[FOM] consistency and completeness in natural language

[Torkel Franzen]

Neil says:

 >Anyone claiming to know that

 >	B(_n) is true if and only if n is a proof in S of (x)A(x)

 >is going to be basing this knowledge-claim on the unprovability-in-S of
 >(x)A(x), which in turn requires the assumption that S is consistent.

  It's still unclear to me how you arrived at this conclusion, but by
all means let us set it aside. Our essential disagreement can be
formulated more directly.

  Suppose we start from the position, as stated on p. 562 of your
paper, that "The semantical argument is designed to help one
*understand why asserting G would be the right thing to do*, rather
than denying G (or: refusing to assert G and refusing to deny G.)".
Now, this does not mean to help us understand why asserting "If S is
consistent then G" would be the right thing to do. After all, "If S is
consistent then G" is provable outright.  Also, it is perfectly
trivial that no "thick" notion of truth is needed to justify "If S is
consistent then G is true". We need merely cite the proof of "If S is
consistent then G" and add the emaciated principle "if G then G is
true".

  Thus the whole point of a semantical argument such as you envisage
must be to help us understand why asserting "S is consistent" would be
the right thing to do (tacitly: in those cases where it is in fact the
right thing to do, rather than asserting "S is inconsistent" or
refusing to either assert or deny that S is consistent). Here it is a
relevant observation that adding suitable axioms for a truth predicate
to the theory S yields a theory in which the consistency of S (and of
course much more) is provable. So one may reasonably suggest that this
shows that the concept of truth is fat and useful, and in
particular leads us to accept G provided we accept the axioms of S as
true. To this you may reasonably respond that truth, however fat and
useful a concept, is not essential to accepting G. In your paper, you
point out that the principle of uniform primitive recursive reflection
allows us to deduce G. Sure it does, just like "S is consistent.  But
the whole question is whether this principle (or, equivalently, "S is
consistent") can be given some justification that does not invoke
truth. Your paper contains some comments about this (I have in mind
the passage about a readiness to accept theorems of S etc), but for
reasons stated earlier I think those comments are inadequate to
provide a justification for the principle.

---
Torkel Franzen