Topics Logic

Question Quine's Paradox (“yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation) doesn't seem to me to be a paradox. Maybe I'm wrong, but it seems to me like it's asserting nothing but the fact that it's false. For something to be true OR false, there must be some other claim made. When I look at the statement, it seems to me that it's not talking about anything but itself -- like an indirect self-reference. It seems to me to have no content but its own claim that it yields to falsehood, and would therefore neither be true nor false. Have I made a mistake in my reasoning/logic?

Accepted:
December 16, 2006

Comments

No, I don't see any mistake---other than that you dismiss the problem simply on the ground that there is self-reference. Self-reference isn't always a problem. In fact, some times it's essential. Consider this phrase:

(*) yields a sentence when preceded by its own quotation

This is a perfectly sensible verb phrase, and some phrases yield a sentence when preceded by their own quotations---e.g., "is a sentence" does so---whereas some others---e.g., "Bill is"---do not. As it happens, (*) too does so. That is,

(**) "yields a sentence when preceded by its own quotation" yields a sentence when preceded by its own quotation

There's self-reference there, too, but what (**) says is true, and I'm not sure how else you'd propose to report that truth. Or should we just not report it? or try very hard not to think about it? or what? Now, among the phrases that do yield a sentence when preceded by their own quotation, some yield true sentences---e.g., "contains three words" does---and some do not---e.g., "is a sentence" does not. So, again, it would seem that "yields a truth when preceded by its own quotation" is just fine, and there are plenty of truths to be stated using it. Presumably, the same is true of the problematic "yields a falsehood when preceded by its own quotation". It's not clear, then, that self-reference, by itself, is so bad.

A more important point, emphasized by Saul Kripke, is that self-reference is, as he puts it, as "incontestably legitimate" as arithmetic. Indeed, all we need is seven very weak assumptions about numbers. Here x and y are "natural" numbers, that is, non-negative integers:

NOT: x + 1 = 0
If x + 1 = y + 1, then x = y
x + 0 = x
x + (y + 1) = (x + y) + 1
x . 0 = 0
x . (y + 1) = (x . y) + x
If x > 0, then, for some y, x = y + 1

Given these seven (completely obvious) assumptions, one can prove Goedel's "diagonal lemma", and the diagonal lemma provides all the resources one needs to construct sentences that are self-referential or, at least, that behave, logically speaking, enough like sentences that are self-referential for present purposes. See any intermediate logic text for an exposition of that result, e.g., Boolos, Burgess, and Jeffrey, Computability and Logic.

Kripke also points out, in his classic paper "Outline of a Theory of Truth", that self-referential paradoxes can arise very easily, and very unintentionally, in ordinary language. Suppose---to take Kripke's example---that Nixon says, "Everything Dean says about Watergate is true", and Dean says, "Most of what Nixon says about Watergate is false". They might well both have said these things, and when they did, they meant to be including everything the other said, including these things. Now suppose that everything Dean had said about Watergate, other than the one thing just mentioned, was true, and that, other than the one thing mentioned, Nixon's Watergate-remarks had split exactly half true, half false. Then it is easy to see that the situation reduces to the so-called Postcard Paradox, which is usually illustrated by a card on one side of which is written:

The sentence on the other side of this card is true.

Of course, the sentence on the other side is:

The sentence on the other side of this card is false.

What Kripke's example points out is that this same situation can, in effect, arise in quite an ordinary case, and that there is nothing intrinsically problematic about the claims being made in these cases. For note: Unless the circumstances were exactly as I described---everything else Dean said was true; the rest Nixon had said split 50-50---then it would be easy to say what truth-values these statements had.