[FOM] Deflationism and the Godel phenomena
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Tue Feb 15 16:30:06 EST 2005
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"? Shapiro
and I think that one needs Tr(PA), or something mathematically at least as
strong (e.g., Feferman's reflective closure; or maybe his unfolding of PA).
Here Tr(PA) is PA + Tarski's inductive definition + full induction for the
extended language. In my paper I formulated it somewhat differently and
called it PA(S), following Richard Kaye, but these theories are equivalent.
I like Feferman's Ref(PA) too. And I think one should iterate that too. One
should reflect as much as one can make sense of, using such truth-theoretic
extensions. All of this is highly non-deflationary (since every reflective
step generates new arithmetic consequences).
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.
> 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. 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. 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. My purposes are
(though I didn't know it at the time) 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.
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.
By this I meant that one consequence of adding the Tarskian truth axioms to
PA was that one thereby had a proof of G, and that this does not happen with
the disquotatonal axioms. In fact, I was reiterating remarks made by Tarski
in 1935/6 (all of which quoted in my article). I also commented on the
Lucas/Penrose argument, since the G-proofs in question are perfectly
formalizable, and do not involve any special spooky insights.
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.
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.
Perhaps nothing to do with that.
It was about a different topic. Namely, the justification of the claim "All
theorems of PA are true" (see the Adequacy Condition (ii) mentioned above,
and the quotes from my article both above and below). A proof of G is a neat
spin-off.
> 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.
In general, you argument is based on attributing to me two false beliefs,
neither of which do I hold:
(B1) In order to prove G, one *must* use Tr(PA);
(B2) The *only* reason for considering Tr(PA) is to prove G.
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.
And you misinterpreted this one sentence that you quoted.
My article was not even about this topic. Like Shapiro's article (and
Feferman's before that), it was about the justification of things like "All
theorems of PA are true" (the global reflection principle for PA).
Anyone who takes a look at what I wrote can easily check this. I wrote:
Any adequate theory of truth should be able to
prove the "equivalence" of a (possibly infinitely
axiomatized) theory and its truth, True(T) (that
is, the metalanguage formula
(forall x)(Prov_T(x) -> Tr(x)).
And Tarski's theory comes up trumps ... it is
possible to show that ...
T + Tarski's theory of satisfaction |- True(T)
(Ketland 1999, p. 90.)
(What I say there is not quite correct, since T must satisfy a certain
finiteness condition. Feferman 1991 has clarified that T must be axiomatized
by finitely many axioms and axiom schemes. The language of T must also
presumably have finitely many non-logical primitives, otherwise there would
be infinitely many basis clauses in the truth definition for these
primitives.)
In fact, the claim (B2)---which you appear to keep attributing to me---just
contradicts everything I wrote in my article. For example, it contradicts
both of the above quotes from my article.
The single sentence you quote may have been slightly ambiguous out of
context; in context it is unambiguous. Indeed, the passage I quoted above
occurs *one line* after the sentence you quoted. The intended meaning of
this sentence was:
(C) Tr(PA) *does* prove G; and (PA + T-sentences) doesn't.
After all, it is obvious there that I am comparing two truth theories. And I
say that one is more powerful than the other. And I am right. This is
*exactly* what Shapiro and I were saying. And you've even conceded it. So,
you admit that truth is (or can be) a powerful notion. (I.e., a Tarskian
inductive definition and you let the truth predicate enter the induction
scheme.) Great. That's exactly what we said. And if Shapiro and I are right,
that probably means that deflationism about truth is refuted.
Of course, Con(PA) implies G. Why would anyone suggest otherwise? If you
think I have suggested otherwise, you are mistaken. You keep attributing to
me views which I don't hold. 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.)
--- Jeff
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