Concept

Quantifier (logic)

Overview

In mathematical logic, quantifiers are formal counterparts of natural-language adjectives like all, some, most, few, etc. which indicate the number of objects satisfying a given property. More precisely, quantifier a is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier

∀

{\displaystyle \forall }

in the first-order formula

∀

≥

{\displaystyle \forall x\;x\geq 0}

expresses that all numbers in the domain are non-negative; this formula is true for the natural numbers domain, but false for the integer domain. On the other hand, the existential quantifier

∃

{\displaystyle \exists }

in the formula

∃

−

{\displaystyle \exists x\;x^{2}-5x+6=0}

expresses that some numbers in the domain satisfy the given quadratic equation; indeed, both 2 and 3 do.

The most commonly used quantifiers are

∀

{\displaystyle \forall }

and

∃

{\displaystyle \exists }

. Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström.

In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement. In contrast, the exchange of quantifications in different types changes the meaning. As an example for the latter, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
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The notion of continuity depends upon that of _order_, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry…
Bertrand Russell, Mysticism and Logic and Other Essays
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On the other hand an event-particle is defined so as to exhibit this character of being a route of approximation marked out by entities posited in sense-awareness. A definite event-particle is defined in reference to a definite punct in the following manner: Let the condition {sigma} mean the property of covering all the abstractive elements which are members of that punct; so that an abstractive set which satisfies…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
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Philosophers who discussed Quantifier (logic)

Related ideas

Opposing perspectives

Higher-order logic

Place in the tradition

Part of Analytic philosophy.

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