Definition:Tableau Extension Rules

The tableau extension rules are a set of rules in the deductive apparatus of propositional calculus.

They can be formulated as follows.

In the below:
 * $$\mathbf{A}$$ and $$\mathbf{B}$$ are general WFFs of propositional calculus;
 * $$T_k$$ is a labeled tree for propositional logic in a propositional tableau chain.

The rules

 * 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 $$\mathbf{A} \and \mathbf{B}$$, extend $$T_k$$ by adding a child $$\mathbf{A}$$ and a grandchild $$\mathbf{B}$$ of $$t$$.


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


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


 * Nor: If $$t$$ has an ancestor $$\neg \left({\mathbf{A} \or \mathbf{B}}\right)$$, 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} \and \mathbf{B} \\ \vdots \\ t \\ \vert \\ \mathbf{A} \\ \Vert \\ \mathbf{B} \end{array} \quad \begin{array}{c} \text{Nor}\\ \hline \vdots \\ \neg \left({\mathbf{A} \or \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} \or \mathbf{B} & & \\ & & \vdots & & \\ & & t & & \\ & \diagup & & \diagdown & \\ \mathbf{A} & & & & \mathbf{B} \\ \end{array} \quad \begin{array}{ccccc} & & \text{Nand} & & \\ \hline & & \vdots & & \\ & & \neg \left({\mathbf{A} \and \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} $$

(If anyone can find a neater way of rendering the above graphs, please go ahead and do so.)

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 set of premises $$\mathbf{H}$$;


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