# 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**, can be any infinite collection $\PP_0$ of arbitrary symbols.

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:

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

## Also defined as

Some sources do not specify that $\PP_0$ be infinite.

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

## Also known as

**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.

## Also see

- 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.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 2.1$: 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 $2.1$