FOM: Franzen and Black re. con-completeness

Torkel Franzen torkel at
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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