[FOM] Deflationism and the Godel phenomena
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Fri Feb 18 00:21:06 EST 2005
Timothy Y. Chow wrote:
> (3) I will not accept 0=1.
>
> So, does this all boil down to a
> complaint that Neil Tennant did not explicitly state (3)?
In a sense, yes. And for reflection principles in general, the required
assumptions must be stronger than "I will not accept 0=1". It's possible
that something like (REF-m) below would be needed.
After reading Neil Tennant's 2002 paper, I considered three kinds of
assumption that might (implicitly) be playing a hidden role.
(A: Self-Consistency)
Possibly: A reasoner may assume his own (arithmetic) consistency (as in (3)
above).
(B: Self-Soundness)
Possibly: A reasoner may assume his own (arithmetic) soundness (roughly, the
principle "If I will accept phi, then phi").
(C: Mental Reflective Closure)
Possibly, Tennant is assuming this. The set Acc(m) of arithmetic sentences
that a mind m will accept is closed under some sort of Reflection Operator
(which maps a formal system S to some reflection principle or scheme).
Strategy (A) was explained by Timothy Chow in the previous message.
Strategy (B) uses just two premises:
(i) Prov_S(phi)-> (I will accept phi)
(ii) (I will accept phi) -> phi
to get the required conclusion:
(iii) Prov_S(phi) -> phi.
But this requires (ii). Can that be right?
To work out (C) a bit more, the required assumption would have to be some
sort of Reflective Closure Principle for the mind. Here is a sketch. Suppose
that S is a formal system for arithmetic, Thm(S) is its set of theorems and
Rfn(S) is its local reflection principle. Then strategy (C) is based on the
assumption:
(REF-m) If Thm(S) subset Acc(m), then Rfn(S) subset Acc(m).
I found these possible strategies (A), (B) and (C) all implausible. I didn't
believe that Neil Tennant had explicitly intended any of these. Some remarks
could be interpreted to involve something like (REF-m), but I don't know.
If Tennant disavows the weakest one (A) (i.e., (3) above), then I don't see
how the reasoner even gets Con(S), given that the reasoner accepts S.
(Unless there is some further thing that I'm missing.)
> Question for Torkel Franzen and Jeffrey Ketland: If Neil Tennant
> explicitly avows (3), does that answer your objection that his
> justification of the reflection principle is incomplete and doesn't
> account for self-contradictory phi?
If Neil Tennant disavows (3), then I think he's failed to justify even
Con(S) (given that the reasoner accepts all the theorems of S).
If he explicitly avows (3), then I think we are into very deep waters
concerning what might be called the "self-reflective properties" of the
human mind.
--- Jeff
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