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I got my midterm back today and I got 0 on the following question. There was no comments on this question at all from the marker.

For propositional formulas A and B, prove (or disprove) that if $A\models \lnot B $ is true then $\vdash_H \lnot(A\rightarrow\lnot B)$

Can someone tell me if the answer is true or not?

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I suspect that the point of it being true or not matters less to your instructor than does HOW you arrived at your answer. Why don't you post your work, and ask whether your proof is sound? – amWhy Nov 27 '12 at 3:16
I'm sure 1 mark is given for saying it's true or not and my proof was quite long. – user44322 Nov 27 '12 at 3:25
up vote 1 down vote accepted

Hint: Put $p$ for $A$ and $\neg p$ for $B$. Now ask: is $A\models \lnot B$ true? Is $\vdash_H \lnot(A\rightarrow\lnot B)$ true?

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That depends on whether $H$ is consistent, it seems. – Henning Makholm Nov 27 '12 at 13:56
Indeed it certainly will! But I thought this would be a good simple question with which to probe $H$ (revealing something significant, whichever answer we get). – Peter Smith Nov 27 '12 at 15:08
@user44322 What more did you write? – Peter Smith Nov 27 '12 at 21:31

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