[FOM] consistency and completeness in natural language

[Neil Tennant]

This is an attempt to "unbaffle" Torkel, with the details of the so-called
"semantical" argument laid out in full. Let me emphasize at the outset
that the argument is given in the metalanguage with regard to the formal
system S. Sentences of the object-language (i.e., the language of the
system S) are named, or mentioned, or quantified over in the course of our
metalinguistic reasoning---but they are not used. The metalinguistic
considerations are syntactic for the first part of the argument, and
become semantic only in the second part.

In the metalanguage, we generalize over natural numbers by saying "for
every/any/each n".  _n is the numeral, in the language of S, for the
natural number n.  

I eschew the use of corner quotes, since the reader will be able to supply
them in context.

	Assume that the formal system S is consistent. Then the
	G"odel-sentence (x)A(x) for S is not provable S. 

	[The proof of this unprovability result makes no mention at all of
	truth. It is a purely syntactic result.]

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

	[Note that we have thus far confined ourselves to syntax, talking 
	only about provability in S. We have not yet made any use of
	the notion of truth (in the intended model). We are arguing in the
	metalanguage with regard to the formal system S. Now, for the
	first time, we bring in the notion of truth:]

	Now assume that every p.r. sentence provable in S is true in N
	(i.e., the intended model of the natural numbers). 
	
	So for every n, A(_n) is true in N. 

	Note further that our metalinguistic quantifications "for every n"
	are intended to range over the natural numbers,	which form the
	domain of N. 

	So now, by the semantical rule for the universal quantifier, it
	follows (in the metalanguage) that the universal sentence (x)A(x)
	is true in N. That is, the independent G"odel-sentence for S is
	true in N. 

Unlike Torkel, I see no vitiating or "baffling" circularity in this piece
of metalinguistic reasoning. In my opinion, it is an accurate and
sympathetic reconstruction, with missing details clarified, of what is
usually understood to be the "semantical" argument for the truth, in N, of
the independent G"odel-sentence for S. 

Torkel may still wish to take issue with the *exegetical* claim that this
is how Dummett's own reasoning in his 1963 paper is best construed. As I
have said in an earlier posting, that does not much concern me---since, if
Dummett fails to be an exemplary "semantical arguer" along the more
detailed lines given above, there are nevertheless others around who do
indeed argue in this way. 

I stress again that, to the extent that Torkel's alternative construal of
Dummett's reasoning makes Dummett out to be innocent of any appeal to a
thick notion of truth, it will be because Dummett is being represented (by
Torkel) as arguing more in the fashion of the deflationist!---for whom, in
my Mind paper, I proposed a truth-eschewing line of justification for the
assertion of the G"odel-sentence for S, using reflection principles in an
extension of S.

Neil Tennant

















___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://www.cohums.ohio-state.edu/philo/people/tennant.html

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