Conservativeness of PAV
This post continues on the theme of the argument against deflationism, concerning the conservativness of truth theories. I want to discuss the role of induction.
It turns out that this is a very complicated issue, and people have slipped up a bit over it. But there are some points that aren't too complicated, and one concerns the result of extending with new vocabulary, and therefore new induction instances, but without new axioms for the new vocabulary. If there are no new axioms, then the result is conservative.
Suppose we let be the usual first-order language of arithmetic (non-logical vocabulary is ) and let
is a predicate symbol (say, of arity ). Let be the extended language. Let be the usual system of Peano arithmetic in , with the induction scheme,
is a formula (possibly with parameters), and is the result of substituting the term for all free occurrences of , relabelling bound variables inside if necessary to avoid collisions).
Definition: is the result of extending with all instances of induction for the extended language .
Theorem: conservatively extends .
Proof. The "proof idea" is to give a translation that maps every proof (derivation) in the "bigger" theory into a proof in the "smaller" theory. We define a translation
symbol, including variables and terms, the translation is the identity mapping. For compounds, we assume just commutes. For a -ary predicate symbol , we translate an atomic formula as follows:
is any -formula. This translation is crazy, of course. But we have no axioms constraining the symbol , so we can translate it any way we like. (If we did have such axioms, say , we would need to try and translate as a theorem of .)
The formula is then an -formula. This means that, for any , is equivalent to some -forrnula, say . Now consider the induction axiom for any :
is equivalent (because commutes with connectives and quantifiers) to
. Hence, it is an axiom of .
Finally, suppose that
. Then applying the translation to all the formulas in the derivation of converts the derivation into a derivation of in (in ), as each induction instance is translated to an induction instance in (which is an axiom of ) and the translation preserves too. So,
This may be applied to the case where the new vocabulary is a predicate , perhaps to be thought of as a truth predicate, but we do not include any new axioms for itself. Halbach 2011 calls the corresponding theory . So, conservatively extends .
It turns out that this is a very complicated issue, and people have slipped up a bit over it. But there are some points that aren't too complicated, and one concerns the result of extending
Suppose we let
be some new vocabulary. Each
(here
Definition:
Theorem:
Proof. The "proof idea" is to give a translation that maps every proof (derivation) in the "bigger" theory into a proof in the "smaller" theory. We define a translation
as follows. For any
where.
The formula
Its translation under
And this is an induction axiom in
Finally, suppose that
where,
QED..
This may be applied to the case where the new vocabulary is a predicate
Comments
Post a Comment