# Definition:Language of Propositional Logic/Formal Grammar/Backus-Naur Form

## Definition

Part of specifying the language of propositional logic is to provide its formal grammar.

In Backus-Naur form, the formal grammar of the language of propositional logic takes the following form:

 $\displaystyle$ ::= $\displaystyle p \ \mid \ \top \ \mid \ \bot$ where $p \in \mathcal P_0$ is a letter $\displaystyle$ ::= $\displaystyle \neg$  $\displaystyle$ ::= $\displaystyle ($ $)$ $\displaystyle$ ::= $\displaystyle \land \ \mid \ \lor \ \mid \implies \mid \iff$

Note that this is a top-down grammar: we start with a metasymbol <formula> and progressively replace it with constructs containing other metasymbols and/or primitive symbols, until finally we are left with a well-formed formula of $\mathcal L_0$ consisting of nothing but primitive symbols.