Definition:Language of Propositional Logic/Alphabet/Letter

Definition

Part of specifying the language of propositional logic $\LL_0$ is to specify its letters.

The letters of $\LL_0$, called propositional symbols, is an arbitrary collection $\PP_0$ of symbols which has no limit on its extent.

It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts.

A typical set of propositional symbols would be, for example:

$\PP_0 = \set {p_1, p_2, p_3, \ldots, p_n, \ldots}$

Some sources do not specify that $\PP_0$ be unlimited in extent.

However, since one can simply "forget to use" all but finitely many letters, this does not provide a more general theory.

Propositional symbols are also known as the propositional variables of $\LL_0$.

Others call them atomic propositions or simply atoms.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$, atom has a broader context, and so is discouraged as an alternative for propositional symbol.

Some sources refer to the collection of letters as the vocabulary of the language.

Definition:Statement Variable: when symbolic logic is presented less precisely than in the context of a formal language, the alphabet from which its symbols may be taken is often not specified.

1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $1$ Formation Rules
1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.2$: Propositional formulas
1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic
2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.5$ First-Order Logic Syntax: Definition $\mathrm{II.5.2}$
2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.1.1$: Definition $2.1$