FOM: reflection principles and the truth-predicate
Volker Halbach
Volker.Halbach at uni-konstanz.de
Mon Jan 11 13:02:53 EST 1999
In a recent posting I claimed that the justification of the reflection
principles requires a truth predicate. In reply to this Neil Tennant wrote:
>There is, of course, the converse view---that of the so-called
>prosententialists (after the paper 'A Prosentential Theory of Truth',
>by Grover, Camp and Belnap, Philosophical Studies 1975). The
>prosententialists would seek to re-analyze every theoretical use of a
>truth-predicate through the use of prosentences, such as are involved
>in the reflection principles.
The proof-theoretic reflection principles are formulated without
prosentences. As far as I remember, a prosentential reformulation would
read as follows:
For all A: If "A" is provable in T, then this sentence.
"this" refers to A. This amounts more or less to a formulation with a
"substitutional" quantifier:
For all A: If "A" is provable in T, then A.
I don't like the term "substitutional", because this suggests that it has a
certain semantics; but we want to have axioms and rules governing the use
of such a quantifier whose variables may occur inside and outside of quotes.
I am not aware of an axiomatization of either the prosentences or the
substitutional quantification, but I would expect that the result would be
equivalent to the addition of a truth predicate with suitable axioms.
Furthermore prosentences and substitutional variables are not contained in
the language of T, because otherwise Tarski's Theorem would apply (This is
again only a guess, because I don't know the appropriate axiomatizations of
these devices).
So I ammend my claim in the following way: The justification of the
proof-theoretic reflection principles requires the use of a truth predicate
or an equivalent device (prosentences, substitutional quantifiers).
>While it is correct to say that the formulation of reflection
>principles does not require an expansion of the language, is not the
>real concern how to get a hold of the extra strength involved in
>extending one's *theory* (with or without the addition of new
>vocabulary)? Adopting a reflection principle is to extend one's theory
>in the original language. Adopting a new predicate (such as a
>truth-predicate) means that existing axiom schemata (such as
>mathematical induction) acquire infinitely many new instances. While
>the "shape" of the theory might not have changed (e.g., it still has
>the same axiom schemes) it will nevertheless have a properly extended
>set of axioms (because of the new instances of those axiom schemes).
This is an important point. Extending PA or ZFC by a new symbol is
straightforward. But how should we extend a restricted induction axiom,
say, of PRA? There are many open questions and I don't have answers.
>Wouldn't a better argument for the use of a truth-predicate rather
>than reflection principles have to consist in some kind of
>demonstration that the use of a truth-predicate allows one to achieve
>some desired result in a way that is theoretically more economical
>than would be the use of reflection principles?
In general, truth predicates as axiomatized by the Tarskian "inductive"
clauses for truth is much stronger than reflection principles. Tarskian
truth for PA gives a theory equivalent to full ACA, while uniform
reflection for PA is equivalent to uniform reflection up to and including
epsilon_0. Thus I would say that the do not fully exploit our assumptions
we make when we justify the reflection principles. What we implicitly
assume is much stronger than even the uniform reflection principle. This is
a good reason to make the assumptions explicit. Again, may be, the
prosentential approach is equivalent.
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