Reduction and Elimination in Philosophy and the Sciences

Deflationism and Conservativity: Who did Change the Subject?
Henri Galinon, Paris, France
1. The Problem
Deflationists about truth hold that truth is not a substantial
property. But what counts as a substantial property? We
shall be interested in the thesis that the following is a necessary
and sufficient condition for the non-substantiality of
truth:
114
(Conservativity) The theory of truth of any given
theory A is a conservative extension of A.
Suppose that (Conservativity) holds, then the deflationist
would have some right to claim that truth is an explanatorily
thin property: for it would show that whenever non semantical
facts can be explained by a theory being true,
they can also be explained by the theory. Suppose (Conservativity)
does not hold; then there is a theory A, and
sentences in the A-vocabulary that witness nonconservativity;
the provability in our theory of truth of those
A-unprovable true LA -sentences would constitute some
evidence against the deflationary thesis that truth is
explanatorily dispensable.
As a matter of logical fact, some putative theories of
truth have the conservativeness property over some given
base theories, whereas others don't. But, the conservativity
argument against deflationism continues, conservative
theories of truth are not acceptable, because they fail to
meet an essential requirement. To be sure, the concept of
truth features in all these theories, that is to say a predicate
satisfying Tarski's convention-T. But having the concept
of truth is not enough for a theory to be the theory of
the truth of a given theory. For truth ascriptions come with
epistemic commitments, and theories of truth must account
for them. Ketland (1999), in particular, has argued that
holding a theory to be true is not only to hold all its theorems
to be true (distributively, so to speak) but also to hold
that all of its theorems are true (resp. collectively). Consequently
the truth theory of A has to prove reflection principles
for A: For all x, if Pr(x) then T(x).
Shapiro (1998) has an argument for a related conclusion.
He offers a perfectly natural explanation (I'll say of
what in a minute) involving the concept of truth, and argues
that in any good theory of truth for A we should be
able to carry out the reasonning. The example, unsurprinsingly,
involves gödelian phenomena: the Gödel sentence
conPA, it is well known, is a true and undecidable statement
of PA; but why is it that conPA is true? A natural
explanation goes like this, according to Shapiro:
[…] all the axioms of PA are true, and inference
rules preserve truth. Thus every theorems of PA are
true. It follows that 0=1 is not a theorem and so PA
is consistent. (Shapiro(1998), p.505. Shapiro uses
« A » where I write « PA »).
As we know, there are arithmetical sentences expressing
(under codings) the consistency of PA, and these sentences
are not theorems of PA. Shapiro's argument is then
that they should be provable in the theory of the truth of
PA.
To sum up: theories of truth come with some epistemic
commitments, and those commitments yield nonconservativity
results of truth theories over their base the-
ory; hence truth is not explanatorily thin: knowledge of the
truth of an arithmetical theory T yields new arithmetical
knowledge beyond T.
We agree that from truth ascriptions consistency ascriptions
should follow. We shall argue, however, that the
conservativity argument against deflationism is flawed 1 .
2. The Fable
Let us go into the fantasy of imagining a concurring civilization
where people call themselves earthlungs.
Earthlungs resemble us in every respect, except that in
mathematics they not only study arithmetic, but also have
come to recognize the interest and significance of arithmutic.
In fact they have come to believe that natural numburs
are the real elementary blocks constituting the universe,
and they are for this reason much interested in studying
them. A partial axiomatization of arithmutic is obtained by
PA + ¬conPA, where ¬conPA denotes the negation of a
given sentence of the language of PA that is true in N (the
standard model of PA) if and only if PA is consistent. Further
axioms have been proposed but they are much debated
at the moment and so we leave them aside. As it
happens, earthlungs mostly use only the PA-part of arithmutic.
Moreover, they use the same conventions for formalism
as we do when doing logic and, believe it or not,
they call PA the partial axiomatisation of arithmutic which
is identical to our PA (a nice starting point for a philosophical
vaudeville). We will sometimes write PA * to denote their
axiomatization and distinguish it from our PA. That is PA *
and PA are formally identical but intentionaly different, the
first being intended as speaking about numburs, while the
second is to be understood as speaking about numbers.
Those people also have two Gudule Theorems that, I have
to admit, are just as good as our Gödel Theorems. They
usually state them as follows:
First theorem: If T is a consistent, recursively enumerable
and sufficiently rich theory, then T is incomplete.
Second theorem If PA is consistent then:
PA does not prove ∀x (¬PrPA (x), ⎡0=1⎤)
All this is standard on Urth. The second theorem has especially
been welcome since it had long been an open
question whether ¬conPA* was independent of PA*. Now
when they hear us say that G-d-l's theorems show that
there are true statements undecidable in PA, they agree,
but they do not agree that conPA is one of them 2 !
Now Peter, a guy from here who doesn't know much
about earthlungs, once decided to engage Puter, one of
theirs, in a conversation about the explanatory power of
truth (it was raining hard that Sunday). Here's the conversation.
(Caveat: I have tried to disambiguate occurences of
1 Field (1999) has an answer to the conservativity argument and we basically
agree with the general lines developped there. We can think of our argument
as a variation on his own. We think our version is worth developing, though,
since it crucially avoids to take a stand on contentious claims about which
axioms are "essential to truth" and which are not (especially in connection with
the problem raised by induction axioms involving the truth predicate).
2 This is just because N is not the salient interpretation of PA in earthlung
conversational contexts.