[FOM] consistency and completeness in natural language
Torkel Franzen
torkel at sm.luth.se
Mon Mar 31 06:35:17 EST 2003
Hartley Slater says:
>Neil Tennant has recently presented the fine details of just how it
>is that we humans can prove what PM, and the like, cannot, in the
>case of Goedel's Theorems (Mind, July 2002).
I take it you have in mind "Deflationism and the Gödel Phenomena",
Mind 443, pp.551-582. Could you elaborate on what you take to be the
illuminating argument or conclusion put forward in that paper? You
mention only that Tennant has argued that we know that reflection
principles hold for the systems in question, but this can hardly be
what you regard as the specific contribution of the paper.
The central argument of the paper turns on whether we make essential
use of a "thick" notion of truth in arriving at the truth of the Godel
sentence for a system S. Tennant argues against what he calls the
"substantialist dogma" according to which "the way in which the
semantical argument establishes the truth of the Gödel sentence
requires that the notion of truth be substantial", by giving a
"deflationary" way of carrying out the argument, using a reflection
principle.
My impression is that the argument of the paper is based on a
misunderstanding. 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. In Tennant's version, it is stated
that every numerical instance A(_n) of the primitive recursive
predicate formalizing "n is not (the Gödel number of) a derivation of
G in S" is provable in S, and since provability in S implies truth, it
follows that every such instance is true. This is not how Dummett
reasons: rather, he observes that every numerical instance A(_n) is
true, not that it is provable in S, and the obvious justification for
claiming that A(_n) is true is that we have shown (assuming S
consistent) that G is not provable in S. It is because we know every
A(_n) to be true that we know every A(_n) to be provable in S. Of
course somebody might attempt to argue instead directly for the
provability of every A(_n) in S, without invoking the result that
every true sentence of this form is provable in S, and then conclude
on the basis of some soundness principle for S (at a minimum
consistency) that every A(_n) is true. I've never seen such an
argument presented, and I can't readily imagine why anybody would want
to do so.
So what the standard semantical argument presented by Dummett shows
is that the Gödel sentence for S is true if S is consistent. Here
"true" is an arithmetical property (of Pi-zero-sentences) and a
further conclusion that the Gödel sentence for S is in fact true,
since S is concistent, will be based on considerations similar to
those that can be invoked to support Tennant's reflection principle.
There is nothing obviously "fat" about the standard semantic argument.
----
Torkel Franzén, Luleå University of Technology
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