On Identity, Necessary and Contingent. Or: How the precision of a formal language can be fool’s gold
Judas in hell would not want to be another in heaven.
Why? Because if he were to become another,
he would have to become nothing in his own being.1placeholder
In Naming and Necessity2placeholder and “Identity and Necessity”,3placeholder Saul Kripke blasts the view, then fairly popular in logic and philosophy, that there be contingent identity statements.4placeholder A lot of verbiage goes with his attack: the two pieces are transcripts of public lectures, and he appends initial notes to both saying that the reader should try to imagine their statements being delivered, with proper pauses and emphases, in order to understand them better. Rhetorical flourishes aside, his clearest argument against that view, and in favor of his own—that identity statements are necessary5placeholder—, is spelled out in the following passage of “Identity and Necessity”:
“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:
- (x)(y) [(x = y) ⊃ (Fx ⊃ Fy)]
On the other hand, every object surely is necessarily self-identical:
- (x) □ (x = x)
But
- (x)(y) (x = y) ⊃ [□ (x = x) ⊃ □ (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:
- (x)(y) ((x = y) ⊃ □ (x = y))
This is because the clause □ (x = x) of the conditional drops out because it is known to be true.” (136)6placeholder
Later he belabors the point by explaining that, as he sees it, (4) “really amounts to something very little different from statement (2)” since, “if x has this property (of necessary identity with x), trivially everything identical with x has it, as (4) asserts” (137-138).
In these passages Kripke is insistent that he is speaking about the objects x and y (or, in case it is one and the same self-identical object, of the single object x, which is also y), though his main concern in both pieces is actually with proper names, and with the radical thesis that they be all rigid designators, naming the very same objects in all possible worlds. But I will leave these momentous developments aside, limiting myself to mentioning that the only evidence he offers in favor of his thesis is its being based on a “natural intuition”—after which he proceeds to bring up problems with different theses.7placeholder My concern here is with what seems to be the unshakable starting point of his adventure, brought out in the passages above. Such passages look unshakable because they use a formal language, in which one is supposed to phrase and prove logical truths, while in fact their apparent force—this is my main point in a nutshell—is based on a trivial ambiguity, which cannot be seen because of the vast expressive limitations of the language itself. The very tool of formality that should give assurance to this “genius” of analytic philosophy8placeholder is actually a trap into which he, and all the throng that enthusiastically followed him, inadvertently fell. It is, indeed, nothing other than fool’s gold.
Take (2) again and, for a more vivid illustration, consider an individual instance of it, constructed with one of Kripke’s beloved proper names:
- □ (Cicero = Cicero).
Using a terminology that goes back to the Middle Ages, (5) is subject to a first kind of ambiguity, in that the modality in it can be read de dicto (relative to a statement; in Latin dictum) or de re (relative to a thing, or object; in Latin res). We could take it as saying that the statement “Cicero = Cicero” is necessary or that the object Cicero has a property necessarily. Since Kripke insists that he is speaking about objects, I will disregard the de dicto reading (and this first kind of ambiguity) and only read (5) de re. But then a second kind of ambiguity looms. I said, cautiously, that in a de re reading (5) is to be taken as saying that the object Cicero has “a property” necessarily. Now, however, the question arises: what property is that?
In a previous quote, Kripke says: “if x has this property (of necessary identity with x).” But, as it turns out, there is not a single property in play here, which could be referred to by the demonstrative “this”: there are at least two. Read de re, (5) could be taken to say one of (at least) the following two things:
- Cicero has, necessarily, the property of being identical to Cicero
- Cicero has, necessarily, the property of being identical to himself.
The formal language used by Kripke—a first-order modal language—is incapable of distinguishing between these two readings. In order to do so, we need to complicate it by introducing analyzed predicates. Years ago I showed how to do this,9placeholder and the tool I used for that purpose was a lambda operator, which allows the construction of expressions such as the following (each of them is followed by an intuitive reading of it):
λxPx (the property of being P)
λx(Px & Qx) (the property of being both P and Q)
λx(∃yRxy) (the property of having the relation R to some y)
λx□Px (the property of being necessarily P).
In this richer language (in which we imagine to have added “Cicero” to the repertory of individual constants), (6)-(7) above could be translated, respectively, as
- λx(□ x = Cicero) (Cicero)
- λx(□ x = x) (Cicero).
If this clear disambiguation is offered, I cannot imagine any of the supporters of contingent identity sharply criticized by Kripke to ever claim that there is a problem with (9)—that Cicero, or any other object, is not necessarily identical to itself. What they would have problems with, rather, is (8)—that Cicero, or any other object, be necessarily identical with Cicero.
Historically, discussion of this topic took a roundabout course. In his (definite) description theory, Bertrand Russell analyzed statements containing descriptions, such as
- The author of Waverley is Scottish,
as conjunctions of three clauses:
- (a) There is at least an author of Waverley
(b) There is at most an author of Waverley
(c) Every author of Waverley is Scottish.10placeholder
In symbols, and more concisely:
- P(ιxQx) = (∃y)(x)((Qx ≡ (x = y)) & Py),
where “P” abbreviates “is Scottish,” “Q” abbreviates “authored Waverley,” and the small Greek iota (which replaces here Russell’s inverted small Greek iota) followed by “x” is read “the x such that.”
But there was a problem as soon as one went further than simple statements like (10). What about, for example,
- The author of Waverley is not Scottish?
Shall we analyze (13) as denying the analysis (11), or as asserting the analysis
- (a) There is at least an author of Waverley
(b) There is at most an author of Waverley
(c) Every author of Waverley is not Scottish?
To address this issue, Russell decided that every description occurring in a complex statement has a scope, and must be analyzed within that scope. Since the scope is indicated by square brackets, the two different readings of (13) must be written, respectively, as
- It is not the case that ([the author of Waverley] the author of Waverley is Scottish)
- [the author of Waverley] (It is not the case that the author of Waverley is Scottish).
When modal logic came along, W. V. O. Quine raised what he thought was a devastating objection against it. The following argument, he claimed, draws a false conclusion from two perfectly legitimate premises:
- Necessarily, 9>7
- 9 = the number of planets
Therefore,
- necessarily, the number of planets>7
(back then, Pluto was still a planet). So, he concluded, there is something irremediably (necessarily?) fishy about the logic of necessity.11placeholder But, by applying Russell’s distinction of scope, Arthur Smullyan showed that there is a reading of (19) which is not false at all, namely
- [the number of planets] (necessarily, the number of planets>7).
In plainer English,
- There is exactly one x that numbers the planets (that is, the number 9, or, if you prefer, 8) and is necessarily greater than 7.12placeholder
What Smullyan does can be attained in my logic of analyzed predicates by distinguishing two predicates, and then the two statements
- λx(□(x>7))(the number of planets)
- □(λx(x>7)(the number of planets)),
of which (22) is true (the number of planets, that is, either 9 or 8, is necessarily greater than 7) whereas (23) is false (it is no necessity that the number of planets be greater than 7). But Smullyan, and Russell, are limited to bringing out such ambiguities for complex statements; so, when someone suspected that similar ambiguities be in play for Kripke’s case, he was quick to point out that what he said held for simple statements too, where no Russellian notion of scope applies, hence the suspicion was based on “a technical error.”13placeholder He was talking there not of the (apparently) straightforward necessity of identity but of the more ambitious rigidity of proper names; it is crucial to notice, however, that for both my language of analyzed predicates is in a position to bring out an ambiguity—not of scope, but of what property is being predicated—and that this ambiguity applies to simple statements as well. It turns out that the whole issue of descriptions and their scope was an awkward epicycle, which allowed people to do justice, in some cases but not others, to an ambiguity that is pervasive and that the language of first-order logic, modal or otherwise, is not able to account for, let alone resolve.
Leaving descriptions aside, discussion of this topic has traditionally centered around identity statements involving distinct proper names, like
- Cicero = Tully.
Is (24) contingent or necessary? Could Cicero not have been Tully? When questions are put that way, Kripkean answers seem inescapable. For, could ever the man who was Cicero, and was also Tully, not have been himself? But this is another red herring, or rather a delusion of brilliant clarity achieved by being blind to the complexities of the issue. Never mind the two distinct names, and return to (6)-(7) (or (8)-(9)). What are we talking about here: of Cicero (that is, Tully) having the property of being identical to himself, or of Cicero (that is, Tully) having the property of being identical to Cicero (that is, to Tully)? The former is undeniable; but the latter? Could an object, in a different possible situation (or world) be another, distinct object (which, of course, would continue to be identical to itself)?
There are any number of contexts where people talk or think about being or becoming other than themselves. Someone who suffers from unrequited love might wish that he exchange places with a more successful rival—that he be the other, no longer himself. Someone who is going up the gallows might wish that he become one of the curious spectators observing his ordeal. And there are those who believe in being born again after death as another creature: an ant, say, or a tiger. These are all situations deserving of serious metaphysical scrutiny: whatever stance one takes with respect to them, it would have to be based on some elaborate theory of what makes for the identity of a person, or in general an object, and how it is, or it is not, possible for one to become another. We should not expect, or want, them to be promptly dismissed out of hand by an elementary application of first-order logic. We should regard that kind of move as a manifestation of elementary arrogance: an arrogance best left to elementary-school kids. What we would want, and should expect, from a formal logical analysis of a philosophical issue is not that it push problems and difficulties away, but that it show, clearly, what the problems and the difficulties are, so that we can address them with the proper philosophical tools. Which in the present case are bound to be, I am afraid, rather more refined than what can be phrased in the rudimentary language of first-order logic.
The Complete Mystical Works of Meister Eckhart, translated by Maurice O’C Walshe (New York: The Crossroads Publishing Company, 2015), 156.
Originally published in Semantics of Natural Language, edited by Donald Davidson and Gilbert Harman (Dordrecht NL: Kluwer, 1973), 253-355. Here quoted from the revised 1980 edition as a book (Cambridge MA: Harvard UP).
In Identity and Individuation, edited by Milton K. Munitz (New York: New York University Press, 1971), 135-164.
See for example the following passage from Naming and Necessity: “Even before I clearly realized the true situation regarding proper names, I felt little sympathy for the dark doctrine of a relation of ‘contingent identity.’ Uniquely identifying properties can coincide contingently, but objects cannot be ‘contingently identical.’” (4-5).
Kripke admits that in some sense identity statements involving definite descriptions (about which more is said later), such as “The inventor of bifocals = the first Postmaster General,” can be contingently true, though he would insist that the object who was the inventor of bifocals (that is, Benjamin Franklin) is necessarily identical with the object who was the first Postmaster General (that is, himself). As I point out in what follows, this entire detour through descriptions is a red herring.
As this is a literal quote, note that in (4) Kripke drops his apparent convention to use square brackets outside ordinary parentheses. Perhaps it has to do with this being a transcript and all.
“Eventually I came to realize […] that the received presuppositions against the necessity of identities between ordinary names were incorrect, that the natural intuition that the names of ordinary language are rigid designators can in fact be upheld” (Naming and Necessity 5).
In New Frontiers in American Philosophy, an article by Taylor Branch published in The New York Times Magazine on August 14, 1977, 12-22, Robert Nozick is reported as saying that Kripke is “the one genius of our profession.”
In “A New Modal Language with the Lambda Operator,” coauthored with Peter W. Woodruff, Studia Logica 40 (1981), 383-389. In two papers published in 1968 (“Modality and Reference,” Noûs 2, 359-372, and “Abstraction in First-Order Modal Logic,” Theoria 34, 203-207), Richmond H. Thomason and Robert C. Stalnaker introduce a similar complication into modal logic, but their interest is limited to definite descriptions and their distinctions of scope (for which see below) and they engage in no discussion of the necessity of identity, except for a passing, and for present purposes inconclusive, mention in footnote 5 of “Modality and Reference.” It is ironical, however, that by the time Kripke gave his lectures, which “stood analytic philosophy on its ear” (from the London Review of Books, cited in the jacket of Naming and Necessity), formal tools were already available that could have made their text less confused, and perhaps leave everyone less “furious, exhilarated, or thoroughly perplexed” (ibid.) and generally unimpressed. Which reminds us that “Kripke’s” modal semantics was given earlier by Stig Kanger in Provability and Logic (Stockholm: Almqvist and Wiksell, 1957); that, as Quentin Smith has claimed (see, for example, “Marcus, Kripke, and the New Theory of Reference,” Synthese 104, 1995, 179-189), Ruth Barcan Marcus was the originator of “Kripke’s” theory of reference, and Kripke even first heard of it from Marcus in a lecture she delivered at the Harvard faculty club in February 1962 (where, as would appear from a question he asked, he seems not to have understood it; see Jim Holt’s Whose Idea Is It, Anyway? A Philosopher’s Feud, in Lingua Franca: A Review of Academic Life 7, 1996, 29-39); that Kripke knew the main result of his “Outline of a Theory of Truth” (Journal of Philosophy 72, 1975, 690-716) to be already present in Robert L. Martin and Peter W. Woodruff’s “On Representing ‘True-in-L’ in L” (Philosophia 5, 1975, 213-217); and that Kripke’s interpretation of Wittgenstein in Wittgenstein on Rules and Private Language (Cambridge MA: Harvard UP, 1982) echoes Robert J. Fogelin’s six years earlier in Wittgenstein (London and Boston: Routledge, 1976). So much for “the one genius of our profession.”
See for example his “On Denoting,” Mind 14 (1905), 479-493.
See “Reference and Modality,” in his From a Logical Point of View (Cambridge MA: Harvard University Press, 1953), 139-159.
See Smullyan’s “Modality and Description,” Journal of Symbolic Logic 13 (1948), 31-37.
See Naming and Necessity 11ff.