[FOM] consistency and completeness in natural language

[Neil Tennant]

On Mon, 31 Mar 2003, Torkel Franzen wrote:
 
> Tennant presents a "Semantical argument for the
> truth of the Godel sentence" in a formulation that he attributes to
> Dummett, and this is the argument he wishes to replace. But this
> "semantical argument" is an odd one, and I don't agree that it is the
> argument put forward by Dummett. 

Let's get clear about the role played by my presentation of Dummett's
argument, in the dialectical context of my paper. I presented Dummett's
argument both by direct quotation (on p555 of my paper 'Deflationism and
the G"odel-Phenomena') and by reconstructive summary (on p556). I also
referred to versions of the argument to be found in works by Kleene and by
J.R.Lucas. All this was by way of introductory and historical background.

Torkel questions how good an example Dummett's argument is, of the kind of
argument (for [the truth of] an independent G"odel-sentence) that I wanted
to call 'semantical'. While I would actually take issue with Torkel's
reading of Dummett's argument, I need not do so here, for the simple
reason that it would be beside the point. If Torkel has a more penetrating
and insightful reading of Dummett's argument, on which that argument turns
out *not* to involve appeal to a thick notion of truth, then so be it! I
shall then stand corrected on a point of exegesis, but will be able to
welcome the extra support now provided by Torkel for the main thesis of
my paper---which is that one does not need to appeal to a thick notion of
truth in order to carry out the argument for [the truth of] an independent
G"odel-sentence. 

The main argument of my paper was directed against Ketland and Shapiro.
These two writers have claimed that the G"odel-phenomena undermine
deflationism. In his now classic 1963 paper, Dummett had of course not
talked about deflationism as such; so my presentation of the semantical
argument, as I took it to be developed in Dummett's paper, served only to
illustrate the vintage of the view against which I was arguing on behalf
of the deflationist.

All that said and done, Torkel still has to square his own understanding
of the deductive details of Dummett's argument with some of the things
Dummett actually says in his 1963 paper. 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.

Torkel claims, in effect, that it should be read, rather, with extra
detail filled in somewhat differently:

	each of the statements A(0), A(1), A(2), ... is true [because the
	falsity of any one of these would imply the provability of
	"for all x A(x)", which is impossible; whence, since each of the
	statements A(0), A(1), A(2), ... is quantifier-free, it is also a 
	theorem of and hence] is true in every model of the formal system.

So the issue is whether the truth of each instance flows from the instance
in question being a theorem of the formal system; or whether [the truth
of] each instance follows from the unprovability of the universal
generalization, in combination with the formal system's completeness on
quantifier-free sentences.

Even if one opts for Torkel's reading, one has to confront a point at
which, pace Dummett, there is still the prima-facie possibility that a
thick notion of truth is doing some work. This is the point at which we
move from

	each instance A(0), A(1), A(2), ... is true

to

	"for all x A(x)" is true.

Dummett calls this move "trivial" (p.192 of Truth and Other Enigmas") and
"quite evident" (p.193). I would disagree with Dummett here. For this is
the truth-theoretic principle of Sigma-zero-one reflection, presupposing
that only the standard numbers fall within the range of the
universal quantifier. Showing that we can dispense both with *this*
prima-facie appeal to a thick notion of truth in 'proving' the [truth of]
the independent G"odel-sentence and with like appeals at any other
juncture in such proof, was the burden of my paper. It does not matter
that the so-called semantical argument *also* appeals to the premise that
the formal system is consistent. The question was rather whether what
appeared to be appeals to the workings of a thick notion of truth were
indispensable. I argued that they were dispensable; and for the details I
have to refer the reader to the original paper. [Mind, Vol. 111, July
2002, pp.551-582.]

Neil Tennant