Definition:Language of Propositional Logic/Formal Grammar

Definition

The formal grammar of the language of propositional logic (and hence its WFFs) can be defined in the following ways.

Backus-Naur Form

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

\(\ds \meta {formula}\)

\(\)::=\(\)

\(\ds p \ \mid \ \top \ \mid \ \bot\)

where $p \in \PP_0$ is a letter

\(\ds \meta {formula}\)

\(\)::=\(\)

\(\ds \neg \meta {formula}\)

\(\ds \meta {formula}\)

\(\)::=\(\)

\(\ds (\meta {formula} \ \meta {op} \ \meta {formula})\)

\(\ds \meta {op}\)

\(\)::=\(\)

\(\ds \land \ \mid \ \lor \ \mid \implies \mid \iff\)

Note that this is a top-down grammar:

we start with a metasymbol $\meta {formula}$

progressively replace it with constructs containing other metasymbols and/or primitive symbols

until finally we are left with a well-formed formula of $\LL_0$ consisting of nothing but primitive symbols.

Bottom-Up Specification

The following rules of formation constitute a bottom-up grammar for the formation of well-formed formulas (WFFs) of the language of propositional logic $\LL_0$.

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

Let $\mathrm {Op} = \set {\land, \lor, \implies, \iff}$.

The rules are:

$\mathbf W: \text {TF}$		$:$	$\top$ is a WFF, and $\bot$ is a WFF.
$\mathbf W: \PP_0$		$:$	If $p \in \PP_0$, then $p$ is a WFF.
$\mathbf W: \neg$		$:$	If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF.
$\mathbf W: \text {Op}$		$:$	If $\mathbf A$ is a WFF and $\mathbf B$ is a WFF and $\circ \in \mathrm {Op}$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF.

Any string which can not be created by means of the above rules is not a WFF.

Well-Formed Formula

Let $\mathbf A$ be approved of by the formal grammar of propositional logic.

Then $\mathbf A$ is called a well-formed formula of propositional logic.

Often, one abbreviates "well-formed formula", speaking of a WFF of propositional logic instead.

Also known as

The formal grammar of propositional logic is also referred to as its syntax.

Also see

• Equivalence of Definitions of Formal Grammar for Propositional Logic

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic