In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, plural quantification is the theory that an individual
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
x may take on ''
plural
The plural (sometimes abbreviated pl., pl, or ), in many languages, is one of the values of the grammatical category of number. The plural of a noun typically denotes a quantity greater than the default quantity represented by that noun. This de ...
'', 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 known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
the power of
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
, 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
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. in ...
, though it is older (see notably
Simons Simons is a surname of Scandinavian origins and a variant of Sigmundsson, a patronymic surname with roots in proto-Germanic ''*segaz'' and ''*mundō'', giving a rough translation of "protection through victory".
Notable people
A
* Alan ...
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, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
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 at least two main versions of nominalism. One version denies the existence of universalsthings th ...
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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
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 phil ...
1895 for a critique of an earlier view defended by
Ernst Schroeder
Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include:
Surname
* Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst"
* Anton Ernst (1975 ...
.
The general idea can be traced back to
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
. (Levey 2011, pp. 129–133)
Interest revived in plurals with work in linguistics in the 1970s by
Remko Scha Remko Jan Hendrik Scha (15 September 1945 – 9 November 2015) was a professor of computational linguistics at the faculty of humanities and Institute for Logic, Language and Computation at the University of Amsterdam. He made important contributio ...
,
Godehard Link
Godehard Link (born 7 July 1944 in Lippstadt) is a professor of logic and philosophy of science at the University of Munich
The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universitä ...
,
Fred Landman
Fred (Alfred) Landman ( he, פרד לנדמן; 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. H ...
,
Friederike Moltmann
Friederike Moltmann is a linguist and philosopher. She has done pioneering work at the intersection of philosophy and linguistics, especially on the interface between metaphysics and natural language semantics, but also on the interface betwee ...
,
Roger Schwarzschild
Roger is a given name, usually masculine, and a surname. The given name is derived from the Old French personal names ' and '. These names are of Germanic languages, Germanic origin, derived from the elements ', ''χrōþi'' ("fame", "renown", " ...
,
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
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
(cf. Linnebo & Nicolas 2008). The notion of multigrade relation/predicate has appeared as early as the 1940s and has been notably used by
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.
Nominalism
Broadly speaking, nominalism denies the
existence of universals (
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.
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 which ...
"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 which ...
(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
2nd-order monadic quantification may be systematically interpreted in terms of plural quantification, and that, therefore, 2nd-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 is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
, 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 premises, leads to a seemingly self-contradictory or a logically u ...
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
,
, etc. (with appropriate arities, which are left implicit)
* Singular variable symbols
,
, etc.
* Plural variable symbols
,
, etc.
Full
sentences
''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology written by Peter Lombard in the 12th century. It is a systematic compilation of theology, written around 1150; it derives its name from the ''sententiae'' o ...
are defined as
* If
is an ''n''-ary predicate symbol, and
are singular variable symbols, then
is a sentence.
* If
is a sentence, then so is
* If
and
are sentences, then so is
* If
is a sentence and
is a singular variable symbol, then
is a sentence
* If
is a singular variable symbol and
is a plural variable symbol, then
is a sentence (where ≺ is usually interpreted as the relation "is one of")
* If
is a sentence and
is a plural variable symbol, then
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 t ...
.
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
where
is the domain,
is a collection of valuations
for each predicate name
in the usual sense, and
is a Tarskian sequence (assignment of values to variables) in the usual sense (i.e. a map from singular variable symbols to elements of
). The new component
is a binary relation relating values in the domain to plural variable symbols.
Satisfaction is given as
*
iff
*
iff
*
iff
and
*
iff there is an
such that
*
iff
*
iff there is an
such that
Where for singular variable symbols,
means that for all singular variable symbols
other than
, it holds that
, and for plural variable symbols,
means that for all plural variable symbols
other than
, and for all objects of the domain
, it holds that
.
As in the syntax, only the last two are truly new in plural logic. Boolos observes that by using assignment ''relations''
, 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
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
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 of ...
*
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 ''varia ...
Notes
References
*
George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. in ...
, 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 phil ...
, 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 ( he, פרד לנדמן; 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. H ...
2000. ''Events and Plurality''. Kluwer.
*
*
David K. Lewis
David Kellogg Lewis (September 28, 1941 – October 14, 2001) was an American philosopher who is widely regarded as one of the most important philosophers of the 20th century. Lewis taught briefly at UCLA and then at Princeton University fr ...
, 1991. ''Parts of Classes''. London: Blackwell.
*
*
*
John Stuart Mill
John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
, 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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
, B., 1903. ''
The Principles of Mathematics
''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical.
The book presents a view of the foundations 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
Adam Morton (1945 – 2020) was a Canadian philosopher. Morton's work focused on how we understand one another's behaviour in everyday life, with an emphasis on the role mutual intelligibility plays in cooperative activity. He also wrote on ethi ...
. "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
* {{cite SEP , url-id=plural-quant , title=Plural quantification , last=Linnebo , first=Øystein
*
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)