Definition:Language of Propositional Logic/Formal Grammar

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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