[FOM] question about Mendelson's Intro to Math Logic

[Robert Lubarsky]

My apologies for the typo in my original question. The new axiom is a form
of contraposition: (not B -> not C) -> (C -> B).

 

Bob Lubarsky

 

From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Robert Lubarsky
Sent: Sunday, February 09, 2014 7:58 PM
To: Foundations of Mathematics
Subject: [FOM] question about Mendelson's Intro to Math Logic

 

There's an exercise from Mendelson's Intro to Mathematical Logic that I
cannot do, and would like an answer to. In the 5th edition, it's 1.59 on p.
39. Namely, he gives three axioms for a Hilbert-style propositional proof
system. The first two are essentially the combinators s and k. The third is 

(not C -> not B) -> [ (not C -> B) -> C ]. The exercise in question is to
show that, if you replace this last schema with (not B -> not C) -> (B -> C)
then the new system proves the same as the old. I don't see how to do this. 

 

My attempt is as follows. We can use the deduction theorem. So assume (not C
-> not B) and (not C -> B), with the goal of proving C. From the new schema,
we get  B -> C. We can compose that with (not C -> B) to get (not C -> C). I
don't see how to go from that to C within this system.

 

Bob Lubarsky

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