Definition:Finished Set of WFFs of Propositional Logic

Definition
Let $\Delta$ be a set of WFFs of propositional logic.

Then $\Delta$ is finished iff:


 * $\Delta$ is not contradictory


 * For each WFF $\mathbf C \in \Delta$, either $\mathbf C$ is basic or one of the following is true:
 * $\mathbf C$ has the form $\neg \neg \mathbf A$ where $\mathbf A \in \Delta$
 * $\mathbf C$ has the form $\left({\mathbf A \land \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$
 * $\mathbf C$ has the form $\neg \left({\mathbf A \land \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\neg \mathbf B \in \Delta$
 * $\mathbf C$ has the form $\left({\mathbf A \lor \mathbf B}\right)$ where either $\mathbf A \in \Delta$ or $\mathbf B \in \Delta$
 * $\mathbf C$ has the form $\neg \left({\mathbf A \lor \mathbf B}\right)$ where both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
 * $\mathbf C$ has the form $\left({\mathbf A \implies \mathbf B}\right)$ where either $\neg \mathbf A \in \Delta$ or $\mathbf B \in \Delta$
 * $\mathbf C$ has the form $\neg \left({\mathbf A \implies \mathbf B}\right)$ where both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
 * $\mathbf C$ has the form $\left({\mathbf A \iff \mathbf B}\right)$ where either:
 * both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$, or:
 * both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$;
 * $\mathbf C$ has the form $\neg \left({\mathbf A \iff \mathbf B}\right)$ where either:
 * both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
 * or:
 * both $\neg \mathbf A \in \Delta$ and $\mathbf B \in \Delta$.

Also see

 * Definition:Tableau Extension Rules: note the similarity.