Soundness and Completeness of Semantic Tableaus/Corollary 1

Corollary to Soundness and Completeness of Semantic Tableaus

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

Let $T$ be a completed semantic tableau for $\mathbf A$.

Then $\mathbf A$ is satisfiable if and only if $T$ is open.

Proof

$\mathbf A$ is satisfiable if and only if it is not unsatisfiable.

By the Soundness and Completeness of Semantic Tableaus, $\mathbf A$ is satisfiable if and only if $T$ is not closed.

Since $T$ is completed, this is the case if and only if $T$ is open.

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.7$: Corollary $2.68$