Quantification
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'''Quantification''' has two distinct meanings.
In [[mathematics]] and [[empirical science]], it refers to human acts, known as [[counting]] and [[measuring]] that map human sense [[observation]]s and [[experience]]s into [[element (mathematics)|member]]s of some [[set]] of [[number]]s. Quantification in this sense is fundamental to the [[scientific method]].
In [[logic]], '''quantification''' refers to 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.
== Natural language ==
All known human languages make use of quantification (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.
The words in italics are called '''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.
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 [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]] and [[Willard Van Orman Quine|Quine]].
==Logic==
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 <em>exist</em> 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 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 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)|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.
=== Mathematics ===
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 [[logical conjunction|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 set|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 (logic)|disjunction]],
: 1 is [[prime number|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.
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:
* [[Relation algebra]], invented by [[DeMorgan]], and developed by [[Ernst Schroder]], [[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]];
* [[Cylindric algebra]], devised by [[Tarski]], [[Henkin]], and others;
*The [[polyadic algebra]] of [[Paul Halmos]].
=== Notation ===
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,
: <math> \exists{x}\, P \quad \forall{x}\, P </math>
where "''P''" denotes a formula. Many variant notations are used, such as
: <math> \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</math>
All of these variations also apply to universal quantification.
Other variations for the universal quantifier are
: <math>(x) \, P \quad \bigwedge_{x} P</math>
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 [[Type system|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 (logic)|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.
=== Nesting ===
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 natural 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 [[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.
To illustrate this, let ''f'' be a real-valued function on '''R'''.
* A: Pointwise continuity of ''f'' on '''R''':
::<math> \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 </math>
interchanging the universal quantifiers over the braces, this is the same as
* A': Pointwise continuity of ''f'' on '''R''':
::<math> \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 </math>
This differs from
* B: Uniform continuity of ''f'' on '''R''':
::<math> \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 </math>
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:
*<math> \exists</math> A:<math> \forall</math> B: C - unambiguous
*there is an A such that <math>\forall</math> 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 <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.
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 axioms|Zermelo Fraenkel]] set theory, variables range over all [[set]]s. 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.
=== Formal semantics ===
[[Mathematical semantics]] is the application of [[mathematics]] to study the meaning of expressions in a [[formal language|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 variable|free]] if it is not within the scope of a quantification for that variable. Thus in
:<math> \forall x (\exists y B(x,y)) \vee C(y,x) </math>
the occurrence of both ''x'' and ''y'' in ''C''(''y'',''x'') is free.
[[Image:IMG Tree.jpg|thumb|323px|left|Syntactic tree illustrating scope and variable capture]]
An interpretation for first-order predicate calculus assumes as given
a domain of individuals ''X''. A formula ''A'' whose free variables
are ''x''<sub>1</sub>, ..., ''x''<sub>n</sub> is interpreted as a
[[boolean]]-valued function ''F''(''v''<sub>1</sub>, ...,
''v''<sub>''n''</sub>) 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
:<math> \forall x_n A(x_1, \ldots , x_n) </math>
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
:<math> \exists x_n A(x_1, \ldots , x_n) </math>
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.
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
:<math> \exists ! x_n A(x_1, \ldots , x_n) </math>
then is the function of ''n''-1 arguments, which is the logical ''and'' of the interpretations of
:<math> \exists x_n A(x_1, \ldots , x_n) </math>
:<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>
=== 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.
Although this article will not treat the semantics of natural language, 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 ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> whose interpretation is
the function ''F'' of variables ''v''<sub>1</sub>,...,''v''<sub>''n''</sub>
then the interpretation of
:<math> \exists^{\mathrm{many}} x_n A(x_1, \ldots, x_{n-1}, x_n) </math>
is the function of ''v''<sub>1</sub>,...,''v''<sub>''n''-1</sub> which is '''T''' if and only if
:<math> \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T} \} \geq b </math>
and '''F''' otherwise. Similarly, the interpretation of
:<math> \exists^{\mathrm{few}} x_n A(x_1, \ldots, x_{n-1}, x_n) </math>
is the function of ''v''<sub>1</sub>,...,''v''<sub>''n''-1</sub> which is '''F''' if and only if
:<math> 0< \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T}\} \leq a </math>
and '''T''' otherwise. We have completely avoided discussion of technical issues regarding [[measurable|measurability]] of the interpretation functions; some of these are technical questions that require [[Fubini's theorem]].
We caution the reader that the ''logic'' corresponding to such semantics is exceedingly complicated.
==Syntax==
Quantification in [[formal language|formal]] and [[natural language]]s falls under [[syntax]] and [[semantics]].
== History ==
[[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]].
[[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]]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''.
In work that culminated in Peirce (1885), [[Charles Sanders Peirce]] and his student O. H. Mitchell independently invented universal and existential qunatifiers, 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 Schroder]], [[Leopold Loewenheim]], [[Thoralf Skolem]], and Polish logicians into the 1950s. Most notably, it is the notation of [[Kurt Goedel]]'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]].
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.
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]].
==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.''<ref>Cattell, J. M., & Farrand, L. (1896) "Physical and mental
measurements of the students of Columbia University," ''The Psychological Review 3'': 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> Wilks, S. S. (1961) "Some Aspects of Quantification in Science," ''Isis 52'': 135-142, p.135.</ref> There is little doubt that ''quantification provided a basis for the objectivity of science.''<ref>Hong, "[http://www.sciencemag.org/cgi/content/full/305/5690/1569 History of Science: Building Circuits of Trust,]" ''Science'', 10 September 2004: 1569-1570. </ref> 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>Crosby (1996: 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 (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 23'': 555-576. </ref> Even today, ''universities use imperfect instruments called '[[exams]]' to indirectly quantify something they call knowledge.''<ref>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 3''.
</ref> This meaning of quantification comes under the heading of [[pragmatics]].
== Development of quantitification both across species and within humans ==
In [[Society for Quantitative Analysis of Behavior|Quantitative analysis of behavior]], [[Evolutionary Psychology]] and Cognitive [[Developmental Psychology]], quantification is studied as behavior.
==See also==
*[[Bound variable]]
*[[Cylindric algebra]]
*[[Determiner]]
*[[Domain of discourse]]
*[[Generalized quantifier]]
*[[Indefinite pronoun]]
*[[Lindström quantifier]]
*[[Montague grammar]]
*[[Relation algebra]]
*[[Variable binding operator]]
==Notes==
<references/>
==References==
*[[Jon Barwise]] and [[John Etchemendy]], 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.
*Crosby, Alfred W. (1996) ''The Measure of Reality: Quantification and Western Society, 1250-1600''. Cambridge University Press.
* [[Gottlob Frege]], 1879. ''[[Begriffsschrift]]''. Translated in [[Jean van Heijenoort]], 1967. ''From Frege to Godel: A Source Book on Mathematical Logic, 1879-1931''. Harvard Univ. Press. The first appearance of quantification.
* [[David Hilbert]] and [[Wilhelm Ackermann]], 1950 (1928). ''[[Principles of Theoretical 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-standar manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of [[first-order logic]].
*[[Charles Peirce]], 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, ''American Journal of Mathematics 7'': 180-202. Reprinted in Kloesel, N. et al, eds., 1993. ''Writings of C. S. Peirce, Vol. 5''. Indiana Univ. Press. The first appearance of quantification in anything like its present form.
* [[Hans Reichenbach]], 1975 (1947). ''Elements of Symbolic Logic'', Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".
* Wiese, 2003. ''Numbers, language, and the human mind''. Cambridge University Press. ISBN 0-521-83182-2.
*Westerstahl, Dag, 2001, "Quantifiers," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.
==External links==
{{Wiktionarypar|quantification}}
* [[Stanford Encyclopedia of Philosophy]]:
** "[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.
**"[http://plato.stanford.edu/entries/generalized-quantifiers/ Generalized quantifiers]" -- by Dag Westerstahl.
* Peter, Stanley, and Dag Westerståhl (2002) "[http://www.stanford.edu/group/nasslli/courses/peters-wes/PWbookdraft2-3.pdf Quantifiers.]"
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