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 $\mathcal L_0$, 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:

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


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:


Sources