Definition:Propositional Tableau/Construction/Finite

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Definition

The finite propositional tableaus are precisely those labeled trees singled out by the following bottom-up grammar:

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

Note how the boxes give an indication of the ancestor WFF mentioned in the clause.

These clauses together are called the tableau extension rules.


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