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⭐ Core Definition: Quantification (logic)

In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier $\forall$ in the first-order formula $\forall xP(x)$ expresses that everything in the domain satisfies the property denoted by $P$ . On the other hand, the existential quantifier $\exists$ in the formula $\exists xP(x)$ expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

The most commonly used quantifiers are $\forall$ and $\exists$ . These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula $\neg \exists xP(x)$ which expresses that nothing has the property $P$ . 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.

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👉 Quantification (logic) in the context of Universal quantification

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier (" $\forall x$ ", " $\forall(x)$ ", or sometimes by " $(x)$ " alone). Universal quantification is distinct from existential quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.

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⭐ Core Definition: Quantification (logic)
👉 Quantification (logic) in the context of Universal quantification
Quantification (logic) in the context of Predicate logic
Quantification (logic) in the context of Existential quantifier

Quantification (logic) in the context of Predicate logic

First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all x, if x is a human, then x is mortal", where "for all x" is a quantifier, x is a variable, and "... is a human" and "... is mortal" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, first-order logic is an extension of propositional logic.

A theory about a topic, such as set theory, a theory for groups, or a formal theory of arithmetic, is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. "Theory" is sometimes understood in a more formal sense as just a set of sentences in first-order logic.

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Quantification (logic) in the context of Existential quantifier

In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier (" $\exists x$ " or " $\exists(x)$ " or " $(\exists x)"$ ), read as "there exists", "there is at least one", or "for some". Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.

Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related formula editors.

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Quantification (logic) in the context of "Universal quantification"

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⭐ Core Definition: Quantification (logic)

👉 Quantification (logic) in the context of Universal quantification

Quantification (logic) in the context of Predicate logic

Quantification (logic) in the context of Existential quantifier