[FOM] Deflationism and the Godel phenomena
Joseph VidalRosset
joseph.vidalrosset at ubourgogne.fr
Thu Feb 17 13:07:35 EST 2005
BEGIN PGP SIGNED MESSAGE
Hash: SHA1
Aatu Koskensilta a écrit :

 On Feb 17, 2005, at 2:51 PM, Joseph VidalRosset wrote:

> Aatu Koskensilta a écrit :
> 
>  On Feb 16, 2005, at 12:58 PM, Joseph VidalRosset wrote:
> 
> > Do we need at this point to grow up to the second order? I believe it.
> 
>  Only if we want to be able to define the truth predicate explicitly.
>
> Thanks.
> But this logicophilosophical debate around deflationism about truth
> tries to define truth explicitly, wondering after its substantial or
> nonsubstantial property.


 It's perfectly acceptable to define the truth predicate implicitly by
 listing the inductive clauses it has to satisfy. A basic fact about first
 order truth is that there is a unique predicate (set) satisfying these
 clauses.
The point is not about knowing if it is logically acceptable or not. My
questions to Jeffrey Ketland on the list were not only technical, but
also philosophical, and I'm waiting a clear philosophical reply on my
question. So I repeat my question: after all, what are the positive
philosophical consequences of this socalled refutation of Deflationism
about truth? Is it a argument on behalf of mathematical realism or not?
 In case you're wondering, the clauses look something like this:

 True('A&B') <=> True('A') & True('B')
 True('~A') <=> ~True('A')
 True('P_1(t_1,...,t_n)') <=> P_1(t_1,...,t_n)
 .
 .
 .
 True('P_i(t_1,...,t_n)') <=> P_i(t_1,...,t_n)
 True('ExA') <=> EyTrue('A'[x/y])

 (where P_i are the predicate symbols of the language in question).

 For example, the theory Tr(PA) Jeffrey Kettland mentioned is obtained
 from PA by
 adding to its language a predicate True, adding the above clauses as
axioms
 (with {P_i} = {+,*,=,S}) and extending the induction schema to cover all
 formulae
 of the extended language.
I'm afraid that this operation of extending the induction schema to
cover all formulae of the extended language is a manner of doing second
order logic under first order notation. Jeffrey Ketland says that set
theory is used, and set theory can't be reduced to first order logic
(predicate calculus).
 The resulting theory is not conservative over PA,
 since e.g. the trivial proof of PA's consistency can be carried out (the
 axioms
 of PA are true and the rules of inference of first order logic preserve
 truth
 hence no contradiction is provable from the axioms of PA).

 Feferman (and others) has pointed out that the most natural
 formalization of
 arithmetic in first order logic uses free predicate variables with a
 substitution
 rule saying that any definable predicate can be substituted for the free
 predicate
 variable. Thus it might be argued that the theory Tr(PA) is the correct
 formalization
 of what one gets by adding an arithmetical truth predicate to PA and
 hence arithmetical
 truth is substantial in the sense that more arithmetical truths can be
 proved by
 using the concept than without it. I'm sure Jeffrey will correct me if I
 have
 manhandled his argument too roughly.
I am afraid of being really idiot or being by nature too skeptic. "More
arithmetical truths"? But what are these arithmetical truths really? The
consistency of PA? The truth of the Gödel sentence saying of itself
being not demonstrable in PA, provided that PA is consistent?
There are many good and impressive logicians on this list who can help
us to reply to this too difficult question for me: is it a difference
between these "arithmetical truths" and the truth of Fermat's theorem
who has been proved by Wiles? It seems to me that these truths are not
at the same level, I mean at the same order. Obviously I do not belong
to the little circle of mathematicians who are not able to follow Wile's
demonstration.
> Gödel believed in truth as substantial, no doubt, but neither Tarski nor
> Carnap did.


 What belief exactly do you have in mind?
See "Some basic theorems on the foundations of mathematics and their
philosophical implications". by Gödel, in Unpublished Philosophical
Essays, ed. Rodriguez Consuegra, Birkäuser, p.140141., against the
conception of mathematical truth as tautologies.
Jo.
BEGIN PGP SIGNATURE
Version: GnuPG v1.2.5 (GNU/Linux)
Comment: Using GnuPG with Thunderbird  http://enigmail.mozdev.org
iD8DBQFCFN1nw/q2qaNF4IARAr4bAJ9RvnwCpbG5s+Tg7yYkpKXALYkSBACeOOoI
nxXiDvSzBESUScx42mmAy0w=
=kSuL
END PGP SIGNATURE
More information about the FOM
mailing list