Definition:Language of Propositional Logic/Labeled Tree

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There are many formal languages expressing propositional logic.

The formal language used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is defined on Definition:Language of Propositional Logic.

This page defines the formal language $\LL_0$ used in:

Explanations are omitted as this is intended for reference use only.



The letters used are an infinite set of symbols $\PP_0$.

This is the same as the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.



The following connectives are used:

\(\ds \bullet \ \ \) \(\ds \neg\) \(:\) \(\ds \)the negation sign\(\)
\(\ds \bullet \ \ \) \(\ds \lor\) \(:\) \(\ds \)the disjunction sign\(\)
\(\ds \bullet \ \ \) \(\ds \land\) \(:\) \(\ds \)the conjunction sign\(\)
\(\ds \bullet \ \ \) \(\ds \implies\) \(:\) \(\ds \)the conditional sign\(\)
\(\ds \bullet \ \ \) \(\ds \iff\) \(:\) \(\ds \)the biconditional sign\(\)
\(\ds \bullet \ \ \) \(\ds \oplus\) \(:\) \(\ds \)the exclusive or sign\(\)
\(\ds \bullet \ \ \) \(\ds \downarrow\) \(:\) \(\ds \)the nor sign\(\)
\(\ds \bullet \ \ \) \(\ds \uparrow\) \(:\) \(\ds \)the nand sign\(\)

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.

Collation System

The collation system used is that of labeled trees and adding ancestors.

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.

Formal Grammar

The following bottom-up formal grammar is used.

Let $\PP_0$ be the vocabulary of $\LL_0$.

The WFFs of $\LL_0$ are the smallest set $\FF$ of labeled trees such that:

\((T1)\)   $:$   For each letter $p \in \PP_0$, the labeled tree with one node, whose label is $p$, is in $\FF$.             
\((T2)\)   $:$   For $T \in \FF$, the labeled tree obtained by adding an ancestor with label $\neg$ to the root node of $T$, is again in $\FF$.             
\((T3)\)   $:$   For $T_1 \in \FF$ and $T_2 \in \FF$ and a binary connective $\mathsf B$, the labeled tree obtained by adding a common ancestor labeled $\mathsf B$ of the root nodes of $T_1$ and $T_2$, is again in $\FF$.             

Graphically, this means one has the following means to construct WFFs:

$\begin{xy}\xymatrix{ p & & \neg \[email protected]{-}[d] & & & \mathsf B \[email protected]{-}[ld] \[email protected]{-}[rd] \\ & & <{\sf WFF}> & & <{\sf WFF}> & & <{\sf WFF}> }\end{xy}$

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.

Also see