Denying the antecedent: is p→q, ¬p ⊨ ¬q valid?
A fallacy: p → q says nothing about what happens when p fails, so ¬p leaves q open. The open branch below shows q true while p is false.
Invalid
p→q, ¬p ⊨ ¬qA branch stays open, so the premises can hold while the conclusion fails.
Proof (semantic tableau)
- 1True: p→qpremise
- 2True: ¬ppremise
- 3False: ¬qnegated conclusion
- 4False: pfrom line 2
- 5True: qfrom line 3
- 6False: pfrom line 1
Branch open.p = ⊥, q = ⊤
- 7True: qfrom line 1
Branch open.p = ⊥, q = ⊤
open branch
Countermodels
Under these assignments every premise is true but the conclusion is false, so the conclusion does not follow.
p = ⊥, q = ⊤