Difference between revisions of "Proposition"
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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 | 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 | 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 ( | + | |
| + | are not closed, i.e. contain parameters ($x$ and $y$ in the first, $z$ in the second). | ||
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | In Western parlance, the term "proposition" tends to be reserved for formulas in a language not involving variables at all (cf. [[Propositional calculus|Propositional calculus]]). The term " | + | In Western parlance, the term "proposition" tends to be reserved for formulas in a language not involving variables at all (cf. [[Propositional calculus|Propositional calculus]]). The term [[sentence|"sentence"]] is used for a formula whose variables are all quantified, as in the examples above. |
Latest revision as of 05:16, 21 June 2013
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).
References
| [1] | A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956) |
Comments
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.
Proposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proposition&oldid=12779