Contraposition: is p→q ⊨ ¬q→¬p valid?

p → q and ¬q → ¬p say the same thing: nothing can make the antecedent true while the consequent fails. A conditional equals its contrapositive.

Validp→q ⊨ ¬q→¬p

Every branch of the tableau closes, so nothing makes the premises true and the conclusion false at once.

Proof (semantic tableau)

  1. 1p→qpremise
    1. 2¬q→¬pnegated conclusion
      1. 3¬q
        1. 4¬p
          1. 5q
            1. 6p
              1. 7p

              2. 8q

closed branch

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