[FOM] Deflationism and the Godel phenomena

Joseph Vidal-Rosset joseph.vidal-rosset at u-bourgogne.fr
Thu Feb 17 13:07:35 EST 2005

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Aatu Koskensilta a écrit :
| On Feb 17, 2005, at 2:51 PM, Joseph Vidal-Rosset wrote:
|> Aatu Koskensilta a écrit :
|> |
|> | On Feb 16, 2005, at 12:58 PM, Joseph Vidal-Rosset 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 logico-philosophical debate around deflationism about truth
|> tries to define truth explicitly, wondering after its substantial or
|> non-substantial 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 so-called 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
| (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

|> 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.140-141., against the
conception of mathematical truth as tautologies.


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