Syntactical proof in a Hilbert system without additional lemmas

Asked Apr 21 '20 at 21:57

Active Apr 22 '20 at 08:59

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How do I proof $\neg \neg p \rightarrow p$ in the following Hilbert system without the use of additional lemmas (derive purely syntactical)?

$A \rightarrow (B \rightarrow C)$
$(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C))$
$(\neg A \rightarrow \neg B) \rightarrow ((\neg A \rightarrow B) \rightarrow A)$

In Logic in action there is a proof for $p \rightarrow p$, which is easy to grasp (pdf-version of the textbook onchapter 2 page 23) and in this pdf file (page 12) there is a proof for the exact problem of this question, except that it uses additional lemmas and transformations.

edited Apr 22 '20 at 08:59

asked Apr 21 '20 at 21:57

John Eren

2

Does this answer your question? Proof of a theorem in Hilbert's system It is enough to copy-paste the "Lemma" into a single derivation: they are not necessary, but used only to improve readibility. – Mauro ALLEGRANZA Apr 22 '20 at 17:16
Hey, thanks for the reference! I am not quite sure where ¬β→¬¬β comes from in the derivation of Lemma 1? (which would pretty much answer the question, but I cant make sense of "assumed [a]" right next to it either) – John Eren Apr 23 '20 at 22:01
Ah, I got it now, thank you very much! – John Eren Apr 23 '20 at 22:06
You are welcome :-) – Mauro ALLEGRANZA Apr 24 '20 at 07:33

Syntactical proof in a Hilbert system without additional lemmas

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