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I know there are plenty of similar posts around, but I could not find an answer to this particular question (and I've been at it for two days now, getting nowhere). The proof I'm trying to construe in Hilbert style system is:

$A \rightarrow (B \rightarrow C) \vdash B \rightarrow (A \rightarrow C)$.

Axioms given:

A1: $(A \rightarrow (B \rightarrow A))$;

A2: $((A \rightarrow(B\rightarrow C))\rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C)))$;

A3: $((\neg B \rightarrow \neg A) \rightarrow ((\neg B \rightarrow A) \rightarrow B))$.

This is the third exercise of Mendelson's book Introduction to Mathematical Logic. This is not for a class; I'm going through the book again, after having discussed many of its topics previously (but never having been required to do Hilbert proofs—luckily). Please don't give an answer which gives a proof in another proof system along with the equivalence of that with a Hilbert style system; I'm trying to be able to do Hilbert proofs. Thanks in advance!

Jacco Oosterhuis
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  • I suggest that if you truly want to understand any proof system, such as this Hilbert-style system, you should try to understand how to translate a proof from one to another. This would enable you to work in the system you find most intuitive, and after that translate it easily to the system you want the proof in. My personal preference is natural deduction (Fitch-style) because it is as natural as its name suggests. The advantage of Hilbert-style systems is not in practical use but in theoretical proofs about proof systems. – user21820 Nov 21 '15 at 10:22
  • Do you have already proved the Deduction Theorem ? If so, it is quite straightforward assuming $A$ and $B$ and using modus ponens twice, followed by the Ded Th twice. – Mauro ALLEGRANZA Nov 21 '15 at 10:23
  • In the book, the deduction theorem is given after this exercise, so I figured the exercise is still to give it without the deduction theorem. It should be rather straightforward still, but I haven't been able to find it.

    Also, I always found other proof systems much easier, too (e.g. Gentzen sequent calculus). But precisely since I always skipped Hilbert proofs, I want to be able to do simple ones to see how they work (and so I know I can do them).

     – Jacco Oosterhuis Nov 21 '15 at 12:05
  • You can see T4 into the answer to this post. – Mauro ALLEGRANZA Nov 21 '15 at 12:34
  • Thanks for that link! I must've missed it. The proof seems rather long, even for a Hilbert style proof, but at least it gives me an idea of how to prove it. – Jacco Oosterhuis Nov 21 '15 at 18:47

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