[FOM] Deflationism and the Godel phenomena

[Torkel Franzen]

This is a follow-up on an earlier discussion on the list of Neil
Tennant's paper "Deflationism and the Gödel Phenomena" (Mind, July
2002). (For the earlier discussion, see the FOM archives for
April, 2003.)

Neil's paper presented a derivation of the Gödel sentence from a
reflection principle, stating that this showed how the deflationist
could embrace a "semantical argument" for the truth of the Gödel
sentence without using a truth predicate. To this it was objected
(by Jeffrey Ketland and myself) that the derivation was well known
and not in question: rather, the issue was how to justify the
reflection principle without invoking truth.

In the latest issue of Mind, there is a paper by Jeffrey presenting
this criticism, and a response by Neil.

In his response, Neil argues that the reflection principle is justified
by "engaging in suitable intellectual reflection", and is "a way of
expressing one's commitment to stand by one's earlier methods for
justifying one's assertions". He formulates this "suitable intellectual
reflection" as applied to the local reflection principle for a theory
(say PA), while emphasizing that only a weaker principle is actually
needed to derive the Gödel sentence. The local reflection principle
states

     Prov_S(phi)->phi

(with corner quotes around the first occurrence of "phi"). The
suitable intellectual reflection goes as follows:

    The deflationist might well wish to adopt all instances of this
    schema. After all, if he was willing to assert any sentence phi
    for which he had furnished an S-proof, why not then also be
    willing to assert any sentence phi for which he can furnish
    a proof to the effect that the sentence phi can be furnished
    with an S-proof?

This justification of the reflection principle is, however, grossly
insufficient.  It does not consider instances of the schema where we
can disprove phi.  In particular, taking phi to be "0=1", the schema
implies the consistency of S (as does the weaker reflection principle
used in Neil's argument). How is this to be justified? Saying that we
are willing to assert 0=1 if we have a proof that 0=1 is provable in
S is not convincing as an argument for the consistency of S.

A technical comment. Neil remarks in a footnote (referring to Rathjen)
that ACA implies every theorem provable in PA, PA+Con_PA,
PA+Con(PA+Con_PA), and so on, iterated to epsilon_0. This is perhaps
a bit misleading, since we only need 1-consistency for this. ACA
proves every theorem of the corresponding sequence of extensions
by uniform reflection. (Whether every arithmetical theorem of ACA
is provable in some theory in this sequence is a question for which
I know of no answer in the literature.)