Friday, January 18, 2008

Dialetheism and Hume's Principle

Suppose the dialetheic treatment of the logical and / or semantic paradoxes is correct; in particular, that sentences of the liar family express propositions which are both true and false. Then consider the following list, which I will call List A:

(1) All whales are mammals.

(2) Nothing can escape a black hole.

(3) (3) is not true.

How many statements on List A are true? Well, both (1) and (2) are true, and under our assumption of dialetheism so is (3). Since these statements are all distinct from each other, it follows that there are three true statements on the list. But (3) is also false, so what it says is not true; hence it is also true that there are only two true statements on the list. So the number of true statements on List A is both two and three. By a parity of reasoning, we can conclude that the number of false statements on the list is both zero and one.

What if we bring Hume’s Principle to bear on this case? According to Hume’s Principle, the number of F’s equals the number of G’s if and only if there is a one-to-one correspondence between them. Suppose, now, that Jones has three books on his coffee table, which we will call Book A, Book B and Book C. Is the number of books on Jones’ coffee table the same as the number of true statements on List A? Invoking Hume’s principle tells us that the number of books is the same as the number of true statements if and only if each book can be paired off with exactly one true statement and vice versa. Can this be done? On the assumption of dialetheism, the answer is “Yes and no”: Book A can be paired off with (1) and Book B with (2), but can Book C be paired with (3)? Insofar as (3) is merely a statement Book C can indeed be paired with it. But in being paired with (3), is Book C paired with a true statement? Since (3) is both true and false, it follows that Book C both is and is not paired with a true statement. Because of this, while it is true that Book A, Book B and Book C are each paired with a different true statement in List A, it is also false that they are thus paired. Hence, if Hume’s Principle holds, we get the result that the number of books on Jones’ table both does and does not equal the number of true statements on List A. What follows from this? If the number of books on the table is three, and the number of books on the table both does and does not equal the number of true statements on List A, shouldn’t it follow that three does not equal three? After all, if there are three books on the table and the number of true statements on List A equals this, there must be three true statements on List A. If the number of true statements on List A is also not equal to three, how can it fail to hold that three, which does in fact number the true statements, is not equal to three? Now, as equality does not depend upon context, if three is not equal to three in this case, it is not equal to three in any case, and I take it that this would be a Very Bad Thing. However, a dialetheist need not embrace this result. Even though, on Hume’s Principle, the number of books on the table is both equal and not equal to three, it does not follow that “the number of books on the table” picks out some one thing that is not equal to itself. Instead, the dialetheist should hold that the phrases “the number of books on the table” and “the number of true statements on List A” are really descriptions which falsely presuppose uniqueness because they contain the definite article ‘the’. In truth, what’s going on here is that there is more than one number which exhaustively numbers the true statements on List A, and this in no way entails that there is some one number which is unequal to itself. All that follows is that the numbers involved are not equal to each other. Yet it remains true that there both is and is not a one-to-one mapping from the books on Jones’ table to the true statements on List A. From this, I think dialetheists should conclude that the number of F’s can be different from the number of G’s even if there is a one-to-one mapping between them, and can be the same even if there is not. They should hold that Hume’s Principle, in dialetheic contexts, is a biconditional which fails in both directions, and as such cannot be used to provide a criterion of identity for numbers.