Definition:Finished Set of WFFs of Propositional Logic

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

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 $$\left({\mathbf A \and \mathbf B}\right)$$ where both $$\mathbf A \in \Delta$$ and $$\mathbf B \in \Delta$$;
 * $$\mathbf C$$ has the form $$\neg \left({\mathbf A \and \mathbf B}\right)$$ where either $$\neg \mathbf A \in \Delta$$ or $$\neg \mathbf B \in \Delta$$;
 * $$\mathbf C$$ has the form $$\left({\mathbf A \or \mathbf B}\right)$$ where either $$\mathbf A \in \Delta$$ or $$\mathbf B \in \Delta$$;
 * $$\mathbf C$$ has the form $$\neg \left({\mathbf A \or \mathbf B}\right)$$ 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 $$\mathbf B \in \Delta$$.

Notice the similarity between these and the tableau expansion rules.