Definition:Propositional Tableau/Identification

Definition
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:


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

$\mathbf C$ is $\neg \neg \mathbf A$
 * $\boxed{\neg \neg}$
 * $t$ has precisely one child $s$

$\Phi \left({s}\right) = \mathbf A$ $\mathbf C$ is $\mathbf A \land \mathbf B$
 * $\boxed \land$
 * $t$ has precisely one child $s$, and one grandchild $r$
 * $t$ has precisely one child $s$, and one grandchild $r$

$\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({r}\right) = \mathbf B$ $\mathbf C$ is $\neg \left({\mathbf A \land \mathbf B}\right)$
 * $\boxed{\land'}$
 * The parent of $t$ satisfies $\boxed\land$
 * $\boxed{\neg \land}$
 * $t$ has precisely two children $s$ and $s'$
 * $\boxed{\neg \land}$
 * $t$ has precisely two children $s$ and $s'$

$\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\mathbf C$ is $\mathbf A \lor \mathbf B$
 * $\boxed \lor$
 * $t$ has precisely two children $s$ and $s'$
 * $t$ has precisely two children $s$ and $s'$

$\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\mathbf C$ is $\neg \left({\mathbf A \lor \mathbf B}\right)$
 * $\boxed{\neg\lor}$
 * $t$ has precisely one child $s$, and one grandchild $r$
 * $t$ has precisely one child $s$, and one grandchild $r$

$\Phi \left({s}\right) = \neg\mathbf A$ and $\Phi \left({r}\right) = \neg\mathbf B$ $\mathbf C$ is $\mathbf A \implies \mathbf B$
 * $\boxed{\neg\lor'}$
 * The parent of $t$ satisfies $\boxed{\neg\lor}$
 * $\boxed \implies$
 * $t$ has precisely two children $s$ and $s'$
 * $\boxed \implies$
 * $t$ has precisely two children $s$ and $s'$

$\Phi \left({s}\right) = \neg\mathbf A$ and $\Phi \left({s'}\right) = \mathbf B$ $\mathbf C$ is $\neg \left({\mathbf A \implies \mathbf B}\right)$
 * $\boxed{\neg\implies}$
 * $t$ has precisely one child $s$, and one grandchild $r$
 * $t$ has precisely one child $s$, and one grandchild $r$

$\Phi \left({s}\right) = \mathbf A$ and $\Phi \left({r}\right) = \neg\mathbf B$ $\mathbf C$ is $\mathbf A \iff \mathbf B$
 * $\boxed \iff$
 * $t$ has precisely two children $s$ and $s'$
 * $t$ has precisely two children $s$ and $s'$

$\Phi \left({s}\right) = \mathbf A \land \mathbf B$ and $\Phi \left({s'}\right) = \neg\mathbf A \land \neg\mathbf B$ $\mathbf C$ is $\neg \left({\mathbf A \iff \mathbf B}\right)$
 * $\boxed{\neg\iff}$
 * $t$ has precisely two children $s$ and $s'$
 * $t$ has precisely two children $s$ and $s'$

$\Phi \left({s}\right) = \mathbf A \land \neg \mathbf B$ 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 $\mathbf C$.