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In natural languages, a quantifier turns a sentence about something having some property into a sentence about the number (quantity) of things having the property. Examples of quantifiers in English are "all", "some", "many", "few", "most", and "no";[1] examples of quantified sentences are "all people are mortal", "some people are mortal", and "no people are mortal", they are considered to be true, true, and false, respectively. In mathematical logic, in particular in first-order logic, a quantifier achieves a similar task, operating on a mathematical formula rather than an English sentence. More precisely, a quantifier specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common formal quantifiers are "for each" (traditionally symbolized by "?"), and "there exists some" ("?").[3] For example, in arithmetic, quantifiers allow one to say that the natural numbers go on forever, by writing that "for each natural number n, there exists some natural number m that is bigger than n"; this can be written formally as "?n?N. ?m?N. m>n".[4] The above English examples could be formalized as "?p?P. m(p)",[5] "?p?P. m(p)", and "¬ ?p?P. m(p)",[6] respectively, when P denotes the set of all people, and m(p) denotes "p is mortal". A formula beginning with a quantifier is called a quantified formula. A formal quantifier requires a variable, which is said to be bound by it, and a subformula specifying a property of that variable. Formal quantifiers have been generalized beginning with the work of Mostowski and Lindström. Contents 1 Relations to logical conjunction and disjunction 1.1 Infinite domain of discourse 2 Algebraic approaches to quantification 3 Notation 4 Order of quantifiers (nesting) 5 Equivalent expressions 6 Range of quantification 7 Formal semantics 8 Paucal, multal and other degree quantifiers 9 Other quantifiers 10 History 11 See also 12 References 13 External links Relations to logical conjunction and disjunction For a finite domain of discourse D = {a1,...an} the universal quantifier is equivalent to a logical conjunction of propositions with singular terms ai having the form Pai for monadic predicates. The existential quantifier is equivalent to a logical disjunction of propositions having the same structure as before. For infinite domains of discourse the equivalences are similar. Infinite domain of discourse Consider the following statement: 1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., 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 syntax rules are expected to generate finite words. The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct equivalent formulation which avoids these problems uses universal quantification: For each natural number n, n · 2 = n + n. A similar analysis applies to the disjunction, 1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc. which can be rephrased using existential quantification: For some natural number n, n is equal to 5+5. Algebraic approaches to quantification It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow[clarification needed] and interest in such algebra has been limited. Three approaches have been devised to date: Relation algebra, invented by Augustus De Morgan, and developed by Charles Sanders Peirce, Ernst Schröder, Alfred Tarski, and Tarski's students. Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the models of relation algebra include the axiomatic set theory ZFC and Peano arithmetic; Cylindric algebra, devised by Alfred Tarski, Leon Henkin, and others; The polyadic algebra of Paul Halmos. Notation The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "?", a rotated letter "A", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is "?", a rotated letter "E", which stands for "there exists" or "exists". An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P(x) the predicate "x likes to dance", and Q(x) the predicate "x likes to go to the beach". Then the above sentence can be written in formal notation as {\displaystyle \forall {x}{\in }X,P(x)\lor Q(x)}{\displaystyle \forall {x}{\in }X,P(x)\lor Q(x)}, which is read, "for every x that is a member of X, P applies to x or Q applies to x". Some other quantified expressions are constructed as follows, {\displaystyle \exists {x}\,P\qquad \forall {x}\,P} \exists{x}\, P \qquad \forall{x}\, P for a formula P. These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance" respectively. Variant notations include, for set X and set members x: {\displaystyle \bigvee _{x}P\qquad (\exists {x})P\qquad (\exists x\ .\ P)\qquad \exists x\ \cdot \ P\qquad (\exists x:P)\qquad \exists {x}(P)\qquad \exists _{x}\,P\qquad \exists {x}{,}\,P\qquad \exists {x}{\in }X\,P\qquad \exists \,x{:}X\,P}{\displaystyle \bigvee _{x}P\qquad (\exists {x})P\qquad (\exists x\ .\ P)\qquad \exists x\ \cdot \ P\qquad (\exists x:P)\qquad \exists {x}(P)\qquad \exists _{x}\,P\qquad \exists {x}{,}\,P\qquad \exists {x}{\in }X\,P\qquad \exists \,x{:}X\,P} All of these variations also apply to universal quantification. Other variations for the universal quantifier are {\displaystyle \bigwedge _{x}P\qquad (x)\,P}{\displaystyle \bigwedge _{x}P\qquad (x)\,P} Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; 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. One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used. Informally or in natural language, the "?x" or "?x" might appear after or in the middle of P(x). Formally, however, the phrase that introduces the dummy variable is placed in front. Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as, For every 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 .... Further, x may be replaced by a pronoun. For example, For every natural number, its product with 2 equals to its sum with itself. Some natural number is prime. Order of quantifiers (nesting) The order of quantifiers is critical to meaning, as is illustrated by the following two propositions: For every natural number n, there exists a natural number s such that s = n2. This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is inversed is different: There exists a natural number s such that for every natural number n, s = n2. This is clearly false; it asserts that there is a single natural number s that is at the same time the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables. A less trivial example from mathematical analysis are the concepts of uniform and pointwise continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called pointwise continuous if {\textstyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}{\textstyle \forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )} uniformly continuous if {\textstyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )}{\textstyle \forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )} In the former case, the particular value chosen for d can be a function of both e and x, the variables that precede it. In the latter case, d can be a function only of e, i.e. it has to be chosen independent of x. For example, f(x) = x2 satisfies pointwise, but not uniform continuity. In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning. The maximum depth of nesting of quantifiers in a formula is called its quantifier rank. Equivalent expressions If D is a domain of x and P(x) is a predicate dependent on object variable x, then the universal proposition can be expressed as {\displaystyle \forall x\!\in \!D\;P(x).}{\displaystyle \forall x\!\in \!D\;P(x).} This notation is known as restricted or relativized or bounded quantification. Equivalently one can write, {\displaystyle \forall x\;(x\!\in \!D\to P(x)).}{\displaystyle \forall x\;(x\!\in \!D\to P(x)).} The existential proposition can be expressed with bounded quantification as {\displaystyle \exists x\!\in \!D\;P(x),}{\displaystyle \exists x\!\in \!D\;P(x),} or equivalently {\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).}{\displaystyle \exists x\;(x\!\in \!\!D\land P(x)).} Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks: {\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),}{\displaystyle \neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),} which shows that to disprove a "for all x" proposition, one needs no more than to find an x for which the predicate is false. Similarly, {\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),}{\displaystyle \neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),} to disprove a "there exists an x" proposition, one needs to show that the predicate is false for all x. 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


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