Completeness Theorem for Semantic Tableaus

Theorem
Let $\mathbf A$ be a WFF of propositional logic.

Let $\mathbf A$ be unsatisfiable for boolean interpretations.

Then every completed tableau for $\mathbf A$ is closed.

Proof
The proof proceeds by contradiction.

That is, we assume that some open tableau $T$ for $\mathbf A$ exists, and prove that $\mathbf A$ must be satisfiable.

This claim is proved by inductively establishing the following claim for all nodes $t$ of $T$:


 * If a descendant leaf node of $t$ is marked open, then $\map U t$ is satisfiable.

By the Semantic Tableau Algorithm, this statement is immediate for leaf nodes.

For, sought satisfiability is the condition for a leaf to be marked open.

Inductively suppose that all children of $t$ satisfy the mentioned condition.

Now suppose that $t$ has a descendant marked open leaf.

Then obviously some child $t'$ of $t$ also has a descendant marked open leaf.

By induction hypothesis, $\map U {t'}$ is satisfiable.

First, the case that the formula $\mathbf B$ used at $t$ by the Semantic Tableau Algorithm is an $\alpha$-formula.

Then $\map U {t'}$ is obtained from $\map U t$ by replacing $\mathbf B$ by $\mathbf B_1$ and $\mathbf B_2$.

Here, $\mathbf B_1$ and $\mathbf B_2$ are the formulas such that $\mathbf B$ is semantically equivalent to $\mathbf B_1 \land \mathbf B_2$.

It follows that if:


 * $v \models_{\mathrm{BI}} \map U {t'}$

for some boolean interpretation $v$, then also:


 * $v \models_{\mathrm{BI}} \map U t$

by the boolean interpretation of $\land$.

Consequently, $\map U t$ is satisfiable.

Next, the case that $\mathbf B$ is a $\beta$-formula.

Then $\mathbf B$ is semantically equivalent to $\mathbf B_1 \lor \mathbf B_2$.

Then, by the Rule of Addition:


 * $\mathbf B_1 \models_{\mathrm{BI}} \mathbf B$
 * $\mathbf B_2 \models_{\mathrm{BI}} \mathbf B$

Consequently, by the definitions of $\map U {t'}$ and $\map U {t''}$:


 * $\map U {t'} \models_{\mathrm{BI}} \map U t$
 * $\map U {t''} \models_{\mathrm{BI}} \map U t$

By hypothesis, one of $\map U {t'}$ and $\map U {t''}$ must be satisfiable.

Hence, so must $\map U t$ be.

This establishes our claim.

Applying the claim to the root node, i.e., to the whole tableau, we obtain the result.