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

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 $$T$$.