# Definition:Propositional Tableau/Construction/Finite

 $\boxed{\mathrm{Root}}$ 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.