Principle of explosion: is p, ¬p ⊨ q valid?
From p and ¬p anything follows: no assignment makes both premises true, so none can make them true while the conclusion fails.
Valid
p, ¬p ⊨ qEvery branch of the tableau closes, so nothing makes the premises true and the conclusion false at once.
Proof (semantic tableau)
- 1True: ppremise
- 2True: ¬ppremise
- 3False: qnegated conclusion
- 4False: pfrom line 2
Branch closed: line 4 contradicts line 1.
closed branch