Definition:Propositional Tableau/Identification

Definition
Let $\struct {T, \mathbf H, \Phi}$ be a labeled tree for propositional logic.

Then $T$ is a propositional tableau 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 $s$

$\map \Phi s = \mathbf A$ $\mathbf C$ 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 $r$

$\map \Phi s = \mathbf A$ and $\map \Phi r = \mathbf B$ $\mathbf C$ is $\neg \paren {\mathbf A \land \mathbf B}$
 * $\boxed{\land'}$
 * 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 $s'$

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

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

$\map \Phi s = \neg \mathbf A$ and $\map \Phi r = \neg \mathbf B$ $\mathbf C$ 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 $s'$

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

$\map \Phi s = \mathbf A$ and $\map \Phi r = \neg \mathbf B$ $\mathbf C$ is $\mathbf A \iff \mathbf B$
 * $\boxed \iff$
 * $t$ has precisely two children $s$ and $s'$
 * $t$ has precisely two children $s$ and $s'$

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

$\map \Phi s = \mathbf A \land \neg \mathbf B$ and $\map \Phi {s'} = \neg \mathbf A \land \mathbf B$
 * }

Note how the boxes give an indication of the shape of the relevant ancestor WFF $\mathbf C$.