Definition:Language of Propositional Logic/Alphabet/Letter

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Part of specifying the language of propositional logic $\mathcal L_0$ is to specify its letters.

The letters of $\mathcal L_0$, called propositional symbols, can be any infinite collection $\mathcal P_0$ 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:

$\mathcal P_0 = \left\{{p_1, p_2, p_3, \ldots, p_n, \ldots}\right\}$

Also defined as

Some sources do not specify that $\mathcal P_0$ 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 $\mathcal L_0$.

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