# Definition:Propositional Tableau/Identification

Let $\left({T, \mathbf H, \Phi}\right)$ be a labeled tree for propositional logic.
Then $T$ is a propositional tableau iff for each node $t$ of $T$ that is not a leaf node:
There exists an ancestor WFF $\mathbf C$ of $t$ such that one of the following conditions holds:
 $\boxed{\neg \neg}$ $t$ has precisely one child $s$ $\mathbf C$ is $\neg \neg \mathbf A$ $\Phi \left({s}\right) = \mathbf A$ $\boxed \land$ $t$ has precisely one child $s$, and one grandchild $r$ $\mathbf C$ is $\mathbf A \land \mathbf B$ $\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({r}\right) = \mathbf B$ $\boxed{\land'}$ The parent of $t$ satisfies $\boxed\land$ $\boxed{\neg \land}$ $t$ has precisely two children $s$ and $s'$ $\mathbf C$ is $\neg \left({\mathbf A \land \mathbf B}\right)$ $\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\boxed \lor$ $t$ has precisely two children $s$ and $s'$ $\mathbf C$ is $\mathbf A \lor \mathbf B$ $\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\boxed{\neg\lor}$ $t$ has precisely one child $s$, and one grandchild $r$ $\mathbf C$ is $\neg \left({\mathbf A \lor \mathbf B}\right)$ $\Phi \left({s}\right) = \neg\mathbf A$ and $\Phi \left({r}\right) = \neg\mathbf B$ $\boxed{\neg\lor'}$ The parent of $t$ satisfies $\boxed{\neg\lor}$ $\boxed \implies$ $t$ has precisely two children $s$ and $s'$ $\mathbf C$ is $\mathbf A \implies \mathbf B$ $\Phi \left({s}\right) = \neg\mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\boxed{\neg\implies}$ $t$ has precisely one child $s$, and one grandchild $r$ $\mathbf C$ is $\neg \left({\mathbf A \implies \mathbf B}\right)$ $\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({r}\right) = \neg\mathbf B$ $\boxed \iff$ $t$ has precisely two children $s$ and $s'$ $\mathbf C$ is $\mathbf A \iff \mathbf B$ $\Phi \left({s}\right) = \mathbf A \land \mathbf B$ and $\Phi \left({s'}\right) = \neg\mathbf A \land \neg\mathbf B$ $\boxed{\neg\iff}$ $t$ has precisely two children $s$ and $s'$ $\mathbf C$ is $\neg \left({\mathbf A \iff \mathbf B}\right)$ $\Phi \left({s}\right) = \mathbf A \land \neg \mathbf B$ and $\Phi \left({s'}\right) = \neg\mathbf A \land \mathbf B$
Note how the boxes give an indication of the shape of the relevant ancestor WFF $\mathbf C$.