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Philosophy of Language > Specific Expressions > Quantifiers > Plural Quantification

Plural Quantification

Edited by Salvatore Florio (University of Birmingham)
About this topic
Summary

Several natural languages contain a grammatical distinction between singular and plural expressions. The distinction also concerns quantification. Alongside singular quantifiers (‘something’, ‘everything’), we can find plural quantifiers (‘some things’, ‘all things’). Plural logic is a formal system that regiments plural quantification as a sui generis form of quantification, distinct from singular quantification. When treated as sui generis, plural quantification and plural logic have been thought to be philosophically significant and have found a number of applications especially in philosophy of mathematics and metaphysics. For the most part, these applications can be traced back to two of the virtues that plural quantification is alleged to have, i.e. ontological innocence and expressive power. On the one hand, it is assumed that plural quantifiers range in a special plural way over the entities in the range of the singular quantifiers and not over special plural entities (e.g. sets, collections, or any kind of set-like entities). Thus they do not incur ontological commitments exceeding those of the singular quantifiers. On the other hand, as shown by Boolos, plural quantification can interpret monadic second-order logic. As a result, plural quantification has been thought to provide more expressive power than singular quantification as captured by first-order logic. While the growing philosophical literature focuses primarily on the logical and foundational features of plural quantification, research in natural language semantics targets the meaning-theoretic and compositional features of plurals, often from an algebraic perspective. These two strands of research appear largely unreconciled.
Key works

Classic papers are Boolos 1984 and Boolos 1985. Yi 1999, Oliver & Smiley 2001, and Rayo 2002 argue forcefully for the significance of plural quantification. The question of logicality is addressed in Linnebo 2003. Full treatments of plural logic are given in Yi 2005 and Yi 2006, McKay 2006, and Oliver & Smiley 2013. Influential contributions in linguistics include Link 1983 and Link 1998, Schein 1993, Schwarzschild 1996, and Landman 2000.
Introductions  Rayo 2007 and Linnebo 2009 provide overviews of the philosophical literature. For a linguistically oriented introduction, see Schein 2006. 
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  1. added 2019-02-01
    Plural Logic, by Alex Oliver and Timothy Smiley: Oxford: Oxford University Press, 2013, Pp. Xiv + 336, £40. [REVIEW]Lloyd Humberstone - 2015 - Australasian Journal of Philosophy 93 (1):192-195.
  2. added 2018-10-04
    We Belong Together? A Plea for Modesty in Modal Plural Logic.Simon Hewitt - manuscript
    No categories
  3. added 2018-09-06
    Logic & Natural Language on Plural Reference and its Semantic and Logical Significance.Hanoch Ben-Yami - 2004 - Routledge.
    Frege's invention of the predicate calculus has been the most influential event in the history of modern logic. The calculus' place in logic is so central that many philosophers think, in fact, of it when they think of logic. This book challenges the position in contemporary logic and philosophy of language of the predicate calculus claiming that it is based on mistaken assumptions. Ben-Yami shows that the predicate calculus is different from natural language in its fundamental semantic characteristics, primarily in (...)
  4. added 2018-08-01
    Frege's Theorem in Plural Logic.Simon Hewitt - manuscript
    A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.
  5. added 2018-06-14
    Modal Structuralism and Reflection.Sam Roberts - forthcoming - Review of Symbolic Logic:1-38.
    Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...)
  6. added 2018-05-06
    We Belong Together.Simon Hewitt - manuscript
    It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.
  7. added 2018-04-03
    Mereological Composition and Plural Quantifier Semantics.Manuel Lechthaler & Ceth Lightfield - 2018 - Philosophia 46 (4):943-958.
    Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity (...)
  8. added 2017-10-14
    Rigid and Flexible Quantification in Plural Predicate Logic.Lucas Champollion, Justin Bledin & Haoze Li - forthcoming - Semantics and Linguistic Theory 27.
    Noun phrases with overt determiners, such as <i>some apples</i> or <i>a quantity of milk</i>, differ from bare noun phrases like <i>apples</i> or <i>milk</i> in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable fixed (...)
  9. added 2017-09-13
    Plurality, Conjunction and Events.Peter Nathan Lasersohn - 1995 - Dordrecht: Kluwer Academic Publishers.
  10. added 2017-02-10
    Semantic Values in Higher-Order Semantics.Stephan Krämer - 2014 - Philosophical Studies 168 (3):709-724.
    Recently, some philosophers have argued that we should take quantification of any (finite) order to be a legitimate and irreducible, sui generis kind of quantification. In particular, they hold that a semantic theory for higher-order quantification must itself be couched in higher-order terms. Øystein Linnebo has criticized such views on the grounds that they are committed to general claims about the semantic values of expressions that are by their own lights inexpressible. I show that Linnebo’s objection rests on the assumption (...)
  11. added 2017-02-01
    Higher-Order Quantification and Ontological Commitment.Peter Simons - 1997 - Dialectica 51 (4):255–271.
    George Boolos's employment of plurals to give an ontologically innocent interpretation of monadic higher‐order quantification continues and extends a minority tradition in thinking about quantification and ontological commitment. An especially prominent member of that tradition is Stanislaw Leśniewski, and shall first draw attention to this work and its relation to that of Boolos. Secondly I shall stand up briefly for plurals as logically respectable expressions, while noting their limitations in offering ontologically deflationary accounts of higher‐order quantification. Thirdly I shall focus (...)
  12. added 2017-01-29
    Strategies for a Logic of Plurals.Timothy Smiley Alex Oliver - 2001 - Philosophical Quarterly 51 (204):289-306.
    English has plural terms as well as singular terms. But our standard formal languages, e.g., the predicate calculus, feature only singular terms. How can the plural idiom be formalized?‘Changing the subject’ is by far the most common plurals strategy among both philosophers and linguists: a plural term is replaced by a singular term standing for some complex object that ‘contains’ the individuals to which the plural term alludes. For example, one might simply replace ‘A, B imply C’ with ‘{A, B} (...)
  13. added 2017-01-22
    Dependent Plurals and Plural Meaning.Eytan Zweig - 2008 - Dissertation, NYU
    While writing this thesis, there were many things I wanted to get right. I wanted to get the data right. I wanted to get my analysis of the data right. I certainly wanted to get all my citations right, which can get pretty tricky when one is trying to finish a chapter at 2am. But if an error did creep in somewhere in the body of the thesis, that is not a disaster. Sooner or later, I will get a chance (...)
  14. added 2017-01-22
    On What There Are.Philippe De Rouilhan - 2002 - Proceedings of the Aristotelian Society 102:183 - 200.
    Is second-order quantification legitimate? For Quine, it was pure non-sense, unless construed as first-order quantification in disguise, ranging over sets. Boolos rightly maintained that it could be interpreted in terms of plural quantification, but claimed that it then ranged over the same individuals as singular, first-order quantification. I protest that plural quantification ranges over what I call multiplicities. But what is a 'multiplicity'? And does this idea itself not fall prey to something like Frege's paradox?
  15. added 2017-01-21
    Plurals.Barry Schein - 2006 - In Ernest Lepore & Barry Smith (eds.), The Oxford Handbook of Philosophy of Language. Oxford University Press. pp. 716--767.
    Extension of the logical language to deliver plural reference and the logical relations that constitute knowledge of the singular and plural acquires empirical bite just in case it conforms with increasing precision to the syntax of the natural language and affords explanation of what speakers know about the distribution and meaning of plural expressions in their language. As for the syntax of natural language, this discussion, being none too precise, is guided throughout by just two considerations and their immediate consequences, (...)
  16. added 2017-01-21
    The Semantics of the Italian Double Plural.Almerindo E. Ojeda - 1995 - Journal of Semantics 12 (3):213-237.
    The purpose of this paper is to endow the Italian double plural with a precise interpretation. Our main point will bethat collective and distributive plurals denote algebras of the same type and that, in certain cases, the algebra denoted by a collective plural is a subalgebra of the one denoted by its distributive counterpart. 1 In such cases, the collective plural will be, semantically, a homomorphic image of its distributive counterpart. If correct, these interpretations will support the claim that nouns (...)
  17. added 2017-01-19
    Thomas McKay. Plural Predication.John P. Burgess - 2008 - Philosophia Mathematica 16 (1):133-140.
    This work, the first book-length study of its topic, is an important contribution to the literature of philosophical logic and philosophy of language, with implications for other branches of philosophy, including philosophy of mathematics. However, five of the book's ten chapters , including many of the author's most original contributions, are devoted to issues about natural language, and lie pretty well outside the scope of this journal, not to mention that of the reviewer's competence. For this reason I will here (...)
  18. added 2017-01-19
    A Note on Plural Pronouns.H. M. Cartwright - 2000 - Synthese 123 (2):227 - 246.
    Gareth Evans'' proposal, as amended by Steven Neale –that a definite pronoun with a quantifiedantecedent that does not bind it has the sense ofa definite description – has been challenged inthe singular case by appeal to counter-examplesinvolving failure of the uniqueness condition forthe legitimacy of a singular description. Thischallenge is here extended to the plural.Counter-examples are provided by cases in which aplural description `the Fs'' does not denote,despite the propriety of the use of `they'' or`them'' it is to replace, because (...)
  19. added 2017-01-19
    Types of Plural Individuals.Roger Schwarzschild - 1992 - Linguistics and Philosophy 15 (6):641 - 675.
  20. added 2017-01-18
    Strategies for a Logic of Plurals. AlexOliver & TimothySmiley - 2001 - Philosophical Quarterly 51 (204):289–306.
  21. added 2017-01-17
    Plural Logic: Second Edition, Revised and Enlarged.Alex Oliver & Timothy Smiley - 2016 - Oxford University Press.
    Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.
  22. added 2016-12-12
    To Be is to Be the Object of a Possible Act of Choice.Massimiliano Carrara & Enrico Martino - 2010 - Studia Logica 96 (2):289-313.
    Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite (...)
  23. added 2016-12-12
    Yi on 2.Joongol Kim - 2010 - Philosophia Mathematica 18 (3):329-336.
    Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two ’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.
  24. added 2016-12-08
    Plural Logic.Alex Oliver & Timothy Smiley - 2013 - Oxford University Press UK.
    Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.
  25. added 2016-12-08
    Plural Grundgesetze.Francesca Boccuni - 2010 - Studia Logica 96 (2):315-330.
    PG (Plural Grundgesetze) is a predicative monadic second-order system which exploits the notion of plural quantification and a few Fregean devices, among which a formulation of the infamous Basic Law V. It is shown that second-order Peano arithmetic can be derived in PG. I also investigate the philosophical issue of predicativism connected to PG. In particular, as predicativism about concepts seems rather un-Fregean, I analyse whether there is a way to make predicativism compatible with Frege’s logicism.
  26. added 2016-12-08
    Mereological Essentialism, Composition, and Stuff: A Reply to Kristie Miller.David Nicolas - 2009 - Erkenntnis 71 (3):425-429.
    In ‘Essential stuff' (2008) and ‘Stuff' (2009), Kristie Miller argues that two generally accepted theses, often formulated as follows, are incompatible: - (Temporal) mereological essentialism for stuff (or matter), the thesis that any portion of stuff has the same parts at every time it exists. - Stuff composition, the thesis that for any two portions of stuff, there exists a portion of stuff that is their mereological sum (or fusion). She does this by considering competing hypotheses about stuff, trying to (...)
  27. added 2016-12-08
    A Modest Logic of Plurals.Alex Oliver & Timothy Smiley - 2006 - Journal of Philosophical Logic 35 (3):317-348.
    We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of (...)
  28. added 2016-12-08
    The Logic and Meaning of Plurals. Part I.Byeong-Uk Yi - 2005 - Journal of Philosophical Logic 34 (5-6):459-506.
    Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result (...)
  29. added 2016-12-05
    Monotonicity and Collective Quantification.Gilad Ben-avi & Yoad Winter - 2003 - Journal of Logic, Language and Information 12 (2):127-151.
    This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and respects theirmonotonicity (...)
  30. added 2016-10-13
    All Things Must Pass Away.Joshua Spencer - 2012 - Oxford Studies in Metaphysics 7:67.
    Are there any things that are such that any things whatsoever are among them. I argue that there are not. My thesis follows from these three premises: (1) There are two or more things; (2) for any things, there is a unique thing that corresponds to those things; (3) for any two or more things, there are fewer of them than there are pluralities of them.
  31. added 2016-10-03
    Plurals and Modals.Øystein Linnebo - 2016 - Canadian Journal of Philosophy 46 (4-5):654-676.
    Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.
  32. added 2016-09-07
    A Note on Plural Logic.G. Fernandéz Díez - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (2):150-162.
    A distinction is introduced between itemized and non-itemized plural predication. It is argued that a full-fledged system of plural logic is not necessary in order to account for the validity of inferences concerning itemized collective predication. Instead, it is shown how this type of inferences can be adequately dealt with in a first-order logic system, after small modifications on the standard treatment. The proposed system, unlike plural logic, has the advantage of preserving completeness. And as a result, inferences such as (...)
  33. added 2016-03-17
    Pluralities and Plural Logic.Michael Rieppel - 2015 - Analysis 75 (3):504-514.
  34. added 2015-12-29
    Logic and Plurals.Salvatore Florio & Øystein Linnebo - 2018 - In Kirk Ludwig & Marija Jankovic (eds.), The Routledge Handbook of Collective Intentionality. Routledge. pp. 451-463.
    This chapter provides an overview of the philosophical and linguistic debate about the logic of plurals. We present the most prominent singularizing analyses of plurals as well as the main criticisms that such analyses have received. We then introduce an alternative approach to plurals known as plural logic, focusing on the question whether plural logic can count as pure logic.
  35. added 2015-12-16
    A Note on Plural Logic.Gustavo Fernández Díez - 2010 - Organon F 17 (2):150-162.
    A distinction is introduced between itemized and non-itemized plural predication. It is argued that a full-fledged system of plural logic is not necessary in order to account for the validity of inferences concerning itemized collective predication. Instead, it is shown how this type of inferences can be adequately dealt with in a first-order logic system, after small modifications on the standard treatment. The proposed system, unlike plural logic, has the advantage of preserving completeness. And as a result, inferences such as (...)
  36. added 2015-12-04
    Introduction.Alessandro Torza - 2015 - In Quantifiers, Quantifiers, and Quantifiers. Themes in Logic, Metaphysics and Language. (Synthese Library vol 373). Springer. pp. 1-15.
    This introductory chapter provides a summary of the contributions to the volume, as well as some critical remarks.
  37. added 2015-11-16
    Composition as Identity and Plural Cantor's Theorem.Einar Duenger Bohn - 2016 - Logic and Logical Philosophy 25 (3).
    I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.
  38. added 2015-09-28
    When Do Some Things Form a Set?Simon Hewitt - 2015 - Philosophia Mathematica 23 (3):311-337.
    This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...)
  39. added 2015-09-28
    Book Review: Alex Oliver and Timothy Smiley, Plural Logic. [REVIEW]Thomas Brouwer & Casper Storm Hansen - 2015 - Studia Logica 103 (5):1095-1100.
  40. added 2015-05-31
    Review of Oliver & Smiley, Plural Logic, 2013. [REVIEW]David Nicolas - 2014 - Notre Dame Philosophical Reviews 2.
  41. added 2015-05-31
    Mass Nouns and Plural Logic.David Nicolas - 2008 - Linguistics and Philosophy 31 (2):211-244.
    A dilemma put forward by Schein (1993) and Rayo (2002) suggests that, in order to characterize the semantics of plurals, we should not use predicate logic, but non-singular logic, a formal language whose terms may refer to several things at once. We show that a similar dilemma applies to mass nouns. If we use predicate logic and sets, we arrive at a Russellian paradox when characterizing the semantics of mass nouns. Likewise, a semantics of mass nouns based upon predicate logic (...)
  42. added 2015-05-31
    Mass Nouns and Plural Logic (Extended Abstract).David Nicolas - 2007 - In Proceedings of the 16th Amsterdam Colloquium. Palteam. pp. 211-244.
  43. added 2015-04-06
    Plural Logic.Michael Rieppel - forthcoming - Analysis:anv010.
  44. added 2015-04-04
    Plural Predication.Agustin Rayo - 2000 - Dissertation, Massachusetts Institute of Technology
    My thesis consists of three self-contained but interconnected papers. In the first one, 'Word and Objects', I assume that it is possible to quantify over absolutely everything, and show that certain English sentences containing collective predicates resist paraphrase in first-order languages and even in first-order languages enriched with plural quantifiers. To capture such sentences I develop a language containing plural predicates . ;The introduction of plural predicates leads to an extension of Quine's criterion of ontological commitment. I argue that theories (...)
  45. added 2015-04-04
    A Curious Plural: T. S. Champlin.T. S. Champlin - 1993 - Philosophy 68 (266):435-455.
    Statements of identity with a plural subject, of the form ‘They are the same person ,’ as illustrated in each of the answers to the above two questions, give rise to a philosophical problem.
  46. added 2015-03-24
    Plural Reference and Syntactic Three-Dimensionality (Book Proposal, Under Contract).Friederike Moltmann - forthcoming - Oxford University Press.
    The book argues for Plural Reference for the semantics of natural language and makes the connection between Plural Reference and Alternative Semantics for the purpose of the interpretation of three-dimensional syntactic structures of coordinate sentences (in the sense of my 1992 MIT Ph D thesis).
  47. added 2015-03-24
    Grounding Megethology on Plural Reference.Massimiliano Carrara & Enrico Martino - 2015 - Studia Logica 103 (4):697-711.
    In Mathematics is megethology Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis’work is very attractive. However, the alleged (...)
  48. added 2015-03-24
    Dependent Plural Pronouns with Skolemized Choice Functions.Yasutada Sudo - 2014 - Natural Language Semantics 22 (3):265-297.
    The present paper discusses two interesting phenomena concerning phi-features on plural pronouns: plural pronouns that denote atomic individuals, and plural pronouns with more than one binder. A novel account of these two phenomena is proposed, according to which all occurrences of phi-features are both semantically and morphologically relevant. For such a ‘uniformly semantic account’ of phi-features, dependent plural pronouns constitute a theoretical challenge, while partial binding is more or less straightforwardly accounted for. In order to make sense of the semantic (...)
  49. added 2015-03-23
    Degenerate Plurals.Miyuki Yamashina & Christopher Tancredi - 2005 - In Emar Maier, Corien Bary & Janneke Huitink (eds.), Proceedings of Sub9. pp. 522--537.
  50. added 2015-03-23
    Plural and Pleonetetic Quantification.J. E. J. Altham - 1991 - In H. G. Lewis (ed.), Peter Geach: Philosophical Encounters. Kluwer Academic Publishers. pp. 105--119.
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