Law of excluded middle: is ⊨ p∨¬p valid?
p ∨ ¬p is true in every row and needs no premises at all: every proposition is either true or false, with no third option.
Valid
⊨ p∨¬pEvery branch of the tableau closes, so nothing makes the premises true and the conclusion false at once.
Proof (semantic tableau)
- 1False: p∨¬pnegated conclusion
- 2False: pfrom line 1
- 3False: ¬pfrom line 1
- 4True: pfrom line 3
Branch closed: line 4 contradicts line 2.
closed branch