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

[Neil Tennant]

1. On Thu, 29 May 2003, Torkel Franzen wrote to inquire how I view the

> minor technical point raised in
> http://www.cs.nyu.edu/pipermail/fom/2003-April/006374.html

I believe---if I recall correctly my train of thought at the time---that I
intended the antecedent "If S is inconsistent" to have within its scope
the rest of what Torkel quoted, namely

> If S is inconsistent, then there will be some n such that n is a proof
> in S of (x)A(x). (Extend the proof of the inconsistency of S with a
> single step of ex falso quodlibet, to obtain (x)A(x) as the conclusion,
> and then find the code number n of the resulting proof.) But B(_n) could
> be false, for all that.

Torkel challenged this modal conclusion, replying that 

>   (x)(B(x) <-> x is a proof in S of (x)A(x))
> is provable in PA, and ... this proof does not assume the consistency 
> of S. The Godel sentence (x)A(x) has the form "for every x, x is not a 
> proof in S of t", where the value of t can be proved to be (x)A(x)
> itself. B(x) is "x is a proof in S of t".

(These predicates A and B had been in play in the preceding discussion,
with which familiarity will here have to be assumed.)

If one uses Torkel's explanations of A(x), B(x) and t, and substitutes, it
might appear that the universally quantified biconditional is logically
true, and not just a non-logical theorem of PA---until one realizes that
what PA would be providing is the proof that the value of t is (x)A(x)
itself. But since, in the overall discussion, PA is one of the values that
can be taken by the system-variable S, one has to ask whether Torkel's
challenge would have any point under the assumption that PA is
inconsistent.

2. Torkel also asked me to comment on another passage from an earlier
email of his, in response to which I shall interpolate my comments in
stages:

>    Certainly there is no (essential) need for talk of truth when proving
>    "if [S] is consistent then G". Talk of truth is here just a matter of
>    convenience. 

Agreed. But the truth-predicate-eschewing proof of G from the assumption
"S is consistent" is very, very long and would not serve as a plausible
regimentation of what is called the "semantical argument" for [the truth
of] G.

>    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. 

Yes; but only if a proof of "if S is consistent then G" is the only way
we have to back our eventual assertion of G. 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.

>    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, 

Again, the deduction of G from "S is consistent" is not what is called
the semantical argument. Remember, I *had* to take *the semantical
argument* as my focus in my Mind paper, in order to challenge the
anti-deflationist's claim that a substantial notion of truth is
essentially involved in the semantical argument itself. My strategy was to
regiment that very argument as faithfully as possible, but in a way that
involved no recourse to a truth-predicate, thereby correcting deceptive
appearances that had taken in the anti-deflationists.

>    and that the real issue concerns
>    how "F is consistent" (or, equivalently, your reflection principle) is
>    to be justified.

Yes, there will be a choice, for anyone interested in whether to assert G,
between two avenues of investigation, which are equivalent, as you
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. In no way was this
intended to rule out the possibility that one might wish, instead, to
follow avenue of inquiry (a), with its very long formal deduction of G
from "[S] is consistent". The latter course would not have engaged the
anti-deflationist *on the semantical argument*.

Neil Tennant