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:
\(\ds \)<formula>\(\) | \(\)::=\(\) | \(\ds p \ \mid \ \top \ \mid \ \bot\) | where $p \in \PP_0$ is a letter | |||||||||||
\(\ds \)<formula>\(\) | \(\)::=\(\) | \(\ds \neg\) <formula>\(\) | ||||||||||||
\(\ds \)<formula>\(\) | \(\)::=\(\) | \(\ds (\) <formula> <op> <formula> \()\) | ||||||||||||
\(\ds \)<op>\(\) | \(\)::=\(\) | \(\ds \land \ \mid \ \lor \ \mid \implies \mid \iff\) |
Note that this is a top-down grammar:
- we start with a metasymbol <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
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic