Definition:Language of Propositional Logic/Basson-O'Connor

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 a non-empty set of symbols $\PP_0$.

Signs

Brackets

The brackets used are round brackets:

 $\displaystyle \bullet \ \$ $\displaystyle ($ $:$ $\displaystyle$the left bracket sign $\displaystyle \bullet \ \$ $\displaystyle )$ $:$ $\displaystyle$the right bracket sign

Connectives

The following connectives are used:

 $\displaystyle \bullet \ \$ $\displaystyle .$ $:$ $\displaystyle$the conjunction sign $\displaystyle \bullet \ \$ $\displaystyle \lor$ $:$ $\displaystyle$the disjunction sign $\displaystyle \bullet \ \$ $\displaystyle \supset$ $:$ $\displaystyle$the conditional sign $\displaystyle \bullet \ \$ $\displaystyle \equiv$ $:$ $\displaystyle$the biconditional sign $\displaystyle \bullet \ \$ $\displaystyle \sim$ $:$ $\displaystyle$the negation sign

Collation System

The collation system used is that of words and concatenation.

Formal Grammar

The following bottom-up formal grammar is used.

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

Let $Op = \left\{{., \lor, \supset, \equiv}\right\}$.

The rules are:

 $\mathbf W: \PP_0$ $:$ If $p \in \PP_0$, then $p$ is a WFF. $\mathbf W: \neg$ $:$ If $\mathbf A$ is a WFF, then $\sim \mathbf A$ is a WFF. $\mathbf W: Op$ $:$ If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF.