Definition:Propositional Tableau

Definition
A propositional tableau with root node $\mathbf H$ is either: whose hypothesis set is $\mathbf H$.
 * A finite propositional tableau, or
 * An infinite propositional tableau

Finite Propositional Tableau
A finite propositional tableau is a labeled tree for propositional logic $T$ which is the last term $T_n$ of some finite propositional tableau chain $T_0, T_1, \ldots, T_n$.

Infinite Propositional Tableau
An infinite propositional tableau is a labeled tree for propositional logic $T$ which is the union of some propositional tableau chain:
 * $T_0, T_1, \ldots, T_k, \ldots$

that is:
 * $\displaystyle T = \bigcup_{k \mathop = 0}^\infty T_k$

That is, $T$ is the infinite labeled tree such that $t \in T$ iff:
 * $t \in T_k$ for some $k \in \N$;
 * whenever $t \in T_k$, the parent $\pi \left({t}\right)$ and WFF $\Phi \left({t}\right)$ are the same in $T$ as in $T_k$.

Finite or Infinite Root
If the hypothesis set $\mathbf H$ at the root is finite, then the propositional tableau has a finite root.

If the hypothesis set $\mathbf H$ at the root is infinite, then the propositional tableau has an infinite root.

Building a Propositional Tableau

 * 1) Start with a rooted tree $T_0$ which consists of a single node, which is the root node, and the hypothesis set $\mathbf H$ at the root.
 * 2) Extend the tableau $T_0$ to a tableau $T_1$, and $T_1$ to a tableau $T_2$ and so on, using one of the tableau extension rules.
 * 3) At each stage a leaf node $t$ of $T_n$ is chosen, and a WFF $\mathbf C$ which appears on the branch through $t$.
 * 4) Build $T_{n+1}$ by adding one, two or four nodes below $t$ according to which rule is determined by the form of $\mathbf C$.

At each stage of the process, we have a finite propositional tableau $T_k$.

If the process continues through all $k \in \N$, the union of the chain of finite tableaus will be an infinite propositional tableau $T$.