Soundness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
If a propositional formula has a tableau proof, then it is a tautology.

That is:
 * If $\vdash \mathbf A$ then $\models \mathbf A$.

Proof
This is a corollary of the Extended Soundness Theorem of Propositional Logic:

Let $\mathbf H$ be a countable set of propositional formulas.

Let $\mathbf A$ be a propositional formula.

If $\mathbf H \vdash \mathbf A$, then $\mathbf H \models \mathbf A$.

In this case, we have $\mathbf H = \varnothing$.

Hence the result.

Also see

 * Completeness Theorem of Propositional Logic in which it is proved that:
 * If $\models \mathbf A$ then $\vdash \mathbf A$.


 * Extended Soundness Theorem of Propositional Logic