[FOM] consistency and completeness in natural language

[Neil Tennant]

Torkel writes:

> By standard properties of "true", (x)A(x) is true if
> and only if every A(_n) is true, A(_n) is true if and only if B(_n) is
> not true, and B(_n) is true if and only if n is a proof in S of
> (x)A(x). Since (x)A(x) is not provable in S given that S is
> consistent, it follows that (x)A(x) is true if S is consistent. We
> can't get more semantic than this.  Why the peculiar added bells and
> whistles?

Because it is too swift to claim, as you do,

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

What the background result of representability-in-S of recursive functions 
ensures is only that (if S is consistent then)

	B(_n) is *provable-in-S*
	if and only if 
	n is a proof in S of (x)A(x).

All that the "bells and whistles" do is take this into account, and shunt
the talk of truth to the very end of the proof, to the point where the
semantic arguer feels forced to resort to it.

Neil