Definition:Finished

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Definition

Finished Set of WFFs of Propositional Logic

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

Then $\Delta$ is finished if and only if:

  • 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 $\paren {\mathbf A \land \mathbf B}$ where both $\mathbf A \in \Delta$ and $\mathbf B \in \Delta$
    • $\mathbf C$ has the form $\neg \paren {\mathbf A \land \mathbf B}$ where either $\neg \mathbf A \in \Delta$ or $\neg \mathbf B \in \Delta$
    • $\mathbf C$ has the form $\paren {\mathbf A \lor \mathbf B}$ where either $\mathbf A \in \Delta$ or $\mathbf B \in \Delta$
    • $\mathbf C$ has the form $\neg \paren {\mathbf A \lor \mathbf B}$ where both $\neg \mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
    • $\mathbf C$ has the form $\paren {\mathbf A \implies \mathbf B}$ where either $\neg \mathbf A \in \Delta$ or $\mathbf B \in \Delta$
    • $\mathbf C$ has the form $\neg \paren {\mathbf A \implies \mathbf B}$ where both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
    • $\mathbf C$ has the form $\paren {\mathbf A \iff \mathbf B}$ 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 $\neg \paren {\mathbf A \iff \mathbf B}$ where either:
      • both $\mathbf A \in \Delta$ and $\neg \mathbf B \in \Delta$
    • or:
      • both $\neg \mathbf A \in \Delta$ and $\mathbf B \in \Delta$.


Finished Branch of Propositional Tableau

Let $T$ be a propositional tableau.

Let $\Gamma$ be a branch of $T$.


Then $\Gamma$ is finished if and only if:

$(1): \quad \Gamma$ is not contradictory
$(2): \quad$ Every non-basic WFF on $\Gamma$ is used at some node of $\Gamma$.


That is, $\Gamma$ is finished if and only if the set $\Delta$ of WFFs of propositional logic which occur along $\Gamma$ is a finished set‎.


Finished Propositional Tableau

Let $T$ be a propositional tableau.


Then $T$ is finished if and only if every branch of $T$ is either finished or contradictory.