De Morgan's law: is ¬(p∧q) ⊨ ¬p∨¬q valid?

¬(p ∧ q) does not say which conjunct fails, only that they cannot both hold — which is exactly ¬p ∨ ¬q. Denying a conjunction spreads the negation.

Valid¬(p∧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. 1¬(p∧q)premise
    1. 2¬p∨¬qnegated conclusion
      1. 3p∧q
        1. 4¬p
          1. 5¬q
            1. 6p
              1. 7q
                1. 8p

                2. 9q

closed branch

How semantic tableaux work →

Try in Calculator

More worked proofs