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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).


[1] A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)


In Western parlance, the term "proposition" tends to be reserved for formulas in a language not involving variables at all (cf. Propositional calculus). The term "sentence" is used for a formula whose variables are all quantified, as in the examples above.

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Proposition. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article