[FOM] Deflationism
A.S.Virdi@lse.ac.uk
A.S.Virdi at lse.ac.uk
Fri Mar 24 13:07:11 EST 2006
On Thurs, 23 Mar 2006 Neil Tennant wrote:
>Here's the rub. There is no such thing as "the" soundness statement for
>PA; for there are various ways to express the thought in question. One
way
>to do so, eschewing the use of a truth-predicate altogether, is to
state a
>a reflection principle involving the provability-predicate for PA. Why
>should that not be a good enough addition to the deflationist's
"arsenal"?
Granted---it is possible to express the thought that all theorems of
Peano Arithemetic are true in a manner that makes no appeal to a truth
predicate. Your local reflection principle would do just as well. So:
(i) What ought the deflationist say in her account of truth? Ok, it is
dispensable in this mathematical case. So, truth is NOT everywhere
indispensable in facilitating the finitary re-expression of potentially
infinite conjunctions (there are cases in which it is clearly
INDISPENSABLE however, such as in blind ascriptions of the "The first
sentence that Tarski uttered on the morning of 22nd March 1960 is true"
ilk). Truth then performs a logical function that is not *uniform*.
(ii) In those cases where truth IS employed, if the deflationist's
perspective in those cases is that its function is NOT to express a
"seriously dyadic" relation (as Crispin Wright once put it, I think in
his "Truth and Objectivity" book) between truth-bearers and their
respective truth-makers, BUT to facilitate the logical role mentioned
then this ought to be a consequence of her theory of truth. What is her
theory of truth? If her theory of truth is parsed axiomatically (quite
reasonably so, given her contention that sentences are cognitively
equivalent to their truth-predicated counterparts) as the (restricted-
for want of her theory's consistency) set of Tarskian T-sentences then
she won't get what wants. For it has been shown (in Tarski's locus
classicus of 1935/36) that you cannot get to the claim that a theory is
true from that theory + the (restricted) set of T-sentences. This is one
way of seeing why it is important to broach the subject of the
justification of the soundness principle for PA, say. In this case where
truth is employed, the justificatory resources required (in the sense of
being both necessary and sufficient) is that we have a theory of truth
that "significantly transcends" those of the so-called diquotational
theory (I am using words that Jeffrey Ketland uses in his Mind 1999
article). Indeed, if I accept PA then I ought to accept that 'PA is
true'. And, contrary to what the deflationist thinks, provably 'PA is
true' is a logically stronger claim than PA (PA is not a 'claim' as
such, but the point still holds)
On Thurs, 23 Mar 2006 Panu Raatikainen wrote:
>> >In this case, there are (at least) two alternative interpretations:
>> >(i) T-sentences say all there is to say about truth;
>> (ii) the truth theory should not entail anything substantial;
>> >i.e., it must be a conservative extension of the base theory.
>> Many thanks for your reply! Firstly, your (i) and (ii) seem not to be
>> mutually exclusive.
>Indeed, the simple addition of T-sentences results to a conservative
>extension. But it is a consistent position just to require (ii) but not
to
>commit oneself to (i) - that was my point.
I accept that "it is a consistent position just to require (ii) but not
to
commit oneself to (i)". I apologise if I gave the impression of
misunderstanding you originally. If my deflationism about truth means
that my theory of truth conservatively extends the base theory then I'll
need a theory of truth that is stronger than that given by the
T-sentences alone. In the case of arithmetic, for example, I will need
Tarski's inductive clauses for arithemetic too. It is the status of
these that I am concerned with. Many think that Tarski's theory of truth
is a type of substantial theory of truth, as opposed to a type of
insubstantialism about truth. The logical facts point in this direction
too (for example, PA + Tarski's inductive clauses + T-sentences + full
induction proves claims (like G, Con(PA)) that PA + Tsentences + full
induction cannot). I look forward to reading your paper on this!
All the best,
Arhat
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