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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

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

are not closed, i.e. contain parameters ( and in the first, 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 "sentencesentence" is used for a formula whose variables are all quantified, as in the examples above.

How to Cite This Entry:
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