[FOM] consistency and completeness in natural language

[Torkel Franzen]

  Neil says:

  >With reference to an independent
  >G"odel-sentence of the form "for all x A(x)", Dummett wrote

  >	each of the statements A(0), A(1), A(2), ... is true in every
  >	model of the formal system.

  >I read this as requiring extra detail to be filled in as follows:

  >	each of the statements A(0), A(1), A(2), ... [is a theorem of and
  >	hence] is true in every model of the formal system.

  Sure. But what is the justification for the claim that each of A(0),
A(1), A(2), ... is a theorem of S? The straightforward justification
is that they are all true, and being p.r. statements are therefore
provable in S.  Your suggestion that Dummett's argument goes "every
A(_n) is provable in S, and is therefore true" presupposes some other
argument for the provability of every A(_n) in S. What is this other
argument, and where in the literature is it to be found?

  The only other argument that occurs to me would be one in which we
note that if every A(_n) is in fact true, then they're all provable
in S, and if some A(_n) is false, S is inconsistent, and they're
all provable in S. In this version we don't need to assume S consistent.
But to cap this argument with "and since every A(_n) is provable in S,
every A(_n) is true" is to make it into a jocular version of the
straightforward justification.

  As for the question of substantialism vs deflationism, I don't think
there is anything to be gained from inquiring into the "thickness" of
the use of "true" in the step from

	each instance A(0), A(1), A(2), ... is true

to

	"for all x A(x)" is true.

  After all, we know that "true" can here (in the context of an
argument for the truth of a Gödel sentence for S) be taken to be an
arithmetically defined predicate on which the above step is provably
correct in S. It doesn't seem to me at all promising to inquire into
whether any "substantial notion of truth" is required in putting forth
or accepting such arithmetically formalizable arguments.

  In a more positive vein, I suggest that the place to look for a
possibly essential use of a substantial notion of truth is in the
justification for those assumptions or premisses in the argument for
the truth of G that are not provable in S itself, such as "S is
consistent" or your reflection principle. In this connection you comment
in your paper:


    One can agree with Shapiro (loc. cit., p. 499) that the 'deflationist
  cannot say that all of the theorems of [S] are true.' But the deflationist
  can instead express (in S*) his willingness, via the soundness principle,
  to assert any theorem of S. The anti-deflationist desires to go one step
  further and embroider upon this same willingness by explicitly using a
  truth predicate.
    One is left wondering, however, whether what the deflationist is
  willing to do in this regard really falls short, in any epistemically
  unsatisfactory way, of what may reasonably be demanded of him. Why
  does one need to *say* at the metalevel what be *shown* instead by
  adopting the inferential norm expressed by the soundness principle at
  the metalevel. 


  Your line of thought here is a natural one, but to justify a
reflection principle, a reference to a willingness to assert any
theorem of S is not sufficient. The statement "if it is provable in S
that there are infinitely many twin primes, then there are infinitely
many twin primes" may seem acceptable merely on the basis that we
expect to be convinced by any actual proof formalizable in S that
there are infinitely many twin primes, but assent to "if it is
provable in S that 0=1, then 0=1" cannot be based on just on a
readiness to accept future theorems proved in S, but requires us to
assert that S is consistent. There is a discussion of these things in
my forthcoming volume in the LNL series.

----
Torkel Franzen