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I have the following proposition:

$$(P_1\leftrightarrow P_2)\leftrightarrow (P_1\rightarrow (P_2\rightarrow \neg P_1))$$ How to that it's not provable in Hilbert system?

Obviously it's not a tautology since if $P_1,P_2$ both get True then $P_1\leftrightarrow P_2$ gets true but $P_1\rightarrow (P_2 \rightarrow \neg P_1)$ gets false. But how to show it by Hilbert system?

Mauro ALLEGRANZA
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MathematicalPhysicist
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  • Similar questions have been asked before, for instance, see. – Tankut Beygu Apr 07 '22 at 07:48
  • With a counter-example i.e. a line of the truth table producing False. – Mauro ALLEGRANZA Apr 07 '22 at 07:53
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    You take your sound and complete system for classical propositional logic. Then treat that formula just like it's an axiom of the system with the same inference rules and other axioms. Then you derive a propositional variable. That shows that the additional axiom is not a tautology, since if it were a tautology deriving a propositional variable would be impossible. This can be useful when working with theorem provers which have an accompanying model checker. This is likely a very fast exercise using a program like Prover9. – Doug Spoonwood Apr 07 '22 at 17:47
  • @MauroALLEGRANZA How was the Hilbert system used with the truth table? – Doug Spoonwood Apr 07 '22 at 17:53

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