# Proposition

The simplest expression of a language. It is a concatenation of words that has an independent meaning, i.e. expresses a complete statement. In formalized languages a proposition is a formula without free variables, i.e. parameters. In formalized languages a proposition is also called a closed formula. E.g., in a first-order language (the language of the narrow predicate calculus) the formulas

$\forall x \forall y \exists z (x \le z \& z \le y), \quad \exists z(1 \le z \& z \le 4), \quad 1 \le 2$

are closed (the first is false, the second and third are true in the domain of natural numbers). The formulas

$\exists z(x \le z \& z \le y), \quad z \le 1$

are not closed, i.e. contain parameters ($x$ and $y$ in the first, $z$ in the second).

How to Cite This Entry:
Proposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proposition&oldid=29867
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article