Uniqueness quantification

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Uniqueness quantification

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and logic

Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
, this sort of quantification

Quantification

Quantification has two distinct meanings. In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into element s of some Set of numbers....
is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "?!" or ?₌₁". For example, the formal statement

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

>x P(x) to mean . An equivalent definition that has the virtue of separating the notions of existence and uniqueness into two clauses, at the expense of brevity, is . Another equivalent definition with the advantage of brevity is .

Generalizations
One generalization of uniqueness quantification is counting quantification

Counting quantification

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Encyclopedia

In mathematics

Mathematics

Logic

Mathematical logic

Quantification

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Reduction to ordinary existential and universal quantification

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic....
by defining the formula ?!x P(x) to mean . An equivalent definition that has the virtue of separating the notions of existence and uniqueness into two clauses, at the expense of brevity, is . Another equivalent definition with the advantage of brevity is .

Generalizations

One generalization of uniqueness quantification is counting quantification

Counting quantification

A counting quantifier is a Mathematics term for a quantifier of the form "there exists at least k elements that satisfy property X".In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand....
. This includes both quantification of the form "exactly k objects exist such that ..." as well as "infinitely many objects exist such that ..." and "only finitely many object exist such that...". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic

First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
.