About this topic
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. For a combination of formal work and cognitive science perspective, see, e.g., Szymanik 2016.

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437 found
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1 — 50 / 437
  1. added 2019-07-05
    Ryle, the Double Counting Problem, and the Logical Form of Category Mistakes.Jonah Goldwater - 2018 - Journal of the History of Philosophy 56 (2):337-359.
    Gilbert Ryle is most famous for accusing the Cartesian dualist of committing a category mistake. Yet the nature of this accusation, and the idea of a category mistake more generally, remains woefully misunderstood. The aim of this paper is to rectify this misunderstanding. I show that Ryle does not conceive of category mistakes as mistakes of predication, as is so widely believed. Instead I show category mistakes are mistakes of conjunction and quantification. This thesis uniquely unifies and explains the wide (...)
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  2. added 2019-06-06
    Comparative and Superlative Quantifiers: Pragmatic Effects of Comparison Type: Articles.Chris Cummins & Napoleon Katsos - 2010 - Journal of Semantics 27 (3):271-305.
    It has historically been assumed that comparative and superlative quantifiers can be semantically analysed in accordance with their core logical–mathematical properties. However, recent theoretical and experimental work has cast doubt on the validity of this assumption. Geurts & Nouwen have claimed that superlative quantifiers possess an additional modal component in their semantics that is absent from comparative quantifiers and that this accounts for the previously neglected differences in usage and interpretation between the two types of quantifier that they identify. Their (...)
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  3. added 2019-06-06
    A Local Normal Form Theorem For Infinitary Logic With Unary Quantifiers.H. Keisler & Wafik Lotfallah - 2005 - Mathematical Logic Quarterly 51 (2):137-144.
    We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.
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  4. added 2019-06-06
    On Quantification with a Finite Universe.Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (3):1055-1075.
    We consider a finite universe $\mathscr U$, second order quantifiers Q$_K$, where for each $\mathscr U$ this means quantifying over a family of n-place relations closed under permuting $\mathscr U$. We define some natural orders and shed some light on the classification problem of those quantifiers.
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  5. added 2019-06-06
    Progress Report on Generalized Functionality.Jonathan P. Seldin - 1979 - Annals of Pure and Applied Logic 17 (1):29.
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  6. added 2019-06-06
    Model-Theoretic Forcing in Logic with a Generalized Quantifier.Kim B. Bruce - 1978 - Annals of Pure and Applied Logic 13 (3):225.
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  7. added 2019-06-06
    Comments on Jaakko Hintikka's Paper “Quantifiers Vs. Quantification Theory”.Erik Stenius - 1976 - Dialectica 30 (1):67-88.
  8. added 2019-06-06
    Δ-Logics and Generalized Quantifiers.J. A. Makowsky - 1976 - Annals of Mathematical Logic 10 (2):155.
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  9. added 2019-06-06
    Forcing and Generalized Quantifiers.J. Krivine - 1973 - Annals of Pure and Applied Logic 5 (3):199.
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  10. added 2019-04-11
    Learnability and Semantic Universals.Shane Steinert-Threlkeld & Jakub Szymanik - forthcoming - Semantics and Pragmatics.
    One of the great successes of the application of generalized quantifiers to natural language has been the ability to formulate robust semantic universals. When such a universal is attested, the question arises as to the source of the universal. In this paper, we explore the hypothesis that many semantic universals arise because expressions satisfying the universal are easier to learn than those that do not. While the idea that learnability explains universals is not new, explicit accounts of learning that can (...)
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  11. added 2018-09-04
    Truth and Generalized Quantification.Bruno Whittle - 2019 - Australasian Journal of Philosophy 97 (2):340-353.
    Kripke [1975] gives a formal theory of truth based on Kleene's strong evaluation scheme. It is probably the most important and influential that has yet been given—at least since Tarski. However, it has been argued that this theory has a problem with generalized quantifiers such as All—that is, All ϕs are ψ—or Most. Specifically, it has been argued that such quantifiers preclude the existence of just the sort of language that Kripke aims to deliver—one that contains its own truth predicate. (...)
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  12. added 2018-04-03
    Some Properties of Iterated Languages.Shane Steinert-Threlkeld - 2016 - Journal of Logic, Language and Information 25 (2):191-213.
    A special kind of substitution on languages called iteration is presented and studied. These languages arise in the application of semantic automata to iterations of generalized quantifiers. We show that each of the star-free, regular, and deterministic context-free languages are closed under iteration and that it is decidable whether a given regular or determinstic context-free language is an iteration of two such languages. This result can be read as saying that the van Benthem/Keenan ‘Frege Boundary’ is decidable for large subclasses (...)
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  13. added 2018-02-16
    Comprehension of Simple Quantifiers: Empirical Evaluation of a Computational Model.Jakub Szymanik & Marcin Zajenkowski - 2010 - Cognitive Science 34 (3):521-532.
    We examine the verification of simple quantifiers in natural language from a computational model perspective. We refer to previous neuropsychological investigations of the same problem and suggest extending their experimental setting. Moreover, we give some direct empirical evidence linking computational complexity predictions with cognitive reality.<br>In the empirical study we compare time needed for understanding different types of quantifiers. We show that the computational distinction between quantifiers recognized by finite-automata and push-down automata is psychologically relevant. Our research improves upon hypothesis and (...)
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  14. added 2017-02-27
    Vagueness and Quantification.Andrea Iacona - 2016 - Journal of Philosophical Logic 45 (5):579-602.
    This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account provided (...)
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  15. added 2017-02-15
    Noun Phrases, Quantifiers, and Generic Names, EJ LOWE Frege and Russell Have Taught Us That Indefinite and Plural Noun Phrases in Natural Language Often Function as Quantifier Expressions Rather Than as Referring Expressions, Despite Possessing Many Syntactical Simi-Larities with Names. But It Can Be Shown That in Some of Their Most Im.Catherine Jl Talmage & Mark Mercer - 1991 - Philosophy 66 (257).
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  16. added 2017-02-15
    Generalized Self-Control of Effort and Stress.R. Eisenberger, Fa Masterson & F. Johnson - 1987 - Bulletin of the Psychonomic Society 25 (5):354-354.
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  17. added 2017-02-14
    Propositions or Choice Functions: What Do Quantifiers Quantify Over.Klaus Abels & Luiza Martí - forthcoming - Natural Language Semantics.
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  18. added 2017-02-14
    Symmetric Propositions and Logical Quantifiers.R. Gregory Taylor - 2008 - Journal of Philosophical Logic 37 (6):575-591.
    Symmetric propositions over domain D and signature Σ = are characterized following Zermelo, and a correlation of such propositions with logical type- quantifiers over D is described. Boolean algebras of symmetric propositions over D and Σ are shown to be isomorphic to algebras of logical type- quantifiers over D. This last result may provide empirical support for Tarski's claim that logical terms over fixed domain are all and only those invariant under domain permutations.
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  19. added 2017-02-14
    Temporal Prepositional Phrases with Quantifiers: Some Additions to Pratt and Francez (2001).A. V. Stechow - 2002 - Linguistics and Philosophy 25 (5-6):755-800.
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  20. added 2017-02-14
    Generalized Dynamic Quantitiers.M. H. van den Berg - 1996 - In J. van der Does & Van J. Eijck (eds.), Quantifiers, Logic, and Language. Stanford University.
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  21. added 2017-02-14
    Repartee, or a Reply to 'Negation, Conjunction and Quantifiers'.George Lakoff - 1970 - Foundations of Language 6 (3):389-422.
  22. added 2017-02-13
    Comparative Quantifiers and Negation: Implications for Scope Economy.N. Fleisher - 2015 - Journal of Semantics 32 (1):139-171.
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  23. added 2017-02-13
    Can the Persistence of Misconceptions Be Generalized and Explained?John Dewey - 1997 - Journal of Thought 32:69-76.
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  24. added 2017-02-13
    Rule Systems Are Not Dead: Existential Quantifiers Are Harder.Richard E. Grandy - 1993 - Behavioral and Brain Sciences 16 (2):351-352.
  25. added 2017-02-12
    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 (...)
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  26. added 2017-02-12
    Presuppositions of Quantified Sentences: Experimental Data. [REVIEW]Emmanuel Chemla - 2009 - Natural Language Semantics 17 (4):299-340.
    Some theories assume that sentences like (i) with a presupposition trigger in the scope of a quantifier carry an existential presupposition, as in (ii); others assume that they carry a universal presupposition, as in (iii). No student knows that he is lucky. Existential presupposition: At least one student is lucky.Universal presupposition: Every student is lucky. This work is an experimental investigation of this issue in French. Native speakers were recruited to evaluate the robustness of the inference from (i) to (iii). (...)
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  27. added 2017-02-12
    Pluractionality and Complex Quantifier Formation.Malte Zimmermann - 2003 - Natural Language Semantics 11 (3):249-287.
    This paper investigates the effects of (surface) DP-internal quantifying expressions on semantic interpretation. In particular, I investigate two syntactic constructions in which an adjective takes scope out of its embedding DP, thus raising an interesting question for strict compositionality. Regarding the first construction, I follow Larson (1999) and assume that the adjective incorporates into the determiner of its DP, forming a complex quantifier [D+A]. I present new evidence in favor of this analysis. Since Larson's semantic analysis of complex quantifiers [D+A] (...)
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  28. added 2017-02-12
    Hierarchies of Partially Ordered Connectives and Quantifiers.Michał Krynicki - 1993 - Mathematical Logic Quarterly 39 (1):287-294.
    Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form equation imagewhat is a strengthening of the Walkoe result. MSC: 03C80.
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  29. added 2017-02-10
    Relational Domains and the Interpretation of Reciprocals.Sivan Sabato & Yoad Winter - 2012 - Linguistics and Philosophy 35 (3):191-241.
    We argue that a comprehensive theory of reciprocals must rely on a general taxonomy of restrictions on the interpretation of relational expressions. Developing such a taxonomy, we propose a new principle for interpreting reciprocals that relies on the interpretation of the relation in their scope. This principle, the Maximal Interpretation Hypothesis (MIH), analyzes reciprocals as partial polyadic quantifiers. According to the MIH, the partial quantifier denoted by a reciprocal requires the relational expression REL in its scope to denote a maximal (...)
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  30. added 2017-02-08
    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.
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  31. added 2017-02-08
    At Least Some Determiners Aren't Determiners.Manfred Krifka - 1999 - In Ken Turner (ed.), The Semantics/Pragmatics Interface From Different Points of View. Elsevier. pp. 1--257.
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  32. added 2017-02-08
    The Semantics of Determiners.Edward L. Keenan - 1996 - In Shalom Lappin (ed.), The Handbook of Contemporary Semantic Theory. Blackwell. pp. 41--64.
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  33. added 2017-02-08
    Quantifiers and Context Dependence.Jason Stanley & Timothy Williamson - 1995 - Analysis 55 (4):291--295.
    Let DDQ be the thesis that definite descriptions are quantifiers. Philosophers often deny DDQ because they believe that quantifiers do not depend on context in certain ways, ways in which definite descriptions do depend on context. In this paper, we examine one such argument, which, if sound, would entail the negation of DDQ.We show that this argument fails, and draw some consequences from its failure.
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  34. added 2017-02-08
    Pronouns, Quantifiers, and Relative Clauses (II).Gareth Evans - 1985 - In Canadian Journal of Philosophy. Clarendon Press. pp. 153--175.
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  35. added 2017-02-08
    Pronouns, Quantifiers, and Relative Clauses (II): Appendix.Gareth Evans - 1977 - Canadian Journal of Philosophy 7 (4):777 - 797.
    It is occasionally tempting, after climbing a mountain, to use the elevation one has gained to dash up to the top of a connected peak which does not have sufficient interest to induce one to climb so high for its sake alone. It is in this spirit that I turn to Geach's Latin Prose theory of relative clauses. The matter itself is of no very great moment, and some new ground will have to be covered in dealing with Geach's arguments. (...)
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  36. added 2017-02-07
    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 (...)
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  37. added 2017-02-02
    Three Quantifier Sentences.Harvey Friedman - manuscript
    We give a complete proof that all 3 quantifier sentences in the primitive notation of set theory (Œ,=), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2 quantifier formulas with one free variable. There is a 5 quantifier sentence that is not decided in ZFC (see [Fr02]).
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  38. added 2017-02-02
    Domain Restriction and the Arguments of Quantificational Determiners.Anastasia Giannakidou - manuscript
    Classical generalized quantifier (GQ) theory posits that quantificational determiners (Q-dets) combine with a nominal argument of type et, a first order predicate, to form a GQ. In a recent paper, Matthewson (2001) challenges this position by arguing that the domain of a Q-det is not of type et, but e, an entity. In this paper, I defend the classical GQ view, and argue that the data that motivated Matthewson’s revision actually suggest that the domain set can, and indeed in certain (...)
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  39. added 2017-02-02
    It's Not Wise to Fool with Mother Nature.Stephen Crain - unknown
    Several recent papers propose that child and adult grammars differ in their underlying representations of universal quantification, e.g., “every” in English. These proposals attempt to explain children’s nonadult responses, in certain circumstances, in response to sentences that contain the universal quantifier. Blaming children’s nonadult behavior on their grammars is questionable, however, in view of the restrictiveness of the theory of Universal Grammar, which tightly constrains the hypothesis space children can navigate in the course of language development. The restrictiveness of the (...)
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  40. added 2017-02-02
    Quantifier Interpretation and Syllogistic Reasoning.Maxwell J. Roberts, Stephen E. Newstead & Richard A. Griggs - 2001 - Thinking and Reasoning 7 (2):173 – 204.
    Many researchers have suggested that premise interpretation errors can account, at least in part, for errors on categorical syllogisms. However, although it is possible to show that people make such errors in simple inference tasks, the evidence for them is far less clear when actual syllogisms are administered. Part of the problem is due to the lack of clear predictions for the solutions that would be expected when using modified quantifiers, assuming that correct inferences are made from them. This paper (...)
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  41. added 2017-02-02
    Supposition as Quantification Versus Supposition as Global Quantificational Effect.Terence Parsons - 1997 - Topoi 16 (1):41-63.
    This paper follows up a suggestion by Paul Vincent Spade that there were two Medieval theories of the modes of personal supposition. I suggest that early work by Sherwood and others was a study of quantifiers: their semantics and the effects of context on inferences that can be made from quantified terms. Later, in the hands of Burley and others, it changed into a study of something else, a study of what I call global quantificational effect. For example, although the (...)
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  42. added 2017-02-02
    Quantifier Meanings: A Study in the Dimensions of Semantic Competence.Steven Cushing - 1982 - Elsevier.
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  43. added 2017-02-01
    Noun Phrases, Quantifiers, and Generic Names.E. J. Lowe - 1991 - Philosophical Quarterly 41 (164):287-300.
  44. added 2017-02-01
    Die Logik der Anführung Und Quasianführung.U. Blau - 1988 - Erkenntnis 29 (2):227 - 268.
    Quine's metalogical 'quasiquotation' is formally added to classical first-Order logic; the resulting system lq is stronger and more natural than all former systems of quotational logic. Lq contains object-Variables ranging over the universe u and expression-Variables ranging over the set e of all expressions of lq; e is a subset of u. Object-Quantifiers are referential, Expression-Quantifiers are substitutional; only the latter ones bind into quasiquotations. Lq contains its own syntactic metatheory and arithmetics. Natural proofs of godel's and tarski's theorems are (...)
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  45. added 2017-01-31
    Quantifiers and Relative Clauses I.Gareth Evans - 1977 - Canadian Journal of Philosophy 7 (3):467-536.
    Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of (...)
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  46. added 2017-01-31
    Prefix Classes of Krom Formulas.Stål O. Aanderaa & Harry R. Lewis - 1973 - Journal of Symbolic Logic 38 (4):628-642.
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  47. added 2017-01-30
    Illusory Inferences with Quantifiers.Salvador Mascarenhas & Philipp Koralus - 2017 - Thinking and Reasoning 23 (1):33-48.
    The psychological study of reasoning with quantifiers has predominantly focused on inference patterns studied by Aristotle about two millennia ago. Modern logic has shown a wealth of inference patterns involving quantifiers that are far beyond the expressive power of Aristotelian syllogisms, and whose psychology should be explored. We bring to light a novel class of fallacious inference patterns, some of which are so attractive that they are tantamount to cognitive illusions. In tandem with recent insights from linguistics that quantifiers like (...)
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  48. added 2017-01-27
    Teória Kvantifikácie a Binárne Predikáty.Miloš Kosterec - 2012 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 19 (3):388-402.
    The paper deals with a problem in formal theory of quan tification. Firstly, by way of examples, I introduce important parts of the theory. Using type analysis, I present a problem which stems from inadequacy of a rule concerning semantic interpretation of sentences involving n-ary predicates and quantifiers. I propose four distinct principles for specific types of sentences. They are generalized into a general semantic rule, which is, finally, applied to particular examples.
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  49. added 2017-01-27
    Generalized Quantifiers, Exception Phrases, and Logicality.Lappin Shalom - 1996 - Journal of Semantics 13 (3).
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  50. added 2017-01-27
    1 993. Questions with Quantifiers.Gennaro Chierchia - 1992 - Natural Language Semantics 1 (1):81-234.
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