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} $$

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.

Basic WFFs
From the above, it can be seen that WFFs which consist of either:


 * A single propositional symbol, or


 * The negation of a propositional symbol

can not be broken down into simpler WFFs by means of these rules.

These are called basic WFFs or literals.

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 $$\mathbf{H}$$.

Justification
The logical justifications for these rules are gathered together in Models for Propositional Calculus‎.