# Finite Main Lemma of Propositional Tableaus

## Lemma

Let $\mathbf H$ be a finite set of WFFs of propositional logic.

Either $\mathbf H$ has a tableau confutation or $\mathbf H$ has a model.

## Proof

Let $\mathbf H$ be a finite set of WFFs of propositional logic which does not have a tableau confutation.

By the Tableau Extension Lemma, the tableau which consists only of a root node with hypothesis set $\mathbf H$ can be extended into a finite finished tableau $T$.

The tableau $T$ still has root $\mathbf H$.

Since $T$ is not a confutation, it has a finished branch $\Gamma$.

By the Corollary to the Finished Branch Lemma, $\Gamma$ has a model, $\mathcal M$, say.

In particular, $\mathcal M$ is a model of $\mathbf H$ as required.

$\blacksquare$

## Comment

From Tableau Confutation implies Unsatisfiable, we already know that $\mathbf H$ can not have both a tableau confutation *and* a model.

This result gives us that $\mathbf H$ has a tableau confutation iff $\mathbf H$ does not have a model.

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.10$: Completeness: Lemma $1.10.1$