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 $\mathcal L_0$ used in:

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


Alphabet

Letters

The letters used are an infinite set of symbols $\mathcal P_0$.

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


Signs

Connectives

The following connectives are used:

\(\displaystyle \bullet \ \ \) \(\displaystyle \neg\) \(:\) \(\displaystyle \)the negation sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \lor\) \(:\) \(\displaystyle \)the disjunction sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \land\) \(:\) \(\displaystyle \)the conjunction sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \implies\) \(:\) \(\displaystyle \)the conditional sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \iff\) \(:\) \(\displaystyle \)the biconditional sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \oplus\) \(:\) \(\displaystyle \)the exclusive or sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \downarrow\) \(:\) \(\displaystyle \)the nor sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \uparrow\) \(:\) \(\displaystyle \)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 $\mathcal P_0$ be the vocabulary of $\mathcal L_0$.


The WFFs of $\mathcal L_0$ are the smallest set $\mathcal F$ of labeled trees such that:

\((T1)\)   $:$   For each letter $p \in \mathcal P_0$, the labeled tree with one node, whose label is $p$, is in $\mathcal F$.             
\((T2)\)   $:$   For $T \in \mathcal F$, the labeled tree obtained by adding an ancestor with label $\neg$ to the root node of $T$, is again in $\mathcal F$.             
\((T3)\)   $:$   For $T_1 \in \mathcal F$ and $T_2 \in \mathcal F$ 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 $\mathcal F$.             

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


Sources