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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.''Cattell, J. M., & Farrand, L. (1896). Physical and mental
measurements of the students of Columbia University, The Psychological Review, 3(6), 618-648, p.648 quoted in James McKeen Cattell (1860-1944) Psychologist, Publisher, and Editor http://www.indiana.edu/~intell/jcattell.shtml It seems to be held as universally true that ''the foundation of quantification is measurement.''Some Aspects of Quantification in Science, S. S. Wilks, Isis, Vol. 52, No. 2 (Jun., 1961), pp. 135-142, p.135 There is little doubt that ''quantification provided a basis for the objectivity of science.''Hong, History of Science: Building Circuits of Trust, Science, 10 September 2004: 1569-1570 http://www.sciencemag.org/cgi/content/full/305/5690/1569 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.''Alfred W. Crosby, The Measure of Reality: Quantification and Western Society, 1250-1600, Cambridge:
Cambridge University Press, 1996, p.201 Any reasonable ''comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification.''Robert Langs, Psychoanalysis as an Aristotelian Science—Pathways to Copernicus and a Modern-Day Approach, Contemporary Psychoanalysis, 23, 1987, pp.555-576 http://www.pep-web.org/document.php?id=cps.023.0555a Even today, ''universities use imperfect instruments called 'exams' to indirectly quantify something they call knowledge.''Aaaron Lynch, Misleading Mix of Religion and Science
http://cfpm.org/jom-emit/1999/vol3/lynch_a.html

More specifically, in 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 Language s 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'' × ''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'' × ''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:





is the function of ''v''1,...,''v''''n''-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 ''v''1,...,''v''''n''-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 '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 Graph s 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 studies as behavior.


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