Wolfgang Schwarz :: Kripke's (Alleged) Argument for the Necessity of Identity Statements

Kripke's (Alleged) Argument for the Necessity of Identity Statements

Posted on Wednesday, 09 Aug 2006.

I have often encountered in articles, talks and classes the following argument for the necessity of true identity statements, always attributed to Kripke:

1) a = b (assumption)
2) $$m[1]$ a = a
3) $$m[1]$ a = b (from 1, 2 by Leibniz' Law)

The argument is no good, and I think it is very doubtful that Kripke ever endorsed it.

A minor problem with the argument is that premise (2) is generally false because things don't exist necessarily. That is easily avoided by making [the identity claims in] (2) and (3) conditional on $$m[1]$ .

The main problem is the move from (1) and (2) to (3) via Leibniz' Law. This move isn't trivial: it is clearly invalid if the box is read as "it is well known that" or "Fred said that". The application of Leibniz' Law is valid iff " $$m[1]$ ... = ..." is an extensional context. That is just what "extensional context" means: a context wherein substitution of co-referential terms preserves truth.

So the question is whether " $$m[1]$ ... = ...", unlike "it is well known that ... = ...", is an extensional context, as the argument presupposes. That surely takes some argument, as sentences of the form " $$m[1]$ ..." are often presented as paradigm examples of non-extensional contexts. In fact, modulo existence, it is necessary that Hesperus = Hesperus, but not that Hesperus = the brightest star in the evening sky, even though "Hesperus" and "the brightest star in the evening sky" co-refer. Perhaps all such cases can be explained away by scope distinctions, but the extensionality of " $$m[1]$ ... = ..." at any rate remains a strong, and prima facie at least questionable, assumption.

The argument does not contain any defense of that assumption. Instead, the assumption is more or less just the conclusion which the argument is supposed to establish. Anybody who doubted that whenever a = b, then $$m[1]$ a = b (or rather, $$m[1]$ ) would certainly have doubted that " $$m[1]$ ... = ..." is extensional. So Kripke's alleged argument relies on a highly suspicious and question-begging assumption.

There are at least three ways how " $$m[1]$ " could fail to be extensional, two of which are often overlooked.

One way is for the context to be intensional: whether " $$m[1]$ Hesperus = the brightest star in the evening sky" is true depends not only on the actual denotation of the two terms involved, but also on what the terms denote at other possible worlds.

A second way is for the context to be hyper-intensional in the way in which "it is well known that ... = ..." are, where we can't even substitute expressions with the same (primary or secondary or 2D) intension (try "3 = 3" and "3 = the smallest prime whose reciprocal has period length 1/2(p-1)").

The division isn't clear. Consider Counterpart Theory, on which "the statue = the clay" may be true and " $$m[1]$ the statue = the clay" false because in normal contexts, different counterpart relations are invoked by the terms "the statue" and "the clay". Do these counterpart relations belong to the meaning of the terms? Arguably not, and certainly not as ordrinary primary or secondary intensions.

Anyway, here is the third, and most curious way. I don't know how to characterize it in general. Here is an example. Suppose it isn't perfectly determinate what "Hesperus" and "Phosphorus" denote: there are some atoms such that it is indeterminate whether the terms denote the planet together with those atoms or without them. Consider the operator "Det" defined as follows: "Det S" is true iff S is true on any resolution of such indeterminacies. Then "Det(Hesperus = Hesperus)" is true, but "Det(Hesperus = Phosophorus)" is false. So "Det(... = ...)" is not extensional. In this case, it doesn't matter whether "Hesperus" and "Phosphorus" have different intensions of whether there is any cognitive difference between them at all (which usually explains hyper-internsionality).

This third way is relevant here, because " $$m[1]$ " is arguably non-extensional in exactly this way. Suppose in a few million years, Venus will fission into two planets. Which of the two, if any, do we now denote by "Venus", "Hesperus" and "Phosphorus"? Suppose that's indeterminate. What then should we say about a sentence like "in 10 million years, Venus will still be part of our solar system"? No answer is entirely satisfactory. Perhaps the best answer is to say that such a sentence is true iff what it says is true of both planets between which "Venus" is indeterminate. Let's accept this proposal; that is, we take "in 10 million years, Phi(Venus)" to be true iff in 10 million years, both planets between which "Venus" is indeterminate satisfy Phi. Likewise for "Hesperus" and "Phosphorus". It follows that "in 10 million years, Hesperus = Phosphorus" is not true, even though "in 10 million years, Hesperus = Hesperus" is true. Hence "in 10 million years, ... = ..." is not extensional. And by exactly parallel reasoning, since Venus could just as well have already fissioned in the past, it is not true that at all worlds, Hesperus = Phosphorus, even though at all worlds, Hesperus = Hesperus (always add: "at all worlds where Hesperus exists").

Now I don't want to insist that this is the best way to handle fission cases. But anyone who claims that " $$m[1]$ " is extensional will have to come up with a better alternative.

Phew, on to the second point, that it is doubtful whether Kripke endorsed the bad argument.

The relevant place is Kripke's paper "Identity and Necessity", which begins with the following argument:

1) $$m[1]$
2) $$m[1]$ (instance of Leibniz' Law)
3) $$m[1]$ (from (1), (2))

(Kripke forgot the pair of brackets in (2), but that's clearly what he meant.)

Kripke says that this argument "has been stated many times in recent philosophy" (p.136), and especially mentions David Wiggins. He then goes on to defend the conclusion and to argue at length that even instances of (3),

4) $$m[1]$ ,

with a and b proper names, are true. Interestingly, his argument for that is not that (4) follows from (3), which he has already defended, by universal instantiation. I'm not quite sure why. Maybe he (rightly!) thought that universal instantiation is very problematic in modal logic. (At every world, $$m[1]$ , but not at every world, $$m[1]$ , unless a necessarily exists. So either what logically follows from a necessary truth is sometimes not itself a necessary truth, or universal instantiation fails.) Or maybe he (again, rightly) thought that since the argument for (3) presupposes the extensionality of " $$m[1]$ ", he'd need an independent argument to show that substitution of co-refering names cannot change the truth value in such a context.

Kripke's argument for (4) is on p.154:

First, recall the remark that I made that proper names seem to be rigid designators, as when we use the name 'Nixon' to talk about a certain man, even in counterfactual situations. [...] If names are rigid designators, then there can be no question about identities being necessary, because 'a' and 'b' will be rigid designators of a certain man or thing x. Then even in every possible world, 'a' and 'b' will both refer to this same object x [...].

More formally, Kripke's argument why identity statements between proper names are necessary goes like this:

1) Proper names are rigid designators;
2) rigid designators denote the same thing at every possible world;
3) if 'a' and 'b' denote the same thing at every possible world, then $$m[1]$ ;
4) for any proper names 'a', 'b', if a = b, then $$m[1]$ (from (1), (2), (3)).

Leibniz' Law doesn't figure in this argument at all. Unlike the original argument, this one is also not entirely suspicious and question-begging: Kripke has made a strong case that names are indeed in some sense rigid, and (2) and (3) are the most straightforward way to analyze rigidity. I prefer the counterpart analysis on which either (2) or (3) or both come out false (depending how they are interpreted). But at least this is something that can be properly called an argument.

Comments

# Aidan McGlynn on 09 August 2006, 16:26

I had a look back at Mark Sainsbury's lecture notes on this topic, and I think you're in agreement on much of this. But just a quick point, I take it you also want a = b in the antecedent of 3?

# wo on 09 August 2006, 19:55

Ah, that's reassuring (being in agreement with Mark Sainsbury). Which 3 do you mean, the one towards the end? Doesn't a=b follow from the fact that 'a' and 'b' corefer at every world? I've noticed that my first comment on the earlier (2) and (3) isn't quite right: what has to be made conditional on (Ex)(x=a) is of course the identity claim in (2) and (3), not (2) and (3) itself. I've corrected this now, but I suppose that's a different point?

# Aidan McGlynn on 10 August 2006, 00:07

Sorry, I got caught out by an ambiguity. 'a' and 'b' denote the same thing at every possible world could mean they corefer at every world. But it could also mean that 'a' denotes the same thing at every world and 'b' denotes the same thing at every world. I'm not sure why I got the unhelpful reading first.

# Tanas on 10 August 2006, 23:11

Not connected to the content of the post, just a notice that the post is dated Thursday, 09 November 2006. Please remove this comment after fixing :)

# wo on 10 August 2006, 23:23

Oops, thanks. I posted that from a computer at Humboldt University that was apparently three months ahead. If you don't mind I'm leaving your comment in case people wonder about the change. (The wrongly dated permalink should still work though.)

# Ross Cameron on 25 August 2006, 15:26

The last argument in the post doesn't seem good to me at all. (I posted about this on metaphysical values before I saw this: http://metaphysicalvalues.blogspot.com/)

As Aidan says, there are two ways to understand 'rigid designators denote the same thing at every possible world'. The way to understand it so that (2) is true and does not beg questions is: if 'a' and 'b' are rigid designators then in every world 'a' refers to what 'a' actually refers to and 'b' refers to what 'b' actually refers to. But understood thus we have no right to (3). If there are contingent identities then 'a' can refer to what it actually refers to and 'b' refer to what *it* actually refers to and 'a=b' be false at a world. 'a' and 'b' will not co-refer at worlds at which the actual referent of 'a' is distinct from the actual referent of 'b'.

The necessity of identity follows if we have (i) in every world 'a' and 'b' refer to what they actually refer to *and* (ii) in every world 'a' and 'b' co-refer. But we have no right to the latter claim: it is not established by considerations of rigidification. It follows from the (i) and the necessity of identity; but of course one can't rely on that without begging the question.

In short, then, if we read 'denote the same thing at every possible world' in (2) in the sense in which we are entitled to, then we have no right to hold (3). To hold (3) with 'denote the same thing at every possible world' read in that sense is to beg the question.

# Robbie Williams on 30 November 2006, 11:51

Hi wo,

Interesting, the point about semantic indeterminacy inducing non-extensionality in modal contexts. I take the general point, but I'm wondering about the issue as it arises within the argument for necessary identity.

"by exactly parallel reasoning, since Venus could just as well have already fissioned in the past, it is not true that at all worlds, Hesperus = Phosphorus, even though at all worlds, Hesperus = Hesperus (always add: "at all worlds where Hesperus exists")."

I'm not sure what position you're adopting here. Suppose that, in fact, Determinately H=P in the actual world. Now, is the worry that Venus could have fissioned? Well, sure it could. But it's not clear why this makes "H=P" indeterminate with respect to such a world. I suppose the thought is that *both* "H" and "P" are referentially indeterminate at that world, and further, there's nothing to suppose there's a penumbral connection between them. So they're not determinately co-referential at that world. So "H=P" is indeterminate at that world.

(Another option: maybe the thought is that there are two intensions that are both equally good candidates to be the intension of "H", say: one of which picks out the left-fission Venus at the fission world, the other the right-fission Venus. And it's indeterminate which of these intensions "H" and "P" has; and as before, there's no reason to suppose the assignments are penumbrally connected.)

But here's one alternative: at the fission world "H" and "P" both pick out the fusion of left-fission Venus and right-fission Venus: the whole caboodle. That's an object that is scattered at some times (it's not a planet at some times, but rather a fusion of two planets). If this is a decent candidate for being the referent of "H" and "P" at some non-actual world, then *in this case at least* we won't get a counterexample to necessary identity.

# wo on 02 December 2006, 09:09

Hi Robbie, yes, my thought was that in fission cases, both "H" and "P" are referentially indeterminate, without penumbral connections.

The fusion account would be a way to circumvent the non-extensionality, as would be an account on which the fission determinately destroys what we call "V", "H" and "P". But do our terms really work that way? I suspect most people don't have clear intuitions on how to apply the names when talking about counterfactual or future fission cases. And it doesn't look like objective joints in nature could come to our rescue here: all candidates are roughly equally natural. So I think the names are really indeterminate between all the possibilities (including denoting nothing and denoting the fusion).

I'm not sure any more why I thought there would be no penumbral connections, though. (Maybe if one of the two successor planets came to play the role of the morning star and the other the role of the evening star, we're more inclined to calling the one "H" and the other "P"?)

# Robbie on 04 December 2006, 11:21

Hi Wo,

Right: the "fission destruction" tactic would also work. And again, thoughts that might rule that out as a response to fission in the actual world case don't easily go over to merely possible cases.

I have some sympathy with the response that these cases just add more candidates for our names to be indeterminate between. A few thoughts however:

(1) Consider the "vagueness" response to the problem of the many: it's vague which of the various agglomerations of material stuff orbiting the sun "H" and "P" refer to. Nevertheless, I guess we want "H=P" to come out determinately true. If so, then we need penumbral connections between "H" and "P", to make sure that they determinately co-refer (that's something that favours the supervaluationist response to POM, since it at least makes room for this). So (a) if you go this way, then the two names are already penumbrally connected, in the actual world (somehow!). (b) spelling out how this penumbral connection gets established may be tricky.

The minimal line is that it's just a matter of matching usage with "H=P". (Compare (2), below). But what about times before the identity was discovered? No usage then to match. Unless future usage is in the supervenience base for meaning, then it looks like the needed penumbral connection wouldn't be established until *after* the identity was discovered. Which is pretty nasty.

There are more substantial ways of accounting for this: e.g. Weatherson's "many many problems" way of appealing to eligibility. That sort of thing should automatically generalize to the transworld case.

(2) Suppose that adding penumbral connections, or making "H", "P" etc designate the fusion, is what it takes to allow us to export them in modal contexts (i.e. to move from Nec (FH) to (Ex)(x=H&FH).) Then we could look at intuitions/arguments in favour of exportation, and appeal to *those* as what makes it the case that "H" has penumbrally constrained intensions or whatever. (But see below for problems with the assumption on which this rests).

(3) Even if we can get "H" and "P" determinately co-referring, we won't get exportation moves to be valid unless something similar holds for variables. Take a variable assignment that assigns Venus to "x". What does that variable denote at the fission world? Looks hard to say anything without either begging the question against those skeptical of determinate transworld identity facts; or developing a pretty non-standard treatment of variables (e.g. allowing them to be penumbrally connected to names).

# wo on 06 December 2006, 09:22

very interesting. I have to think more about the variable cases. Just a quick thought on your first point.

One way to get the dssired penumbral connections between "H", "P" and maybe even "x" would be a restriction on resolutions of indeterminacy to the effect that whenever A and B are indeterminate between exactly the same things, then they corefer on every resolution. Like this: let < be a well-ordering of all things; take as determinate interpretations the functions f_0,...,f_n,... that assign to each term the candidate referent which is nth w.r.t <. (There should be a way of putting this more simply: the particular well-ordering will never matter.) Alternatively, one could have some restriction to the effect that if f_n assigns candidate referent x to A, and x is also a candidate referent for B, then f_n must also assign x to B. But that (the alternative) would appear to exclude all cases of indeterminate identity.

# Robbie on 06 December 2006, 17:01

What about a situation where N is indeterminate between 2 things, and M is indeterminate between 3 (different) things? Looks in danger of leaving out the precisification where M refers to the third thing wrt < among its candidates, and N refers to the first thing wrt < among its candidates (an unwanted penumbral connection).

It's interesting to me, because ruling out co-indeterminate names that don't determinately corefer, excludes at least one natural response to Brandom's about the indiscernable complex numbers "i" and "j" (the idea is that each of these names is indeterminate between the conjugate roots of -1, but they determinately fail to corefer). Similarly, extreme inscrutability as generated by permutation argument has all names indeterminate between every object: but of course they don't determinately corefer.

# wo on 07 December 2006, 04:09

right, the ordering will create too many penumbral connections. Also (as you say), sometimes terms can be determinately not co-referential even though they are indeterminate between exactly the same candidates. On many structuralist accounts, all number terms are indeterminate between the same sets, but still it is determinate that 0 != 1 (due to the penumbral connections imposed by the structuralist conditions: any eligible assignment must respect the Peano axioms). Likewise arguably for "very flat" and "somewhat flat": they both can be resolved to any degree of flatness, but whenever something comes out as "very flat" on a resolution, it cannot come out as "slightly flat" (and vice versa).

I just notice that I don't have any clear grasp of how penumbral connections work. Are pragmatic considerations involved here: if somebody claims that A != B (or A = B) and everything else they say leaves A and B indeterminate between the same things, interpret them as meaning different (or the same) things, because otherwise what they claim would be false?

# Robbie on 12 December 2006, 17:02

It's a bit up in the air what it takes to create a penumbral connection. I guess this is coordinated with it being up in the air what a "precisification" and even what "supervaluationism" itself is.

I think people who look to Fine as the paradigm of supervaluationism often think you get penumbral connections only where you have "conceptual truths": (there are multiple readings of that: on one, every precisfication must satisfy all the conceptual truths. On another, you only get those penumbral connections that are required to make supertrue the conceptual truths. Cf. Fodor and Lepore "What cannot be valuated..." and ensuing discussion.)

Incidentally, people in the Fine tradition typically will look askance at using "supervaluationism" in connection with putative indeterminacy of proper names, since the Fine story about what precisifications is something like: ways of filling in the gap between the extension and anti-extension of a predicate that respects the conceptual truths. Not obvious how to extend that to the name-case, where you don't have extensions and anti-extensions (disclaimer: the Fine article is long and subtle and he may have some story somewhere that deals with this). Would be interesting to think about applying a Fine-style story to the case of names.

The most liberal version of "supervaluationism" in the literature I know of is one where you get penumbral connections whenever they're required to make sense of the conventions of truthfulness and trust that prevail. That's the general background I was assuming earlier in thinking that inferential practice might generate penumbral connections, if it was suitably entrenched.

There's also other ways: e.g. Weatherson's proposal that you get precisifications corresponding to the meanings determined by each possible linearizations of the "more eligible than" relation, given a Lewisian story about meaning-fixing.

By the way, although it *seems* possible to get cases where you want to say that we have terms indeterminate between the same range of candidates without them being determinately coreferential, I think there's reason to think that the case is philosophically problematic, which would then constitute an *objection* to the forms of structuralism that predicate it (for example).

Wolfgang Schwarz

Kripke's (Alleged) Argument for the Necessity of Identity Statements

Comments

Add a comment