Soundness and Completeness of Semantic Tableaux/Corollary 2

Corollary to Soundness and Completeness of Semantic Tableaux

Let $\mathbf A$ be a WFF of propositional logic.

Then $\mathbf A$ is a tautology if and only if $\neg \mathbf A$ has a closed tableau.

Proof

By Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology if and only if $\neg \mathbf A$ is unsatisfiable.

By the Soundness and Completeness of Semantic Tableaux, this amounts to the existence of a closed tableau for $\neg \mathbf A$.

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.7$: Corollary $2.69$