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Sunday, December 30, 2007

On an Attempted Refutation of Leibniz's Law

At Inconsistent Thoughts, Colin Caret has a post which links to a post of Brian Rabern's over at armchair investigations which challenges Leibniz's Law. In case you're not familiar with it, Leibniz's Law states that if x is identical to y then x and y have all their properties in common. Brian's argument is intriguing, but as he suggests there is a premise in the argument which can plausibly be denied. I'm not sure if this is what he had in mind, but I posted my own diagnosis of what goes wrong with the argument in a comment on Colin's post, which I will reproduce here:

Hi Colin,

I think that attempted refutation you linked to is confusing (at least, it confuses me) because it expresses the property G (i.e., "if x had quotes around its last word, then x would have been true") counterfactually and then asks us to evaluate the truth of the statements A and B respectively in those counterfactual circumstances, without specifying clearly whether the letters refer to the sentences as they actually are or as they are in the counterfactual circumstances we are considering. The cruicial part of the proof is the following:

"A is not such that if it had quotes around its last word it would have been true (since if A had quotes around its last word its last word would have been “obscene” not ‘obscene’). Hence, ~G(B). But B is such that if it had quotes around its last word it would have been true (since if B had quotes around its last word it would have rightly said of A that its last word is ‘obscene’)."

I agree with the conclusion that A does not exemplify G. For if A had had quotes around its last word, it would have looked like this: 'The last word of A is 'obscene'.' and in those circumstances its last word is indeed ''obscene'' and not 'obscene'. But note that this is because when we evaluate the truth of A in those circumstances, we are taking A to refer to itself as it is in those counterfactual circumstances, not how it actually is. For as A actually is its last word *is* 'obscene', and if we evaluated A in the counterfactual scenario as referring to itself as it actually is (i.e., as 'The last word of A is obscene') then it would have been true, since it would have referred to itself as it is in our possible world and not the possible world in which its truth is evaluated. However, I think that B also does not exemplify G. For B, to recall, says 'The last word of A is obscene.' Now if B exemplifies G, it would have been true if it had quotes around its last word, in which case it would have looked like this: 'The last word of A is 'obscene'.' But if it had said that it would not have been true, for in those counterfactual circumstances A would, as above, have looked like this: 'The last word of A is 'obscene'.', and so in those circumstances A's last word would have been ''obscene'' and not 'obscene'.

Does this solution make sense to you, or am I still confused?


So what do you think? Is my diagnosis correct? Or is there some other way the argument goes wrong?

3 comments:

Richard said...

Yeah, that was my first thought: if A and B name the same sentence, then changing B would also change A.

However, it may be that A and B are meant to denote different sentence tokens. We could add quotes to the second sentence-token, i.e. B, without changing the first (i.e. A) in any way. But then A and B are not numerically identical in the first place, so the argument wouldn't go through for that reason.

Brian Rabern said...

I agree with Richard but let me break it down again just to be sure.

Assume (to reach a contradiction) that LL is true. Let 'A' designate the sentence 'The last word of A is obscene' and let 'B' designate the sentence 'The last word of A is obscene'.

Now we could take this to mean (at least) two different things.

(1) Let 'A' and 'B' both designate the sentence type 'The last word of A is obscene'.

(1*) Let 'A' designate the following sentence token: 'The last word of A is obscene'. And let 'B' designate the following (distinct) sentence token: 'The last word of A is obscene'.

Clearly, A = B only follows from (1). Now consider the property G (i.e. the the property of being an x such that if x had quotes around its last word, then x would have been true). Going with interpretation (1) it follows that G(A) ↔ G(B) by LL.

A is not such that if it had quotes around its last word it would have been true (since if A had quotes around its last word its last word would have been "obscene" not 'obscene'). Hence, ~G(B). But B is such that if it had quotes around its last word it would have been true (since if B had quotes around its last word it would have rightly said of A that its last word is 'obscene'). Thus, G(B).

This seems right but if we are going with interpretation (1), this must be read as saying that sentence type A is not G but sentence type B is. I am not sure if sentence types have properties like G. What sense does it make to attribute such properties to sentence types? (Kaplan's got a lot of stuff to say here about the metaphysics of sentences.) But, regardless of that tricky issue, if A has the property then so does B. A just is B! When we are pulled to think that only one of them has the property we are considering the sentence token (or the thing written on the screen, etc.). And, as Richard said, if you go with this interpretation (i.e. (1*)), then A and B are distinct, so we can't derive a contradiction.

So, yeah, thats why it doesn't work. If you are talking about sentence types, then there is no reason to think A lacks a property that B has. If you are talking about sentence tokens then A doesn't equal B.

This is all based off of "Problem 4" in the final section of The Journal of Philosophy, Vol. 68, No. 3. (Feb. 11, 1971), pp. 82-86. I think you can find it on JSTOR. The problem is attributed to Richard Cartwright, who originally posed the problem to his class at MIT. David Kaplan proposed several solutions in "Bob and Carol and Ted and Alice" (1973), in Approaches to Natural Language, J.Hintikka et al., eds.

Quirinius_Quine said...

Hi guys,

Sorry for the delay; I was waiting to reply until I got more responses, but it looks like that's not going to happen. I don't have much to add except thanks for your replies. Brian, I'll check out those references if I can track them down.