[FOM] Deflationism and the Godel phenomena

[Timothy Y. Chow]

On Thu, 17 Feb 2005, Torkel Franzen wrote:
> (1)     Prov_S(psi&~psi)->psi&~psi
[...]
>  But then I must object that "if I am willing to assert ..." etc is
> not at all what is stated in (1). So just what do you take to be the
> actual argument? How do you get from "if i am willing to assert ..."
> to (1)?

O.K., from this and from other messages on and off the list, I think I 
understand the issue better now.  The following seem *not* to be under 
dispute:

(a) (1) does not follow logically from the axioms of S, so any argument
     for (1) must involve some idea that goes beyond what the axioms of
     PA themselves assert.

(b) Various theories of truth can yield (1).

(c) These theories of truth go *substantially* beyond S itself.

And so what remains under dispute seems to be:

(d) There is a plausible way to go from S to (1) without invoking a
     theory of truth that goes *substantially* beyond S, or indeed any
     theory of truth at all.

The reason this appears to be so hotly disputed is that Neil Tennant
has not, to my knowledge, *formalized* the "plausible way" of (d).
Therefore, anyone who doesn't find his informal arguments plausible
can always find gaps or imprecisions in them.

Formalizing the "plausible way" seems like an interesting task, and
in the spirit of f.o.m.  On the other hand, I'm not sure why Jeffrey
Ketland and Torkel Franzen are denying (d), or something like (d).
(Maybe they're not denying it, but they at least give me that impression.)
After all, all that stuff about theories of truth and reflective closure
presumably stems from some informal intuition that it makes sense to go
from S to (1), and this informal intuition obviously doesn't *explicitly*
contain all the formal apparatus of theories of truth.

But never mind that; let me get back to the more limited question that
I was asking about.  The question seems to be, can we get from

   (2) If I had a proof that there is an S-proof of 0=1, then I would
       be willing to assert 0=1

to "S is consistent"?  The "proof that there is a proof" part of (2)
is a little awkward, and although that is indeed the wording in the
passage quoted by Torkel Franzen, perhaps all parties concerned would
be satisfied if one could get from the simpler assertion

   (2') If there is an S-proof of 0=1, then I will accept 0=1

to "S is consistent".  (The sentence (2') also has the advantage of 
getting rid of the subjunctive and therefore being more plausibly
interpreted as a material conditional.  At the back of my mind,
though, I'm still wondering if the pesky intrusion of the subjunctive
mood every time we try to talk about "willingness to assert" isn't
telling us something.)

Now, it seems to me that all parties concerned agree that there isn't,
and can't be, a trivial *logical* derivation of "S is consistent" from 
(2'), if for no other reason than that the notion of "willingness" is not 
explicated logically here.  But we can also make the simple observation 
that (2') together with

   (3) I will not accept 0=1

*does* yield "S is consistent."  So, does this all boil down to a 
complaint that Neil Tennant did not explicitly state (3)?  Let me
ask directly:

Question for Neil Tennant: Is (3) indeed part of your "intellectual
reflection" argument?  Your remark about modus tollens seems to
suggest so.

Question for Torkel Franzen and Jeffrey Ketland: If Neil Tennant
explicitly avows (3), does that answer your objection that his
justification of the reflection principle is incomplete and doesn't
account for self-contradictory phi?

Tim