Soundness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
Tableau proofs (in terms of propositional tableaus) are a sound proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:


 * $\vdash_{\mathrm{PT}} \mathbf A$ implies $\models_{\mathrm{BI}} \mathbf A$

Proof
This is a corollary of the Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations:

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 for Propositional Tableaus and Boolean Interpretations in which it is proved that:
 * If $\models \mathbf A$ then $\vdash \mathbf A$.


 * Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations