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I'm looking at Wikipedia's article on Intuitionist logic, and I can't figure out how it would prove $(p \rightarrow p)$.

Does it prove $(p \rightarrow p)$? If yes, how?

If no, is there a correct (or reasonable, if there is no "correct") axiomatization of intuitionistic logic available anywhere online?

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As you can see from the answer below, the "standard" derivation, in an Hilbert-style proof system, of $p \rightarrow p$ use only the propositional axioms THEN-1 and THEN-2. They are both intuitionistically available. In classical logic, where $p \rightarrow q$ is equivalent to $\lnot p \lor q$, the tautology $p \rightarrow p$ is equivalent to $\lnot p \lor p$, i.e. to Excluded Middle; this does not work in intuitionistic logic, where the connectives are not interdefinable. –  Mauro ALLEGRANZA Mar 29 at 11:38

1 Answer 1

up vote 7 down vote accepted

  • $p→(p→p)$ (THEN-1)
  • $p→((p→p)→p)$ (THEN-1)
  • $(p→((p→p)→p))→((p→(p→p))→(p→p))$ (THEN-2)
  • $(p→(p→p))→(p→p)$ (MP from lines 2,3)
  • $(p→p)$ (MP from lines 1,4)
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