FOM: reflection principles and the truth-predicate
neilt at mercutio.cohums.ohio-state.edu
Mon Jan 4 20:24:28 EST 1999
Volker Halbach writes:
> I conclude that what is actually going on is the addition of a truth
> predicate; the consistency statement and the proof-theoretic reflection
> principles are only consequences of this. In the literature peopl have
> preferred to suppress the use of the truth predicate and to jump directly
> from the theory itself to the reflections principles. I guess the main
> reason is that the formulation of the latter does not require an expansion
> of the language, whereas a truth predicate is a new symbol in the language.
> Therefore we should focus on the addition of truth predicates (and suitable
> axioms for it) rather than consistency statements and the proof-theoretic
> reflection principles.
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.
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).
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?
Otherwise it would seem that prosententialist reflection on a given
system is philosophically preferable to the importation of a
truth-predicate. By remaining expressively conservative, reflection
harnesses that inner drive of theories dealing with indefinitely
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