Definition:Propositional Tableau/Construction/Finite

Definition

The finite propositional tableaus are precisely those labeled trees singled out by the following bottom-up grammar:

$\boxed{\mathrm{Root}}$	A labeled tree whose only node is its root node is a finite propositional tableau.
For the following clauses, let $t$ be a leaf node of a finite propositional tableau $T$.
$\boxed{\neg \neg}$	If $\neg \neg \mathbf A$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$ is a finite propositional tableau.
$\boxed \land$	If $\mathbf A \land \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$ a child $r$ to $s$, with $\Phi \left({r}\right) = \mathbf B$ is a finite propositional tableau.
$\boxed{\neg \land}$	If $\neg \left({\mathbf A \land \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$ another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg\mathbf B$ is a finite propositional tableau.
$\boxed \lor$	If $\mathbf A \lor \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$ another child $s'$ to $t$, with $\Phi \left({s'}\right) = \mathbf B$ is a finite propositional tableau.
$\boxed{\neg\lor}$	If $\neg \left({\mathbf A \lor \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$ a child $r$ to $s$, with $\Phi \left({r}\right) = \neg \mathbf B$ is a finite propositional tableau.
$\boxed \implies$	If $\mathbf A \implies \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \neg\mathbf A$ another child $s'$ to $t$, with $\Phi \left({s'}\right) = \mathbf B$ is a finite propositional tableau.
$\boxed{\neg\implies}$	If $\neg \left({\mathbf A \implies \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A$ a child $r$ to $s$, with $\Phi \left({r}\right) = \neg \mathbf B$ is a finite propositional tableau.
$\boxed \iff$	If $\mathbf A \iff \mathbf B$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A \land \mathbf B$ another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg \mathbf A \land \neg\mathbf B$ is a finite propositional tableau.
$\boxed{\neg\iff}$	If $\neg \left({\mathbf A \iff \mathbf B}\right)$ is an ancestor WFF of $t$, the labeled tree obtained from $T$ by adding: a child $s$ to $t$, with $\Phi \left({s}\right) = \mathbf A \land \neg \mathbf B$ another child $s'$ to $t$, with $\Phi \left({s'}\right) = \neg \mathbf A \land \mathbf B$ is a finite propositional tableau.

Note how the boxes give an indication of the ancestor WFF mentioned in the clause.

These clauses together are called the tableau extension rules.

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.7$: Tableaus: Definition $1.7.1$, $1.7.2$