In an indirect proof, is it possible to reject the assumption based on contradiction of a premise?

Question

Irene Simon · Accepted Answer

This is a proof that   x  ∈  A    ⟹    ¬  P. It can be simplified, since everything between &quot;Assume   P&quot; and &quot;Derive   x  ∉  A&quot; is just a proof that   P    ⟹    x  ∉  A, from which by contrapositive we get   x  ∈  A    ⟹    ¬  P, which was to be demonstrated.You can&#039;t in general conclude from this that   P is always false, of course.

brasocas6 · Accepted Answer

Yes. Deriving a contradiction from the assumption of   P is a proof for   ¬  P.Although apparently similar, this is not actually Reduction ad Absurdum.This is the Rule of Negation Introduction.   Unlike RAA this is an intuitionistically valid rule of inference.                                                        Σ              ,              ¬              P              ⊢              ⊥                                      Σ              ⊢              ¬              ¬              P                                                            ¬            I                                      Σ        ⊢        P                        ¬      ¬      E

In an indirect proof, is it possible to reject the assumption based on contradiction of a premise?

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