Definition:Propositional Tableau/Construction/Finite

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


 * {| style="border-spacing:20px;"


 * $\boxed{\mathrm{Root}}$
 * A labeled tree whose only node is its root node is a finite propositional tableau.
 * colspan=2 | 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 $\Phi \left({s}\right) = \mathbf A$
 * 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 $\Phi \left({r}\right) = \mathbf B$
 * 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 $\Phi \left({s'}\right) = \neg\mathbf B$
 * 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 $\Phi \left({s'}\right) = \mathbf B$
 * 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 $\Phi \left({r}\right) = \neg \mathbf B$
 * 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 $\Phi \left({s'}\right) = \mathbf B$
 * 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 $\Phi \left({r}\right) = \neg \mathbf B$
 * 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 $\Phi \left({s'}\right) = \neg \mathbf A \land \neg\mathbf B$
 * 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 $\Phi \left({s'}\right) = \neg \mathbf A \land \mathbf B$
 * 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.