Propositional function

2020 Mathematics Subject Classification: Primary: 03B [MSN][ZBL]

A function whose arguments and values are truth values. This term is used when the discussion is about interpretations of a formalized logical language.

If $\Omega$ is the set of truth values of formulas of a given language, then a propositional function is any expression of the type $\Omega^n\to\Omega$ ($n\geq0$). These functions are interpreted as propositional connectives that allow one to form new statements or formulas. In the classical two-valued interpretation of the set of truth values, i.e. when $\Omega=\{0,1\}$, such functions are also called functions of the algebra of logic.

Comments

Propositional functions are also called truth functions. When $\Omega=\{0,1\}$, they are also called Boolean functions.

More or less equivalently, propositional functions are functions whose arguments and values are propositions.

References

[a1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1959) pp. 144; 226