[FOM] Deflationism and the Godel phenomena
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Mon Feb 14 22:35:20 EST 2005
On Mon, 14 Feb 2005, Torkel Franzen wrote:
> ... 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.
On the contrary; the justification is given for arbitrary 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?
By means of the very obvious proof below.
> 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.
Really? Here is the argument that I would give for the consistency of S:
Reflection________________ ___________(1)
Prov_S(0=1)->0=1 Prov_S(0=1)
_______________________________ _____axiom
0=1 ~0=1
________________________
#
____________(1)
~Prov_S(0=1)
"Saying that we are willing to assert 0=1 if we have a proof that 0=1 is
provable" involves a big "if". This man's modus ponens is actually his
modus tollens.
> 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.
Actually, I don't say this. The iteration is to epsilon_(epsilon_0).
> This is perhaps
> a bit misleading, since we only need 1-consistency for this.
What is misleading here is taking my footnoted claim out of the
dialectical context of my disagreement with Ketland over how strong one's
justificatory resources should be. Ketland had admitted that
his preferred truth-theory (for the purposes of justifying the claim
that the G"odel sentence for PA is true) was intertranslatable with ACA.
The purpose of my footnote was to point out how much further (i.e.
higher in the hierarchy of consistency strengths) than PA+Con(PA) this
would take us, even though PA+Con(PA) is all that one needs.
Neil Tennant
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