[FOM] consistency and completeness in natural language

[Neil Tennant]

On Wed, 2 Apr 2003, Torkel Franzen wrote:

>   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.  

This is not the only straightforward justification. Here is another:

	Each instance A(n) is of the form not-B(n), where B(n) represents
	"n is a proof, in the formal system, of (x)A(x)". If, for any n,
	B(n) were a theorem of the formal system, then there would be a
	proof, in the formal system, of (x)A(x). But (assuming the system
	is consistent) there is no proof, in the system, of (x)A(x). Hence
	B(n) is not a theorem (for any n). But every p.r. statement is
	provable or refutable in the system. Hence, for every n, B(n) is
	refutable; whence, A(n) [i.e., not-B(n)] is provable.

> 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?

	This other argument is given above.

> 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.

A jocular version of "the straightforward justification" that you took to
be the only available one; but it is not jocular if you take, instead, the
alternative argument that I have just given.
 
>   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. 

The only interpretation of your claim that is consistent with G"odel's
incompleteness theorem would require the correctness of the step in
question to be proved in some *extension* of S. I believe it is at this
point that the appeal is made to a thick notion of truth, by those who
think that the semantical argument for the truth of the G"odel-sentence
requires a thick notion.

> 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.

Here again, by objecting to my diagnosis of the semantic arguers'
semantical argument, you support, in effect, the deflationary thesis of my
paper. The principle that I called U.R._p.r. (uniform reflection on
primitive recursive predicates) would be an adequate arithmetical
formalization of the allegedly "semantical" step above. 

Neil