) inference step (3) as an instance of the assumption of Leibniz's identity principle in assumption (1). Then we simply do not get the needed substitution. If a = b, then a and b have all of their properties in common. If we mean literally all properties, then the principle is already sunk by referentially opaque or intensional propositional attitude counterexamples, in which it is true that a = b, where we believe (doubt, fear, hope, etc.) that Fa, but we do not believe (etc.) that Fb. The property of being believed by me to be the first bright star in the evening does not logically carry over to the property of being believed by me to be the last bright star in the morning, if I do not already believe that The Evening Star = The Morning Star. Leibnizian indiscernibility of identicals is, accordingly, beset from the outset by needed qualifications. Such objections further open the door to the question whether Leibniz's principle should be understood to include modal properties as among the identifying or distinguishing properties of things in identity and nonidentity determinations, based on an object's possession of properties.

Kripke seems to reason as follows. Building on the reflexivity of identity, in the passage quoted above, one might project that he proposes first that identity relations in which a property holds of an object involve nothing more than the object's having the property. This explication at a distance seems unobjectionable enough. The universal material equivalence is then formalized by Kripke, here in more contemporary notation, as:

(K1) [for all]x[[for all]y[y = x [right arrow] Fy] [left and right arrow] Fx]

From which, in the same style of symbolization, it trivially follows that, necessarily, as a theorem of predicate-quantificational logic:

(K2) [for all]x,y[y = x [right arrow] Fx] [left and right arrow] [for all]xFx

This part of Kripke's presumed reasoning is nevertheless more murky than may at first appear. The logical inference is straightforwardly deductively valid. The question is rather why Kripke would choose to proceed in this two-step way in his informal discussion, rather than immediately producing (K2). Certainly, (K2) is a necessary theorem of logic, while (K1), true or false, is at best logically contingent. The triviality of inferring (K2) from (K1) is reflected in the fact that (K1) does not play the formal role that Kripke tries to impose on it informally, in what is supposed to be equivalent colloquial reasoning. We can verify that (K2) follows logically from (K1), but Kripke seems to want the availability of a valid derivation to imply that property F referred to in the equivalents of (K1) and (K2) can be any property, including the property of being necessarily identical to any chosen rigidly designated object like a. This is why he needs (K2), and why he may believe he needs to argue for the proposition's truth, by first invoking (K1). Kripke does not explicitly quantify over all properties in a higher-order logic as he proceeds informally from the equivalent of (K1) to (K2). Rather, in the colloquial equivalents of the proposed formalizations (K1) and (K2) of the relevant propositions under discussion, he first-order quantifies universally only over all objects.

The question to be addressed is whether either proposition (K1) or (K2) is true, for which the answer would appear in both cases to be no. Consider, to be casual about free variables for the moment, what happens to (K1) or (K2) if x [not equal to] y. Then y = x [right arrow] Fy is trivially true, making Fx trivially true. The problem is that if Tom [not equal to] Mary, then it follows logically from both Kripkean (K1) and (K2) that Mary now has any property we please, including such properties as being a prime number, that Mary manifestly does not possess. Nor is the problem avoided if we water-down (K1) and (K2) to conditional rather than biconditional or material equivalence formulations. If we rewrite the propositions in this direction:

(K1C1) [for all]x,y[[y = x [right arrow] Fy] [right arrow] Fx]

(K2C1) [for all]x,y[y = x [right arrow] Fx] [right arrow] [for all]xFx

Then, once again, we can validly detach the proposition that either a certain specific object x not identical to y has any property F, or that everything in the logic's semantic domain once again has any property F we choose. In either case, we can then validly deduce from the true assumption that Tom [not equal to] Mary, together with (K1C1) or (K1C2), the false proposition that Mary has the property of being a prime number. The derivation goes like this:

1. t [not equal to] m (Tom [not equal to] Mary) Assumed fact

2. [t = m [right arrow] Ft] [right arrow] Fm Instantiating (K1C1)

3. t = m [right arrow] Ft 1 Material implication

4. Fm 2,3 Conditional detachment

It follows classically from the supposition that t [not equal to] m that t = m [right arrow] Ft, and hence from the above instantiation of (K1C1), that Fm. Mary thereby has property F, for any property F, including the property she most definitely does not have of being a prime number. We invalidly infer falsehoods from truths, when we try more generally to apply the basis of Kripke's mode of reasoning in the argument.

Similarly for (K2C1):

Suppose Tom [not equal to] Mary (t [not equal to] m)

[t = m [right arrow] Ft] [right arrow] [for all]xFx Instantiating (K1C1)

Where (K2C1) is concerned, not only does it follow that Fm, that Mary, particular object that she is, has any property F, including the property of being a prime number, but more generally now (K2C1) must extravagantly hold, regardless of the extra-logical facts, for any object x and any property F. These absurd consequences discredit both interpretations of Kripke's (K1) as (K1C1) or (K2C1). We consider next the converse conditionals:

(K1C2) [for all]x,y[Fx [right arrow] [y = x [right arrow] Fy]]

(K2C2) [for all]xFx [right arrow] [for all]x,y[y = x [right arrow] Fx]

It is hard on this interpretation to understand the relationship that is supposed to obtain between these propositions. How exactly could Kripke move from one of these formulas to the other, and what could their progression be meant to show? (K1C2), true or false, is a logically contingent proposition, whereas (K2C2) is an empty tautology, a theorem of predicate-quantificational logic. Moreover, (K1C2) is logically equivalent to the Leibnizian intensional or property-determined indiscernibility of identicals. Kripke, therefore, does not gain anything by considering (K1C2) that is logically not already present in his argument's original assumption (1). The proposition does not tell us anything new about whether we are entitled to quantify over any and every property in applying Leibniz's principle, or about what instances or kinds of property F can or cannot be included in a permissible instantiation of the indiscernibility of identicals.

The fact that (K2C2) is a tautology does not necessarily make it useless for Kripke's purposes in trying to invoke Leibniz's Law to establish the necessity of identity relations. Syntactically, we note that the consequent of (K2C2) can only be validly detached when F is a universal property, a property that is true universally of absolutely every object in the logic's semantic domain. If every object has property F, then it is logically trivial to interpolate the condition that object y is identical to object x, in order for object x in particular to have property F. Nor is this Kripke's intention. The K(Q-W) inference depends essentially instead on understanding the apparent universal quantification over all properties in Leibniz's principle as including the modally loaded property of being necessarily identical to a certain specific rigidly designated object. If we again permit an unrestricted open sentence free variable formulation of the required abstracted property, then Kripke appears to have in mind the property [F.sub.k] = [lambda]x[[]x = y]. (6)

Adopting Kripke's method of argument in K(Q-W), it would then appear to follow on grounds of syntactical parity that we should be equally justified to include and hence instantiate among the properties shared by any x and y, when x = y, the more frontally paradoxical property [F.sup.*] = [lambda]x[[]x [not equal to] b]. If we do not simply beg the question in favor of K(Q-W) assumption that all identity relations are necessary, then from his assumption (1) and in conformity with the same structure of inference, we immediately arrive at the false conditional, where a = b, that [lambda]x[[]x [not equal to] b]a [right arrow] [lambda]x[[]x [not equal to] b]b. Assuming the analytic reflexivity of identity, and avoiding circular reasoning again by not simply presupposing that [logical not][a = b [conjunction] [lambda]x[[]x [not equal to] b]a], we will then have introduced a designer property into another presumably unanticipated application of Leibniz's principle that (logically) falsifies Kripke's own K(Q-W) assumption (1), thereby logically undermining his formal application of the Q-W predecessor argument in support of the necessity of identity.

Another way to appreciate the logical situation in K(Q-W) is to observe that (K1) alone and its auxiliary assumptions, including the necessary reflexivity of identity and the definition of a property instance as F = [lambda]x[[]x = a], with "a" understood as rigidly designative, is logically insufficient to prove that []a = b.

0. a [not equal to] b

1. [for all]x,y[[y = x [right arrow] Fy] [left and right arrow] Fx]

2. [for all]x[x = x]

3. F = [lambda]x[[]x = a]

4. [for all]y[y = b [right arrow] Fy] [left and right arrow] Fb

5. [for all]y[y = b [right arrow] [lambda]x[[]x = a]y] [left and right arrow] [lambda]x[[]x = a]b

6. []a = b [left and right arrow] [a = b [right arrow] []a = a]

7. []a = b

8. a [not equal to] b [conjunction] []a = b

The argument appears to deliver exactly what Kripke needs in steps (1)-(7). The problem is seen when we notice that the detachment from proposition (6) in step (7) follows equally validly from everything that has proceeded, even if we assume as background proposition (0) a [not equal to] b, resulting in the logically inconsistent conjunction (8), a [not equal to] b [conjunction] []a = b. The counter-inference shows that the same argument structure with different assumptions makes it trivial to conclude a = b [right arrow] []a = b. If the inference were correct, then the necessary identity []a = b would not follow deductively both when a = b and when a [not equal to] b.

Kripke himself does not speak explicitly of logical necessity in this context, and nor have we. It is proposed by some of Kripke's readers that he may have implicitly in mind something more like metaphysical rather than logical necessity of identity. If true, then when Kripke uses the modal operator for necessity, "[]", in formalizing inference assumption (2) as (x) [] (x = x), he must mean something like metaphysical necessity. Unless Kripke equivocates in his use of the symbol "[]" between these two essays, then his groundbreaking possible worlds semantics in his earliest essays would also presumably need to be interpreted as involving metaphysical rather than logical necessity. Or modalities more generally, or anyway as a weaker modality about which Kripke develops no distinctive formal semantic structure. I find this farfetched, partly because it requires casting Kripke as a metaphysician using logic as a tool to express ideas arrived at independently of logic, rather than, as I read the essays, a logician rigorously pursuing logic's metaphysical possibilities and implications.

It is more in keeping with the spirit of Kripke as logician in this sense to suppose that, when Kripke does not specify logical necessity in speaking of necessity without further qualification, it might be, not that he thereby vaguely intends a weaker or anyway different modality than logical necessity, but rather that he might consider "necessity" alternatively as abbreviating "logical necessity." Equivalently, the latter expression can be regarded as redundant for "necessity" in Kripke's study of the modality of identity, at least whenever in these early writings a univocally intended modal operator "[]" appears. Having expressed my own philosophical preference for the strongest flavor of logical necessity in Kripke's "Identity and Necessity," I continue below the previous practice of referring only to necessity, without qualification as to logical, metaphysical, or in any other specific sense. The intention is to do justice to Kripke's use of terminology, in order, first, to correctly understand what he means to say and to make any criticisms that might be advanced relevant to his efforts to prove the []-necessity of identity.

If we are permitted to instantiate the unqualified universal quantification over "all" properties in Leibniz's indiscernibility of identicals by Kripke's property [F.sub.k], then we should equally be able to include [F.sup.*] = [lambda]x[[]x [not equal to] b]. After all, surely some things are possibly not identical to object b, and those things have the property [F.sup.*] of possibly not being identical to b. Unless there is a non-ad hoc basis for discriminating the two applications of the supposedly universal quantification over all properties in the indiscernibility of identicals, then both properties must be admitted as legitimate instantiations of the fully generalized any property quantification that Kripke claims follows from the sequence of propositions we have formalized in (K1) and (K2), or conceivably in the weaker versions (K1C1), (K2C2), (K1C2), (K2C2). If [F.sup.*] property instantiations specially constructed to cause logical havoc are permitted in applications of Leibniz's principle, then the indiscernibility of identicals is itself shown to be false. It will then be internally logically inconsistent or logically incompatible with properties that the identity relation is supposed to possess in every (we do not say, logically) possible world. On the other hand, if we close the portal to exclude by fiat such instantiations as [F.sup.*], by qualifying the sense in which any x and y have all properties in common when x = y, then it appears that we may be stymied by lack of an independently justifiable way to draw the line between permissible and impermissible property instantiations. We cannot do so in that case without in the same stroke undermining K(Q-W), by excluding Kripke's choice of any property belonging to x and y when x = y, and allowing him to introduce what then appears to be the modality-of-identity-question-begging property [F.sub.k] = [lambda]x[[]x = a].

The admissibility of such a property shared by the same object under multiple iterative or distinct designations is nevertheless essential to Kripke's derivation of argument step (3) from the indiscernibility of identicals principle in his assumption (1). If, on the other hand, [F.sup.*], syntactically modeled on Kripke's [F.sub.k], makes K(Q-W) assumption (1) false, then the effort to apply the indiscernibility of identicals in proving the necessity of identity relations, the style of inference Kripke adopts, and the overall strategy on which the Q-W argument essentially depends, is explicitly self-defeating. If the indiscernibility of identicals is (necessarily) false, then any inference based on K(Q-W) assumption (1) is rendered trivially deductively valid, and in particular capable of equally logically supporting the very opposite proposition, the negation of the K(Q-W) conclusion that identity is necessary.

Not that anyone wants to defend the position that no identity statements are necessary, but only to temper Kripke's counterintuitive extreme general claim that all identities are necessary. Even if the thesis is true, the question is whether Kripke's thumbnail proof of the proposition that a = b [right arrow] []a = b in "Identity and Necessity" actually proves what it purports to prove. The important question to ask at this juncture is what if anything could possibly justify Kripke in inferring without further ado as he does from his discussion of (K1), (K2) or their jointly logically equivalent conditional variants, that F can be any property, including that of being necessary, even logically necessary, to some chosen rigidly designated object. A general reference is thereby made to the total sharing of type-undiscriminated properties by an identical object referred to in different ways that already appears in the statement of Leibniz's indiscernibility of identicals in assumption (1) of K(Q-W).

3. Rigidly and Nonrigidly Designative Identity

Kripke offers no argument or attempt at justification for limiting the indiscernibility of identicals to rigidly designated objects. Instead, he assumes that "x" and "y" are bound variables whose substituends are all and only proper names, and hence rigid designators, appearing in the logical notation as object constants.

If we understand identity as a relation expressible only by proper names, and if we assume that proper names are rigid designators, characterized in Kripke's way, overlooking the fact that objects can also be nonrigidly designated with very different implications for K(Q-W), then it is logically uninteresting to conclude that identity is always necessary and never merely contingent. Why not, in lieu of a sufficient argument to the contrary, expect instead that Leibniz's principle applies to all identities, regardless of how they are formulated, including those expressed by means of rigidly designative proper names or object constants and those expressed by means of nonrigidly designative definite descriptors? Why not countenance and consider the modality of such identities as []xFx = []xGx as well as a = b?

We can, after all, nonrigidly say []xFx = []xGx [right arrow] [for all]H[H[]xFx [right arrow] H[]xGx], with the same presumption of truth, as we can say in a rigidly designative idiom that a = b [right arrow] [for all]F[Fa [right arrow] Fb]. It appears philosophically unobjectionable to instantiate the universal reflexivity of identity in K(Q-W) assumption (2), [for all]x[]x = x, in the counterpart definite descriptor expression, [][]xFx = []xFx. Consider, on such an interpretation, that if a = b, then a = []xx = a and b = []xx = b. By transitivity of identity, it follows that []xx = a = []xx = b, and by the indiscernibility of identicals applied to identities formulated by means of nonrigid designators, it further follows that F[]xx = a [right arrow] F[]xx = b. Whereupon, if Kripke's modally loaded property [F.sub.k] is uniformly substituted for the universal any and every property instantiation that Kripke takes to be mandated by Leibniz's principle, then we obtain the obviously false conclusion, reflecting back on a falsehood lurking somewhere in the premises, if there is no logical misstep, that [][]xx = a [right arrow] [][]xx = b. The antecedent we may assume to be true, that, say, Kokomo Joe = the tallest person in Bangkok in 2013, but the consequent is presumably false, where a = []xx = a, and even where, contrary to K(Q-W) conclusion (4), as first assumed, a = b.

The difference, to put the same point in slightly different terms, is that if "a" and "b" are rigid designators, then a = b [right arrow] []a = b, given the transworld modality of identity that is built into Kripke's concept of rigid designation. This trivializes either Kripke's K(Q-W) as a proof of the necessity of rigidly designative identities and identity statements, Kripke's unspecified concept of []-necessity, or his category of rigid designation. Whereas, intuitively, contrarily, [logical not][[]xFx = []xGx [right arrow] [][]xFx = []xGx]. Indeed, some logicians have proposed that we syntactically and semantically reduce proper names to definite descriptors as a way of unpacking their referential meaning. Wittgenstein, in the Tractatus, under the special conditions of the logic he discusses there, writes:
   5.526: One can describe the world completely by completely
   generalized propositions, i.e. without from the outset
   co-ordinating any name with a definite object.

   In order then to arrive at the customary way of expression we need
   simply say after an expression 'there is one and only one x, which
   ...': and this x is a. (7)

This is a thesis Kripke vigorously opposes as an inapplicable theory of naming and the reference of proper names. (8) The general point is that by not taking nonrigid designation into account, Kripke does not demonstrate the necessity of identity relations as such, but only of those identity relations expressed by means of rigid designators. To unpack the proposition that rigidly designative identity relations are necessary is to arrive at nothing philosophically more interesting than an empty tautology. The argument ignores the contrary implications of identity relations expressed by means of nonrigid designators such as definite descriptors.

If we expand K(Q-W) assumption (1) to take nonrigidly designative identity relations also into account, then we might substitute the otherwise redundant but in this context more informatively explicit proposition:

([1.sup.+]) [for all]x,y[x = y [right arrow] [for all]F[[Fx [right arrow] Fy] [conjunction] [F[]z[z = x] [right arrow] F[]z[z = y]]]]

K(Q-W) assumption (1) in comparison with ([1.sup.+]) does not reflect all the presumed available applications of the indiscernibility of identicals, indifferent to the rigid or nonrigid designation of objects in identity statements. If ([1.sup.+]) is true, then K(Q-W) assumption (1), in a way that seems vital to Kripke's conclusion about the modality of identity, says something true, but does not reflect the whole truth of the indiscernibility of identicals. Moreover, the argument's implicit limitation of applications by universal instantiation of Leibniz's principle to rigidly designative substituends for universally bound object variables, as already suggested, begs the interesting question as to the modality of identity statements generally. Significantly, the part Kripke leaves out of consideration, nonrigidly designative identities expressed by means of definite descriptors rather than proper names or object constants or bound variables, supports a counterargument whose conclusion immediately contradicts Kripke's claim that identity relations generally are necessary.

The indiscernibility of identicals by itself does not imply or otherwise encourage us to accept the necessity of all identity relations, but only of those in which objects are considered to be rigidly designated. If we understand rigid designation as reference to the numerically identical object in every possible world in which the object exists, and if we agree that rigid designation is ordinarily accomplished by means of proper names or the object constants in a predicate logic, then the conclusion that identity relations expressed exclusively and entirely by means of rigid designators and the identity relation sign "=" are necessary is guaranteed to disappoint. The problem arises for the K(Q-W) conclusion if (A) we keep sight of the fact that there are other kinds of identity relations besides those in which an identity relation holds between rigidly designated objects; (B) that these identity relations are standardly expressed by means of the identity relational predicate and nonrigid designators, such as definite descriptors; (C) that there is no reason to suppose that identity relations do not hold between nonrigidly designated objects, just as they do in the case of rigidly designated objects; (D) that, therefore, there is no reason to suppose that Leibniz's principle of the indiscernibility of identicals does not equally apply to the nonrigidly designated objects in nonrigidly designative identity relations, just as Kripke supposes the principle applies in the case of rigidly designated objects in rigidly designative identity relations, of the form a = b; (E) the only difference, which is in fact the crux of K(Q-W), lies in what can or cannot be concluded without circularity or logical triviality concerning the alethic or other modality generally of identity, of all identities, all existent identity relations and all true identity statements. K(Q-W) loses philosophical significance if it can only most charitably be construed as proving that some identities are necessary, since we know that already in the case of any iteratively reflexive identity, []a = a.

We may learn something about the logic of rigid designation in relation to identity from Kripke's K(Q-W) argument, and perhaps that is the right explanation of the role the argument plays in Kripke's thought. However, with these acknowledged qualifications, we do not learn anything definite from Kripke's reconstruction concerning the modality of identity relations in the more general sense of the word.

Although identity relations expressed exclusively as holding between a (distinctly or iteratively) rigidly designated object are supposed to be necessary in the sense of existing in every possible world in which the respective rigidly designated object exists, true nonrigidly expressed identity relations are generally not supposed to be necessary, but contingently dependent on prevailing facts. We do not ordinarily

expect to be able to derive necessities from nonrigid definitely descriptive identities. The circumscribed application of Leibniz's principle in Kripke's K(Q-W) makes it unexciting as a result to conclude what is strictly false and valid only under an implicit question-begging qualification, that all identities are necessary. If we replace K(Q-W) assumption (1) with ([1.sup.+]), as arguably we should, then, when we follow Kripke's K(Q-W) substitution instance precedent for the open sentence free variable property abstraction [F.sub.k] = [lambda]x[[]x = y], we obtain, instead of K(Q-W) conclusions (3) and (4):

([3.sup.+]) [for all]x,y[x = y [right arrow] [[[]x = x [conjunction] [][]z[z = x]] [right arrow] [][]z[z = y]]]

([4.sup.+]) [for all]x,y[x = y [right arrow] [[]x = y [conjunction] [][]z[z = x]]]

The problem is that the second conjunct of the antecedent of the nested conditional ([3.sup.+]), and of the consequent of ([4.sup.+]), instantiated by []xFx = []xGx in [][]xFx = []xFx [right arrow] [][]xFx = []xGx, is patently false. The antecedent is necessarily true, allowing K(Q-W) to proceed in all respects, except where the exclusive reference to rigidly designative identity relations is concerned. The consequent, however, is contingently false. Since ([3.sup.+]) and ([4.sup.+]) are false, the general Kripkean thesis that identity relations are always necessary and never merely contingent is also false, provided that identity relations are not limited only to those expressed by means of rigid designators, but include definite descriptors as well as proper names or object constants in expressing all of the full range of available identity relations.

4. Critique of Kripke's Formalization

We have seen that K(Q-W) proposition (3) as a valid substitution instance of (1) is mediated by property abstraction. Abstraction makes it possible to define the property of being necessarily identical to entity x as bound by the universal quantifier in K(Q-W) steps (1) and (3). Such an abstract is then supposed to be taken as a legitimate substitutend for predicate "F" in K(Q-W) assumption (1). This step completes a permitted instantiation of the higher-order universal quantification over all properties in Leibniz's indiscernibility of identicals principle, in order to derive K(Q-W) conclusion (3).

Previously, we had not questioned the exact logical formulation of prospective Kripkean property [F.sub.k], as the property of being logically necessarily identical to a designated object. The issue now becomes unavoidable, as we consider whether Kripke's reconstructed key inference of (3) as a substitution instance of the original unqualified assumption (1) in K(Q-W) is deductively valid. The abstract [F.sub.k] = [lambda]x[[]x = y] in just this open sentence free object variable form cannot be accepted without scruple, because it is not well-formed, containing as it does the free variable "y." (9) The abstract with variable "y" unbound by any particular object quantifier is equivocal with respect to expressing the property of being necessarily identical to every or merely some object, or even exclusively to the object having certain specific properties. As such, it is in a sense not really a property abstraction at all, but merely a non-designating syntax string or incomplete unsaturated schema for a property that has yet to be fully specified.

The only way to convert the expression into a proper abstract is by somehow binding the variable by a quantifier or [lambda],-operator, or replacing it with a designating singular term, proper name, object constant, or definite descriptor. There are only a handful of possibilities for redefining the relational property of being necessarily identical to a bound variable x, and none that is sufficient for Kripke's purposes in trying to prove that all identity relations are necessary. We briefly consider these alternatives, and show that none presents the right logical structure to derive conclusion (3) in K(Q-W), as a valid substitution instance of the Leibnizian indiscernibility of identicals principle in Kripke's proof step (1).

(i) [F.sub.k] = [lambda]x,y[[]x = y]

The definition in this first proposal will not do, because it requires simultaneous satisfaction by two object terms. K(Q-W) involves the substitution of a predicate in identity principle (1) to one of two different object terms. If we begin with arbitrary proper names and try to work toward a conclusion that can then be universally quantified to derive K(Q-W) conclusion (3) as a substitution instance of (1), then we begin with the instantiation, a = b [right arrow] [[lambda]x,y[[]x = y]a [right arrow] [lambda]x,y[[]x = y]b]. Such a construction is nevertheless not well-formed. From it, unfortunately, we are not even authorized to deduce: a = b [right arrow] [[]a = a [right arrow] []b = b], which conditional could only follow from a = b [right arrow] [[lambda]x,y[[]x = y]a,a [right arrow] [lambda]x,y[[]x = y]b,b]. This instantiation is not even as strong as conclusion (3) in Kripke's K(Q-W) argument for the necessity of identity, where the equivalent proposition follows trivially in any case from K(Q-W) assumption (2). The best we could obtain would then be: (a) a = b [right arrow] [[lambda]y[[]a = y] [right arrow] [lambda]x,y[[]b = y]]; (b) a = b [right arrow] [[lambda]x[[]x = a] [right arrow] [lambda]x[[]x = b]]; (c) a = b [right arrow] [[lambda]y[[]a = y] [right arrow] [lambda]x[[]x = b]], or (d) a = b [right arrow] [[lambda]x[[]x = a] [right arrow] [lambda]y[[]b = y]]. None of these syntax strings is well-formed, taking only abstracted predicates rather than predications as antecedents and consequents of the relevant material conditional.

(ii) [F.sub.k] = [lambda]y[[for all]x[]x = y]

This redefinition is also inadequate. Substitution of the defmiens in (ii) for predicate "[F.sub.k]" in Kripke's identity principle (1) by standard logic yields: a = b [right arrow] [[for all]x[]x = a [right arrow] [for all]x[]x = b]. The resulting proposition once again does not validly entail K(Q-W) conclusion (3).

(iii) [F.sub.k] = [lambda]y [[][for all]xx = y]

Nor is proposition (3) in K(Q-W) validly derivable as a substitution instance of (1) by the property abstract in (iii).

(iv) [F.sub.k] = [lambda]y[[there exists]x[]x = y] and (v) [F.sub.k] = [lambda]y[[][there exists]xx = y]

The existential counterparts of abstracts (ii) and (iii) in abstracts (iv) and (v) above attempt to define the abstract required for Kripke's proof of necessary identity by existentials are also certain to be inadequate, because they are quantificationally even weaker than their universal counterparts (ii) and (iii). Standard decision methods readily establish that they are unable together with K(Q-W) assumptions (1) and (2) to entail the constant term precursor form of conclusion (3) in his K(Q-W) argument for the necessity of identity.

Because (iv) and (v) are existential, they cannot support the universal generalization of a = b [right arrow] [[]a = a [right arrow] []a = b] as proposition (3) requires in [for all]x,y[x = y [right arrow] [[]x = x [right arrow] []x = y]]. The abstracts formalize the property of being that to which something is necessarily identical or of being necessarily identical to something. As such, the abstracts at most support only the conclusion that something is necessarily identical to everything (any x). This conclusion is trivially true, since anything is obviously necessarily identical to itself. The proposition follows directly from Kripke's proof assumption (2), as an expression of the reflexivity of identity, without invoking abstracts (iv) or (v), but it does not entail K(Q-W) conclusion (3). It appears, then, that the property on which K(Q-W) logically depends cannot be adequately formulated by embedding either a universal or existential quantifier within the abstract with either internal or external scope relative to the modal operator.

(vi) [for all]x[[F.sub.k] = [lambda]y[[]x = y]]

The alternative is to allow the quantifier to have scope over the entire abstract. This proposal at last makes it possible validly to derive K(Q-W) conclusion (3) from the indiscernibility of identicals principle in assumption (1). Here, what makes abstract (vi) strong enough to deduce (3), unfortunately, also makes it too powerful, implying outright logical contradictions, and thereby logically trivializing not only K(Q-W) assumption (3), derivable ex quodlibet falsum, independently of assumptions (1) and (2), in company, classically, with any inference whatsoever as equally deductively valid.

Such an abstract, somewhat in the manner of (ii) and (iii), defines [F.sub.k] as the property of being necessarily identical to everything (any x). From (vi), we validly obtain from identity principle (1): a = b [right arrow] [[lambda]y [[]a = y]a [right arrow] [lambda]y [[]a = y]b]. It then follows as above by abstraction equivalence that: a = b [right arrow] [[]a = a [contains] []a = b]. This proposition, assuming that constant terms a and b are arbitrary, can then be twice universally generalized, to derive K(Q-W) assumption (3). So far, so good. A problem nevertheless results in the extension of predicate '[F.sub.k]', if a [not equal to] b, when multiple instantiations of (vi) entail both, [F.sub.k] = [lambda]y[[]y = a] [conjunction] [F.sub.k] = [lambda]y[[]y = b]. A contradiction immediately follows, because it is true that [F.sub.k]a for [F.sub.k] as it is instantiated in the first conjunct, but equally true that [logical not][F.sub.k]a for the same property [F.sub.k] as instantiated in the second conjunct. The opposite holds for [F.sub.k]b, as [F.sub.k] is instantiated in the second conjunct, and for [logical not][F.sub.k]b as [F.sub.k] is instantiated in the first conjunct. The inconsistency makes it hopeless to define the necessary identity property Kripke needs for the derivation of conclusion (3) as a substitution instance of identity principle (1) by means of abstract (vi). The same problem occurs if we try to bring the abstract within the scope of a single universal quantifier, by embedding it in a restatement of K(Q-W) assumption (1) with the form, [for all]x,y[[x = y [conjunction] [F.sub.k] = [lambda]y [[]x = y]] [right arrow] [[F.sub.k]X [right arrow] [F.sub.k]y]].

(vii) [there exists]x[[F.sub.k] = [lambda]y[[]y = x]]

This option is not subject to the confusion of the previous definition. It formalizes only the property of being necessarily identical to at least one entity. This is also logically inadequate to derive K(Q-W) assumption (3) precisely as Kripke formulates it in the argument. If the existential in (vii) is introduced as an implicit or hidden assumption in the proof, then the conclusion on pain of deductive invalidity cannot be universally, but itself at most only existentially, quantified. The problem of how to close the equivocal open abstraction [lambda]x[[]x = y] stubbornly remains. We cannot use (vii) to derive Kripke's interesting result that all identity relations are necessary, but only the vastly less stirring proposition that some identity relations are necessary, [there exists]x[for all]y[x = y [right arrow] [[]x = x [right arrow] []x = y]], and hence [there exists]x[for all]y[x = y [right arrow] []x = y]. We already know that some identity relations are necessary, in particular those that express the equivalent of K(Q-W) assumption (2), that are in effect instantiations and existential implications of the proposition that [for all]x[]x = x. The same problem occurs if we try to bring the abstract within the scope of a single existential quantifier by embedding it in a restatement of K(Q-W) assumption (1) in the expression, [there exists]x[for all]y[[x = y [conjunction] [F.sub.k] = [lambda]y [[]x = y]] [right arrow] [[F.sub.k]x [right arrow] [F.sub.k]y]].

(viii) [F.sub.k] = [lambda]y[[]y = a]

Finally, we consider the possibility of defining K(Q-W) property F as the property of being necessarily identical to a rigidly designative proper name or constant designated individual object a. The conspicuous fact about interpretation (viii) is that it gives up on quantification altogether to close the open proto-abstraction, and resorts to the explicit introduction of a rigidly designative object constant to specify the property of being necessarily identical to that same object.

If the property of being necessarily identical to a particular object can only be expressed as in (viii), by means of a succession of rigidly designative object constants, then there exists no sound recursion to elevate the series (for a, b, c, etc.) to the universal generalization demanded by conclusions (3) and (4) in K(Q-W) for the necessity of all identities. Since we already know that some identities are necessary, if K(Q-W) is to command interest, it must establish a more unexpected universally generalized conclusion concerning the nature or necessary properties of every identity relation. We cannot logically reduce K(Q-W) conclusion (4) even to a countably infinite run of instantiations of the Leibnizian assumption in K(Q-W) step (1), involving rigidly designated object a, b, c, etc., constants. The reason is that we are generally prevented from logically reducing any universal quantification to a countably infinite conjunction of relevant predications to the individual objects in a logic's referential domain; or, similarly, to a countably infinite run of disjunctions in the case of existential quantifications. Quantifiers, contrary to Wittgenstein's Tractatus, are not merely convenient abbreviations for propositional logical operations to which they are logically reducible.

With no further available interpretations to consider, it seems unavoidable to conclude that there is no property that satisfies the needs of Kripke's K(Q-W) argument for the necessity of identity relations. There is no permissibly formalizable property possessing the right logical structure to derive conclusion (3) in K(Q-W), as involving an acceptable abstract substitution instance for F, in applying Leibnizian indiscernibility of identicals as Kripke proposes in K(Q-W) assumption (1). (10)


University of Bern


(1.) Saul A. Kripke, "Identity and Necessity," in Identity and Individuation, edited by Milton K. Munitz (New York: New York University Press, 1971), pp. 135-136. Reprinted in Saul Kripke, Philosophical Troubles, Collected Papers, Volume I (Oxford: Oxford University Press, 2011), pp. 1-2 (primary reference). After citing Kripke's discussion in its original form, I introduce minor self-explanatory stylistic revisions to contemporize Kripke's notation, and to clearly distinguish my discussion of Kripke from what Kripke is quoted as writing.

(2.) Ruth Barcan, "The Identity of Individuals in a Strict Functional Calculus of Second Order," Journal of Symbolic Logic, 12, 1947, 15, proposition 2.31. A.N. Prior, Formal Logic (Oxford: The Clarendon Press, 1962), pp. 205-206. David Wiggins, 'Identity-Statements', in R. J. Butler, ed., Analytical Philosophy (Oxford: Basil Blackwell, 1965), 2nd series, pp. 40-41. More recently, Nicholas Griffin has revived the argument as a disproof of absolute identity. Griffin, Relative Identity (Oxford: The Clarendon Press, 1977), p. 3. Griffin observes that David Wiggins in Identity and Spatio-Temporal Continuity (Oxford: Basil Blackwell, 1967, p. 68, note 26), disowns most of his 1965 article, including presumably his version of the proof that all identities are necessary. See Wiggins, "Identity-Statements," pp. 42-46, where he speaks of "the only part of that article of which I should now wish to offer much defence." I discuss Griffin's use of the argument as justification for rejecting absolute in favor of relative identity in Dale Jacquette, "Modal Objection to Naive Leibnizian Identity," History and Philosophy of Logic, 32, 2011, pp. 107-118.

(3.) Kripke, Naming and Necessity, pp. 43-49; see especially pp. 48-49: "... a designator rigidly designates a certain object if it designates that object wherever the object exists; if, in addition, the object is a necessary existent, the designator can be called strongly rigid." Quine's famous "Three Grades of Modal Involvement" anti-quantified-modal-logic set piece can be reconstructed in these terms:

1. 9 = Number of planets

2. [] 9 > 7

3. [] Number of planets > 7.

Substitution in the punch-line conclusion is supported only if = contexts are assumed to be extensional. Quine does not countenance even to exclude the alternative, that identity is instead intensional or referentially opaque. The essential pressupposition in Quine's neat three-step argument against quantified modal logic can nevertheless be turned around, in yet another way than Kripke's K(Q-W) formalization and subsequent philosophical interpretation of its consequences, as proving the intensionality of identity relation statements. We agree with Quine that the conclusion is false, that the number of planets is a contingent truth about the world. However, we avoid Quine's conclusion in the example by paraphrasing identity out of the first assumption, reinterpreting it as the predication to the number of planets of the property of being ninefold in number. True predications generally, non-identity predications, unlike true identity statements, are not supposed to sustain the inter-substitution of co-referential terms or logically equivalent expressions in purely extensional contexts. Quine takes substitution failure salva veritate as a sign that modal contexts are intensional, that we cannot freely inter-substitute identicals within them as in proceeding from the second to the third step of the inference on the strength of the first assumption. The inference is intriguing in part because it upholds a number of conflicting possibilities, including a reductio ad absurdum instead against the presupposition that identity contexts are extensional. We may have no better choice in considering the full range of Quine-inspired substitution failures, entirely outside of modal context applications and the problem of quantifying into supposedly "intensional" modal contexts, assuming the extensionality of identity and the reflexivity of identity applied to propositions as objects. As we see here, for any distinct propositions p and q that happen to be materially equivalent (Snow is white; Grass is green; [the proposition that] Snow is white [not equal to] [the proposition that] Grass is green; Snow is white iff Grass is green [the two numerically distinct propositions are materially equivalent]:

1. p [conjunction] q Assumption

2. p [not equal to] q Assumption

3. p [left and right arrow] q (1) Elementary logic

4. p [not equal to] p [conjunction] q [not equal to] q (2),(3) Intersubstitution of [left and right arrow] Equivalents

(4.) Kripke, "Identity and Necessity" (2011), p. 3 [Munitz, p. 137].

(5.) I examine Kripke's arguments for the necessary a posteriori in Jacquette, "Kripkean Epistemically Possible Worlds," in Possible Worlds: Logic, Semantics and Ontology, edited by Guido Imaguire and Dale Jacquette (Munich: Philosophia Verlag, 2010), pp. 99-140. Kripke presumably understands The Morning Star = The Evening Star or Hesperus = Phosphorus case as typifying all semantically necessary epistemically a posteriori noniterative or cognitively informative identity statements, and considers himself to have established this category for all (at least rigidly designative) experientially discoverable identities.

(6.) Alonzo Church, The Calculi of Lambda Conversion (Princeton University Press, 1941). H.P. Barendregt, The Lambda Calculus: Its Syntax and Semantics (Amsterdam: North-Holland Press, 1984). See Felice Cardone and J. Roger Hindley, "History of Lambda-Calculus and Combinatory Logic," in Dov Gabbay and John Woods, eds., Handbook of the History of Logic, volume 5, Logic from Russell to Church (Amsterdam: North-Holland Press (Elsevier), 2006), pp. 723-818.

(7.) Ludwig Wittgenstein, Tractatus Logico-Philosophicus, C. K. Ogden, ed. (London: Routledge & Kegan Paul, 1922).

(8.) Kripke, Naming and Necessity, especially pp. 5-15, 27-34, 53-77.

(9.) Kripke's informal gloss obscures the problem, when he writes, "Identity and Necessity," p. 3 [pp. 137-138]: "Anyone who believes formula (2) is, in my opinion, committed to formula (4) ... Since x, by definition of identity, is the only object identical with x, '(y)(y = x [contains] Fy)' seems to me to be little more than a garrulous way of saying 'Fx', and thus (x)(y)(y = x [contains] Fx) says the same as (x)Fx no matter what 'F' is --in particular, even if 'F stands for the property of necessary identity with x. So if x has this property (of necessary identity with x), trivially everything identical with x has it, as (4) asserts. But, from statement (4) one may apparently be able to deduce [that] various particular statements of identity must be necessary and this is then supposed to be a very paradoxical consequence." A definition of property F, as a syntactical matter, might be incorporated into a restatement of (1), in which both the Leibnizian indiscernibility of identicals and the property of necessary identity share the same variable-binding quantifier. The possibility is considered below in [section]2 (vi)-(vii), where a universally quantified abstract defining F as the necessary identity of x embedded in (1) is rejected as implying an inconsistency in the extension of predicate "F," and the existential formulation is rejected as too weak to entail K(Q-W) conclusions (3) and (4). For inspiring criticisms see especially David Bostock, "Kripke on Identity and Necessity," The Philosophical Quarterly, 109, 1977, pp. 313-324.

(10.) I discuss some of these topics in Kripke's modal defense of mind-body property dualism at greater length in Jacquette, "Kripke and the Mind-Body Problem," Dialectica, 41, 1987, pp. 293-300, and in the second expanded edition of my Philosophy of Mind: The Metaphysics of Consciousness (Continuum Books, 2009), pp. 24-31. For a critique of Kripke's use of necessary identities in refuting the description theory of reference, paving the way for his preferred description-theory-objection-avoiding account of rigid designators, the causal-historical "alternative picture" of reference, and the use of rigid designation in stipulating transworld identities for objects with distinct accidental or inessential property variances from world to world, see Jacquette, "Kripke's Modal Objection to the Description Theory of Reference," in We Will Show Them! Essays in Honour of Dov Gabbay, Sergei Artemov, Howard Barringer, Artur d'Avila Garcez, Luis C. Lamb and John Woods, eds. (London: King's College Publications, 2005), pp. 143-168. Also, Jacquette, "Kripke on Identity and the Description Theory of Reference," The Logica Yearbook 2002, Timothy Childers and Ondrej Majer, eds. (Prague: Filosofia, 2003) (Institute of Philosophy, Academy of Sciences of the Czech Republic), pp. 109-116.
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Date:Jan 1, 2014
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