Finite Main Lemma of Propositional Tableaus

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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