Kvantifikatora

Kvantifikatora, in many languages referred to as the quantifier, is a logical or linguistic operator that specifies how many elements of a domain satisfy a given predicate. In formal logic, a quantifier binds a variable within a formula and determines the scope of the statement about the domain under discussion.

The two primary quantifiers are the universal quantifier and the existential quantifier. The universal quantifier is written as for all x, P(x) and expresses that P(x) holds for every element x in the domain. The existential quantifier, written as there exists x, P(x), asserts that at least one element x satisfies P. These quantifiers are duals: not all x P(x) is equivalent to there exists x not P(x), and not there exists x P(x) is equivalent to for all x not P(x).

Beyond the basics, there are additional and generalized forms of quantification. The unique quantifier ∃!x P(x) means exactly one x satisfies P. Counting quantifiers such as ∃≥k x P(x) express that at least k elements meet the condition. Generalized quantifiers extend the notion further to capture statements like “most,” “few,” or “infinitely many.” Quantifiers also feature in natural language semantics, where words such as every, some, no, many convey similar meanings.

In formal reasoning, quantifiers have scope, can be moved to prenex form, and may be eliminated or transformed through procedures such as Skolemization. Historically, the modern notation of quantifiers emerged in the development of predicate logic in the late 19th and early 20th centuries.