Modus ponens: is p→q, p ⊨ q valid?

If p → q holds and p is true, q follows. Modus ponens is the rule most proofs lean on, and the tableau below closes on every branch.

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. 2ppremise
      1. 3qnegated conclusion
        1. 4p

        2. 5q

closed branch

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