[FOM] consistency and completeness in natural language

[Neil Tennant]

On Fri, 4 Apr 2003, Torkel Franzen wrote:

> exactly the same proof that shows, in (an extension by definitions of) PA, 
> 
>    (x)(B(x) <-> x is a proof in S of (x)A(x))
> 
> shows, given standard properties of "true", that for every n, B(_n) is
> true if and only if n is a proof in S of (x)A(x).

But note that the result

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

is provable in strict subsystems of PA; you do not have to resort to a
system as strong as an extension by definitions of PA.
 
>   Second, suppose we don't want to use this proof, but instead, for
> whatever reason, in our semantic argument want to use
> 
> (2)  B(x) represents "x is a proof in S of (x)A(x)" in T, and T
>      is consistent.
> 
>   You seem to have the idea that the T here should be the same theory
> as S. Why? 

Because of the dialectical context of my Mind paper, which was to explore
the minimum extra needed, on the part of the user of S, to be justified in
asserting the G"odel-sentence for S. [See the last paragraph below.]

> After all, whether or not we know that S is consistent, we
> know that B(_n) is true if and only if n is a proof in S of (x)A(x).

This is incorrect. Suppose we don't know that S is consistent. I take 
it, then, that you are countenancing the possibility that S is
inconsistent. But if S is inconsistent, the "if" part of your claim fails.

If you mean, rather, that (we assume that) S is consistent but do not
actually *know* that S is consistent, then your knowledge-claim

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

is still conditional on the assumption that S is consistent.

> If we wish to stick to your tortured and peculiar argument, we don't
> even know that this is the case unless we know that S is consistent.

That is no problem peculiar to my position. 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.

One person's "tortured and peculiar" argument is another person's argument
with due attention to detail. When you give your swift-looking semantical
argument liberally speckled with occurrences of "true", it rapidly grows
into something tortured and peculiar (at least, by your own
cognitive/aesthetic criteria) as soon as you are pressed to provide the
nitty-gritty justificatory detail for each step.

You have now introduced the system T, possibly distinct from S, in which
one proves the S-representability of all recursive functions. For any
system S in which all recursive functions are representable, and which
happens to be weaker than the weakest such T (if such exists), the user of
S will be loath to appeal to T when justifying the assertion of the
G"odel-sentence for S. There will be a premium on using the weakest
possible additional assumptions in order to justify the assertion of the
G"odel-sentence for S; and S plus these additional assumptions could well 
be weaker than T. [Bear in mind, though, that not a single writer
putting forward anything resembling a "semantical" argument has taken the
care to distinguish (or gone to the bother of distinguishing) between S
and T.] On the other hand, if S is stronger than some system T proving the
S-representability of all recursive functions, then there is no need to
distinguish S from T, since we know that we need more than S anyway in
order to justify the assertion of the G"odel-sentence for S.

Neil Tennant