Definition:Finished Set of WFFs of Propositional Logic

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.