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| '''Quantification''' has several distinct senses. In [[mathematics]] and [[empirical science]], it is the act of [[counting]] and [[measuring]] that maps human sense [[observation]]s and [[experience]]s into [[element (mathematics)|member]]s of some [[Set (mathematics)|set]] of [[number]]s. Quantification in this sense is fundamental to the [[scientific method]].
| | The author is recognized by the title of Numbers Wunder. Hiring is my profession. Years ago we moved to North Dakota and I adore each day residing here. To perform baseball is the hobby he will never stop doing.<br><br>Also visit my web site ... at home std testing ([http://www.jeepshop.asia/gal_offroad/1658289 Keep Reading]) |
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| In [[logic]], '''quantification''' is the binding of a variable ranging over a [[domain of discourse]]. The variable thereby becomes [[bound variable|bound]] by an [[variable binding operator|operator]] called a '''quantifier'''. Academic discussion of quantification refers more often to this meaning of the term than the preceding one.
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| In grammar, a '''quantifier''' is a type of [[Determiner (class)|determiner]], such as ''all'' or ''many'', that indicates quantity. These items have been argued to correspond to logical quantifiers at the semantic level.
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| ==Natural language==
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| {{See also|Number names}}
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| All known human languages make use of quantification (Wiese 2004). For example, in English:
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| * ''Every'' glass in my recent order was chipped.
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| * ''Some of'' the people standing across the river have white armbands.
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| * ''Most of'' the people I talked to didn't have a clue who the candidates were.
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| * ''A lot of'' people are smart.
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| The words in italics are quantifiers. | |
| 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.
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| 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. For example the sentence ″''Someone gets mugged in New York every 10 minutes''″ does not identify whether it is the same person getting mugged every 10 minutes.
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| [[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 [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]] and [[Willard Van Orman Quine|Quine]].
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| ==Logic==
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| {{see also|generalized quantifier|Lindström quantifier}}
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| In [[language]] and [[logic]], quantification is a construct that specifies the quantity of specimens in 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. A language element which generates a quantification is called a quantifier. The resulting expression is a quantified expression; and the expression is said to be quantified over the predicate or function expression whose [[Free variables and bound variables|free variable]] is [[bound variable|bound]] by the quantifier. Quantification is used in both [[natural language]]s and [[formal language]]s. Examples of quantifiers in English 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.
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| The two fundamental kinds of quantification in [[predicate (logic)|predicate logic]] are [[universal quantification]] and [[existential quantification]]. The traditional symbol for the universal quantifier "all" is "∀", an inverted letter "[[A]]", and for the [[existential quantification|existential quantifier]] "exists" is "∃", a rotated letter "[[E]]". These quantifiers have been generalized beginning with the work of [[Andrzej Mostowski|Mostowski]] and [[Per Lindström|Lindström]].
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| ===Mathematics===
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| Consider the following statement:
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| : 1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 · 2 = 3 + 3, ...., and ''n'' · 2 = ''n'' + ''n'', etc.
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| This has the appearance of an ''infinite [[logical conjunction|conjunction]]'' of propositions. From the point of view of [[formal language]]s this is immediately a problem, since [[syntax]] rules are expected to generate [[finite set|finite]] objects. The example above is fortunate in that there is a [[Algorithm|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 formulation which avoids these problems uses ''universal quantification'':
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| : For any [[natural number]] ''n'', ''n''·2 = ''n'' + ''n''.
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| A similar analysis applies to the [[disjunction (logic)|disjunction]],
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| : 1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or n is equal to 5 + 5, etc.
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| which can be rephrased using ''existential quantification'':
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| : For some [[natural number]] ''n'', ''n'' is equal to 5 + 5.
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| It is possible to devise [[abstract algebra]]s whose [[model theory|models]] include [[formal language]]s with quantification, but progress has been slow and interest in such algebra has been limited. Three approaches have been devised to date:
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| * [[Relation algebra]], invented by [[De Morgan]], and developed by [[Charles Sanders Peirce]], [[Ernst Schröder]], [[Tarski]], and Tarski's students. Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the [[model theory|models]] of relation algebra include the [[axiomatic set theory]] [[ZFC]] and [[Peano arithmetic]];
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| * [[Cylindric algebra]], devised by [[Tarski]], [[Henkin]], and others;
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| * The [[polyadic algebra]] of [[Paul Halmos]].
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| ===Notation===
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| There are two quantifiers, the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "∀", an inverted 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".
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| An example of quantifying an English statement would be as follows. Given the statement, "All of Peters friends either like to dance or like to go to the beach", we can identify key aspects and rewrite using symbols including quantifiers. So, let x be any one particular friend of Peter, X the set of all Peter's friends, P(x) be the [[predicate]] (mathematical logic) "x likes to dance", and lastly Q(x) the predicate "x likes to go to the beach". Then we have, : <math> \forall{x}{\in}X, P(x) \or Q(x) </math>. Which is read, "for all x that's a member of X, P of x or Q of x."
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| Some other quantified expressions are constructed as follows,
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| : <math> \exists{x}\, P \qquad \forall{x}\, P </math>
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| for a formula ''P''. Variant notations include, for set ''X'' and set members ''x'':
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| : <math> (\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</math>
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| All of these variations also apply to universal quantification.
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| Other variations for the universal quantifier are
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| : <math>(x) \, P \qquad \bigwedge_{x} P</math>
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| 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:
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| * Assume a fixed domain of discourse for every quantification, as is done in [[Zermelo–Fraenkel set theory]],
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| * 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 [[Type system|statically typed]] [[computer programming]] languages, where variables have declared types.
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| * Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain or the [[type (logic)|type]] of the objects in that domain.
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| 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.
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| 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.
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| Mathematical formulas mix symbolic expressions for quantifiers, with natural language quantifiers such as
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| : For any natural number ''x'', ....
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| : There exists an ''x'' such that ....
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| : For at least one ''x''.
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| Keywords for [[uniqueness quantification]] include:
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| : For exactly one natural number ''x'', ....
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| : There is one and only one ''x'' such that ....
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| Further, ''x'' may be replaced by a [[pronoun]]. For example,
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| :For any natural number, its product with 2 equals to its sum with itself
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| :Some natural number is prime.
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| ===Equivalent expressions===
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| If ''X'' is a domain of ''x'' and ''P''(''x'') is a predicate dependent on ''x'', then the universal proposition is expressed in [[Boolean algebra]] terms as<br />
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| :<math>\forall x\in X, P(x) \equiv \{x\in X\} \rightarrow P(x) \equiv \{x\notin X\} \or P(x),</math>
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| which equivalently reads "if ''x'' is in ''X'', then ''P''(''x'') is true." If ''x'' is not in ''X'', then ''P''(''x'') is indeterminate. Note that the truth of the expression requires only that ''x'' be in ''X'', so it can be ''any x'' in ''X'', independent of ''P''(''x''), whereas the ''falsity'' of the expression, or the truth of
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| :<math>\{x\in X\} \and \neg P(x),</math>
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| additionally requires that ''x'' be such that ''P''(''x'') evaluates to false; this is the reason behind calling ''x'' a "bound variable." This last expression can thus be read as "for some ''x'' in ''X'', ''P''(''x'') is false," or "there exists an ''x'' in ''X'' such that ''P''(''x'') is false." So, we now have the equivalent Boolean expression for the existential proposition:<br />
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| :<math>\exists x\in X : P(x) \equiv \{x\in X\} \and P(x).</math>
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| Thus, together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:
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| :<math>\neg (\forall x\in X, P(x)) \equiv \exists x\in X : \neg P(x), </math>
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| 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,<br />
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| :<math>\neg (\exists x\in X : P(x)) \equiv \forall x\in X, \neg P(x), </math>
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| to disprove a "there exists an ''x''" proposition, one needs to show that the predicate is false for all ''x''.
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| ===Nesting===
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| Consider the following statement:
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| :For any natural number <math>n</math>, there is a natural number <math>s</math> such that <math>s = n^2 </math>.
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| This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the quantifiers are turned around is different:
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| : There is a natural number <math>s</math> such that for any natural number <math>n</math>, <math>s = n^2</math>.
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| 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 is because the syntax directs that any newly introduced variable cannot be a function of subsequently introduced variables.
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| This illustrates that the order of quantifiers is critical to meaning.
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| A less trivial example is the important concept of [[uniform continuity]] from [[mathematical analysis|analysis]], which differs from the more familiar concept of [[continuous function|pointwise continuity]] only by an exchange in the positions of two quantifiers.
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| To illustrate this, let ''f'' be a real-valued function on '''R'''.
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| * A: Pointwise continuity of <math>f</math> on '''R''':
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| ::<math> \underbrace{\forall x \in \mathbb{R} \, \, \forall \epsilon >0} \, \exists \delta > 0 \, \forall h \in \mathbb{R} \, \left( \, |h| < \delta \, \to \, |f(x) - f(x+h)| < \epsilon \, \right) </math>
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| interchanging the universal quantifiers over the braces, this is the same as
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| * A': Pointwise continuity of ''f'' on '''R''':
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| ::<math> \forall \epsilon >0 \, \underbrace{\forall x \in \mathbb{R} \, \, \exists \delta > 0} \, \forall h \in \mathbb{R} \, \left( \, |h| < \delta \, \to \, |f(x) - f(x+h)| < \epsilon \, \right)</math>
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| Thus, it is implied that the particular value chosen for ''δ'' can only be a function of ''ε'' and ''x'', the variables that precede it; whereas in
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| * B: Uniform continuity of ''f'' on '''R''':
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| ::<math> \forall \epsilon >0 \, \underbrace{\exists \delta > 0 \, \, \forall x \in \mathbb{R}} \, \forall h \in \mathbb{R} \, \left( \, |h| < \delta \, \to \, |f(x) - f(x+h)| < \epsilon \, \right) </math>
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| by interchanging the existential and universal quantifiers over the braces in A', ''δ'' is asserted to be independent of ''x''.
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| Ambiguity is avoided with the quantifiers in front:
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| *<math> \exists</math> A:<math> \forall</math> B: C – unambiguous
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| *there is an A such that <math>\forall</math> B: C – unambiguous
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| *there is an A such that for all B, C – unambiguous, provided that the separation between B and C is clear
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| *there is an A such that C for all B – it is often clear that what is meant is
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| :: there is an A such that (C for all B)
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| :but it could be interpreted as
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| :: (there is an A such that C) for all B
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| *there is an A such that C <math>\forall</math> B — suggests more strongly that the first is meant; this may be reinforced by the layout, for example by putting "C <math>\forall</math> B" on a new line.
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| The maximum depth of nesting of quantifiers inside a formula is called its [[Quantifier rank]].
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| ===Range of quantification===
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| 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.
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| A more natural way to restrict the domain of discourse uses ''guarded quantification''. For example, the guarded quantification
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| :For some natural number ''n'', ''n'' is even and ''n'' is prime
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| means
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| :For some [[even number]] ''n'', ''n'' is prime.
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| In some mathematical theories a single domain of
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| discourse fixed in advance is assumed. 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
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| above to express
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| :For any natural number ''n'', ''n''·2 = ''n'' + ''n''
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| in Zermelo–Fraenkel set theory, it can be said
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| :For any ''n'', if ''n'' belongs to '''N''', then ''n''·2 = ''n'' + ''n'',
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| where '''N''' is the set of all natural numbers.
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| ===Formal semantics===
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| Mathematical Semantics is the application of [[mathematics]] to study the meaning of expressions in a formal 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. This article only addresses the issue of how quantifier elements are interpreted.
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| Given a [[model theory|model theoretical]] logical framework, the syntax of a formula can be given by a syntax tree. Quantifiers have scope and a variable ''x'' is [[free variable|free]] if it is not within the scope of a quantification for that variable. Thus in
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| :<math> \forall x (\exists y B(x,y)) \vee C(y,x) </math>
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| the occurrence of both ''x'' and ''y'' in ''C''(''y'',''x'') is free.
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| [[File:IMG Tree.gif|thumb|230px|right|Syntactic tree illustrating scope and variable capture]]
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| An interpretation for first-order predicate calculus assumes as given
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| a domain of individuals ''X''. A formula ''A'' whose free variables
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| are ''x''<sub>1</sub>, ..., ''x''<sub>n</sub> is interpreted as a
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| [[Boolean function|boolean]]-valued function ''F''(''v''<sub>1</sub>, ...,
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| ''v''<sub>''n''</sub>) of ''n'' arguments, where each argument ranges
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| 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
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| :<math> \forall x_n A(x_1, \ldots , x_n) </math>
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| is the function ''G'' of ''n''-1 arguments such that ''G''(''v''<sub>1</sub>, ...,''v''<sub>''n''-1</sub>) = '''T''' if and only if ''F''(''v''<sub>1</sub>, ..., ''v''<sub>''n''-1</sub>, ''w'') = '''T''' for every ''w'' in ''X''. If ''F''(''v''<sub>1</sub>, ..., ''v''<sub>''n''-1</sub>, ''w'') = '''F''' for at least one value of ''w'', then ''G''(''v''<sub>1</sub>, ...,''v''<sub>''n''-1</sub>) = '''F'''. Similarly the interpretation of the formula
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| :<math> \exists x_n A(x_1, \ldots , x_n) </math>
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| is the function ''H'' of ''n''-1 arguments such that ''H''(''v''<sub>1</sub>, ...,''v''<sub>''n''-1</sub>) = '''T''' if and only if ''F''(''v''<sub>1</sub>, ...,''v''<sub>''n''-1</sub>, ''w'') = '''T''' for at least one ''w'' and ''H''(''v''<sub>1</sub>, ..., ''v''<sub>''n''-1</sub>) = '''F''' otherwise.
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| 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
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| :<math> \exists ! x_n A(x_1, \ldots , x_n) </math>
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| then is the function of ''n''-1 arguments, which is the logical ''and'' of the interpretations of
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| :<math> \exists x_n A(x_1, \ldots , x_n) </math>
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| :<math> \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\}.</math>
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| ===Paucal, multal and other degree quantifiers===
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| {{See also|Fubini's theorem|measurable}}
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| None of the quantifiers previously discussed apply to a quantification such as
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| :There are many integers ''n'' < 100, such that ''n'' is divisible by 2 or 3 or 5.
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| One possible interpretation mechanism can be 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 ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> whose interpretation is
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| the function ''F'' of variables ''v''<sub>1</sub>,...,''v''<sub>''n''</sub>
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| then the interpretation of
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| :<math> \exists^{\mathrm{many}} x_n A(x_1, \ldots, x_{n-1}, x_n) </math>
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| is the function of ''v''<sub>1</sub>,...,''v''<sub>''n''-1</sub> which is '''T''' if and only if
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| :<math> \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T} \} \geq b </math>
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| and '''F''' otherwise. Similarly, the interpretation of
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| :<math> \exists^{\mathrm{few}} x_n A(x_1, \ldots, x_{n-1}, x_n) </math>
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| is the function of ''v''<sub>1</sub>,...,''v''<sub>''n''-1</sub> which is '''F''' if and only if
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| :<math> 0< \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T}\} \leq a </math>
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| and '''T''' otherwise.
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| ===Other quantifiers===
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| A few other quantifiers have been proposed over time. In particular, the solution quantifier,<ref>[[E. C. R. Hehner|Hehner, Eric C. R.]], 2004, [http://www.cs.utoronto.ca/~hehner/aPToP ''Practical Theory of Programming''], 2nd edition, p. 28</ref> noted § ([[section sign]]) and read "those". For example:
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| :<math> \left [ \S n \in \mathbb{N} \quad n^2 \leq 4 \right ] = \left\{0, 1, 2\right\}</math>
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| is read "those ''n'' in '''N''' such that ''n''<sup>2</sup> ≤ 4 are in {0,1,2}." The same construct is expressible in [[set-builder notation]]:
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| :<math>\{n \in \mathbf N: n^2 \le 4\} = \left\{0, 1, 2\right\} \,\!</math>
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| ===History===
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| [[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.
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| [[Gottlob Frege]], in his 1879 ''[[Begriffsschrift]]'', was the first to employ a quantifier to bind a variable ranging over a [[domain of discourse]] and appearing in [[Predicate (mathematical logic)|predicate]]s. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀''x''~, or [[contraposition]]. Frege's treatment of quantification went largely unremarked until [[Bertrand Russell]]'s 1903 ''Principles of Mathematics''.
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| In work that culminated in Peirce (1885), [[Charles Sanders Peirce]] and his student [[Oscar Howard Mitchell]] independently invented universal and existential quantifiers, and [[bound variable]]s. Peirce and Mitchell wrote Π<sub>x</sub> and Σ<sub>x</sub> where we now write ∀''x'' and ∃''x''. Peirce's notation can be found in the writings of [[Ernst Schröder]], [[Leopold Loewenheim]], [[Thoralf Skolem]], and Polish logicians into the 1950s. Most notably, it is the notation of [[Kurt Gödel]]'s landmark 1930 paper on the [[Gödel's completeness theorem|completeness]] of [[first-order logic]], and 1931 paper on the [[Gödel's incompleteness theorem|incompleteness]] of [[Peano arithmetic]].
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| Peirce's approach to quantification also influenced [[William Ernest Johnson]] and [[Giuseppe Peano]], who invented yet another notation, namely (''x'') for the universal quantification of ''x'' and (in 1897) ∃''x'' for the existential quantification of ''x''. Hence for decades, the canonical notation in philosophy and mathematical logic was (''x'')''P'' to express "all individuals in the domain of discourse have the property ''P''," and "(∃''x'')''P''" for "there exists at least one individual in the domain of discourse having the property ''P''." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the ''[[Principia Mathematica]]'' of [[Alfred North Whitehead|Whitehead]] and [[Bertrand Russell|Russell]], [[Willard Van Orman Quine|Quine]], and [[Alonzo Church]]. In 1935, [[Gentzen]] introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.
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| Around 1895, Peirce began developing his [[existential graph]]s, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching [[heterogeneous reasoning]] and [[logical graph|diagrammatic inference]].
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| ==Natural science==
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| Some measure of the undisputed general importance of quantification in the [[natural science]]s can be gleaned from the following comments: "these are mere facts, but they are quantitative facts and the basis of science."<ref>[[James McKeen Cattell|Cattell, James McKeen]]; and [[Livingston Farrand|Farrand, Livingston]] (1896) "Physical and mental measurements of the students of Columbia University", ''The Psychological Review'', Vol. 3, No. 6 (1896), pp. 618-648; p. 648 quoted in ''[http://www.indiana.edu/~intell/jcattell.shtml James McKeen Cattell (1860-1944) Psychologist, Publisher, and Editor.]''</ref> It seems to be held as universally true that "the foundation of quantification is measurement."<ref>[[Samuel Stanley Wilks|Wilks, Samuel Stanley]] (1961) "Some Aspects of Quantification in Science", ''Isis'', Vol. 52, No. 2 (1961), pp. 135-142; p. 135</ref> There is little doubt that "quantification provided a basis for the objectivity of science."<ref>Hong, Sungook (2004) "[http://www.sciencemag.org/cgi/content/full/305/5690/1569 History of Science: Building Circuits of Trust]", ''Science'', Vol. 305, No. 5690 (10 September 2004), pp. 1569-1570</ref> [The context of these quotes is slightly misleading - most scientists use quantification as related to quantities, which is categorically different than quantifier as used in this article. Most would probably article that the word is Qualifier.] 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."<ref>[[Alfred W. Crosby|Crosby, Alfred W.]] (1996) ''The Measure of Reality: Quantification and Western Society'', Cambridge University Press, 1996, p. 201</ref> Any reasonable "comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification."<ref>Langs, Robert J. (1987) "[http://www.pep-web.org/document.php?id=cps.023.0555a Psychoanalysis as an Aristotelian Science—Pathways to Copernicus and a Modern-Day Approach]", ''Contemporary Psychoanalysis'', Vol. 23 <!-- one vol. per year -->(1987), pp. 555-576</ref> Even today, "universities use imperfect instruments called '[[exams]]' to indirectly quantify something they call knowledge."<ref>[[Aaron Lynch|Lynch, Aaron]] (1999) "[http://cfpm.org/jom-emit/1999/vol3/lynch_a.html Misleading Mix of Religion and Science,]" ''Journal of Memetics: Evolutionary Models of Information Transmission'', Vol. 3, No. 1<!-- because there are only two issues, and there is Gatherer, Derek G. (1999) "Reply to Commentaries", ''Journal of Memetics: Evolutionary Models of Information Transmission, Vol. 3 [No. 2] (1999) http://jom-emit.cfpm.org/1999/vol3/gatherer_reply.html --> (1999)</ref> This meaning of quantification comes under the heading of [[Pragmatism|pragmatics]].
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| In some instances in the natural sciences a seemingly intangible concept may be quantified by creating a scale—for example, a [[pain scale]] in medical research, or a discomfort scale at the intersection of [[meteorology]] and [[human physiology]] such as the [[heat index]] measuring the combined perceived effect of heat and [[humidity]], or the [[wind chill factor]] measuring the combined perceived effects of cold and wind.
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| ==Social sciences==
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| {{See also|Society for Quantitative Analysis of Behavior|Quantitative psychological research}}
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| In the [[social sciences]], quantification is an integral part of [[economics]] and [[psychology]]. Both disciplines gather data—economics by [[empirical observation]] and psychology by [[experimentation]], and both use [[statistics|statistical techniques]] such as [[regression analysis]] to draw conclusions from it.
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| In some instances a seemingly intangible property may be quantified by asking subjects to rate something on a [[Scale (social sciences)|scale]]—for example, a [[Happiness economics#Methodology|happiness scale]] or a [[Self-Perceived Quality of Life Scale|quality of life scale]]—or by the construction of a scale by the researcher, as with the [[index of economic freedom]]. In other cases, an unobservable variable may be quantified by replacing it with a [[proxy (statistics)|proxy variable]] with which it is highly correlated—for example, per capita [[gross domestic product]] is often used as a proxy for [[standard of living]] or [[quality of life]].
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| Frequently in the use of regression, the presence or absence of a trait is quantified by employing a [[dummy variable (statistics)|dummy variable]], which takes on the value 1 in the presence of the trait or the value 0 in the absence of the trait.
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| [[Quantitative linguistics]] is an area of [[linguistics]] that relies on quantification. For example,<ref>Bybee, Joan; Perkins, Revere; and Pagliuca, William. (1994) ''The Evolution of Grammar'', Univ. of Chicago Press: ch. 4.</ref> indices of [[grammaticalization]] of [[morpheme]]s, such as [[phonology|phonological]] shortness, dependence on surroundings, and fusion with the verb, have been developed and found to be significantly correlated across languages with stage of evolution of function of the morpheme.
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| ==Hard versus soft science==
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| The ease of quantification is one of the features used to distinguish [[hard and soft science]]s from each other. Hard sciences are often considered to be more scientific, rigorous, or accurate. In some [[social sciences]] such as [[sociology]], specific accurate data are difficult to obtain, either because laboratory conditions are not present or because the issues involved are conceptual but not directly quantifiable.{{facts|date=December 2013}}
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| ==See also==
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| {{Portal|Logic}}
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| *[[Indefinite pronoun]]
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| *[[Quantifier elimination]]
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| ==References==
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| <references/>
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| * [[Jon Barwise|Barwise, Jon]]; and [[John Etchemendy|Etchemendy, John]], 2000. ''Language Proof and Logic''. CSLI (University of Chicago Press) and New York: Seven Bridges Press. A gentle introduction to [[first-order logic]] by two first-rate logicians.
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| * Crosby, Alfred W. (1996) ''The Measure of Reality: Quantification and Western Society, 1250-1600''. Cambridge University Press.
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| * [[Gottlob Frege|Frege, Gottlob]], 1879. ''[[Begriffsschrift]]''. Translated in [[Jean van Heijenoort]], 1967. ''From Frege to Gödel: A Source Book on Mathematical Logic, 1879-1931''. Harvard University Press. The first appearance of quantification.
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| * [[David Hilbert|Hilbert, David]]; and [[Wilhelm Ackermann|Ackermann, Wilhelm]], 1950 (1928). ''[[Principles of Mathematical Logic]]''. Chelsea. Translation of ''Grundzüge der theoretischen Logik''. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standard manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of [[first-order logic]].
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| * [[Charles Sanders Peirce|Peirce, C. S.]], 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, ''American Journal of Mathematics'', Vol. 7, pp. 180–202. Reprinted in Kloesel, N. ''et al.'', eds., 1993. ''Writings of C. S. Peirce, Vol. 5''. Indiana University Press. The first appearance of quantification in anything like its present form.
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| * [[Hans Reichenbach|Reichenbach, Hans]], 1975 (1947). ''Elements of Symbolic Logic'', Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".
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| * Westerståhl, Dag, 2001, "Quantifiers," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.
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| * Wiese, Heike, 2003. ''Numbers, language, and the human mind''. Cambridge University Press. ISBN 0-521-83182-2.
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| ==External links==
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| {{Wiktionary|quantification}}
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| * {{springer|title=Quantifier|id=p/q076260}}
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| * [[Stanford Encyclopedia of Philosophy]]:
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| ** "[http://plato.stanford.edu/entries/logic-classical/ Classical Logic] — by Stewart Shapiro. Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.
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| **"[http://plato.stanford.edu/entries/generalized-quantifiers/ Generalized quantifiers]" — by Dag Westerståhl.
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| * Peters, Stanley; Westerståhl, Dag (2002). "[http://www.stanford.edu/group/nasslli/courses/peters-wes/PWbookdraft2-3.pdf Quantifiers.]"
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| {{Formal Fallacy}}
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| {{Logic}}
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| {{lexical categories|state=collapsed}}
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| [[Category:Logic]]
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| [[Category:Predicate logic]]
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| [[Category:Quantification]]
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| [[Category:Philosophical logic]]
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| [[Category:Semantics]]
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