FOM: Franzen and Black re. con-completeness
Torkel Franzen
torkel at sm.luth.se
Tue Feb 10 02:19:48 EST 1998
Mic Detlefsen says:
>It seems that Joe Shipman and Torkel Franzen want to deny (I).
I'm not denying anything, as far as I am aware. Your statement (I) is
(I) (neg Con(PA)-#) is true and assertable.
and neg Con(PA)-#, if I understand your notation, is
If neg-Con(PA) is provable in PA, then PA is inconsistent.
Now as a material conditional this statement is trivially true,
since neg-Con(PA) is not in fact provable in PA. If you formulate it
as a counterfactual conditional
If neg-Con(PA) were provable in PA, then PA would be inconsistent
I still see no reason to deny it.
My one and only objection to your line of thought has been that you
haven't explained how
(1) If S were provable in PA, then PA would be inconsistent
is to be understood. In particular, I'm wondering what sort of proof
of the corresponding material conditional
(2) If S is provable in PA, then PA is inconsistent
would establish (1). (I'm not assuming that a proof in any particular
formal system is required.) This is a pressing question, since you apparently
do not want to accept a proof of (2) of the following form as
establishing (1):
S is not provable in PA. So, if S is provable in PA, then
(by ordinary propositional logic), PA is inconsistent.
---
Torkel Franzen
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