Definition:Language of Propositional Logic/Alphabet

Definition

The alphabet $\AA$ of the language of propositional logic $\LL_0$ is defined as follows:

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:

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

Signs

The signs of the language of propositional logic come in two categories:

Brackets

\(\ds \bullet \ \ \)

\(\ds (\)

\(:\)

\(\ds \)the left bracket sign\(\)

\(\ds \bullet \ \ \)

\(\ds )\)

\(:\)

\(\ds \)the right bracket sign\(\)

Connectives

\(\ds \bullet \ \ \)

\(\ds \land\)

\(:\)

\(\ds \)the conjunction sign\(\)

\(\ds \bullet \ \ \)

\(\ds \lor\)

\(:\)

\(\ds \)the disjunction sign\(\)

\(\ds \bullet \ \ \)

\(\ds \implies\)

\(:\)

\(\ds \)the conditional sign\(\)

\(\ds \bullet \ \ \)

\(\ds \iff\)

\(:\)

\(\ds \)the biconditional sign\(\)

\(\ds \bullet \ \ \)

\(\ds \neg\)

\(:\)

\(\ds \)the negation sign\(\)

\(\ds \bullet \ \ \)

\(\ds \top\)

\(:\)

\(\ds \)the tautology sign\(\)

\(\ds \bullet \ \ \)

\(\ds \bot\)

\(:\)

\(\ds \)the contradiction sign\(\)

These comprise:

• The nullary connectives $\top$ and $\bot$, representing the canonical tautology and contradiction, respectively

• The unary connective $\neg$, representing negation

• The binary connectives $\land, \lor, \implies$ and $\iff$, representing, respectively, conjunction, disjunction, implication and biconditional.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 2.1$: Formation Rules
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems: $\S 1.3$