# Definition:Language of Propositional Logic/Labeled Tree

## Contents

## 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:

\(\displaystyle \bullet \ \ \) | \(\displaystyle \neg\) | \(:\) | \(\displaystyle \)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 $\PP_0$ 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

## Sources

- 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.3$: Propositional logic as a formal language - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.1.1$: Definition $2.2$