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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, plural quantification is the theory that an individual variable x may take on ''
plural In many languages, a plural (sometimes list of glossing abbreviations, abbreviated as pl., pl, , or ), is one of the values of the grammatical number, grammatical category of number. The plural of a noun typically denotes a quantity greater than ...
'', 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 stories. The point of the theory is to give
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
the power of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, but without any " existential commitment" to such objects as sets. The classic expositions are Boolos 1984 and Lewis 1991.


History

The view is commonly associated with George Boolos, though it is older (see notably Simons 1982), and is related to the view of classes defended by
John Stuart Mill John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, politician and civil servant. One of the most influential thinkers in the history of liberalism and social liberalism, he contributed widely to s ...
and other
nominalist In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be inst ...
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). A similar position was also discussed by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
in chapter VI of Russell (1903), but later dropped in favour of a "no-classes" theory. See also
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
1895 for a critique of an earlier view defended by Ernst Schroeder. The general idea can be traced back to
Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
. (Levey 2011, pp. 129–133) Interest revived in plurals with work in linguistics in the 1970s by Remko Scha, Godehard Link,
Fred Landman Fred (Alfred) Landman (; born October 28, 1956) is a Dutch-born Israeli professor of semantics. He teaches at Tel Aviv University has written a number of books about linguistics. Biography Fred Landman was born in Holland. He immigrated to Israe ...
, Friederike Moltmann, Roger Schwarzschild, Peter Lasersohn and others, who developed ideas for a semantics of plurals.


Background and motivation


Multigrade (variably polyadic) predicates and relations

Sentences like : Alice and Bob cooperate. : Alice, Bob and Carol cooperate. are said to involve a ''multigrade'' (also known as ''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 In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...
(cf. Linnebo & Nicolas 2008). The notion of multigrade relation/predicate has appeared as early as the 1940s and has been notably used by
Quine Quine may refer to: * Quine (computing), a program that produces its source code as output * Quine's paradox, in logic * Quine (surname), people with the surname ** Willard Van Orman Quine (1908–2000), American philosopher and logician See al ...
(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.


Nominalism

Broadly speaking, nominalism denies the existence of universals ( abstract entities), like sets, classes, relations, properties, etc. Thus the plural logics 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. Standard first-order logic has difficulties in representing some sentences with plurals. Most well-known is the
Geach–Kaplan sentence In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic whic ...
"some critics admire only one another". Kaplan proved that it is
nonfirstorderizable In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic whic ...
(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. Boolos argued that
second-order Second-order may refer to: Mathematics * Second order approximation, an approximation that includes quadratic terms * Second-order arithmetic, an axiomatization allowing quantification of sets of numbers * Second-order differential equation, a d ...
monadic quantification may be systematically interpreted in terms of plural quantification, and that, therefore, second-order monadic quantification is "ontologically innocent".. Later, Oliver & Smiley (2001), Rayo (2002), Yi (2005) and McKay (2006) argued that sentences such as :They are shipmates :They are meeting together :They lifted a piano :They are surrounding a building :They admire only one another 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. 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). Several writers have suggested that plural logic opens the prospect of simplifying the
foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
, avoiding the
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
es of set theory, and simplifying the complex and unintuitive axiom sets needed in order to avoid them. Recently, Linnebo & Nicolas (2008) have suggested that natural languages often contain superplural variables (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".


Formal definition

This section presents a simple formulation of plural logic/quantification approximately the same as given by Boolos in ''Nominalist Platonism'' (Boolos 1985).


Syntax

Sub-sentential units are defined as * Predicate symbols F, G, etc. (with appropriate arities, which are left implicit) * Singular variable symbols x, y, etc. * Plural variable symbols \bar, \bar, etc. Full
sentences The ''Sentences'' (. ) is a compendium of Christian theology written by Peter Lombard around 1150. It was the most important religious textbook of the Middle Ages. Background The sentence genre emerged from works like Prosper of Aquitaine's ...
are defined as * If F is an ''n''-ary predicate symbol, and x_0, \ldots, x_n are singular variable symbols, then F(x_0, \ldots, x_n) is a sentence. * If P is a sentence, then so is \neg P * If P and Q are sentences, then so is P \land Q * If P is a sentence and x is a singular variable symbol, then \exists x.P is a sentence * If x is a singular variable symbol and \bar is a plural variable symbol, then x \prec \bar is a sentence (where ≺ is usually interpreted as the relation "is one of") * If P is a sentence and \bar is a plural variable symbol, then \exists \bar.P is a sentence 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. This logic turns out to be equi-interpretable with
monadic second-order logic In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's ...
.


Model theory

Plural logic's model theory/semantics is where the logic's lack of sets is cashed out. A model is defined as a tuple (D,V,s,R) where D is the domain, V is a collection of valuations V_F for each predicate name F in the usual sense, and s is a Tarskian sequence (assignment of values to variables) in the usual sense (i.e. a map from singular variable symbols to elements of D). The new component R is a binary relation relating values in the domain to plural variable symbols. Satisfaction is given as * (D,V,s,R) \models F(x_0, \ldots, x_n) iff (s_, \ldots, s_) \in V_F * (D,V,s,R) \models \neg P iff (D,V,s,R) \nvDash P * (D,V,s,R) \models P \land Q iff (D,V,s,R) \models P and (D,V,s,R) \models Q * (D,V,s,R) \models \exists x.P iff there is an s' \approx_x s such that (D,V,s',R) \models P * (D,V,s,R) \models x \prec \bar iff s_xR\bar * (D,V,s,R) \models \exists \bar.P iff there is an R' \approx_\bar R such that (D,V,s,R') \models P Where for singular variable symbols, s \approx_x s' means that for all singular variable symbols y other than x, it holds that s_y = s'_y, and for plural variable symbols, R \approx_\bar R' means that for all plural variable symbols \bar other than \bar, and for all objects of the domain d, it holds that dR\bar = dR'\bar. As in the syntax, only the last two are truly new in plural logic. Boolos observes that by using assignment ''relations'' R, 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 \exists \bar. \forall y. y \prec \bar \leftrightarrow F(y) 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 F(\bar) are not well-formed: predicate names can only combine with singular variable symbols, not plural variable symbols. This can be taken as the simplest, and most obvious argument that plural logic as Boolos defined it is ontologically innocent.


See also

*
Generalized quantifier In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets o ...
* Homogeneity (linguistics) *
Variadic function In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages. The term ''var ...


Notes


References

* 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. * --------, 1985, "Nominalist platonism." ''Philosophical Review'' 94: 327–344. In Boolos 1998, 73–87. * --------, 1998. ''Logic, Logic, and Logic''. Harvard University Press. * Burgess, J.P., "From Frege to Friedman: A Dream Come True?" * --------, 2004, “E Pluribus Unum: Plural Logic and Set Theory,” ''Philosophia Mathematica'' 12(3): 193–221. * Cameron, J. R., 1999, "Plural Reference," ''Ratio''. * * De Rouilhan, P., 2002, "On What There Are," ''Proceedings of the Aristotelian Society'': 183–200. *
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
, 1895, "A critical elucidation of some points in E. Schroeder's ''Vorlesungen Ueber Die Algebra der Logik''," ''Archiv für systematische Philosophie'': 433–456. *
Fred Landman Fred (Alfred) Landman (; born October 28, 1956) is a Dutch-born Israeli professor of semantics. He teaches at Tel Aviv University has written a number of books about linguistics. Biography Fred Landman was born in Holland. He immigrated to Israe ...
2000. ''Events and Plurality''. Kluwer. * * David K. Lewis, 1991. ''Parts of Classes''. London: Blackwell. * * *
John Stuart Mill John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, politician and civil servant. One of the most influential thinkers in the history of liberalism and social liberalism, he contributed widely to s ...
, 1904, ''A System of Logic'', 8th ed. London: . * Moltmann, Friederike, 1997, ''Parts and Wholes in Semantics''. Oxford University Press, New York. * Moltmann, Friederike, 'Plural Reference and Reference to a Plurality. Linguistic Facts and Semantic Analyses'. In M. Carrara, A. Arapinis and F. Moltmann (eds.): Unity and Plurality. Logic, Philosophy, and Semantics. Oxford University Press, Oxford, 2016, pp. 93–120. * * * * * --------, 2006, “Beyond Plurals,” in Rayo and Uzquiano (2006). * --------, 2007, “Plurals,” forthcoming in ''Philosophy Compass''. * --------, and Gabriel Uzquiano, eds., 2006. ''Absolute Generality'' Oxford University Press. *
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
, B., 1903. ''
The Principles of Mathematics ''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented Russell's paradox, his famous paradox and argued his thesis that mathematics and logic are identical. The book presents a view of ...
''. Oxford Univ. Press. * Peter Simons, 1982, “Plural Reference and Set Theory,” in Barry Smith, ed., ''Parts and Moments: Studies in Logic and Formal Ontology''. Munich: Philosophia Verlag. * --------, 1987. ''Parts''. Oxford University Press. * * * --------, 2005, “The Logic and Meaning of Plurals, Part I,” ''Journal of Philosophical Logic'' 34: 459–506. * Adam Morton. "Complex individuals and multigrade relations." Noûs (1975): 309-318. * Samuel Levey (2011) "Logical theory in Leibniz" in Brandon C. Look (ed.) ''The Continuum Companion to Leibniz'', Continuum International Publishing Group,


External links

* * Moltmann, Friederike. (August 2012)
Plural Reference and Reference to a Plurality. A Reassessment of the Linguistic Facts


* https://web.archive.org/web/20150211224457/http://lumiere.ens.fr/~amari/genius/PapersSeminar/Nicolas-Semantics-for-plurals-Handout-0110.pdf Quantifier (logic) {{Formal semantics