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| In [[mathematics]] and [[mathematical logic|logic]], '''plural quantification''' is the theory that an individual [[Variable (mathematics)|variable]] x may take on ''[[plural]]'', as well as singular, values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 storeys.
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| The point of the theory is to give [[First-order predicate calculus|first-order logic]] the power of [[set theory]], but without any "[[existential commitment]]" to such objects as sets. The classic expositions are Boolos 1984, and Lewis 1991.
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| == History ==
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| The view is commonly associated with [[George Boolos]], though it is older (see notably [[Peter Simons (academic)|Simons]] 1982), and is related to the view of classes defended by [[John Stuart Mill]] and other [[Nominalism|nominalist]] philosophers. Mill argued that universals or "classes" are not a peculiar kind of thing, having an objective existence distinct from the individual objects that fall under them, but "is neither more nor less than the individual things in the class". (Mill 1904, II. ii. 2,also I. iv. 3).
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| A similar position was also discussed by [[Bertrand Russell]] in chapter VI of Russell (1903), but later dropped in favour of a "no-classes" theory. See also [[Gottlob Frege]] 1895 for a critique of an earlier view defended by [[Ernst Schroeder]].
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| The general idea can be traced back to [[Leibniz]]. (Levey 2011, pp. 129–133)
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| Interest revived in plurals with work in linguistics in the 1970s by [[Remko Scha]], [[Godehard Link]], [[Fred Landman]], [[Roger Schwarzschild]], [[Peter Lasersohn]] and others, who developed ideas for a semantics of plurals.
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| ==Background and motivation==
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| === Multigrade (variably polyadic) predicates and relations ===
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| Sentences like
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| : Alice and Bob cooperate.
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| : Alice, Bob and Carol cooperate.
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| are said to involve a '''multigrade''' (aka '''variably polyadic''', also '''anadic''') predicate or relation ("cooperate" in this example), meaning that they stand for the same concept even though they don't have a fixed [[arity]] (cf. Linnebo & Nicolas 2008). The notion of multigrade relation/predicate has appeared as early as the 1940s and has been notably used by [[Willard Van Orman Quine|Quine]] (cf. Morton 1975). Plural quantification deals with formalizing the quantification over the variable-length arguments of such predicates, e.g. "''xx'' cooperate" where ''xx'' is a plural variable. Note that in this example it makes no sense, semantically, to instantiate ''xx'' with the name of a single person.
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| ===Nominalism===
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| {{main|Nominalism}}
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| Broadly speaking, nominalism denies the [[problem of universals|existence of universals]] ([[Abstract entity|abstract entities]]), like sets, classes, relations, properties, etc. Thus the plural logic(s) were developed as an attempt to formalize reasoning about plurals, such as those involved in multigrade predicates, apparently without resorting to notions that nominalists deny, e.g. sets.
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| == Plural quantification ==
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| Standard first order logic has difficulties in representing some sentences with plurals. Most well-known is the [[Geach–Kaplan sentence]] "some critics admire only one another". Kaplan proved that it is [[nonfirstorderizable]] (the proof can be found in that article). Hence its paraphrase into a formal language commits us to quantification over (i.e. the existence of) sets. But some{{who|date=November 2012}} find it implausible that a commitment to sets is essential in explaining these sentences.
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| Note that an individual instance of the sentence, such as "Alice, Bob and Carol admire only one another", need not involve sets and is equivalent to the conjunction of the following first-order sentences:
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| :∀x(if Alice admires x, then x = Bob or x = Carol)
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| :∀x(if Bob admires x, then x = Alice or x = Carol)
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| :∀x(if Carol admires x, then x = Alice or x = Bob)
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| where x ranges over all critics [it being taken as read that critics cannot admire themselves]. But this seems to be an instance of "some people admire only one another", which is nonfirstorderizable.
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| Boolos argued that [[second-order logic|2nd-order]] [[monadic logic|monadic]] quantification may be systematically interpreted in terms of plural quantification, and that, therefore, 2nd-order monadic quantification is "ontologically innocent".{{citation needed|date=November 2012}}
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| Later, Oliver & Smiley (2001), Rayo (2002), Yi (2005) and McKay (2006) argued that sentences such as
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| :They are shipmates
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| :They are meeting together
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| :They lifted a piano
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| :They are surrounding a building
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| :They admire only one another
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| also cannot be interpreted in monadic second order logic. This is because predicates such as "are shipmates", "are meeting together", "are surrounding a building" are not ''distributive''. A predicate F is distributive if, whenever some things are F, each one of them is F. But in standard logic, ''every monadic predicate is distributive''. Yet such sentences also seem innocent of any existential assumptions, and do not involve quantification.
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| So one can propose a unified account of plural terms that allows for both distributive and non-distributive satisfaction of predicates, while defending this position against the "singularist" assumption that such predicates are predicates of sets of individuals (or of mereological sums).
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| Several writers{{who|date=November 2012}} have suggested that plural logic opens the prospect of simplifying the [[foundations of mathematics]], avoiding the [[paradox]]es of set theory, and simplifying the complex and unintuitive axiom sets needed in order to avoid them.{{clarify|date=November 2012}}
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| Recently, Linnebo & Nicolas (2008) have suggested that natural languages often contain [[superplural variable]]s (and associated quantifiers) such as "these people, those people, and these other people compete against each other" (e.g. as teams in an online game), while Nicolas (2008) has argued that plural logic should be used to account for the semantics of mass nouns, like "wine" and "furniture".
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| == Formal definition ==
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| This section presents a simple formulation of plural logic/quantification approximately the same as given by Boolos in ''Nominalist Platonism'' (Boolos 1985).
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| === Syntax ===
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| Sub-sentential units are defined as
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| * Predicate symbols <math>F</math>, <math>G</math>, etc. (with appropriate arities, which are left implicit)
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| * Singular variable symbols <math>x</math>, <math>y</math>, etc.
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| * Plural variable symbols <math>\bar{x}</math>, <math>\bar{y}</math>, etc.
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| Full [[sentence (mathematical logic)|sentences]] are defined as
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| * If <math>F</math> is an ''n''-ary predicate symbol, and <math>x_0, \ldots, x_n</math> are singular variable symbols, then <math>F(x_0, \ldots, x_n)</math> is a sentence.
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| * If <math>P</math> is a sentence, then so is <math>\neg P</math>
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| * If <math>P</math> and <math>Q</math> are sentences, then so is <math>P \land Q</math>
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| * If <math>P</math> is a sentence and <math>x</math> is a singular variable symbol, then <math>\exists x.P</math> is a sentence
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| * If <math>x</math> is a singular variable symbol and <math>\bar{y}</math> is a plural variable symbol, then <math>x \prec \bar{y}</math> is a sentence
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| * If <math>P</math> is a sentence and <math>\bar{x}</math> is a plural variable symbol, then <math>\exists \bar{x}.P</math> is a sentence
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| The last two lines are the only essentially new component to the syntax for plural logic. Other logical symbols definable in terms of these can be used freely as notational shorthands.
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| This logic turns out to be equi-interpretable with [[monadic second order logic]].
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| === Model theory ===
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| Plural logic's model theory/semantics is where the logic's lack of sets is cashed out. A model is defined as a tuple <math>(D,V,s,R)</math> where <math>D</math> is the domain, <math>V</math> is a collection of valuations <math>V_F</math> for each predicate name <math>F</math> in the usual sense, and <math>s</math> is a Tarskian sequence (assignment of values to variables) in the usual sense (i.e. a map from singular variable symbols to elements of <math>D</math>). The new component <math>R</math> is a binary relation relating values in the domain to plural variable symbols.
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| Satisfaction is given as
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| * <math>(D,V,s,R) \models F(x_0, \ldots, x_n)</math> iff <math>(s_{x_0}, \ldots, s_{x_n}) \in V_F</math>
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| * <math>(D,V,s,R) \models \neg P</math> iff <math>(D,V,s,R) \nvDash P</math>
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| * <math>(D,V,s,R) \models P \land Q</math> iff <math>(D,V,s,R) \models P</math> and <math>(D,V,s,R) \models Q</math>
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| * <math>(D,V,s,R) \models \exists x.P</math> iff there is an <math>s' \approx_x s</math> such that <math>(D,V,s',R) \models P</math>
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| * <math>(D,V,s,R) \models x \prec \bar{y}</math> iff <math>s_xR\bar{y}</math>
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| * <math>(D,V,s,R) \models \exists \bar{x}.P</math> iff there is an <math>R' \approx_\bar{x} R</math> such that <math>(D,V,s,R') \models P</math>
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| Where for singular variable symbols, <math>s \approx_x s'</math> means that for all singular variable symbols <math>y</math> other than <math>x</math>, it holds that <math>s_y = s'_y</math>, and for plural variable symbols, <math>R \approx_\bar{x} R'</math> means that for all plural variable symbols <math>\bar{y}</math> other than <math>\bar{x}</math>, and for all objects of the domain <math>d</math>, it holds that <math>dR\bar{y} = dR'\bar{y}</math>.
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| As in the syntax, only the last two are truly new in plural logic. Boolos observes that by using assignment ''relations'' <math>R</math>, the domain does not have to include sets, and therefore plural logic achieves ontological innocence while still retaining the ability to talk about the extensions of a predicate. Thus, the plural logic comprehension schema <math>\exists \bar{x}. \forall y. y \prec \bar{x} \leftrightarrow F(y)</math> does not yield Russell's paradox because the quantification of plural variables does not quantify over the domain. Another aspect of the logic as Boolos defines it, crucial to this bypassing of Russell's paradox, is the fact that sentences of the form <math>F(\bar{x})</math> are not well-formed: predicate names can only combine with singular variable symbols, not plural variable symbols.
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| This can be taken as the simplest, and most obvious argument that plural logic as Boolos defined it is ontologically innocent. It should be observed that in some sense, Boolos' logic is at least as innocent as non-plural logic with no sets in the domain, because any Tarskian sequence <math>s</math> is also a relation that relates values to variables, but which relates precisely one value to each variable (i.e. it is a function, and all functions are relations). Likewise, it is at most as innocent as set theory, because any set can be identified with its [[indicator function]].
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| == Criticism ==
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| Philippe de Rouilhan (2000) has argued that Boolos relied on the assumption, never defended in detail, that plural expressions in ordinary language are "manifestly and obviously" free of existential commitment. But when I utter "there are critics who admire only one another" is it manifest and obvious that I am only committing myself with respect to critics? Or is Boolos victim of a "grammatical illusion" (p. 10)? Consider
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| :There is at least one critic who admires only himself.
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| :There are critics who admire only one another
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| The first case is clearly "innocent". But what about the second? There is an obvious logical difference, since in the first case the plural is distributive, in the second, it is collective, and irreducibly so. How is it obvious that this difference is innocent? Also, the second is equivalent to
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| :Some group (or collection) of critics is such that they admire only one another
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| But what is a "group" or "collection" in this sense? "That is the whole problem". Perhaps Boolos has accorded a kind of innocence to [the second] that would actually belong only to the first.
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| == See also ==
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| * [[variadic function]]
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| * [[generalized quantifier]]
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| == References ==
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| * [[George Boolos]], 1984, "To be is to be the value of a variable (or to be some values of some variables)," ''Journal of Philosophy'' 81: 430–449. In Boolos 1998, 54–72.
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| * --------, 1985, "Nominalist platonism." ''Philosophical Review'' 94: 327–344. In Boolos 1998, 73–87.
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| * --------, 1998. ''Logic, Logic, and Logic''. Harvard University Press.
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| * Burgess, J.P., "From Frege to Friedman: A Dream Come True?"
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| * --------, 2004, “E Pluribus Unum: Plural Logic and Set Theory,” ''Philosophia Mathematica'' 12(3): 193–221.
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| * Cameron, J. R., 1999, "Plural Reference," ''Ratio''.
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| * {{Cite journal | doi = 10.1023/A:1015190829525 | last1 = Cocchiarella | first1 = Nino | authorlink = Nino Cocchiarella | year = 2002 | title = On the Logic of Classes as Many | url = | journal = Studia Logica | volume = 70 | issue = | pages = 303–338 }}
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| * De Rouilhan, P., 2002, "On What There Are," ''Proceedings of the Aristotelian Society'': 183–200.
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| * [[Gottlob Frege]], 1895, "A critical elucidation of some points in E. Schroeder's ''Vorlesungen Ueber Die Algebra der Logik''," ''Archiv fur systematische Philosophie'': 433–456.
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| * Landman, F., 2000. ''Events and Plurality''. Kluwer.
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| *{{citation
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| | last = Laycock
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| | first = Henry
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| | title = Words without Objects
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| | publisher = Clarendon Press
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| | location = Oxford
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| | year = 2006
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| | isbn = 9780199281718
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| | doi = 10.1093/0199281718.001.0001}}
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| * [[David K. Lewis]], 1991. ''Parts of Classes''. London: Blackwell.
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| * {{Cite journal | last1 = Linnebo | first1 = Øystein | last2 = Nicolas | first2 = David | year = | title = Superplurals in English | url = http://d.a.nicolas.free.fr/research/Linnebo-Nicolas-Superplurals.pdf | journal = Analysis | volume = 68 | issue = 3| pages = 186–97 }}
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| *{{citation
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| | last = McKay | first = Thomas J.
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| | author-link = Thomas J. McKay
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| | title = Plural Predication
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| | publisher = Oxford University Press
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| | location = New York
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| | isbn = 978-0-19-927814-5
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| | year = 2006}}
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| * [[John Stuart Mill]], 1904, ''A System of Logic'', 8th ed. London: .
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| * {{Cite journal | doi = 10.1007/s10988-008-9033-2 | last1 = Nicolas | first1 = David | year = 2008 | title = Mass nouns and plural logic | url = http://d.a.nicolas.free.fr/Nicolas-Mass-nouns-and-plural-logic-Revised-2.pdf | journal = Linguistics and Philosophy | volume = 31 | issue = 2| pages = 211–244 }}
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| * {{Cite journal | doi = 10.1111/j.0031-8094.2001.00231.x | last1 = Oliver | first1 = Alex | last2 = Smiley | first2 = Timothy | year = 2001 | title = Strategies for a Logic of Plurals | url = | journal = Philosophical Quarterly | volume = 51 | issue = 204| pages = 289–306 }}
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| * {{Cite journal | doi = 10.1093/mind/113.452.609 | last1 = Oliver | first1 = Alex | year = 2004 | title = Multigrade Predicates | url = | journal = Mind | volume = 113 | issue = | pages = 609–681 }}
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| * {{Cite journal | last1 = Rayo | first1 = Agustín | year = 2002 | title = Word and Objects | url = | journal = Noûs | volume = 36 | issue = | pages = 436–64 }}
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| * --------, 2006, “Beyond Plurals,” in Rayo and Uzquiano (2006).
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| * --------, 2007, “Plurals,” forthcoming in ''Philosophy Compass''.
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| * --------, and Gabriel Uzquiano, eds., 2006. ''Absolute Generality'' Oxford University Press.
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| * [[Bertrand Russell]], B., 1903. ''[[The Principles of Mathematics]]''. Oxford Univ. Press.
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| * [[Peter Simons (academic)|Peter Simons]], 1982, “Plural Reference and Set Theory,” in [[Barry Smith (academic and ontologist)|Barry Smith]], ed., ''Parts and Moments: Studies in Logic and Formal Ontology''. Munich: Philosophia Verlag.
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| * --------, 1987. ''Parts''. Oxford University Press.
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| * {{Cite journal | last1 = Uzquiano | first1 = Gabriel | year = 2003 | title = Plural Quantification and Classes | url = | journal = Philosophia Mathematica | volume = 11 | issue = 1| pages = 67–81 }}
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| * {{Cite journal | last1 = Yi | first1 = Byeong-Uk | year = 1999 | title = Is two a property? | url = | journal = Journal of Philosophy | volume = 95 | issue = | pages = 163–190 }}
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| * --------, 2005, “The Logic and Meaning of Plurals, Part I,” ''Journal of Philosophical Logic'' 34: 459–506.
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| * [[Adam Morton]]. "Complex individuals and multigrade relations." Noûs (1975): 309-318. {{JSTOR|2214634}}
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| * Samuel Levey (2011) "Logical theory in Leibniz" in Brandon C. Look (ed.) ''The Continuum Companion to Leibniz'', Continuum International Publishing Group, ISBN 0826429750
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| == External links ==
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| * {{sep entry|plural-quant|Plural quantification|Øystein Linnebo}}
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| * Moltmann, Friederike. (August 2012) "[http://semantics.univ-paris1.fr/pdf/plural-reference-paper-2012.pdf Plural Reference and Reference to a Plurality. A Reassessment of the Linguistic Facts]"
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| * [http://www.st-andrews.ac.uk/arche/twiki/~ahwiki/bin/view/Arche/MathPlurals/index.html A more extensive bibliography]
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| * http://lumiere.ens.fr/~amari/genius/PapersSeminar/Nicolas-Semantics-for-plurals-Handout-0110.pdf
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| [[Category:Quantification]]
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