Natural deduction proof that ( P ↔ ¬ P ) is a contradiction, without first deriving ( P ∨ ¬ P )

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Natural deduction proof that   (  P  ↔  ¬  P  ) is a contradiction, without first deriving   (  P  ∨  ¬  P  )

helsedel1v · Accepted Answer

One possible route is proving   ¬  (  P  ↔  ¬  P  ) without premises. Following that path, this would be a Gentzen-style proof                                                        [              P              ]                            [              P              ↔              ¬              P              ]                                                                                            [                    P                    ]                                        ¬                    P                                                        ¬                    P                                                              ¬              I              n              t              r              o                                      ↔        E        l        i        m                                                    [              ¬              P              ]                            [              P              ↔              ¬              P              ]                                                                                            [                    ¬                    P                    ]                                        P                                    P                                            ¬              E              l              i              m                                      ↔        E        l        i        m                    ¬        (        P        ↔        ¬        P        )              ¬  E  l  i  m

Natural deduction proof that (P harr not P) is a contradiction, without first deriving (P vee not P)

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