## Friday, August 28, 2009

### Liar and Curry

Consider this sentence:

1. No true sentence (i.e., sentence that is true) satisfies F.
Observe that for some F's, this is quite unproblematic, and even true. For instance, no true sentence is written on the surface of Io, no true sentence is self-contradictory, no true sentence is such that it is both original and was uttered in the Honorable Member's speech. Now, if (1) is the one and only sentence that satisfies F, then (1) is a liar sentence. So some instances of (1) are paradoxical, and some are not.

In general, as in the Io case, it seems that:

1. When F does not have semantic vocabulary and no sentence satisfies F, then (1) is unproblematically true.

If we accept (2), we get an interesting result. Suppose that F is the predicate "is identical to (1) and not-p", where "p" is any non-semantic statement. Then, (1) says that no true sentence is identical to (1) and is such that not-p. This is equivalent to the claim that if (1) is true, then p. In other words, in this case, (1) is equivalent to a Curry sentence just like:

1. If (3) is true, then p.
Now suppose that p is in fact true. Then no sentence satisfies F, and by (2), it follows that (1) is unproblematically true. Therefore, it seems that (3) is also unproblematically true. But if (3) is unproblematically true, then we have established something quite interesting: Curry sentences with true non-semantic consequents are true.

No Curry sentence can be false. For if it's false, then it's true, because any material conditional with false antecedent is true. But whenever a Curry sentence is true, its consequent is true as well, by modus ponens (the antecedent is true because the Curry sentence is true!). So a Curry sentence is true when its consequent is true (assuming the consequent lacks semantic vocabulary) and is nonsense when its consequent is false.

Or so this argument shows. My own intuition is that (3) is nonsense even when p is true. But then I need to get out of the above arguments...