Double negation: is ¬¬p ⊨ p valid?
¬¬p and p are true in the same rows, so a doubled negation can be dropped wherever it stands. Two negations cancel.
Valid
¬¬p ⊨ pEvery branch of the tableau closes, so nothing makes the premises true and the conclusion false at once.
Proof (semantic tableau)
- 1True: ¬¬ppremise
- 2False: pnegated conclusion
- 3False: ¬pfrom line 1
- 4True: pfrom line 3
Branch closed: line 4 contradicts line 2.
closed branch