Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations

Theorem
Let $\mathrm{PT}$ be the proof system of propositional tableaus.

Let $\mathrm{BI}$ be the formal semantics of boolean interpretations.

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

Let $\mathbf A$ be a propositional formula.

If $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$, then $\mathbf H \models_{\mathrm{BI}} \mathbf A$.

That is, if $\mathbf A$ is a $\mathrm{PT}$-provable consequence of $\mathbf H$, then $\mathbf A$ is also a $\mathrm{BI}$-semantic consequence of $\mathbf H$.

Proof
We are given $\mathbf H$ and $\mathbf A$.

By definition, $\mathbf H \vdash \mathbf A$ means there exists a tableau proof of $\mathbf A$ from $\mathbf H$.

Thus we have a tableau confutation of $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$.

So, by Tableau Confutation implies Unsatisfiable, $\mathbf H \cup \left\{{\neg \mathbf A}\right\}$ has no model.

That is, no model of $\mathbf H$ is also a model of $\neg \mathbf A$.

So if $\mathcal M$ is a model of $\mathbf H$, $\mathcal M$ is also a model of $\mathbf A$.

Thus, by definition of semantic consequence, $\mathbf H \models \mathbf A$.

Also see

 * Extended Completeness Theorem of Propositional Logic, which proves:
 * If $\mathbf H \models \mathbf A$, then $\mathbf H \vdash \mathbf A$.