structural:scholar_edition
Semantic argument

An alternate argument for the principle stems from model theory. A sentence P {\displaystyle P} is a semantic consequence of a set of sentences Γ {\displaystyle \Gamma } only if every model of Γ {\displaystyle \Gamma } is a model of P {\displaystyle P}. However, there is no model of the contradictory set (P ∧ ¬ P) {\displaystyle (P\wedge \lnot P)}. A fortiori, there is no model of (P ∧ ¬ P) {\displaystyle (P\wedge \lnot P)} that is not a model of Q {\displaystyle Q}. Thus, vacuously, every model of (P ∧ ¬ P) {\displaystyle (P\wedge \lnot P)} is a model of Q {\displaystyle Q}. Thus Q {\displaystyle Q} is a semantic consequence of (P ∧ ¬ P) {\displaystyle (P\wedge \lnot P)}.