Definition:Tableau Extension Rules

Definition
The tableau extension rules are a set of rules for manipulating labeled trees for propositional logic.

In the below:


 * $\mathbf A$ and $\mathbf B$ are general WFFs of propositional logic;


 * $T_k$ is a labeled tree for propositional logic in a propositional tableau chain.


 * Double Negation: If $t$ has an ancestor $\neg \neg \mathbf A$, extend $T_k$ by adding the child $\mathbf A$ of $t$.


 * And: If $t$ has an ancestor $\mathbf A \land \mathbf B$, extend $T_k$ by adding a child $\mathbf A$ and a grandchild $\mathbf B$ of $t$.


 * Nand: If $t$ has an ancestor $\neg \left({\mathbf A \land \mathbf B}\right)$, extend $T_k$ by adding two children $\neg \mathbf A$ and $\neg \mathbf B$ of $t$.


 * Or: If $t$ has an ancestor $\mathbf A \lor \mathbf B$, extend $T_k$ by adding two children $\mathbf A$ and $\mathbf B$ of $t$.


 * Nor: If $t$ has an ancestor $\neg \left({\mathbf A \lor \mathbf B}\right)$, extend $T_k$ by adding a child $\neg \mathbf A$ and a grandchild $\neg \mathbf B$ of $t$.


 * Implies: If $t$ has an ancestor $\mathbf A \implies \mathbf B$, extend $T_k$ by adding two children $\neg \mathbf A$ and $\mathbf B$ of $t$.


 * Not Implies: If $t$ has an ancestor $\neg \left({\mathbf A \implies \mathbf B}\right)$, extend $T_k$ by adding a child $\mathbf A$ and a grandchild $\neg \mathbf B$ of $t$.


 * Iff: If $t$ has an ancestor $\mathbf A \iff \mathbf B$, extend $T_k$ by adding two children $\mathbf A$ and $\neg \mathbf A$ of $t$, a child $\mathbf B$ of $\mathbf A$ and a child $\neg \mathbf B$ of $\neg \mathbf A$.


 * Exclusive Or: If $t$ has an ancestor $\neg \left({\mathbf A \iff \mathbf B}\right)$, extend $T_k$ by adding two children $\mathbf A$ and $\neg \mathbf A$ of $t$, a child $\neg \mathbf B$ of $\mathbf A$ and a child $\mathbf B$ of $\neg \mathbf A$.

In each case:
 * the ancestor is said to be used at $t$;
 * the other WFFs mentioned are said to be added at $t$.

Graphical Representation
These rules can be represented graphically as follows.


 * $\begin{array}{c}

\text{Double Negation}\\ \hline \vdots \\ \neg \neg \mathbf A \\ \vdots \\ t \\ \vert \\ \mathbf A \\ {} \\ {} \\ \end{array} \quad \begin{array}{c} \text{And}\\ \hline \vdots \\ \mathbf A \land \mathbf B \\ \vdots \\ t \\ \vert \\ \mathbf A \\ \Vert \\ \mathbf B \end{array} \quad \begin{array}{c} \text{Nor}\\ \hline \vdots \\ \neg \left({\mathbf A \lor \mathbf B}\right) \\ \vdots \\ t \\ \vert \\ \neg \mathbf A \\ \Vert \\ \neg \mathbf B \end{array} \quad \begin{array}{c} \text{Not Implies}\\ \hline \vdots \\ \neg \left({\mathbf A \implies \mathbf B}\right) \\ \vdots \\ t \\ \vert \\ \mathbf A \\ \Vert \\ \neg \mathbf B \end{array}$


 * $\begin{array}{ccccc}

& & \text{Or} & & \\ \hline & & \vdots & & \\ & & \mathbf A \lor \mathbf B & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \mathbf A & & & & \mathbf B \\ \end{array} \quad \begin{array}{ccccc} & & \text{Nand} & & \\ \hline & & \vdots & & \\ & & \neg \left({\mathbf A \land \mathbf B}\right) & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \neg \mathbf A & & & & \neg \mathbf B \\ \end{array} \quad \begin{array}{ccccc} & & \text{Implies} & & \\ \hline & & \vdots & & \\ & & \mathbf A \implies \mathbf B & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \neg \mathbf A & & & & \neg \mathbf B \\ \end{array}$


 * $\begin{array}{ccccc}

& & \text{Iff} & & \\ \hline & & \vdots & & \\ & & \mathbf A \iff \mathbf B & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \mathbf A & & & & \neg \mathbf A \\ \Vert & & & & \Vert \\ \mathbf B & & & & \neg \mathbf B \\ \end{array} \quad \begin{array}{ccccc} & & \text{Exclusive Or} & & \\ \hline & & \vdots & & \\ & & \neg \left({\mathbf A \iff \mathbf B}\right) & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \mathbf A & & & & \neg \mathbf A \\ \Vert & & & & \Vert \\ \neg \mathbf B & & & & \mathbf B \\ \end{array}$

Note that when both a child and a grandchild are added at the same node, the child and grandchild are connected by a double line.

How To Use
Tableaus can be used in two ways:


 * To build a formal proof of a WFF from a hypothesis set $\mathbf H$;


 * To build a model of a set of WFFs $\mathbf H$.

Also see

 * Models for Propositional Logic, which gathers the logical justifications for these rules.