Logic, Language, and Existence

I have been thinking a lot about the argument of an earlier post (I Necessarily Exist), due to some excellent comments on the post and because I have been having some discussion via email with Kent Bach about it, and I think I understand what the argument is supposed to look like now. So what I want to do is take some time to show how this argument for frigidity goes and how it ultimately supports what I say about What Kripke Really Thinks.

The argument, to remind you, is one that David Rosenthal presented in a Quine class I had with him and and is a proof by reductio that the existence of any object that one desires is a theorem of first-order logic. All that one has to do to get the proof going is to agree that to say that it exists is to say something with the logical form Ex (x=c) where ‘c’ is a singular terms that refers to the object in question. Here is a version that prooves that Saul Kripke’s existence is a theorem. Let ‘SK’ name the actual Saul Kripke.

1. –(Ex) (x=SK)                     assumption for reductio
2. (x) –(x=SK)                       equivalent to 1.
3. (x) (x=x)                            axiom of identity
4. (SK=SK)                             UI of 3.
5. –(SK=SK)                           UI of 2.
6. (SK=SK) & -(SK=SK)       4, 5

This argument is valid and is supposed to illustrate the problems that Quine discussed in his famous article ‘On What There Is’ involving existence statements. Some people have objected that since the first premises assumes that SK does not exist then he is not in the domain of the quantifier and so something fishy is going on in step 5 (and possibly step 4. as well). But this is not right because the argument is supposed to illustrate that something funny happens when you try to say that something doesn’t exist and you use a logic with singular terms. So, SK must refer (in first-order logic) and it does refer. We then show that since it refers it is a theorem of first-order logic that SK exists. So the ultimate aim of Rosenthal’s argument is to show that if we have singular terms in our logic, as opposed to just variables, then it turns out that it is a theorem of first-order logic that Saul Kripke exists, or that you do, or that I do, or that unicorns do…something has gone wrong and the natural candidate is the use of the singular term.

Quine’s solution to this problem is to suggest that we use Russell’s theory of descriptions so that when we analyze sentences like ‘Saul Kripke Exists’ we get a logical statement free of singular terms. He, of course, recommended that we invent a description like ‘the thing that Kripkisizes’, or ‘the Kripkisizer’ so that we would render ‘Kripke exists’ as Ex (Kx) where ‘K’ stands for the invented description. This is kind of weird and off-putting but the argument is good and so we should see if there is some more natural way to treat (linguistic) names as descriptions.

The Bachian strategy that I endorse is to use the description that mentions the name. So according to this view the linguistic name ‘Saul Kripke’ is semantically equivelent to “The bearer of ‘Saul Kripke'”. So we render ‘Kripke exists’ into first-order logic as Ex (Kx), where ‘K’ stands for the description that mentions the name (Bach calls this a nominal description). So, this part of the argument shows that we should rid first-order logic of singular terms and if one takes first-order logic to be in the business of giving a formal semantics then we should rid our semantic theory of singular terms, and this is just what frigidity does.

Now in the earlier post I suggested that we could adapt Rosenthal’s proof to a modal proof that Kripke (or you, or me, or unicorns) necessarily exists which to remind you again went as follows.

(2) Saul Kripke necessarily exists: □Ex (x=SK))
1. ◊ –Ex (x=SK)           assumption for reductio
2. ◊ (x) –(x=SK)          equivalent to 1.
3. (x)□ (x=x)                modal axiom of identity
4. □ (SK=SK)                UI of 3.
5. ◊ -(SK=SK)               UI of 2.
6. –□ (SK=SK)              equivalent to 5.
7. □ (SK=SK) & -□ (SK=SK)           4,6

Now, in the course of doing some research about this I made an interesting discovery.

It turns out that the problem of necessary existence has some history in modal logic. In fact it turns out that Kripke is famous for formulating a system of quantified modal logic that is supposed to block proofs of necessary existence (as well as some other pesky things like the Barcan formula). So how does Kripke do this? Well, in his 1963 paper “Semantical Considerations on Modal Logic” he modifies standard quanitified modal logic in two ways. The first is by requiring that there be no free variables in any of the axioms or theroems that we use.

The Stanford Encyclopedia entry on actualism has a nice Proof of necessary existence in S5 if one wants to look at it and the same article has some discussion of how Kripke’s move blocks the inference, but as is usually the case with papers in html the quantifiers do not show up and so it is hard to follow the discussion (in the article that is, the proof above is an image and so one can see the quantifiers)…so I will reproduce the proof with the ‘typwritter notation’ that I have been using here.

So the claim of necessary existence is taken to be the claim that everything that exists necessarily exists or, (x)□Ey (y=x) the proof of this proceeds as follows

1. x=x axiom of identity
2. (y) -(y=x) –> -(x=x) instance of quantifier axiom
3. (x=x) –> -(y) -(y=x) from 2 by contraposition
4. (x=x) –> Ey (y=x) from 3 quantifier exchange
5. Ey (y=x) from 1 &4 by modus Ponens
6. □Ey (y=x) from 5 by rule of necessitation
7. (x)□Ey (y=x) from 6 by rule of universal generalization

Ok, so now notice that the axioms 1 and 2 above have free variables which have to be bound in Kripke’s system. So we get 1′. (x) (x=x) and 2′. (x) ((y) -(y=x) –> -(x=x)) and so we cannot derive the problematic theorem. Instead we get the following.

1′. (x) (x=x)
2′. (x) ((y) -(y=x) –> -(x=x)
3′. (x) ((x=x) –> Ey (y=x) From 2′ by contraposition and quantifier exchange
4′. (x) (x=x) –> (x)Ey (y=x) From 3′ by quantifier distribution rule
5′. (x)Ey (y=x) From 1′ & 4′ by modus ponens
6′. □(x)Ey (y=x) From 5′ by rule of necessitation

But 6′ is harmless as it just says that necessarily, everything that exists is self identical. In order to get the pesky result that everything that exists necessarily exist we need a theorem that says □(x)Ey (y=x) –> (x)□Ey (y=x) (which is the so-called converse Barcan formula). If we had this we could derive the offending theorem from 6′ and the converse Barcan formula by modus ponens. “But,” the article continues,

as Kripke points out, the usual…proof of [the converse Barcan formula] also depends essentially on an application of Necessitation to an open formula derived by universal instantiation — the same “flaw” that infects the proof of [necessary existence]. (See the inference from line 1 to line 2 in the supplementary document Proof of the Converse Barcan Formula in S5.) Hence, it, too, fails under the generality interpretation of free variables.

But notice that the modal proof that I gave does not fail under the generality constraint.  The axiom of identity that I appeal to contains no free variables.

So what is going on here? Well, as the article continues by pointing out that we can still prove the offending theorems simply by replacing the free variables in the original proof by constants (this is in effect what the proof I offered did), and so,

The second element of Kripke’s solution, therefore, is to banish constants from the language of quantified modal logic; that is, to specify the language of quantified modal logic in such a way that variables are the only terms.

In other words Kripke thinks that we should eliminate singular terms from our quantified modal logic and so by extension from our semantical theory; in other words it looks like this is further support for my claim that Kripke really has something like frigidity in mind rather than rigidity.

Now there is more that needs to be said here, but this post is already way too long so I will save it for another time…

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