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

[Torkel Franzen]

  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.

---

Torkel Franzen