Reduction and Elimination in Philosophy and the Sciences

"PA" by using "PA* ", at least at the begining of the conversation.
I wrote PA(*) when I was not sure which one
one had in mind.)
Peter: Do you believe that the axioms of PA are
true?
Deflationism and Conservativity: Who did Change the Subject? — Henri Galinon
Puter: Yes, I believe that the axioms of PA* are true.
Peter: And do you believe inference rules to be
truth-preserving?
Puter: I do.
Peter: You believe then that all of PA's theorems
are true?
Puter: Indeed.
Peter (getting excited): Hence, since PA proves
0≠1, you believe it is true, and thus you believe that
PA does not prove 0=1, that is you believe PA to be
consistent.
Puter: Yes, I do!
Peter: You agree, then, that your commitment to the
truth of PA is a commitment to its consistency, and
that a good theory of truth for PA should account for
that?
Puter: Yes, it would be nice.
Peter (proud) : Look, Tarski's theory of truth for PA,
T(PA), is doing precisely this 3 .
Puter (sincerly): I know, that's great indeed! 4
Peter: But look: PA does not prove the consistency
of PA, while T(PA) does. Truth has an explanatory
power, it explains new facts that PA can't account
for, facts that are expressible in the language of PA.
My acceptance of PA does not logically commit me
to the acceptance of conPA, but once I have
acknowledged the truth of PA the acceptance of
conPA is forced upon me. There's a new purely
arithmetical fact which is explained by my truthattribution
to PA.
Puter (embarrassed): But the consistency of PA (*) is
not an arithmutical fact.
Peter: I beg you pardon?
Puter: Well, I agree that your argument is a sound
arithmetic reasoning, but it is not an arithmutic
reasoning. First, it is false that the sentence conPA
expresses the consistency of PA * in arithmutic. And
second, you cannot carry your inductive reasoning
out in any sound axiomatization of arithmutic, be it
PA * or another theory. This is fortunate, since the
negation of conPA is a true arithmutical fact!
Peter: I'm not with you here.
3 For reference, T(PA) is the theory obtained by extending PA with the Tarkian
recursive axioms for truth, letting the the truth predicate enter the induction
scheme.
Equivalently one could get a theory of satisfaction. Moreover, such recursive
definitions can be turned into explicit definitions provided that we ascend to a
richer theory (when it exists) allowing higher-order variables, or proving existence
of sets of higher rank than than any set the existence of which is provable
in the base theory . See for instance McGee (1991). It is not very important
here which of those strong « theory of truth » for PA one has in mind
4 Of course he had understood Peter's claim as: T(PA* ) proves the consistency
of PA* .
Puter: Well, ok. First things first: the sentence conPA,
it is well-known, expresses the consistency of $PA*
$ in the sense that it is true in N if and only if PA(*)
is consistent. But it is of course not the case that
conPA is true in arithmutic if and only if PA is
consistent! Now the second point. In your argument
you apply induction in the following manner: axioms
are true, rules are truth preserving, hence all
theorems are true. This is a perfectly correct
inference of course, but it belongs to arithmetic, not
arithmutic, since the induction involves vocabulary
beyond the language of PA. To carry this argument
out on the theory PA*, we usually use an axiomatic
metatheory containing PA-arithmetic to talk about
strings of PA*, plus a recursive truth theory (Tarski
style), and we let the truth predicate appear in the
meta-induction scheme. Sometimes we also enrich
the logical-mathematical part of our metatheory so
as to be able to explicitely define the truth predicate
for our base theory. In any case, there is in the
metalanguage a sentence conPA* which expresses
the consistency of PA* in the sense of being true in
the intended model of the metatheory if and only if
PA* is consistent, which is provable in the
metatheory.
Peter: I'm not sure that I have understood your point
correctly. For arithmetic, arithmutic, or what have
you, it remains true that the truth-theory of PA nonconservatively
extends PA, doesn't it?
Puter: As you can see from my example, T(PA* ) is
not conservative over PA*. But my point is that in
this case it does not mean anything interesting.
Although non-conservative over PA*, T(PA* ) does
not explain anything more than PA* in the sense
that arithmutic is left the same before and after we
ascend to its theory of truth. It is easy to see that
under disambiguation of the vocabulary of PA
between arithmutic in the base theory and arithmetic
in the metatheory, the non-conservativity
phenomenon vanishes.
Peter: Ok. There may have been a
misundertanding. I agree with you that the nonconservativity
of T(PA* ) over PA* is meaningless
and may just result from a fallacy of equivocation
between PA and PA*. But the point remains that the
theory of the truth of PA (and by this I now mean
explicitely arithmetic PA) should prove the
consistency of PA (idem), which PA does not. And
in that case, since PA is thought of as a theory of
arithmetic, and since you admit that consistency of a
formal system is an arithmetical fact, you have to
admit that the non-conservativity of T(PA) over PAarithmetic
shows truth to have an explanatory power
after all!
Puter: I don't think so, for the situation is in fact
exactly the same as before. Suppose I'm told that
the theory PA is true, but that I'm not sure whether it
is PA or PA* which is under discussion. I will in any
case be able to prove that the theory is consistent in
my truth-theoretic metatheory. For not only is T(PA)
non-conservative over PA, but so is T(PA* ). It is as
it should be since the consistency of PA has nothing
to do with the the way we think of PA, it has to do
only with its formal features. So whether I interpret
PA as arithmetic, arithmutic, or whatever in the
metatheory, consistency will follow from truth. Now
the further claim that we have so derived a truth
pertaining to the field of investigation of our base
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