Definition:Language of Propositional Logic/Alphabet
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Definition
The alphabet $\mathcal A$ of the language of propositional logic $\mathcal L_0$ 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
\(\displaystyle \bullet \ \ \) | \(\displaystyle (\) | \(:\) | \(\displaystyle \)the left bracket sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle )\) | \(:\) | \(\displaystyle \)the right bracket sign\(\) |
Connectives
\(\displaystyle \bullet \ \ \) | \(\displaystyle \land\) | \(:\) | \(\displaystyle \)the conjunction sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \lor\) | \(:\) | \(\displaystyle \)the disjunction sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \implies\) | \(:\) | \(\displaystyle \)the conditional sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \iff\) | \(:\) | \(\displaystyle \)the biconditional sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \neg\) | \(:\) | \(\displaystyle \)the negation sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \top\) | \(:\) | \(\displaystyle \)the tautology sign\(\) | ||||||||||
\(\displaystyle \bullet \ \ \) | \(\displaystyle \bot\) | \(:\) | \(\displaystyle \)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: $\S 1.3$