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Summary

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.

Introductions

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 2009.

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  1. Varol Akman (1998). Book Review--Jaap Van der Does and Jan Van Eijk, Eds., Quantifiers, Logic, and Language. [REVIEW] 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.
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  2. Natasha Alechina (1995). On a Decidable Generalized Quantifier Logic Corresponding to a Decidable Fragment of First-Order Logic. 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 (...)
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  3. Natasha Alechina & Michiel van Lambalgen (1996). Generalized Quantification as Substructural Logic. 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 (...)
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  4. Alon Altman, Ya'Acov Peterzil & Yoad Winter (2005). Scope Dominance with Upward Monotone Quantifiers. 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.
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  5. Jouko Va An Anen (2004). Barwise: Abstract Model Theory and Generalized Quantifiers. Bulletin of Symbolic Logic 10 (1):37-53.
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  6. G. Aldo Antonelli (2013). On the General Interpretation of First-Order Quantifiers. 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.
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  7. J. Atlas (1996). 'Only' Noun Phrases, Pseudo-Negative Generalized Quantifiers, Negative Polarity Items, and Monotonicity. 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 (...)
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  8. Gilad B. Avi & Yoad Winter (2003). Monotonicity and Collective Quantification. Journal of Logic, Language and Information 12 (2):127--151.
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  9. Jody Azzouni (2008). Anaphorically Unrestricted Quantifiers and Paradoxes. In J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox. Oup Oxford.
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  10. Kent Bach (1982). Semantic Nonspecificity and Mixed Quantifiers. Linguistics and Philosophy 4 (4):593 - 605.
  11. Mark C. Baker (1995). On the Absence of Certain Quantifiers in Mohawk. In Emmon Bach, Eloise Jelinek, Angelika Kratzer & Barbara Partee (eds.), Quantification in Natural Languages. Kluwer. 21--58.
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  12. John T. Baldwin & Douglas E. Miller (1982). Some Contributions to Definability Theory for Languages with Generalized Quantifiers. Journal of Symbolic Logic 47 (3):572-586.
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  13. Thomas Baldwin (1979). Interpretations of Quantifiers. Mind 88 (350):215-240.
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  14. Jon Barwise (1979). On Branching Quantifiers in English. Journal of Philosophical Logic 8 (1):47 - 80.
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  15. Jon Barwise & Robin Cooper (1981). Generalized Quantifiers and Natural Language. Linguistics and Philosophy 4 (2):159--219.
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  16. Andreas Baudisch (1984). Magidor-Malitz Quantifiers in Modules. 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.
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  17. Andreas Baudisch (ed.) (1980). Decidability and Generalized Quantifiers. Akademie-Verlag.
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  18. Irena Bellert (1985). Interpretative Model for Linguistic Quantifiers. In G. Dorn & P. Weingarten (eds.), Foundations of Logic and Linguistics. Problems and Solutions. Plenum. 503--541.
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  19. Dorit Ben Shalom (2003). One Connection Between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers. 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 (...)
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  20. Gilad Ben-Avi & Yoad Winter (2004). Scope Dominance with Monotone Quantifiers Over Finite Domains. 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 (...)
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  21. Gilad Ben-Avi & Yoad Winter (2003). Monotonicity and Collective Quantification. 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 (...)
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  22. Hanoch Ben-Yami (2014). Bare Quantifiers? 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 (...)
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  23. Hanoch Ben-Yami (2012). Response to Westerstahl. Logique Et Analyse 55 (217):47-55.
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  24. Hanoch Ben-Yami (2009). Generalized Quantifiers, and Beyond. 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.
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  25. Johan Benthem (1989). Polyadic Quantifiers. Linguistics and Philosophy 12 (4):437 - 464.
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  26. Johan Van Benthem (1984). Questions About Quantifiers. Journal of Symbolic Logic 49 (2):443 - 466.
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  27. Johan Van Benthem, Sujata Ghosh & Fenrong Liu (2008). Modelling Simultaneous Games in Dynamic Logic. 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.
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  28. Rajesh Bhatt & Roumyana Pancheva (2007). Degree Quantifiers, Position of Merger Effects with Their Restrictors, and Conservativity. In Chris Barker & Pauline I. Jacobson (eds.), Direct Compositionality. Oxford University Press. 14--306.
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  29. Belinda Blevins-Knabe, Robert G. Cooper, Prentice Starkey, Patty Goth Mace & Ed Leitner (1987). Preschoolers Sometimes Know Less Than We Think: The Use of Quantifiers to Solve Addition and Subtraction Tasks. Bulletin of the Psychonomic Society 25 (1):31-34.
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  30. Reinhard Blutner (1993). Dynamic Generalized Quantifiers and Existential Sentences in Natural Languages. Journal of Semantics 10 (1):33-64.
    The central topic to be discussed in this paper is the definiteness restriction in there-insertion contexts. Various attempts to explain this definiteness restriction using the standard algebraic framework are discussed (Barwise & Cooper 1981; Keenan 1978; Milsark 1974; Higginbortham 1987; Lappin 1988) and the shortcomings of these attempts are demonstrated. Finally, a new approach to the interpretation of existential there be-sentences is developed within the framework of Groenendijk & Stokhof's (1990) Dynamic Montague Grammar. This approach makes use of a variant (...)
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  31. Ludwik Borkowski (1964). Correction to the Paper “on Proper Quantifiers II”. Studia Logica 15 (1):272 -.
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  32. Ludwik Borkowski (1960). On Proper Quantifiers II. Studia Logica 10 (1):7 - 28.
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  33. Ludwik Borkowski (1958). On Proper Quantifiers I. Studia Logica 8 (1):65 - 130.
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  34. Oliver Bott, Fabian Schlotterbeck & Jakub Szymanik (2011). Tractable Versus Intractable Reciprocal Sentences. In J. Bos & S. Pulman (eds.), Proceedings of the International Conference on Computational Semantics 9.
    In three experiments, we investigated the computational complexity of German reciprocal sentences with different quantificational antecedents. Building upon the tractable cognition thesis (van Rooij, 2008) and its application to the verification of quantifiers (Szymanik, 2010) we predicted complexity differences among these sentences. Reciprocals with all-antecedents are expected to preferably receive a strong interpretation (Dalrymple et al., 1998), but reciprocals with proportional or numerical quantifier antecedents should be interpreted weakly. Experiment 1, where participants completed pictures according to their preferred interpretation, provides (...)
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  35. Denis Bouyssou & Jean-Claude Vansnick (1986). Noncompensatory and Generalized Noncompensatory Preference Structures. Theory and Decision 21 (3):251-266.
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  36. E. P. Brandon (1982). Quantifiers and the Pursuit of Truth. Educational Philosophy and Theory 14 (1):51–58.
  37. Mark Brown (1984). Generalized Quantifiers and the Square of Opposition. Notre Dame Journal of Formal Logic 25 (4):303-322.
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  38. Mark A. Brown (1995). Some Remarks on Zawadowski's Theory of Preordered Quantifiers. In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers. 255--264.
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  39. Mark A. Brown (1990). Questions and Quantifiers. Theoria 56 (1-2):62-84.
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  40. Samuel R. Buss & Alan S. Johnson (2010). The Quantifier Complexity of Polynomial‐Size Iterated Definitions in First‐Order Logic. Mathematical Logic Quarterly 56 (6):573-590.
    We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier-free case and in the case of purely existential or universal quantifiers, we show that Ω quantifiers (...)
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  41. Xavier Caicedo (1995). Hilbert's Ε-Symbol in the Presence of Generalized Quantifiers. In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers. 63--78.
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  42. Xavier Caicedo (1991). Hilbert∈-Symbol in the Presence of Generalized Quantifiers. Bulletin of the Section of Logic 20 (3/4):85-86.
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  43. Xavier Caicedo (1990). Definability Properties and the Congruence Closure. Archive for Mathematical Logic 30 (4):231-240.
    We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (...)
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  44. Herman Cappelan & Ernest Lepore (2002). Insensitive Quantifiers. In Joseph K. Campbell, Michael O'Rourke & David Shier (eds.), Meaning and Truth - Investigations in Philosophical Semantics. Seven Bridges Press. 197--213.
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  45. Janis Cırulis (2012). Orthoposets with Quantifiers. Bulletin of the Section of Logic 41 (1/2):1-12.
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  46. Robin Clark (2011). Generalized Quantifiers and Number Sense. Philosophy Compass 6 (9):611-621.
    Generalized quantifiers are functions from pairs of properties to truth-values; these functions can be used to interpret natural language quantifiers. The space of such functions is vast and a great deal of research has sought to find natural constraints on the functions that interpret determiners and create quantifiers. These constraints have demonstrated that quantifiers rest on number and number sense. In the first part of the paper, we turn to developing this argument. In the remainder, we report on work in (...)
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  47. Robin Clark (2010). On the Learnability of Quantifiers. In Johan Van Benthem & Alice Ter Meulen (eds.), Handbook of Logic and Language, 2nd Edition.
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  48. Robin Clark & Murray Grossman (2007). Number Sense and Quantifier Interpretation. Topoi 26 (1):51--62.
    We consider connections between number sense—the ability to judge number—and the interpretation of natural language quantifiers. In particular, we present empirical evidence concerning the neuroanatomical underpinnings of number sense and quantifier interpretation. We show, further, that impairment of number sense in patients can result in the impairment of the ability to interpret sentences containing quantifiers. This result demonstrates that number sense supports some aspects of the language faculty.
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  49. Robin Cooper (1996). The Role of Situations in Generalized Quantifiers. In Shalom Lappin (ed.), The Handbook of Contemporary Semantic Theory. Blackwell. 65--86.
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  50. Robin Cooper (1987). Preliminaries to the Treatment of Generalized Quantifiers in Situation Semantics. In Peter Gärdenfors (ed.), Generalized Quantifiers. Reidel Publishing Company. 73--91.
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