Disjunctive syllogism: is p∨q, ¬p ⊨ q valid?

A disjunction needs at least one true side, so p ∨ q with ¬p leaves q. Ruling out one disjunct leaves the other standing.

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¬ppremise
      1. 3qnegated conclusion
        1. 4p
          1. 5p

          2. 6q

closed branch

How semantic tableaux work →

Try in Calculator

More worked proofs