Extended Soundness Theorem for Propositional Tableaus and Boolean Interpretations

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

That is, for every collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$:


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

Proof
By definition of tableau proof, $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ means:


 * There exists a tableau confutation of $\mathbf H \cup \left\{{\neg\mathbf A}\right\}$.

By Tableau Confutation implies Unsatisfiable, it follows that $\mathbf H \cup \left\{{\neg\mathbf A}\right\}$ is unsatisfiable for boolean interpretations.

Therefore, if some boolean interpretation $v$ models $\mathbf H$:


 * $v \models_{\mathrm{BI}} \mathbf H$

then since $\mathbf H \cup \left\{{\neg\mathbf A}\right\}$ is unsatisfiable:


 * $v \not\models_{\mathrm{BI}} \neg\mathbf H$

Now by definition of the relation $\models_{\mathrm{BI}}$, it must be that:


 * $v \left({\neg \mathbf A}\right) = F$

By the truth table for $\neg$, this implies:


 * $v \left({\mathbf A}\right) = T$

which is to say $v \models_{\mathrm{BI}} \mathbf A$.

Hence:


 * $v \models_{\mathrm{BI}} \mathbf H$ implies $v \models_{\mathrm{BI}} \mathbf A$

that is, $\mathbf A$ is a $\mathrm{BI}$-semantic consequence of $\mathbf H$:


 * $\mathbf H \models_{\mathrm{BI}} \mathbf A$

which was to be shown.

Also see

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