Definition:Propositional Tableau/Construction/Finite
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 $\Phi \left({s}\right) = \mathbf A$  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'''.