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 $\mathbf H = \O$.

Hence the result.

$\blacksquare$

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

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.8$: Soundness