Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification

Definition

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

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.

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

Examples

Example 1

The following is a WFF of propositional logic:

$\paren {\paren {p \land q} \implies \paren {\lnot \paren {q \lor r} } }$

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 1$
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $1$ Formation Rules
• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Definition $1.2$
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic: Definition $1.2.1$
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.3$: Propositional logic as a formal language: Definition $1.27$