Material implication: is p→q ⊨ ¬p∨q valid?

p → q holds in exactly the rows where ¬p ∨ q holds: either the antecedent fails, or the consequent holds. A conditional is a disjunction in disguise.

Validp→q ⊨ ¬p∨q

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¬p∨qnegated conclusion
      1. 3¬p
        1. 4q
          1. 5p
            1. 6p

            2. 7q

closed branch

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