About this topic

Generalized quantifier theory studies the semantics of quantifier expressions, like, `every’, `some’, `most’, ‘infinitely many’, `uncountably many’, etc. The classical version was developed in the 1980s, at the interface of linguistics, mathematics and philosophy. In logic generalized quantifier are often defined as classes of models closed on isomorphism (topic neutral). For instance, quantifier “infinitely many” may be defined as a class of all infinite models. Equivalently, in linguistics generalized quantifiers are formally treated as relations between subset of the universe. For instance, in sentence `Most of the students are smart”, quantifier `most’ is a binary relation between the set of students and the set of smart people. The sentence is true if and only if the cardinality of the set of smart students is greater than the cardinality of the set of students who are not smart. 

Key works

Gottlob Frege was one of the major figures in forming the modern concept of quantification. In Begriffsschrift (1879) he made a distinction between bound and free variables and treated quantifiers as well-defined, denoting entities. However, historically speaking the notion of a generalized quantifier was formulated for the first time in a seminal paper of Andrzej Mostowski 1957, where the notions of existential and universal quantification were extended to the concept of a monadic generalized quantifier binding one variable in one formula, and later this was generalized to arbitrary types by Per Lindström 1966. Soon it was realized by Richard Montague 1970 that the notion can be used to model the denotations of noun phrases in natural language. Jon Barwise and Robin Cooper (1981) introduced the apparatus of generalized quantifiers as a standard semantic toolbox and started the rigorous study of their properties from the linguistic perspective.


For an encyclopedia article see Westerståhl 2008. For a survey of classical results we recommend: Keenan & Westerstahl 2011. Peters & Westersthl 2006 is a thorough handbook treatment focused on definability questions and their applications in model theory and linguistics. For more computer science results consult, e.g., Makowsky & Pnueli 1995 . For a psychological perspective, see, e.g. Moxey & Sanford 1993. For a combination of formal work and cognitive science perspective, see, e.g., Szymanik 2016.

Related categories

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  1. Conservative Reduction Classes of Krom Formulas.Stål O. Aanderaa, Egon Börger & Harry R. Lewis - 1982 - Journal of Symbolic Logic 47 (1):110-130.
    A Krom formula of pure quantification theory is a formula in conjunctive normal form such that each conjunct is a disjunction of at most two atomic formulas or negations of atomic formulas. Every class of Krom formulas that is determined by the form of their quantifier prefixes and which is known to have an unsolvable decision problem for satisfiability is here shown to be a conservative reduction class. Therefore both the general satisfiability problem, and the problem of satisfiability in finite (...)
  2. Prefix Classes of Krom Formulas.Stål O. Aanderaa & Harry R. Lewis - 1973 - Journal of Symbolic Logic 38 (4):628-642.
  3. A Unified Approach to Split Scope.Klaus Abels & Luisa Martí - 2010 - Natural Language Semantics 18 (4):435-470.
    The goal of this paper is to propose a unified approach to the split scope readings of negative indefinites, comparative quantifiers, and numerals. There are two main observations that justify this approach. First, split scope shows the same kinds of restrictions across these different quantifiers. Second, split scope always involves low existential force. In our approach, following Sauerland, natural language determiner quantifiers are quantifiers over choice functions, of type <<,t>,t>. In split readings, the quantifier over choice functions scopes above other (...)
  4. Propositions or Choice Functions: What Do Quantifiers Quantify Over.Klaus Abels & Luiza Martí - forthcoming - Natural Language Semantics.
  5. Some Proof Theoretical Remarks on Quantification in Ordinary Language.Michele Abrusci & Christian Retoré - manuscript
    This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's oper- ator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to (...)
  6. Quantifiers, Games and Inductive Definitions.Peter Aczel - 1975 - In Stig Kanger (ed.), Journal of Symbolic Logic. Elsevier. pp. 1--14.
  7. A New Theory of Quantifiers and Term Connectives.Ken Akiba - 2009 - Journal of Logic, Language and Information 18 (3):403-431.
    This paper sets forth a new theory of quantifiers and term connectives, called shadow theory , which should help simplify various semantic theories of natural language by greatly reducing the need of Montagovian proper names, type-shifting, and λ-conversion. According to shadow theory, conjunctive, disjunctive, and negative noun phrases such as John and Mary , John or Mary , and not both John and Mary , as well as determiner phrases such as every man , some woman , and the boys (...)
  8. Book Review--Jaap Van der Does and Jan Van Eijk, Eds., Quantifiers, Logic, and Language. [REVIEW]Varol Akman - 1998 - Philosophical Explorations.
    This is a review of Quantifiers, Logic, and Language, edited by Jaap van der Does and Jan van Eijk, published by CSLI (Center for the Study of Language and Information) Publications in 1996.
  9. On a Decidable Generalized Quantifier Logic Corresponding to a Decidable Fragment of First-Order Logic.Natasha Alechina - 1995 - Journal of Logic, Language and Information 4 (3):177-189.
    Van Lambalgen (1990) proposed a translation from a language containing a generalized quantifierQ into a first-order language enriched with a family of predicatesR i, for every arityi (or an infinitary predicateR) which takesQxg(x, y1,..., yn) to x(R(x, y1,..., y1) (x,y1,...,yn)) (y 1,...,yn are precisely the free variables ofQx). The logic ofQ (without ordinary quantifiers) corresponds therefore to the fragment of first-order logic which contains only specially restricted quantification. We prove that it is decidable using the method of analytic tableaux. Related (...)
  10. Generalized Quantification as Substructural Logic.Natasha Alechina & Michiel van Lambalgen - 1996 - Journal of Symbolic Logic 61 (3):1006-1044.
    We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...)
  11. Correspondence and Completeness for Generalized Quantifiers.Natasha Alechina & Michiel van Lambalgen - 1995 - Logic Journal of the IGPL 3 (2-3):167-190.
  12. No More Shall We Part: Quantifiers in English Comparatives.Peter Alrenga & Christopher Kennedy - 2014 - Natural Language Semantics 22 (1):1-53.
    It is well known that the interpretation of quantificational expressions in the comparative clause poses a serious challenge for semantic analyses of the English comparative. In this paper, we develop a new analysis of the comparative clause designed to meet this challenge, in which a silent occurrence of the negative degree quantifier no interacts with other quantificational expressions to derive the observed range of interpretations. Although our analysis incorporates ideas from previous analyses, we show that it is able to account (...)
  13. Scope Dominance with Upward Monotone Quantifiers.Alon Altman, Ya'Acov Peterzil & Yoad Winter - 2005 - Journal of Logic, Language and Information 14 (4):445-455.
    We give a complete characterization of the class of upward monotone generalized quantifiers Q1 and Q2 over countable domains that satisfy the scheme Q1 x Q2 y φ → Q2 y Q1 x φ. This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff Q1 is ∃ or Q2 is ∀ (excluding trivial cases). Our result shows that in infinite domains, there are more general types of quantifiers that support these entailments.
  14. On Second-Order Generalized Quantifiers and Finite Structures.Anders Andersson - 2002 - Annals of Pure and Applied Logic 115 (1--3):1--32.
    We consider the expressive power of second - order generalized quantifiers on finite structures, especially with respect to the types of the quantifiers. We show that on finite structures with at most binary relations, there are very powerful second - order generalized quantifiers, even of the simplest possible type. More precisely, if a logic is countable and satisfies some weak closure conditions, then there is a generalized second - order quantifier which is monadic, unary and simple, and a uniformly obtained (...)
  15. Barwise: Abstract Model Theory and Generalized Quantifiers.Jouko Va An Anen - 2004 - Bulletin of Symbolic Logic 10 (1):37-53.
  16. Life on the Range.G. Aldo Antonelli - forthcoming - In A. Torza (ed.), Quantifiers, Quantifiers, and Quantifiers. Synthese LIbrary.
  17. On the General Interpretation of First-Order Quantifiers.G. Aldo Antonelli - 2013 - Review of Symbolic Logic 6 (4):637-658.
    While second-order quantifiers have long been known to admit nonstandard, or interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretationgeneral” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.
  18. Numerical Abstraction Via the Frege Quantifier.G. Aldo Antonelli - 2010 - Notre Dame Journal of Formal Logic 51 (2):161-179.
    This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
  19. On Propositional Quantifiers in Provability Logic.Sergei N. Artemov & Lev D. Beklemishev - 1993 - Notre Dame Journal of Formal Logic 34 (3):401-419.
  20. Determiners and Resource Situations.Nicholas Asher & Daniel Bonevac - 1987 - Linguistics and Philosophy 10 (4):567 - 596.
  21. Multiple Quantification and the Use of Special Quantifiers in Early Sixteenth Century Logic.E. J. Ashworth - 1978 - Notre Dame Journal of Formal Logic 19 (4):599-613.
  22. 'Only' Noun Phrases, Pseudo-Negative Generalized Quantifiers, Negative Polarity Items, and Monotonicity.J. Atlas - 1996 - Journal of Semantics 13 (4):265-328.
    The theory of Generalized Quantifiers has facilitated progress in the study of negation in natural language. In particular it has permitted the formulation of a DeMorgan taxonomy of logical strength of negative Noun Phrases (Zwarts 1996a,b). It has permitted the formulation of broad semantical generalizations to explain grammatical phenomena, e.g. the distribution of Negative Polarity Items (Ladusaw 1980; Linebarger 1981, 1987, 1991; Hoeksema 1986, 1995; Zwarts 1996a,b; Horn 1992, 1996b). In the midst of this theorizing Jaap Hoepelman invited me to (...)
  23. Monotonicity and Collective Quantification.Gilad B. Avi & Yoad Winter - 2003 - Journal of Logic, Language and Information 12 (2):127--151.
  24. Anaphorically Unrestricted Quantifiers and Paradoxes.Jody Azzouni - 2008 - In J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox. Oxford University Press.
  25. Semantic Nonspecificity and Mixed Quantifiers.Kent Bach - 1980 - Linguistics and Philosophy 4 (4):593 - 605.
  26. On the Absence of Certain Quantifiers in Mohawk.Mark C. Baker - 1995 - In Emmon Bach, Eloise Jelinek, Angelika Kratzer & Barbara Partee (eds.), Quantification in Natural Languages. Kluwer Academic Publishers. pp. 21--58.
  27. Some Contributions to Definability Theory for Languages with Generalized Quantifiers.John T. Baldwin & Douglas E. Miller - 1982 - Journal of Symbolic Logic 47 (3):572-586.
  28. Restricted Quantifiers and Logical Theory.Thomas Baldwin - 2010 - In T. J. Smiley, Jonathan Lear & Alex Oliver (eds.), The Force of Argument: Essays in Honor of Timothy Smiley. Routledge. pp. 18--19.
  29. Interpretations of Quantifiers.Thomas Baldwin - 1979 - Mind 88 (350):215-240.
  30. Generalized Net Structures of Empirical Theories. II.Wolfgang Balzer & Joseph D. Sneed - 1978 - Studia Logica 37 (2):167 - 194.
  31. Generalized Net Structures of Empirical Theories. I.Wolfgang Balzer & Joseph D. Sneed - 1977 - Studia Logica 36 (3):195 - 211.
  32. The Control of Attributional Patterns by the Focusing Properties of Quantifying Expressions.S. B. Barton & A. J. Sanford - 1990 - Journal of Semantics 7 (1):81-92.
    Recent evidence has shown that certain quantifiers (few, only a few) and quantifying adverbs (seldom, rarely) when used tend to make people think of reasons for the small proportions or low frequencies which they denote. Other expressions single out small proportions or low frequences, but do not lead to a focus on reasons (e. g. a few; occasionally). In the present paper, these observations are applied to the attribution of cause in short two–line vignettes which make reference to situations, and (...)
  33. On Branching Quantifiers in English.Jon Barwise - 1979 - Journal of Philosophical Logic 8 (1):47 - 80.
  34. Generalized Quantifiers and Natural Language.Jon Barwise & Robin Cooper - 1981 - Linguistics and Philosophy 4 (2):159--219.
  35. Magidor-Malitz Quantifiers in Modules.Andreas Baudisch - 1984 - Journal of Symbolic Logic 49 (1):1-8.
    We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain Q 2 α -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ 0 -interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.
  36. Decidability and Generalized Quantifiers.Andreas Baudisch (ed.) - 1980 - Akademie Verlag.
  37. Are Quantifier Phrases Always Quantificational? The Case of 'Every F'.Pierre Baumann - 2013 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 20 (2):143-172.
    This paper argues that English quantifier phrases of the form ‘every F’ admit of a literal referential interpretation, contrary to the standard semantic account of this expression, according to which it denotes a set and a second-order relation. Various arguments are offered in favor of the referential interpretation, and two likely objections to it are forestalled.
  38. Peirce's Development of Quantifiers and of Predicate Logic.Richard Beatty - 1969 - Notre Dame Journal of Formal Logic 10 (1):64-76.
  39. Quantifier Elimination in Valued Ore Modules.Luc Bélair & Françoise Point - 2010 - Journal of Symbolic Logic 75 (3):1007-1034.
    We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.
  40. Feature System for Quantification Structures in Natural Language.I. Bellert - 1989 - Foris Publications.
  41. Interpretative Model for Linguistic Quantifiers.Irena Bellert - 1985 - In G. Dorn & P. Weingarten (eds.), Foundations of Logic and Linguistics. Problems and Solutions. Plenum. pp. 503--541.
  42. One Connection Between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers.Ben Shalom Dorit - 2003 - Journal of Logic, Language and Information 12 (1):47-52.
    The language of standard propositional modal logic has one operator (? or ?), that can be thought of as being determined by the quantifiers ? or ?, respectively: for example, a formula of the form ?F is true at a point s just in case all the immediate successors of s verify F.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized (...)
  43. Monotonicity and Collective Quantification.Ben-Avi Gilad & Winter Yoad - 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 (...)
  44. Scope Dominance with Monotone Quantifiers Over Finite Domains.Gilad Ben-Avi & Yoad Winter - 2004 - Journal of Logic, Language and Information 13 (4):385-402.
    We characterize pairs of monotone generalized quantifiers Q1 and Q2 over finite domains that give rise to an entailment relation between their two relative scope construals. This relation between quantifiers, which is referred to as scope dominance, is used for identifying entailment relations between the two scopal interpretations of simple sentences of the form NP1–V–NP2. Simple numerical or set-theoretical considerations that follow from our main result are used for characterizing such relations. The variety of examples in which they hold are (...)
  45. Bare Quantifiers?Hanoch Ben-Yami - 2014 - Pacific Philosophical Quarterly 95 (2):175-188.
    In a series of publications I have claimed that by contrast to standard formal languages, quantifiers in natural language combine with a general term to form a quantified argument, in which the general term's role is to determine the domain or plurality over which the quantifier ranges. In a recent paper Zoltán Gendler Szabó tried to provide a counterexample to this analysis and derived from it various conclusions concerning quantification in natural language, claiming it is often ‘bare’. I show that (...)
  46. Response to Westerstahl.Hanoch Ben-Yami - 2012 - Logique Et Analyse 55 (217):47-55.
  47. 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.
  48. Polyadic Quantifiers.Johan Benthem - 1989 - Linguistics and Philosophy 12 (4):437 - 464.
  49. Modelling Simultaneous Games in Dynamic Logic.Johan Van Benthem, Sujata Ghosh & Fenrong Liu - 2008 - Synthese 165 (2):247 - 268.
    We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.
  50. Rings Which Admit Elimination of Quantifiers.Chantal Berline - 1981 - Journal of Symbolic Logic 46 (1):56-58.
    The aim of this paper is to provide an addendum to a paper by Rose with the same title which has appeared in an earlier issue of this Journal [2]. Our new result is: Theorem. A ring of characteristic zero which admits elimination of quantifiers in the language {0, 1, +, ·} is an algebraically closed field.
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