# [FOM] Consistency v. reflection, and the truth of the Godel-sentence

Torkel Franzen torkel at sm.luth.se
Fri May 30 01:04:28 EDT 2003

Neil says:

>If one uses Torkel's explanations of A(x), B(x) and t, and substitutes, it
>might appear that the universally quantified biconditional is logically
>true, and not just a non-logical theorem of PA---until one realizes that
>what PA would be providing is the proof that the value of t is (x)A(x)
>itself. But since, in the overall discussion, PA is one of the values that
>can be taken by the system-variable S, one has to ask whether Torkel's
>challenge would have any point under the assumption that PA is
>inconsistent.

My "challenge" consisted in the simple observation that

(x)( B(x) <-> x is a proof in S of G)

is provable in (a weak subsystem of) PA. Hence, when you claim that
B(_n) could be false even if n is a proof in S of G, you are rejecting
this theorem.  You now add that on the assumption that PA is
inconsistent, the point of [something] becomes moot. If we assume that
PA is inconsistent, we must indeed question a great many arguments.

>Agreed. But the truth-predicate-eschewing proof of G from the assumption
>"S is consistent" is very, very long and would not serve as a plausible
>regimentation of what is called the "semantical argument" for [the truth
>of] G.

Questions of length aside, no proof of G from the assumption "S is
consistent" can amount to an argument for the truth of G, since it
only shows "if S is consistent then G". Such proofs apply equally to
theories S for which we have no idea whether or not G is true.

>I am suggesting that the
>semantical argument provides (for those who go in for it) an alternative
>way, and that it is best regimented by use of the uniform reflection
>principle for p.r. sentences, this principle being adopted in an extension
>of S.

Again, this only amounts to a proof of "if A then G", where A is some
conjunction of instances of your reflection principle. It does not yield
any semantical argument, in your sense of an argument "designed to help
one understand why asserting G would be the right thing to do".

>In my Mind paper, I was urging (b) as a route that could be taken by a
>deflationist who wished to capture the essential structure of the
>so-called semantical argument for [the truth of] G.

A proof of "if [a conjunction of instances of your reflection
principle] then G" by itself yields no grounds whatever for accepting
G. So the question then, from a deflationist point of view, is how the
reflection principle can be justified without invoking the notion of
truth.

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Torkel Franzen