[FOM] Diagonalization Does Not Deliver Self-Reference

[Richard G Heck]

A recent message circulated by Harvey Friedman regarding 
self-referential systems of propositional logic reminded me of the 
following two examples. They arose naturally in the context of some 
informal investigations of deflationism.  But I realized later that they 
can be used to show that standard methods of treating self-reference 
don't actually give you self-reference. The point, I think, will be 
familiar to most readers of FOM, but I'm not sure it has really been 
appreciated as it might be. First the examples, then some commentary.

Consider:
(1)	The right-hand side of (1) is true iff 2+2=4.
The technique used here can also be used this way:
(2) 	The rhs of (2) is true iff the lhs of (2) is false.
That delivers the postcard paradox in one sentence.

Now, presumably, there are various ways to formalize these kinds of 
examples. But the straightforward use of diagonalization is not the way. 
Let rhs(x,y) formaliaze: y is the rhs of x. Then we might try to 
formalize (1) by diagonalizing on:
	(Ey)(rhs(x,y) & [T(y) <--> 2+=4])
That will deliver a sentence A such that Q proves:
(3)	A <--> (Ey)(rhs(*A*,y) & [T(y) <--> 2+=4])
However, as is obvious from quick consideration of the usual proofs of 
the diagonal lemma, A is not a biconditional. It therefore has no rhs, 
and since such syntactic facts are provable, A itself is refutable. The 
case is, obviously, similar with (2).

The reason we fail to formalize (1) and (2) using diagonalization is 
that diagonalization does not produce a self-referential sentence. The 
sentence A just mentioned does not actually "say of itself" that it has 
a rhs that is true iff 2+2=4. Rather, it is provably equivalent to a 
sentence that says of A that it has rhs, etc. In many contexts, of 
course, this difference does not matter. We naturally speak of the 
Goedel sentence G for PA, say, "saying of itself" that it is not 
provable. But, in fact, the sentence says, in the strict sense, no such 
thing. It too is simply provably equivalent to a sentence that says of G 
that it is not provable.

Now, presumably there is some way to formalize (2), but I do not myself 
know how to do it. I'd appreciate hearing from anyone who does.

Richard Heck
Professor of Philosophy
Harvard University