Definition:Language of Propositional Logic/Labeled Tree

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Definition

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.

Alphabet

Letters

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

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

Signs

Connectives

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

Collation System

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

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}$