Affirming a disjunct: is p∨q, p ⊨ ¬q valid?
A fallacy: p ∨ q is inclusive, so both sides may hold at once. Knowing p tells you nothing about q, as the open branch below shows.
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
- 4True: qfrom line 3
- 5True: pfrom line 1
Branch open.p = ⊤, q = ⊤
- 6True: 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 = ⊤