About this topic

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 1999Oliver & 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 2006McKay 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
Related categories

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  1. 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} (...)
  2. Strategies for a Logic of Plurals. AlexOliver & TimothySmiley - 2001 - Philosophical Quarterly 51 (204):289–306.
  3. Plural and Pleonetetic Quantification.J. E. J. Altham - 1991 - In H. G. Lewis (ed.), Peter Geach: Philosophical Encounters. Kluwer Academic Publishers. pp. 105--119.
  4. Plural and Conflicting Values.Elizabeth S. Anderson & Michael Stocker - 1992 - Philosophical Review 101 (4):931.
  5. 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 (...)
  6. Higher‐Level Plurals Versus Articulated Reference, and an Elaboration of Salva Veritate.Hanoch Ben-Yami - 2013 - Dialectica 67 (1):81-102.
    In recent literature on plurals the claim has often been made that the move from singular to plural expressions can be iterated, generating what are occasionally called higher-level plurals or superplurals, often correlated with superplural predicates. I argue that the idea that the singular-to-plural move can be iterated is questionable. I then show that the examples and arguments intended to establish that some expressions of natural language are in some sense higher-level plurals fail. Next, I argue that these and some (...)
  7. Response to Westerstahl.Hanoch Ben-Yami - 2012 - Logique Et Analyse 55 (217):47-55.
  8. Generalized Quantifiers, and Beyond.Hanoch Ben-Yami - 2009 - Logique Et Analyse (208):309-326.
    I show that the contemporary dominant analysis of natural language quantifiers that are one-place determiners by means of binary generalized quantifiers has failed to explain why they are, according to it, conservative. I then present an alternative, Geachean analysis, according to which common nouns in the grammatical subject position are plural logical subject-terms, and show how it does explain that fact and other features of natural language quantification.
  9. Plural Quantification Logic: A Critical Appraisal.Hanoch Ben-Yami - 2009 - Review of Symbolic Logic 2 (1):208-232.
    I first show that most authors who developed Plural Quantification Logic (PQL) argued it could capture various features of natural language better than can other logic systems. I then show that it fails to do so: it radically departs from natural language in two of its essential features; namely, in distinguishing plural from singular quantification and in its use of an relation. Next, I sketch a different approach that is more adequate than PQL for capturing plural aspects of natural language (...)
  10. A Formal Semantics for Plural Quantification, Intersentential Binding and Anaphoric Pronouns as Rigid Designators.Alan Berger - 2002 - Noûs 36 (1):50–74.
  11. Quantification as Reference: Evidence From Q-Verbs.Maria Bittner & Naja Trondhjem - 2008 - In Lisa Matthewson (ed.), Quantification: A Cross-Linguistic Perspective. Emerald. pp. 7-66.
    Formal semantics has so far focused on three categories of quantifiers, to wit, Q-determiners (e.g. 'every'), Q-adverbs (e.g. 'always'), and Q-auxiliaries (e.g. 'would'). All three can be analyzed in terms of tripartite logical forms (LF). This paper presents evidence from verbs with distributive affixes (Q-verbs), in Kalaallisut, Polish, and Bininj Gun-wok, which cannot be analyzed in terms of tripartite LFs. It is argued that a Q-verb involves discourse reference to a distributive verbal dependency, i.e. an episode-valued function that sends different (...)
  12. Plural Logicism.Francesca Boccuni - 2013 - Erkenntnis 78 (5):1051-1067.
    PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically (...)
  13. On the Consistency of a Plural Theory of Frege’s Grundgesetze.Francesca Boccuni - 2011 - Studia Logica 97 (3):329-345.
    PG (Plural Grundgesetze) is a predicative monadic second-order system which is aimed to derive second-order Peano arithmetic. It exploits the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. In this paper, a model-theoretical consistency proof for the system PG is provided.
  14. 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.
  15. 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.
  16. Monism, Emergence, and Plural Logic.Einar Duenger Bohn - 2012 - Erkenntnis 76 (2):211-223.
    In this paper I argue that we need to take irreducibly plural logic more seriously in metaphysical debates due to the fact that the verdict of many metaphysical debates hangs on it. I give two examples. The main example I focus on is the debate recently revived by Jonathan Schaffer over the fundamental cardinality of the world. I show how the three main arguments provided by Schaffer are unsound in virtue of an employment of plural logic. The second example I (...)
  17. Reply to Charles Parsons' ``Sets and Classes''.George Boolos - 1998 - In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. pp. 30-36.
  18. Nominalist Platonism.George Boolos - 1985 - Philosophical Review 94 (3):327-344.
  19. To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables).George Boolos - 1984 - Journal of Philosophy 81 (8):430-449.
  20. Quantified Modal Logic and the Plural De Re.Phillip Bricker - 1989 - Midwest Studies in Philosophy 14 (1):372-394.
    Modal sentences of the form "every F might be G" and "some F must be G" have a threefold ambiguity. in addition to the familiar readings "de dicto" and "de re", there is a third reading on which they are examples of the "plural de re": they attribute a modal property to the F's plurally in a way that cannot in general be reduced to an attribution of modal properties to the individual F's. The plural "de re" readings of modal (...)
  21. Book Review: Alex Oliver and Timothy Smiley, Plural Logic. [REVIEW]Thomas Brouwer & Casper Storm Hansen - 2015 - Studia Logica 103 (5):1095-1100.
  22. 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 (...)
  23. E Pluribus Unum: Plural Logic and Set Theory.John P. Burgess - 2004 - Philosophia Mathematica 12 (3):193-221.
    A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.
  24. Plural Reference.J. R. Cameron - 1999 - Ratio 12 (2):128–147.
  25. A Unified Analysis of the English Bare Plural.Greg N. Carlson - 1977 - Linguistics and Philosophy 1 (3):413 - 456.
    It is argued that the English bare plural (an NP with plural head that lacks a determiner), in spite of its apparently diverse possibilities of interpretation, is optimally represented in the grammar as a unified phenomenon. The chief distinction to be dealt with is that between the generic use of the bare plural (as in Dogs bark) and its existential or indefinite plural use (as in He threw oranges at Alice). The difference between these uses is not to be accounted (...)
  26. Unity and Plurality. Philosophy, Logic, and Semantics.Massimiliano Carrara, Alessandra Arapinis & Friederike Moltmann (eds.) - 2016 - Oxford University Press.
    This volume brings together new work on the logic and ontology of plurality and a range of recent articles exploring novel applications to natural language semantics. The contributions in this volume in particular investigate and extend new perspectives presented by plural logic and non-standard mereology and explore their applications to a range of natural language phenomena. Contributions by P. Aquaviva, A. Arapinis, M. Carrara, P. McKay, F. Moltmann, O. Linnebo, A. Oliver and T. Smiley, T. Scaltsas, P. Simons, and B.-Y. (...)
  27. 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 (...)
  28. On the Infinite in Mereology with Plural Quantification.Massimiliano Carrara & Enrico Martino - 2011 - Review of Symbolic Logic 4 (1):54-62.
    In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification (...)
  29. 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 (...)
  30. On the Ontological Commitment of Mereology.Massimiliano Carrara & Enrico Martino - 2009 - Review of Symbolic Logic 2 (1):164-174.
    In Parts of Classes (1991) and Mathematics Is Megethology (1993) David Lewis defends both the innocence of plural quantification and of mereology. However, he himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out as a whole. In the case of plural quantification . Instead, in the mereological case: (Lewis, 1991, p. 87). The aim of the paper is to (...)
  31. 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 (...)
  32. On Plural Reference and Elementary Set Theory.Helen Morris Cartwright - 1993 - Synthese 96 (2):201 - 254.
    The view that plural reference is reference to a set is examined in light of George Boolos's treatment of second-order quantification as plural quantification in English. I argue that monadic second-order logic does not, in Boolos's treatment, reflect the behavior of plural quantifiers under negation and claim that any sentence that properly translates a second-order formula, in accordance with his treatment, has a first-order formulation. Support for this turns on the use of certain partitive constructions to assign values to variables (...)
  33. 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.
  34. 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 (...)
  35. Denoting Concepts, Reference, and the Logic of Names, Classes as Many, Groups, and Plurals.Nino Cocchiarella - 2005 - Linguistics and Philosophy 28 (2):135 - 179.
    Bertrand Russell introduced several novel ideas in his 1903 Principles of Mathematics that he later gave up and never went back to in his subsequent work. Two of these are the related notions of denoting concepts and classes as many. In this paper we reconstruct each of these notions in the framework of conceptual realism and connect them through a logic of names that encompasses both proper and common names, and among the latter, complex as well as simple common names. (...)
  36. The Problem of Existence in Quantification Logic.K. Das - 2000 - Indian Philosophical Quarterly 27 (3):229-246.
  37. On What There Are.Philippe de Rouilhan - 2002 - Proceedings of the Aristotelian Society 102 (2):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?
  38. Word Order and Quantification Over Times.Denis Delfitto & Pier Marco Bertinetto - 2000 - In Achille Varzi, James Higginbotham & Fabio Pianesi (eds.), Speaking of Events. Oxford University Press.
  39. A Note on Plural Logic.Gustavo Fernández 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 (...)
  40. On Passing From Singular to Plural Consequences.K. Došen - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg. pp. 533--547.
  41. 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 (...)
  42. 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 (...)
  43. Semantics and the Plural Conception of Reality.Salvatore Florio - 2014 - Philosophers' Imprint 14:1-20.
    According to the singular conception of reality, there are objects and there are singular properties, i.e. properties that are instantiated by objects separately. It has been argued that semantic considerations about plurals give us reasons to embrace a plural conception of reality. This is the view that, in addition to singular properties, there are plural properties, i.e. properties that are instantiated jointly by many objects. In this article, I propose and defend a novel semantic account of plurals which dispenses with (...)
  44. Untyped Pluralism.Salvatore Florio - 2014 - Mind 123 (490):317-337.
    In the semantic debate about plurals, pluralism is the view that a plural term denotes some things in the domain of quantification and a plural predicate denotes a plural property, i.e. a property that can be instantiated by many things jointly. According to a particular version of this view, untyped pluralism, there is no type distinction between objects and properties. In this article, I argue against untyped pluralism by showing that it is subject to a variant of a Russell-style argument (...)
  45. Logic and Plurals.Salvatore Florio & Øystein Linnebo - forthcoming - In Kirk Ludwig & Marija Jankovic (eds.), The Routledge Handbook of Collective Intentionality.
    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.
  46. On the Innocence and Determinacy of Plural Quantification.Salvatore Florio & Øystein Linnebo - 2016 - Noûs 50 (3):565–583.
    Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic (...)
  47. Plural Logic and Sensitivity to Order.Salvatore Florio & David Nicolas - 2015 - Australasian Journal of Philosophy 93 (3):444-464.
    Sentences that exhibit sensitivity to order (e.g. 'John and Mary arrived at school in that order' and 'Mary and John arrived at school in that order') present a challenge for the standard formulation of plural logic. In response, some authors have advocated new versions of plural logic based on fine-grained notions of plural reference, such as serial reference (Hewitt 2012) and articulated reference (Ben-Yami 2013). The aim of this article is to show that sensitivity to order should be accounted for (...)
  48. A Treatment of Plurals and Plural Quantifications Based on a Theory of Collections.Enrico Franconi - 1993 - Minds and Machines 3 (4):453-474.
    Collective entities and collective relations play an important role in natural language. In order to capture the full meaning of sentences like The Beatles sing Yesterday, a knowledge representation language should be able to express and reason about plural entities — like the Beatles — and their relationships — like sing — with any possible reading (cumulative, distributive or collective).In this paper a way of including collections and collective relations within a concept language, chosen as the formalism for representing the (...)
  49. Quantification Theory and Objects of Reference.P. T. Geach - 1972 - In Logic Matters. Blackwell.
  50. Knowing the Past: Plural Identities and the Antinomies of Loss in Highland Bolivia.Olivia Harris - 1995 - In Richard Fardon (ed.), Counterworks: Managing the Diversity of Knowledge. Routledge. pp. 105--123.
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