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Philosophy of Language > Specific Expressions > Quantifiers > Generalized Quantifiers

Generalized Quantifiers

Edited by Jakub Szymanik (University of Amsterdam)
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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 & Westerståhl 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.

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  1. Varol Akman (1998). Book Review--Jaap Van der Does and Jan Van Eijk, Eds., Quantifiers, Logic, and Language. [REVIEW] .
    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).
    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. J. Atlas (1996). 'Only' Noun Phrases, Pseudo-Negative Generalized Quantifiers, Negative Polarity Items, and Monotonicity. Journal of Semantics 13 (4):265-328.
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  6. Gilad B. Avi & Yoad Winter (2003). Monotonicity and Collective Quantification. Journal of Logic, Language and Information 12 (2):127--151.
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  7. Kent Bach (1982). Semantic Nonspecificity and Mixed Quantifiers. Linguistics and Philosophy 4 (4):593 - 605.
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  8. 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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  9. Jon Barwise (1979). On Branching Quantifiers in English. Journal of Philosophical Logic 8 (1):47 - 80.
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  10. Jon Barwise & Robin Cooper (1981). Generalized Quantifiers and Natural Language. Linguistics and Philosophy 4 (2):159--219.
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  11. 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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  12. Andreas Baudisch (ed.) (1980). Decidability and Generalized Quantifiers. Akademie-Verlag.
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  13. 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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  14. Gilad Ben-Avi & Yoad Winter (2004). Scope Dominance with Monotone Quantifiers Over Finite Domains. Journal of Logic, Language and Information 13 (4).
    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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  15. 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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  16. Hanoch Ben-Yami (2012). Response to Westerstahl. Logique Et Analyse 55 (217):47-55.
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  17. 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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  18. Johan Benthem (1989). Polyadic Quantifiers. Linguistics and Philosophy 12 (4):437 - 464.
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  19. Johan Van Benthem (1984). Questions About Quantifiers. Journal of Symbolic Logic 49 (2):443 - 466.
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  20. 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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  21. Reinhard Blutner (1993). Dynamic Generalized Quantifiers and Existential Sentences in Natural Languages. Journal of Semantics 10 (1):33-64.
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  22. 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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  23. Mark Brown (1984). Generalized Quantifiers and the Square of Opposition. Notre Dame Journal of Formal Logic 25 (4):303-322.
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  24. Robin Clark (2011). Generalized Quantifiers and Number Sense. Philosophy Compass 6 (9):611-621.
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  25. 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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  26. 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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  27. Robin Cooper (1996). The Role of Situations in Generalized Quantifiers. In Shalom Lappin (ed.), The Handbook of Contemporary Semantic Theory. Blackwell.
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  28. Mary Dalrymple, Makoto Kanazawa, Yookyung Kim, Sam McHombo & Stanley Peters (1998). Reciprocal Expressions and the Concept of Reciprocity. Linguistics and Philosophy 21 (2):159-210.
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  29. Paul Dekker (2003). Meanwhile, Within the Frege Boundary. Linguistics and Philosophy 26 (5):547-556.
    In this paper, I want to contribute to understanding and improving on Keenan'sintriguing equivalence result about reducible type quantifiers (Keenan, 1992).I give an alternative proof of his result which generalizes to type quantifiers, andI show how the reduction of a reducible type quantifier to (the composition of) ntype quantifiers can be effected.
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  30. Małgorzata Dubiel (1977). Generalized Quantifiers and Elementary Extensions of Countable Models. Journal of Symbolic Logic 42 (3):341-348.
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  31. Fredrik Engström (2012). Generalized Quantifiers in Dependence Logic. Journal of Logic, Language and Information 21 (3):299-324.
    We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent (...)
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  32. Solomon Feferman, Which Quantifiers Are Logical?
    ✤ It is the characterization of those forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter.
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  33. Tim Fernando, Conservative Generalized Quantifiers and Presupposition.
    Conservativity in generalized quantifiers is linked to presupposition filtering, under a propositions-as-types analysis extended with dependent quantifiers. That analysis is underpinned by modeltheoretically interpretable proofs which inhabit propositions they prove, thereby providing objects for quantification and hooks for anaphora.
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  34. Jörg Flum, Matthias Schiehlen & Jouko Väänänen (1999). Quantifiers and Congruence Closure. Studia Logica 62 (3):315-340.
    We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finite structures, we define (...)
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  35. Peter Fritz (forthcoming). Modal Ontology and Generalized Quantifiers. Journal of Philosophical Logic.
    Timothy Williamson has argued that in the debate on modal ontology, the familiar distinction between actualism and possibilism should be replaced by a distinction between positions he calls contingentism and necessitism. He has also argued in favor of necessitism, using results on quantified modal logic with plurally interpreted second-order quantifiers showing that necessitists can draw distinctions contingentists cannot draw. Some of these results are similar to well-known results on the relative expressivity of quantified modal logics with so-called inner and outer (...)
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  36. D. M. Gabbay & J. M. E. Moravcsik (1974). Branching Quantifiers, English and Montague Grammar. Theoretical Linguistics 1:140--157.
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  37. Peter Gärdenfors (ed.) (1987). Generalized Quantifiers. Reidel Publishing Company.
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  38. B. Geurts (2005). Monotonicity and Processing Load. Journal of Semantics 22 (1):97-117.
    Starting out from the assumption that monotonicity plays a central role in interpretation and inference, we derive a number of predictions about the complexity of processing quantified sentences. A quantifier may be upward entailing (i.e. license inferences from subsets to supersets) or downward entailing (i.e. license inferences from supersets to subsets). Our main predictions are the following: If the monotonicity profiles of two quantifying expressions are the same, they should be equally easy or hard to process, ceteris paribus. Sentences containing (...)
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  39. Bart Geurts (2003). Reasoning with Quantifiers. Cognition 86 (3):223--251.
    In the semantics of natural language, quantification may have received more attention than any other subject, and one of the main topics in psychological studies on deductive reasoning is syllogistic inference, which is just a restricted form of reasoning with quantifiers. But thus far the semantical and psychological enterprises have remained disconnected. This paper aims to show how our understanding of syllogistic reasoning may benefit from semantical research on quantification. I present a very simple logic that pivots on the monotonicity (...)
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  40. Nina Gierasimczuk & Jakub Szymanik (2011). Invariance Properties of Quantifiers and Multiagent Information Exchange. In M. Kanazawa (ed.), Proceedings of the 12th Meeting on Mathematics of Language, Lecture Notes in Artificial Intelligence 6878. Springer.
    The paper presents two case studies of multi-agent information exchange involving generalized quantifiers. We focus on scenarios in which agents successfully converge to knowledge on the basis of the information about the knowledge of others, so-called Muddy Children puzzle and Top Hat puzzle. We investigate the relationship between certain invariance properties of quantifiers and the successful convergence to knowledge in such situations. We generalize the scenarios to account for public announcements with arbitrary quantifiers. We show that the Muddy Children puzzle (...)
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  41. Nina Gierasimczuk & Jakub Szymanik (2011). A Note on a Generalization of the Muddy Children Puzzle. In K. Apt (ed.), Proceeding of the 13th Conference on Theoretical Aspects of Rationality and Knowledge. ACM.
    We study a generalization of the Muddy Children puzzle by allowing public announcements with arbitrary generalized quantifiers. We propose a new concise logical modeling of the puzzle based on the number triangle representation of quantifi ers. Our general aim is to discuss the possibility of epistemic modeling that is cut for specifi c informational dynamics. Moreover, we show that the puzzle is solvable for any number of agents if and only if the quanti fier in the announcement is positively active (...)
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  42. Nina Gierasimczuk & Jakub Szymanik (2007). Hintikka's Thesis Revisited. The Bulletin of Symbolic Logic 13:273.
    We discuss Hintikka’s Thesis [Hintikka 1973] that there exist natural language sentences which require non–linear quantification to express their logical form.
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  43. Daniel Gogol (1975). Formulas with Two Generalized Quantifiers. Notre Dame Journal of Formal Logic 16 (1):133-136.
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  44. Joanna Golinska-Pilarek & Konrad Zdanowski (2003). Spectra of Formulae with Henkin Quantifiers. In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers.
    It is known that various complexity-theoretical problems can be translated into some special spectra problems. Thus, questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quantifiers in the empty vocabulary.
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  45. Georg Gottlob (1997). Relativized Logspace and Generalized Quantifiers Over Finite Ordered Structures. Journal of Symbolic Logic 62 (2):545-574.
    We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized to an oracle in C. We show that this is not always (...)
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  46. Erich Grädel, Phokion Kolaitis, Libkin G., Marx Leonid, Spencer Maarten, Vardi Joel, Y. Moshe, Yde Venema & Scott Weinstein (2007). Finite Model Theory and its Applications. Springer.
    This book gives a comprehensive overview of central themes of finite model theory – expressive power, descriptive complexity, and zero-one laws – together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and hierarchies within first-order, second-order, fixed-point, and infinitary logics (...)
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  47. J. A. G. Groenendijk, Dick de Jongh & M. J. B. Stokhof (eds.) (1986/1987). Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers. Foris Publications.
    Semantic Automata Johan van Ben them. INTRODUCTION An attractive, but never very central idea in modern semantics has been to regard linguistic expressions ...
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  48. Martin Hackl (2009). On the Grammar and Processing of Proportional Quantifiers: Most Versus More Than Half. Natural Language Semantics 17 (1):63--98.
    Abstract Proportional quantifiers have played a central role in the development of formal semantics because they set a benchmark for the expressive power needed to describe quantification in natural language (Barwise and Cooper Linguist Philos 4:159–219, 1981). The proportional quantifier most, in particular, supplied the initial motivation for adopting Generalized Quantifier Theory (GQT) because its meaning is definable as a relation between sets of individuals, which are taken to be semantic primitives in GQT. This paper proposes an alternative analysis of (...)
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  49. I. Heim, H. Lasnik & R. May (1991). Reciprocity and Plurality. Linguistic Inquiry 22 (1):63--101.
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  50. Lauri Hella, Phokion G. Kolaitis & Kerkko Luosto (1996). Almost Everywhere Equivalence of Logics in Finite Model Theory. Bulletin of Symbolic Logic 2 (4):422-443.
    We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (C)=1 (...)
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  51. Lauri Hella & Kerkko Luosto (1992). The Beth-Closure of L(Qα) is Not Finitely Generated. Journal of Symbolic Logic 57 (2):442 - 448.
    We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).
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  52. Lauri Hella, Kerkko Luosto & Jouko Väänänen (1996). The Hierarchy Theorem for Generalized Quantifiers. Journal of Symbolic Logic 61 (3):802-817.
    The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...)
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  53. Lauri Hella, Merlijn Sevenster & Tero Tulenheimo (2008). Partially Ordered Connectives and Monadic Monotone Strict Np. Journal of Logic, Language and Information 17 (3).
    Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is (...)
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  54. Lauri Hella, Jouko Väänänen & Dag Westerståhl (1997). Definability of Polyadic Lifts of Generalized Quantifiers. Journal of Logic, Language and Information 6 (3):305-335.
    We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.
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  55. Jaakko Hintikka (1976). Partially Ordered Quantifiers Vs. Partially Ordered Ideas. Dialectica 30:89--99.
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  56. Jaakko Hintikka (1974). Quantifiers Vs. Quantificational Theory. Linguistic Inquiry 5:153--77.
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  57. Tapani Hyttinen & Gabriel Sandu (2000). Henkin Quantifiers and the Definability of Truth. Journal of Philosophical Logic 29 (5):507-527.
    Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L * 1(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L * 1(H) with respect to Boolean operations, and obtain the language L 1(H). At the next level, we consider an extension L * 2(H) of L 1(H) in which every sentence (...)
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  58. A. A. Ivanov (1999). Generic Expansions of Ω-Categorical Structures and Semantics of Generalized Quantifiers. Journal of Symbolic Logic 64 (2):775-789.
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  59. Ed Keenan (1999). Quantification in English is Inherently Sortal. History and Philosophy of Logic 20 (3-4):251-265.
    Within Linguistics the semantic analysis of natural languages (English, Swahili, for example) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well as the discovery of new types (...)
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  60. Edward Keenan (2002). Some Properties of Natural Language Quantifiers: Generalized Quantifier Theory. Linguistics and Philosophy 25 (5-6):627-654.
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  61. Edward L. Keenan, Further Excursions in Natural Logic: The Mid—Point Theorems.
    Pursuing a study begun in (Keenan 2004) this note investigates inference patterns in natural language which proportionality quantifiers enter. We desire to identify such patterns and to isolate any such which are specific to proportionality quantifiers.
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  62. Edward L. Keenan (2009). Some Logical Properties of Natural Language Quantifiers. In Joseph Almog & Paolo Leonardi (eds.), The Philosophy of David Kaplan. Oxford University Press.
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  63. Edward L. Keenan (1993). Natural Language, Sortal Reducibility and Generalized Quantifiers. Journal of Symbolic Logic 58 (1):314-325.
    Recent work in natural language semantics leads to some new observations on generalized quantifiers. In § 1 we show that English quantifiers of type $ $ are booleanly generated by their generalized universal and generalized existential members. These two classes also constitute the sortally reducible members of this type. Section 2 presents our main result--the Generalized Prefix Theorem (GPT). This theorem characterizes the conditions under which formulas of the form Q1x 1⋯ Qnx nRx 1⋯ xn and q1x 1⋯ qnx nRx (...)
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  64. Edward L. Keenan (1992). Beyond the Frege Boundary. Linguistics and Philosophy 15 (2):199 - 221.
    In sentences likeEvery teacher laughed we think ofevery teacher as aunary (=type ) quantifier — it expresses a property ofone place predicate denotations. In variable binding terms, unary quantifiers bind one variable. Two applications of unary quantifiers, as in the interpretation ofNo student likes every teacher, determine abinary (= (type ) quantifier; they express properties oftwo place predicate denotations. In variable binding terms they bind two variables. We call a binary quantifierFregean (orreducible) if it can in principle be expressed by (...)
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  65. Edward Keenan & Denis Paperno (2010). Stanley Peters and Dag Westerståhl: Quantifiers in Language and Logic. Linguistics and Philosophy 33 (6):513-549.
    Quantifiers in Language and Logic (QLL) is a major contribution to natural language semantics, specifically to quantification. It integrates the extensive recent work on quantifiers in logic and linguistics. It also presents new observations and results. QLL should help linguists understand the mathematical generalizations we can make about natural language quantification, and it should interest logicians by presenting an extensive array of quantifiers that lie beyond the pale of classical logic. Here we focus on those aspects of QLL we judge (...)
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  66. Ruth M. Kempson & Annabel Cormack (1981). Ambiguity and Quantification. Linguistics and Philosophy 4 (2):259 - 309.
    In the opening sections of this paper, we defined ambiguity in terms of distinct sentences (for a single sentence-string) with, in particular, distinct sets of truth conditions for the corresponding negative sentence-string. Lexical vagueness was defined as equivalent to disjunction, for under conditions of the negation of a sentence-string containing such an expression, all the relevant more specific interpretations of the string had also to be negated. Yet in the case of mixed quantification sentences, the strengthened, more specific, interpretations of (...)
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  67. Juha Kontinen (2006). The Hierarchy Theorem for Second Order Generalized Quantifiers. Journal of Symbolic Logic 71 (1):188 - 202.
    We study definability of second order generalized quantifiers on finite structures. Our main result says that for every second order type t there exists a second order generalized quantifier of type t which is not definable in the extension of second order logic by all second order generalized quantifiers of types lower than t.
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  68. Juha Kontinen & Jakub Szymanik (2011). Characterizing Definability of Second-Order Generalized Quantifiers. In L. Beklemishev & R. de Queiroz (eds.), Proceedings of the 18th Workshop on Logic, Language, Information and Computation, Lecture Notes in Artificial Intelligence 6642. Springer.
    We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier $\sQ_1$ is definable in terms of another quantifier $\sQ_2$, the base logic being monadic second-order logic, reduces to the question if a quantifier $\sQ^{\star}_1$ is definable in $\FO(\sQ^{\star}_2,<,+,\times)$ for certain first-order quantifiers $\sQ^{\star}_1$ and $\sQ^{\star}_2$. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier $\most^1$ is not definable (...)
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  69. Juha Kontinen & Jakub Szymanik (2008). A Remark on Collective Quantification. Journal of Logic, Language and Information 17 (2):131-140.
    We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize all collective (...)
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  70. Kepa Korta (1990). Generalized Quantifiers: Linguistic and Logical Approaches. Theoria 5 (1):266-268.
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  71. Michał Krynicki, Alistair Lachlan & Jouko Väänänen (1984). Vector Spaces and Binary Quantifiers. Notre Dame Journal of Formal Logic 25 (1):72-78.
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  72. Fred Landman (2000). Against Binary Quantifiers. In Events and Plurality. Kluwer Academic Publisher.
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  73. S. Lappin (1996). Generalized Quantifiers, Exception Phrases, and Logicality. Journal of Semantics 13 (3):197-220.
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  74. Per Lindström (1966). First Order Predicate Logic with Generalized Quantifiers. Theoria 32:186--195.
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  75. Kerkko Luosto (2000). Hierarchies of Monadic Generalized Quantifiers. Journal of Symbolic Logic 65 (3):1241-1263.
    A combinatorial criterium is given when a monadic quantifier is expressible by means of universe-independent monadic quantifiers of width n. It is proved that the corresponding hierarchy does not collapse. As an application, it is shown that the second resumption (or vectorization) of the Hartig quantifier is not definable by monadic quantifiers. The techniques rely on Ramsey theory.
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  76. A. Macintyre (1980). Ramsey Quantifiers in Arithmetic. In L. Pacholski, J. Wierzejewski & A. J. Wilkie (eds.), Model Theory of Algebra and Arithmetics. Springer--Verlag.
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  77. M. Magidor & J. I. Malitz (1977). Compact Extensions of L(Q). Annals of Mathematical Logic 11:217--261.
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  78. C. T. Mcmillan, R. Clark, P. Moore, C. Devita & M. Grossman (2005). Neural Basis for Generalized Quantifiers Comprehension. Neuropsychologia 43:1729--1737.
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  79. C. T. Mcmillan, R. Clark, P. Moore & M. Grossman (2006). Quantifiers Comprehension in Corticobasal Degeneration. Brain and Cognition 65:250--260.
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  80. Richard Montague (1973). The Proper Treatment of Quantification in Ordinary English. In Patrick Suppes, Julius Moravcsik & Jaakko Hintikka (eds.), Approaches to Natural Language. Dordrecht.
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  81. Carl Morgenstern (1982). On Generalized Quantifiers in Arithmetic. Journal of Symbolic Logic 47 (1):187-190.
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  82. Andrzej Mostowski (1957). On a Generalization of Quantifiers. Fundamenta Mathematicae 44:12--36.
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  83. Marcin Mostowski (1998). Computational Semantics for Monadic Quantifiers. Journal of Applied Non--Classical Logics 8:107--121.
    The paper gives a survey of known results related to computational devices (finite and push–down automata) recognizing monadic generalized quantifiers in finite models. Some of these results are simple reinterpretations of descriptive—feasible correspondence theorems from finite–model theory. Additionally a new result characterizing monadic quantifiers recognized by push down automata is proven.
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  84. Marcin Mostowski & Jakub Szymanik (2012). Semantic Bounds for Everyday Language. Semiotica 188 (1/4):363-372.
    We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second–order logic. Two arguments for this thesis are formulated. Firstly, we show that so–called Barwise's test of negation normality works properly only when assuming our main thesis. Secondly, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they are bounded by second–order (...)
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  85. Marcin Mostowski & Jakub Szymanik (2007). Computational Complexity of Some Ramsey Quantifiers in Finite Models. The Bulletin of Symbolic Logic 13:281--282.
    The problem of computational complexity of semantics for some natural language constructions – considered in [M. Mostowski, D. Wojtyniak 2004] – motivates an interest in complexity of Ramsey quantifiers in finite models. In general a sentence with a Ramsey quantifier R of the following form Rx, yH(x, y) is interpreted as ∃A(A is big relatively to the universe ∧A2 ⊆ H). In the paper cited the problem of the complexity of the Hintikka sentence is reduced to the problem of computational (...)
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  86. Linda M. Moxey & Anthony J. Sanford (1993). Communicating Quantities: A Psychological Perspective (Essays in Cognitive Psychology). Psychology Press.
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  87. Linda M. Moxey & Anthony J. Sanford (1986). Quantifiers and Focus. Journal of Semantics 5 (3):189-206.
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  88. John Nerbonne (1995). Nominal Comparatives and Generalized Quantifiers. Journal of Logic, Language and Information 4 (4):273-300.
    This work adopts the perspective of plural logic and measurement theory in order first to focus on the microstructure of comparative determiners; and second, to derive the properties of comparative determiners as these are studied in Generalized Quantifier Theory, locus of the most sophisticated semantic analysis of natural language determiners. The work here appears to be the first to examine comparatives within plural logic, a step which appears necessary, but which also harbors specific analytical problems examined here.Since nominal comparatives involve (...)
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  89. Thomas E. Patton (1991). On the Ontology of Branching Quantifiers. Journal of Philosophical Logic 20 (2):205 - 223.
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  90. Renato H. L. Pedrosa & Antonio M. A. Sette (1988). A Representation Theorem for Languages with Generalized Quantifiers Through Back-and-Forth Methods. Studia Logica 47 (4):401 - 411.
    We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.
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  91. Stanley Peters & Dag Westerståhl (2006). Quantifiers in Language and Logic. Clarendon Press.
    Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, and many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role. Quantifiers in Language and Logic is intended for everyone with a scholarly interest in the exact treatment (...)
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  92. Philip L. Peterson (1991). Complexly Fractionated Syllogistic Quantifiers. Journal of Philosophical Logic 20 (3):287 - 313.
    Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantifiers, and then include further quantifiers which are modifications of such fractions (such as almost 1/2 the S are P and Much more than 1/2 the S are P). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes & Peterson (forthcoming) for proofs of the unsoundness and (...)
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  93. Niki Pfeifer (2006). Contemporary Syllogistics: Comparative and Quantitative Syllogisms. In G. Kreuzbauer & G. J. W. Dorn (eds.), Argumentation in Theorie Und Praxis: Philosophie Und Didaktik des Argumentierens. Lit.
    Traditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All… are…” and “At least one… is…” and their respective negated versions. Unfortunately, the practical use of traditional syllogisms is quite restricted. On the one hand, the “All…” propositions are too strict, since a single counterexample suffices for falsification. On the other hand, the “At least one …” propositions are too weak, since a single example suffices for verification. The present contribution studies (...)
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  94. Paul Pietroski, Jeffrey Lidz, Tim Hunter & Justin Halberda (2009). The Meaning of 'Most': Semantics, Numerosity and Psychology. Mind and Language 24 (5):554-585.
    The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...)
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  95. Charles C. Pinter (1975). Algebraic Logic with Generalized Quantifiers. Notre Dame Journal of Formal Logic 16 (4):511-516.
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  96. Ian Pratt & Nissim Francez (2001). Temporal Prepositions and Temporal Generalized Quantifiers. Linguistics and Philosophy 24 (2):187-222.
    In this paper, we show how the problem of accounting for the semanticsof temporal preposition phrases (tPPs) leads us to some surprisinginsights into the semantics of temporal expressions ingeneral. Specifically, we argue that a systematic treatment of EnglishtPPs is greatly facilitated if we endow our meaning assignments with context variables, a device which allows a tPP to restrict domainsof quantification arising elsewhere in a sentence. We observe that theuse of context variables implies that tPPs can modify expressions intwo ways, and (...)
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  97. Ian Pratt-Hartmann (2005). Complexity of the Two-Variable Fragment with Counting Quantifiers. Journal of Logic, Language and Information 14 (3).
    The satisfiability and finite satisfiability problems for the two-variable fragment of first-order logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are coded succinctly.
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  98. Ian Pratt-Hartmann (2004). Fragments of Language. Journal of Logic, Language and Information 13 (2):207-223.
    By a fragment of a natural language we mean a subset of thatlanguage equipped with semantics which translate its sentences intosome formal system such as first-order logic. The familiar conceptsof satisfiability and entailment can be defined for anysuch fragment in a natural way. The question therefore arises, for anygiven fragment of a natural language, as to the computational complexityof determining satisfiability and entailment within that fragment. Wepresent a series of fragments of English for which the satisfiabilityproblem is polynomial, NP-complete, EXPTIME-complete,NEXPTIME-complete (...)
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  99. Erich Rast (2011). On Contextual Domain Restriction in Categorial Grammar. Synthese (Online First) 2011 (June).
    Abstract -/- Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that depend on a previously introduced interpreter and interpretation time in a two-dimensional semantic framework on the basis of simple type theory and categorial grammar. Against Stanley (2002) it is argued that a suitable version of QDR can deal with superlatives like tallest. However, it is shown that NR is needed to account for utterances when the speaker intends to convey different restrictions for (...)
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  100. Livio Robaldo (2010). Independent Set Readings and Generalized Quantifiers. Journal of Philosophical Logic 39 (1):23-58.
    Several authors proposed to devise logical structures for Natural Language (NL) semantics in which noun phrases yield referential terms rather than standard Generalized Quantifiers. In this view, two main problems arise: the need to refer to the maximal sets of entities involved in the predications and the need to cope with Independent Set (IS) readings, where two or more sets of entities are introduced in parallel. The article illustrates these problems and their consequences, then presents an extension of the proposal (...)
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