[FOM] Deflationism and the Godel phenomena

[Neil Tennant]

On Tue, 15 Feb 2005, Jeffrey Ketland wrote:

> Neil Tennant wrote:
> 
> > What is misleading here is taking my footnoted claim out of the
> > dialectical context of my disagreement with Ketland over how strong one's
> > justificatory resources should be.
> 
> Do you mean "how strong one's justificatory resources should be" in order to 
> prove the global reflection principle "All theorems of PA are true"? 

No, I do not mean that. I mean in order to prove the G"odel sentence for
the system PA. The whole debate sparked by my Mind 2002 paper was about
justifying one's assertion of that sentence, rather than justifying the
claim that all theorems of PA are true.

> What do you think the justificatory resources should be for proving "All 
> theorems of PA are true"? After all, Stewart Shapiro and I both wrote long 
> articles about precisely this topic.

Yes, I know. I addressed both those papers, taking issue, on behalf of the
deflationist, with the claimed need to prove "All theorems of PA are true"
in order to prove the G"odel sentence for PA.  I cannot speak for the
deflationist on all matters, but I think the deflationist should simply
say "I am prepared to assert whatever can be proved in PA". The reflection
principle Prov_PA(phi)->phi would capture this.
 
> > Ketland had admitted that
> > his preferred truth-theory (for the purposes of justifying the claim
> > that the G"odel sentence for PA is true) was intertranslatable with ACA.
> 
> Right. I noted that Tr(PA) is equivalent to ACA.
> 
> Let me describe again what I *actually* did in my 1999 article, rather than 
> what you appear to think I did.

In my most recent article in Mind, which was a reply to your reply (in the
same issue), I was referring only to your reply, not to your 1999 article,
except by way of repeating a quotation from it which, I believed, the
considerations adduced in my original 2002 paper "Deflationism and the
G"odel Phenomena" showed to be in error.

> The 1999 article compared the disquotational 
> theory and the Tarskian truth theory, in connection with two central 
> conditions on a theory of truth:
> 
> (i) The Conservation Condition (a deflationary truth theory should be 
> conservative);
> (ii) The Adequacy Condition (adding truth axioms should give a proof of 
> reflection principles, and in particular "All theorems of S are true").
> 
> (Shapiro gave an almost identical line of argument.)
> 
> I stated there that the Tarskian theory Tr(PA) satisfies the Adequacy 
> Condition, but violated the Conservation Condition. I concluded that Tr(PA) 
> was not deflationary. I said that any adequate theory would be 
> non-deflationary (see quote below).
> Furthermore, I said that Tr(PA) "significantly transcends" the 
> disquotational theory. This is an important fact. Why do you think I 
> "admitted" something? How could I "admit" something that I actually *stated* 
> 6 years earlier (and which is, in fact, true)? That isn't what the word 
> "admit" means.
> 
> More importantly, why do you think that I have a "preferred truth theory for 
> the purposes of justifying the claim that the G"odel sentence is true"? This 
> is a misinterpretation of what I wrote. I have not made any such claim. 

Why, then, do you continue to take issue with the deflationist's claim
that there is a deflationist way (appealing to a suitable, weak
reflection principle) to justify the assertion of the G"odel sentence? If
we set ourselves just that limited task, why resort to these much stronger
truth theories? In the context of our disagreement over whether the
deflationist has weaker resources that will still justify the assertion of
the G"odel sentence, what you said counts as an admission of extra,
unnecessary strength in the resources to which you resort.

> I 
> don't have any such "purposes" in mind at all. My actual purposes (like 
> Shapiro's and Feferman's) concerned the justification of reflection 
> principles, as is clear from actually reading my article. 

And what I am saying is that the reflection principles do not themselves
need that kind of underpinning. They become evident (G"odel's own term)
upon suitable reflection.

> My purposes are 
> (though I didn't know it at the time) 

---yes, as I recall emailing you on one occasion, to point this out--=

> pretty much the same as Feferman's 
> purposes in his 1991 JSL article, "Reflecting on Incompleteness". Namely, 
> how do we explain/justify reflection? The plan---common to Feferman, Shapiro 
> and me---is to use a truth-theoretic extension.

It would be most helpful, and perhaps advance the debate, if you could
first provide some reason *why* reflection principles need to be justified
in this way. What do you say to the reflecter who claims to have an
immediate insight into such matters, upon suitable reflection?

> My idea was to compare a couple of theories of truth in relation to my 
> conditions (i) and (ii) above: in particular, the disquotational theory and 
> the Tarskian theory. I noted that the disquotational theory was conservative 
> (i.e., deflationary), and I noted that much more preferable theory Tr(PA) 
> proves reflection principles (including "All theorems of PA are true"). So, 
> Tr(PA) is adequate, but non-deflationary.
> I wrote a subsection noting that Tr(PA) also proves G. In this section, I 
> closed with a discussion of the adequacy condition, and why it contradicts 
> the conservation condition:
> 
>    To summarize, an *adequate* theory of truth looks as if it must
>    be non-conservative. Indeed, it is bound to be non-conservative
>    if it satisfies the "equivalence principle" [i.e., adequacy condition
>    (ii) above] above. Tarski's theory does the job nicely. But the
>    deflationary theories are conservative. So they are inadequate.
>    (Ketland 1999, p. 88.)
> 
> One line above this, I had stated that "our ability to recognize the truth 
> of Goedel sentences involves" the Tarskian theory, as opposed to the 
> disquotational theory. I said that the Tarskian theory "significantly 
> transcends" the disquotational theory. And this is exactly right.

No! The demonstrative "this" in your last claim points to too much.
You are exactly right that the Tarskian theory "significantly
transcends" the disquotational theory. No one (myself included) has
ever taken issue with that. But you are exactly wrong in claiming that
"our ability to recognize the truth of Goedel sentences involves" the
Tarskian theory.  It doesn't! And I showed that it doesn't.

> I made no claim concerning whether this was the only way to do it (the 
> thought didn't even occur to me). That particular claim is a 
> misinterpretation.

In "Deflationism and the G"odel Phenomena" I considered two different
readings of your claim, in context. It is clear that when one says
something like "our ability to X involves theory Y" one is
saying, among other things, that we cannot X unless we are committed to
theory Y.
 
> This claim is not of central relevance, in any case. The paper that I wrote 
> had little to do with the "justificatory resources" needed to prove G. 

Except, of course, for your incorrect claim that our ability to recognize
the truth of Goedel sentences involves the Tarskian theory of truth.

> It was about a different topic. 

In that case, you strayed off topic to make a false claim.

> > The purpose of my footnote was to point out how much further (i.e.
> > higher in the hierarchy of consistency strengths) than PA+Con(PA) this
> > would take us, even though PA+Con(PA) is all that one needs.
> 
> I stated in my article that the Tarskian truth theory Tr(PA) was quite 
> strong ("significantly transcends" the disquotational theory). Your footnote 
> concedes that I was right. Good.

But that has never been in contention.

> In your 2002 Mind article you quoted exactly one sentence from my article. 
> There are, I'd guess, around 300 sentences in my article (25 pages; 40 lines 
> per page; 3 lines per sentence). You quoted just one sentence. So, you 
> ignored the other 99.7% of the article. Great.

This is churlish. May I remind you of Geach's anecdote about Wittgenstein?
When asked why Wittgenstein attacked a particular point made by an
opponent, rather than considering the opponent's other points as well,
Wittgenstein growled back that that particular point was the weakest one.
"Ce n'est pas la quantite', mais la qualitate', qui compte." (Apologies to
native French speakers who might spot errors in my French.)

> And you misinterpreted this one sentence that you quoted.

No I did not. I took it on the obvious reading that any competent reader
would give it, in the context in which it occurred.

> My article was not even about this topic. 

Too bad, then, that you strayed off topic.

> The intended meaning of 
> this sentence was:
> 
> (C) Tr(PA) *does* prove G; and (PA + T-sentences) doesn't.

In that case, why didn't you write exactly that, rather than the
misleadidng, unambiguous, false claim that you wrote instead?

> Of course, Con(PA) implies G. Why would anyone suggest otherwise? 

I have never said or written that anyone has suggested otherwise.

> You keep attributing to 
> me views which I don't hold. 

No, I just keep quoting a sentence that you have written, which is
unambiguous, important, relevant to my topic, and false.

> In fact, in my Mind 2005 reply to you, I 
> pointed out that one might have *empirical reasons* for accepting Con(PA), 
> and thus G. (These empirical reasons would be defeasible of course.)

Forgive me if I don't follow your lead into this territory. I find any
version of non-(a priorism) in the foundations of mathematics dubious, to
say the least. Note also that G"odel's own suggestions as to how
set-theoretical axioms might derive some justification from the "lower
level" mathematical claims that they help to prove remains an a priorist
position. It does not commit one at all to looking at empirical facts in
order to support a mathematical claim.

Neil Tennant