[FOM] Consistency v. reflection, and the truth of theGodel-sentence

[Jeffrey Ketland]

Torkel (Franzen) to Neil (Tennant):

>   Certainly there is no (essential) need for talk of truth when proving
>   "if F is consistent then G". Talk of truth is here just a matter of
>   convenience. But in your paper you are concerned with a "semantical
>   argument...designed to help one understand why asserting G would be
>   the right thing to do". If a proof of "if F is consistent then G"
>   is to prompt us to assert G, we must take the view that asserting "F
>   is consistent" is the right thing to do. It is here that any "need"
>   for talk of truth must be located. Hence my complaint about your
>   paper, that we already knew that G can be deduced from "F is
>   consistent" without invoking truth, and that the real issue concerns
>   how "F is consistent" (or, equivalently, your reflection principle) is
>   to be justified.

Yes, this seems right. Neil's paper is partly a response to papers by
Stewart Shapiro (98) and myself (99). Let's call the argument that Shapiro
and myself gave the "Reflection Argument" (against deflationism about
truth).

Shapiro and I were not interested in the two-line argument from "F is
consistent" to G. Con(F)->G is already a theorem of very weak theories like
PRA, say, but this is not relevant to what we argued. We were interested in
the justification of Con(F), given F---which (we argue) involves
truth-theoretic reasoning. The justification of G given Con(F) is rather
trivial, and non-truth-theoretic. Some posters here have been unclear on
this point.

Neil Tennant:

>I am suggesting that the
>semantical argument provides (for those who go in for it) an alternative
>way, and that it is best regimented by use of the uniform reflection
>principle for p.r. sentences, this principle being adopted in an extension
>of S.

I'm very unclear what Neil means by "semantical argument". It doesn't seem
to be what I (or Stewart) mean by "semantical argument". The semantical
argument that I'm referring to is one which discusses the proof of the
reflection principle, and not the consequences derivable from it. Torkel has
pointed this out too. Of course, it's true that G is provable from some
scheme of the form Prov([A])->A (say, with A as any Pi_1 sentence). Sure,
but that's not the issue. This has nothing to do with the semantical
argument. The question just becomes: why should anyone accept the scheme
Prov([A])->A? That's the issue that Shapiro and I were addressing.

Neil continues:

>Yes, there will be a choice, for anyone interested in whether to assert G,
>between two avenues of investigation, which are equivalent, as you [Torkel]
>yourself point out:
>
>(a)  Is [S] consistent?---and, if so, how do we know that?
>(b)  Is uniform reflection for p.r. sentences in S a warranted
>principle?---and, if so, how do we know that?
>
>In my Mind paper, I was urging (b) as a route that could be taken by a
>deflationist who wished to capture the essential structure of the
>so-called semantical argument for [the truth of] G.

This appears to me to be ad hoc and ignores the "essential structure" of the
argument given by Shapiro and myself, an argument which concerns the proof
of the reflection principles. The question becomes: how would the
deflationist prove or justify the reflection principle(s)? The argument that
Shapiro and I gave concerned the truth-theoretic proof of these reflection
principle(s). Let me now speak for myself and not implicate Stewart
furthermore in this.

The paper I wrote was an attempt to argue that "F is consistent" (or the
reflection principles) should be
accepted by someone who accepts F, and that this conditional acceptance is
justified truth-theoretically. If you accept F, then you should accept "F is
true". Briefly, accepting a theory commits you to its reflection
principle(s). And this conditional acceptance is justified
truth-theoretically.

If this Reflection Argument is right, then it spells trouble for
deflationism. Because the reflection principles yield new theorems (the
truth-theoretic extension of PA is non-conservative). A deflationist (so
both I and Shapiro argued) should impose a constraint of conservation on
their truth-theoretical axioms. How could a deflationary theory be
non-conservative? If one supposes that truth is merely a "useful
instrument", then one should _not_ be able to invoke it in non-trivial
explanations, yielding new theorems, etc.

(C.f., if Gell-Mann's SU(3) quark model were merely a useful instrument,
then how come it predicted the Omega-minus, which was later observed? If the
Salam-Weinberg Electroweak model is merely a useful instrument, then how
come it predicts the W and Z particles, later observed? And so on. Useful
instruments should be conventional ways of organizing our thoughts, not
substantial in their novel predictions, explanations, and so on).

Rather crudely, here's the reasoning we wish to understand better:

(*)      PA
          Therefore, "PA is true"

This is rather odd reasoning, since the assumption is a whole theory.
Nonetheless, there's something right about it. Someone who accepted PA
should also accept "PA is true". The problem is to understand the "should"
here.

In particular, since "PA is true" is stronger than PA (if PA is consistent),
the argument (*) seems to be justified by something implicit or hidden. What
is this hidden something? I argued that it is: our understanding of truth
for the language of PA. And if we add the inductive axioms for truth to PA,
we can prove "PA is true". That is the semantical argument. The truth axioms
are the inferential glue that legitimate the passage from accepting PA to
accepting "PA is true" (reflection principle).

More generally, for a large class of theories F, the truth-theoretic
extension Tr(F) of F proves the (global) reflection principle for F
(Feferman 1991 JSL, "Reflecting on Incompleteness", has more details).
Feferman talks about the "reflective closure" of a theory. This is
formalized by taking the theory F and adding certain truth
axioms. This (non-conservative) extension is meant to generate all the
"extra" consequences of accepting F, given that you also accept the theory
of truth for the language of F. There are constraints on the base theory F
for this to work (F must be axiomatized by a finite number of axioms and a
finite number of schemes).

Whether the Reflection Argument works or not (i.e., against deflationism) is
a different matter. But at least this is roughly the argument that we gave.
Here I'm not discussing the other counter-arguments against Shapiro and
myself (e.g., given by Halbach, Field, Azzouni).

There is also something rather "natural" about proving G in the
truth-theoretic extension of PA, rather than in some second-order extension
(like ACA with induction on all formulas). We know that G is a theorem of
second-order arithmetic, but the proof of this involves formalizing the
truth-definition for L_{PA} in the second-order language. The proof, as it
were, piggybacks on the formalization of semantics within the second-order
theory. This point goes right back to Tarski's original paper _Der
Wahrheitsbegriff_.

So: given that I accept PA, why should I accept G? Because I accept the
inductive definition of truth for L_{PA} and thus equipped I can prove "PA
is true" (and thus G, in a couple of lines). If this is right, then truth is
therefore not a "deflationary" notion. It can be used for substantial
purposes, of explanation and so on.

--- Jeff

~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy
University of Leeds
LEEDS LS2 9JT
U.K.
j.j.ketland at leeds.ac.uk
ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~