Quantification

The term quantification has several meanings, general and specific. Primarily it covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.

Some measure of the undisputed general importance of quantification in the natural sciences can be gleaned from the following comments: these are mere facts, but they are quantitative facts and the basis of science. It seems to be held as universally true that the foundation of quantification is measurement. There is little doubt that quantification provided a basis for the objectivity of science. In ancient times, musicians and artists...rejected quantification, but merchants, by definition, quantified their affairs, in order to survive, made them visible on parchment and paper. Any reasonable comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification. Even today, universities use imperfect instruments called 'exams' to indirectly quantify something they call knowledge.

More specifically, in language and logic, quantification is a construct that specifies the quantity of individuals of the domain of discourse that apply to (or satisfy) an open formula. For example, in arithmetic, it allows the expression of the statement that every natural number has a successor, and in logic, that something (at least one thing) in the domain of discourse has a certain property, i.e., there exist things with that property in the domain. A language element which generates a quantification is called a quantifier. The resulting expression is a quantified expression, and we say we have quantified over the predicate or function expression whose free variable is bound by the quantifier. Quantification is used in both natural languages and formal languages. Examples of quantifiers in a natural language are: for all, for some, many, few, a lot, and no. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted as an extent of validity. Quantification is an example of a variable-binding operation.

The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. These concepts are covered in detail in their individual articles; here we discuss features of quantification that apply in both cases. Other kinds of quantification include uniqueness quantification.

The traditional symbol for the universal quantifier "all" is "∀", an inverted letter "A", and for the existential quantifier "exists" is "∃", a rotated letter "E". These quantifiers have been generalized beginning with the work of Mostowski and Lindström. See generalized quantifier and Lindström quantifier for further details.

Quantification in natural language
All known human languages make use of quantification, even languages without a fully fledged number system (Wiese 2004). For example, in English:
 * Every glass in my recent order was chipped.
 * Some of the people standing across the river have white armbands.
 * Most of the people I talked to didn't have a clue who the candidates were.
 * Everyone in the waiting room had at least one complaint against Dr. Ballyhoo.
 * There was somebody in his class that was able to correctly answer every one of the questions I submitted.
 * A lot of people are smart.

There exists no simple way of reformulating any one of these expressions as a conjunction or disjunction of sentences, each a simple predicate of an individual such as That wine glass was chipped. These examples also suggest that the construction of quantified expressions in natural language can be syntactically very complicated. Fortunately, for mathematical assertions, the quantification process is syntactically more straightforward.

The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. This comes in part from the fact that the grammatical structure of natural language sentences may conceal the logical structure. Moreover, mathematical conventions strictly specify the range of validity for formal language quantifiers; for natural language, specifying the range of validity requires dealing with non-trivial semantic problems.

Montague grammar gives a novel formal semantics of natural languages. Its proponents argue that it provides a much more natural formal rendering of natural language than the traditional treatments of Frege, Russell and Quine.

Need for quantifiers in mathematical assertions
We will begin by discussing quantification in informal mathematical discourse. Consider the following statement
 * 1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 · 2 = 3 + 3, ...., and n · 2 = n + n, etc.

This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since we expect syntax rules to generate finite objects. Putting aside this objection, also note that in this example we were lucky in that there is a procedure to generate all the conjuncts. However, if we wanted to assert something about every irrational number, we would have no way enumerating all the conjuncts since irrationals cannot be enumerated. A succinct formulation which avoids these problems uses universal quantification:
 * For any natural number n, n·2 = n + n.

A similar analysis applies to the disjunction,
 * 1 is prime, or 2 is prime, or 3 is prime, etc.

which can be rephrased using existential quantification:
 * For some natural number n, n is prime.

Nesting of quantifiers
Consider the following statement:
 * For any natural number n, there is a natural number s such that s = n &times; n.

This is clearly true; it just asserts that every number has a square.

The meaning of the assertion in which the quantifiers are turned around is quite different:
 * There is a natural number s such that for any natural number n,  s = n &times; n.

This is clearly false; it asserts that there is a single natural number s that is at once the square of every natural number.

This illustrates a fundamentally important point when quantifiers are nested: The order of alternation of quantifiers is of absolute importance.

A less trivial example is the important concept of uniform continuity  from analysis, which differs from the more familiar concept of pointwise continuity only by an exchange in the positions of two quantifiers. To illustrate this, let f be a real-valued function on R.


 * A: Pointwise continuity of f on R:
 * $$ \underbrace{\forall x \in \mathbb{R}, \ \forall \epsilon >0}, \exists \delta > 0, \forall h \in \mathbb{R}, \quad |h| < \delta \implies |f(x) - f(x+h)| < \epsilon $$

interchanging the universal quantifiers over the braces, this is the same as
 * A': Pointwise continuity of f on R:
 * $$ \forall \epsilon >0, \ \underbrace{\forall x \in \mathbb{R}, \exists \delta > 0}, \  \forall h \in \mathbb{R}, \quad |h| < \delta  \implies |f(x) - f(x+h)| < \epsilon $$

This differs from
 * B: Uniform continuity of f on R:
 * $$ \forall \epsilon >0, \underbrace{\exists \delta > 0, \forall x \in \mathbb{R}}, \forall h \in \mathbb{R}, \quad |h| < \delta\implies |f(x) - f(x+h)| < \epsilon $$

by interchanging the existential and universal quantifiers over the braces in A'.

Ambiguity is avoided by putting the quantifiers (in symbols or words) in front:
 * $$ \exists$$ A:$$ \forall$$ B: C - unambiguous
 * there is an A such that $$\forall$$ B: C - unambiguous
 * there is an A such that for all B, C - unambiguous, provided that the separation between B and C is clear
 * there is an A such that C for all B - it is often clear that what is meant is
 * there is an A such that (C for all B)
 * but it could be interpreted as
 * (there is an A such that C) for all B


 * there is an A such that C $$\forall$$ B - suggests more strongly that the first is meant; this may be reinforced by the layout, for example by putting "C $$\forall$$ B" on a new line.

See also below.

Range of quantification
Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as "n" for natural numbers and "x" for real numbers, although relying exclusively on naming conventions cannot work in general since ranges of variables can change in the course of a mathematical argument.

A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification
 * For some natural number n, n is even and n is prime

means
 * For some even number n, n is prime.

In some mathematical theories one assumes a single domain of discourse fixed in advance. For example, in Zermelo Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above to express
 * For any natural number n, n·2 = n + n

in Zermelo-Fraenkel set theory, one can say
 * For any n, if n belongs to N, then n·2 = n + n,

where N is the set of all natural numbers.

Notation for quantifiers
The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows,
 * $$ \exists{x}\, P \quad \forall{x}\, P $$

where "P" denotes a formula. Many variant notations are used, such as
 * $$ \exists{x}\, P \quad (\exists{x}) P \quad (\exists x \ . \ P) \quad (\exists x : P) \quad \exists{x}(P) \quad \exists_{x}\, P \quad \exists{x}{,}\, P \quad \exists{x}{\in}\mathbb{N}\, P \quad \exists\, x{:}\mathbb{N}\, P$$

All of these variations also apply to universal quantification. Other variations for the universal quantifier are
 * $$(x) \, P \quad \bigwedge_{x} P$$

Early 20th century documents do not use the ∀ symbol. The typical notation was (x)P to express "for all x, P", and "(∃x)P" for "there exists x such that P". The ∃ symbol was coined by Giuseppe Peano around 1890. Later, around 1930, Gerhard Gentzen introduced the ∀ symbol to represent universal quantification. Frege's Begriffsschrift used an entirely different notation, which did not include an existential quantifier at all; ∃x P was always represented instead with the Begriffsschrift equivalent of ¬∀x ¬P.

Note that some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified, but for a given mathematical theory, this can be done in several ways:
 * Assume a fixed domain of discourse for every quantification, as is done in Zermelo Fraenkel set theory,
 * Fix several domains of discourse in advance and require that each variable have a declared domain, which is the type of that variable. This is analogous to the situation in statically typed computer programming languages, where variables have declared types.
 * Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain or the type of the objects in that domain.

Also note that one can use any variable as a quantified variable in place of any other, under certain restrictions, that is in which variable capture does not occur. Even if the notation uses typed variables, one can still use any variable of that type. The issue of variable capture is exceedingly important, and we discuss that in the formal semantics below.

Informally, the "∀x" or "∃x" might well appear after P(x), or even in the middle if P(x) is a long phrase. Formally, however, the phrase that introduces the dummy variable is standardly placed in front. See also above.

Note that mathematical formulas mix symbolic expressions for quantifiers, with natural language quantifiers such as
 * For any natural number x, ....
 * There exists an x such that ....
 * For at least one x.

Keywords for uniqueness quantification include:
 * For exactly one natural number x, ....
 * There is one and only one x such that ....

One might even avoid variable names such as x using a pronoun. For example,
 * For any natural number, its product with 2 equals to its sum with itself
 * Some natural number is prime.

Formal semantics
Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal—that is, mathematically specified—language. It has three elements: A mathematical specification of a class of objects via syntax, a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. In this article, we only address the issue of how quantifier elements are interpreted.

In this section we only consider first-order logic with function symbols. We refer the reader to the article on model theory for more information on the interpretation of formulas within this logical framework. The syntax of a formula can be given by a syntax tree. Quantifiers have scope and a variable x is free if it is not within the scope of a quantification for that variable. Thus in
 * $$ \forall x (\exists y B(x,y)) \vee C(y,x) $$

the occurrence of both x and y in C(y,x) is free.

An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x1, ..., xn is interpreted as a boolean-valued function F(v1, ..., vn) of n arguments, where each argument ranges over the domain X. Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood). The interpretation of the formula
 * $$ \forall x_n A(x_1, \ldots, x_n) $$

is the function G of n-1 arguments such that G(v1, ...,vn-1) = T if and only if F(v1, ..., vn-1, w) = T for every w in X. If F(v1, ..., vn-1, w) = F for at least one value of w, then G(v1, ...,vn-1) = F. Similarly the interpretation of the formula
 * $$ \exists x_n A(x_1, \ldots, x_n) $$

is the function H of n-1 arguments such that H(v1, ...,vn-1) = T if and only if F(v1, ...,vn-1, w) = T for at least one w and H(v1, ..., vn-1) = F otherwise.

The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X. The interpretation of
 * $$ \exists !  x_n A(x_1, \ldots, x_n) $$

then is the function of n-1 arguments, which is the logical and of the interpretations of
 * $$ \exists x_n A(x_1, \ldots, x_n) $$
 * $$ \forall y,z \left\{ A(x_1, \ldots ,x_{n-1}, y) \wedge  A(x_1, \ldots ,x_{n-1}, z) \implies y = z \right\}$$

Paucal, multal and other degree quantifiers
So far we have only considered universal, existential and uniqueness quantification as used in mathematics. None of this applies to a quantification such as


 * There were many dancers out on the dance floor this evening.

Though we will not consider semantics of natural language in this article, we will attempt to provide a semantics for assertions in a formal language of the type


 * There are many integers n < 100, such that n is divisible by 2 or 3 or 5.

One possible interpretation mechanism can obtained as follows: Suppose that in addition to a semantic domain X, we have given a probability measure P defined on X and cutoff numbers 0 < a ≤ b ≤ 1. If A is a formula with free variables x1,...,xn whose interpretation is the function F of variables v1,...,vn then the interpretation of
 * $$ \exists^{\mathrm{many}} x_n A(x_1, \ldots, x_{n-1}, x_n) $$

is the function of v1,...,vn-1 which is T if and only if
 * $$ \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T} \} \geq b $$

and F otherwise. Similarly, the interpretation of
 * $$ \exists^{\mathrm{few}} x_n  A(x_1, \ldots, x_{n-1}, x_n) $$

is the function of v1,...,vn-1 which is F if and only if
 * $$ 0< \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T}\} \leq a $$

and T otherwise. We have completely avoided discussion of technical issues regarding measurability of the interpretation functions; some of these are technical questions that require Fubini's theorem.

We also caution the reader that the corresponding logic for such a semantics is exceedingly complicated.

History of formalization
Term logic treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Aristotelian logic treated All', Some and No in the 1st century BC, in an account also touching on the alethic modalities.

The first variable-based treatment of quantification was Gottlob Frege's 1879 Begriffsschrift. To universally quantify a variable, Frege would make a dimple in an otherwise straight line appearing in his diagrammatic formulas, then write the quantified variable over the dimple. Frege did not have a specific notation for existential quantification, instead using the equivalent of $$\sim\forall x:\sim\ldots$$. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics.

Meanwhile, Charles Sanders Peirce and his student O. H. Mitchell independently invented the existential as well as the universal quantifier, in work culminating in Peirce (1885). Peirce and Mitchell wrote Πx and Σx where we now write ∀x and ∃x. This notation can be found in the writings of Ernst Schroder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. It is the notation of Kurt Goedel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic. Peirce's later existential graphs can be seen as featuring tacit variables whose quantification is determined by the shallowest instance. Peirce's approach to quantification influenced Ernst Schroder, William Ernest Johnson, and all of Europe via Giuseppe Peano. Peirce's logic has attracted fair attention in recent decades by those interested in heterogeneous reasoning and diagrammatic inference.

Peano notated the universal quantifier as (x). Hence "(x)φ" indicated that the formula φ was true for all values of x. He was the first to employ, in 1897, the notation (∃x) for existential quantification. The Principia Mathematica of Whitehead and Russell employed Peano's notation, as did Quine and Alonzo Church throughout their careers. Gentzen introduced the ∀ symbol in 1935 by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1950s.

The Development of Quantitification both across Species and Within Humans
In Quantitative analysis of behavior, Evolutionary Psychology and Cognitive Developmental Psychology, quantification is studied as behavior.