[FOM] Diagonalization Does Not Deliver Self-Reference
Richard G Heck
heck at fas.harvard.edu
Mon Dec 23 12:00:55 EST 2002
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.
(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.
Professor of Philosophy
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