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Quantifier (Logic)

Identifier used to quantify a statement. In natural language these are words like “some” or “all”, in formal logic symbols like “” or “” are used.

In propositional logic, as used in the context of this site, only two quantifiers, each in a positive and a negative variant, are of interest: the universal (“all”) and the existential quantifier (“some”).

Universal quantifier

The universal quantifier refers to all elements of a set. In spoken language, this is usually expressed by an indefinite pronoun such as “all”, “every” (positive) or “none”, “no” (negative).

As a logical formula, this can be represented as follows:

∀ 𝑠 : P
(For all 𝑠, it is true that P)

or the negative variant:

∄ 𝑠 : P
(There exists no 𝑠, for which P is true)

For more information, please see Universal quantification.

Existential quantifier

An existential quantifier refers only to a part of the propositional set. Strictly interpreted, it only says that at least one element exists for which the statement is true.

In spoken language this can be expressed by pronouns like “some”, or by phrases like “there exist …”.

As a logical formula, an existential proposition may be expressed e.g. like the following:

∃ 𝑠 : P
(there exists [at least] one 𝑠, for which P is true)

A negative existential proposition can be expressed in a number of ways, including the following:

∃ 𝑠 : ¬P
(there exists [at least] one 𝑠, for which not-P is true)

For more information, please see: Existential quantification.

Generalized quantifiers

The above-mentioned quantifiers were already used by Aristotle more or less with the same meaning that we still use them today. They are still relevant – not least to explain categorical statements and syllogisms as well as their related fallacies, without adding unnecessary ballast – but, of course, logic has moved on in the millennia that have passed.

The need for change has often come from mathematics, which requires expressions beyond those that can be easily described by means of categorical statements. For example, a mathematical proof might contain a statement such as, “There exist infinitely many X for which holds …”, or, “There exists exactly one X for which holds …”. These can not easily be expressed only with the universal or existential quantifiers alone.

In order to be able to deal with these and similar problems, the generalized quantifiers were developed in modern logic. In this system, the above-mentioned universal and existential quantifiers are only specific use cases within a more general system.

This (quite fascinating!) subject is outside the scope of this website and is mentioned here only for the sake of completeness. If you are interested, you can find more information in the article Generalized Quantifiers on the website of the Stanford Encyclopedia of Philosophy.

See also

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