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Summary

Semantic theories of natural and formal languages often appeal to the notion of domain of quantification in specifying the interpretations and truth conditions of sentences of the object language. In natural language, quantificational expressions, such as ‘every’, ‘some’, ‘most’, are routinely evaluated with respect to a salient and typically restricted range of entities  (e.g. an ordinary utterance of ‘she knew everything’ can be true despite the fact that the person referred to is not omniscient). In formal languages, standard model-theoretic semantics specify the interpretations of the object language by fixing a domain of quantification and assigning semantic values constructed from that domain to non-logical expressions of the language. A question that has received much attention of late is whether there is an unrestricted domain of quantification, a domain containing absolutely everything there is. Is there a discourse or inquiry that has absolute generality? Prima facie examples of sentences that quantify over an all-inclusive domain abound (e.g. 'everything is self-identical' or ‘the empty set contains no element’).  However, a number of philosophical arguments have been offered in support of the view that absolutely unrestricted quantification cannot be achieved. The growing body of literature on the issue has ramifications for semantics, metaphysics, and logic.

Key works Williamson 2003 contains an influential discussion and defense of absolute generality. Many of the main contributions to date are in found in Rayo & Uzquiano 2006.
Introductions Rayo and Uzquiano's Introduction to Rayo & Uzquiano 2006 and Florio 2014
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67 found
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  1. Modal Realism and the Meaning of 'Exist'.T. Parent - manuscript
    Here I first raise an argument purporting to show that Lewis’ Modal Realism ends up being entirely trivial. But although I reject this line, the argument reveals how difficult it is to interpret Lewis’ thesis that possibilia “exist.” Five natural interpretations are considered, yet upon reflection, none appear entirely adequate. On the three different “concretist” interpretations of ‘exist’, Modal Realism looks insufficient for genuine ontological commitment. Whereas, on the “multiverse” interpretation, Modal Realism acknowledges physical possibilities only--and worse, (assuming either axiom (...)
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  2. Review of Absolute Generality. [REVIEW]G. Priest - forthcoming - Notre Dame Philosophical Reviews.
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  3. Necessitism and Unrestricted Quantification.Violeta Conde - 2023 - Teorema: International Journal of Philosophy 42 (2):7 - 24.
    As Williamson puts it, ‘necessitism’ is the metaphysical view that claims that “necessarily everything is necessarily something”. As that claim involves modal unrestricted quantification, the necessitist must accept it as a part of an intelligible discourse. Here, I present one of the main objections that have been presented against the intelligibility of unrestricted quantification: the objection based on the so-called All-in-One Principle. I then propose possible strategies that the necessitist could adopt to shield themselves from the objection.
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  4. On Type Distinctions and Expressivity.Salvatore Florio - 2023 - Proceedings of the Aristotelian Society 123 (2):150-172.
    Quine maintained that philosophical and scientific theorizing should be conducted in an untyped language, which has just one style of variables and quantifiers. By contrast, typed languages, such as those advocated by Frege and Russell, include multiple styles of variables and matching kinds of quantification. Which form should our theories take? In this article, I argue that expressivity does not favour typed languages over untyped ones.
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  5. Two conceptions of absolute generality.Salvatore Florio & Nicholas K. Jones - 2023 - Philosophical Studies 180 (5-6):1601-1621.
    What is absolutely unrestricted quantification? We distinguish two theoretical roles and identify two conceptions of absolute generality: maximally strong generality and maximally inclusive generality. We also distinguish two corresponding kinds of absolute domain. A maximally strong domain contains every potential counterexample to a generalisation. A maximally inclusive domain is such that no domain extends it. We argue that both conceptions of absolute generality are legitimate and investigate the relations between them. Although these conceptions coincide in standard settings, we show how (...)
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  6. Unrestricted quantification and extraordinary context dependence?Michael Glanzberg - 2023 - Philosophical Studies 180 (5):1491-1512.
    This paper revisits a challenge for contextualist approaches to paradoxes such as the Liar paradox and Russell’s paradox. Contextualists argue that these paradoxes are to be resolved by appeal to context dependence. This can offer some nice and effective ways to avoid paradox. But there is a problem. Context dependence is, at least to begin with, a phenomenon in natural language. Is there really such context dependence as the solutions to paradoxes require, and is it really just a familiar linguistic (...)
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  7. Strengthening the Russellian argument against absolutely unrestricted quantification.Laureano Luna - 2022 - Synthese 200 (3):1-13.
    The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are (...)
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  8. Unrestricted quantification and ranges of significance.Thomas Schindler - 2022 - Philosophical Studies 180 (5):1579-1600.
    Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, (...)
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  9. Unrestricted Quantification and the Structure of Type Theory.Salvatore Florio & Nicholas K. Jones - 2021 - Philosophy and Phenomenological Research 102 (1):44-64.
    Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and the structure of (...)
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  10. The Many and the One: A Philosophical Study of Plural Logic.Salvatore Florio & Øystein Linnebo - 2021 - Oxford, England: Oxford University Press.
    Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...)
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  11. Unrestricted quantification and extraordinary context dependence?Michael Glanzberg - 2021 - Philosophical Studies 180 (5-6):1-22.
    This paper revisits a challenge for contextualist approaches to paradoxes such as the Liar paradox and Russell’s paradox. Contextualists argue that these paradoxes are to be resolved by appeal to context dependence. This can offer some nice and effective ways to avoid paradox. But there is a problem. Context dependence is, at least to begin with, a phenomenon in natural language. Is there really such context dependence as the solutions to paradoxes require, and is it really just a familiar linguistic (...)
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  12. Everything, More or Less: A Defence of Generality Relativism.James Studd - 2019 - Oxford, England: Oxford University Press.
    Almost no systematic theorizing is generality-free. Scientists test general hypotheses; set theorists prove theorems about every set; metaphysicians espouse theses about all things of any kind. But do we ever succeed in theorizing about absolutely everything? Not according to generality relativism, which J.P. Studd defends in this book.
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  13. Quantification and Paradox.Edward Ferrier - 2018 - Dissertation, University of Massachusetts Amherst
    I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...)
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  14. What Russell Should Have Said to Burali–Forti.Salvatore Florio & Graham Leach-Krouse - 2017 - Review of Symbolic Logic 10 (4):682-718.
    The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...)
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  15. 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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  16. Can the Cumulative Hierarchy Be Categorically Characterized?Luca Incurvati - 2016 - Logique Et Analyse 59 (236):367-387.
    Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist (...)
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  17. Indefinite Divisibility.Jeffrey Sanford Russell - 2016 - Inquiry: An Interdisciplinary Journal of Philosophy 59 (3):239-263.
    Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...)
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  18. Reply to Bacon, Hawthorne and Uzquiano.Timothy Williamson - 2016 - Canadian Journal of Philosophy 46 (4-5):542-547.
  19. Reply to Fine.Timothy Williamson - 2016 - Canadian Journal of Philosophy 46 (4-5):571-583.
  20. Introduction.Alessandro Torza - 2015 - In Quantifiers, Quantifiers, and Quantifiers. Themes in Logic, Metaphysics and Language. (Synthese Library vol 373). Springer. pp. 1-15.
    This introductory chapter provides a summary of the contributions to the volume, as well as some critical remarks.
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  21. Quantifiers, Quantifiers, and Quantifiers. Themes in Logic, Metaphysics, and Language. (Synthese Library vol. 373).Alessandro Torza (ed.) - 2015 - Springer.
    This volume covers a wide range of topics that fall under the 'philosophy of quantifiers', a philosophy that spans across multiple areas such as logic, metaphysics, epistemology, and even the history of philosophy. It discusses the import of quantifier variance in the model theory of mathematics. It advances an argument for the uniqueness of quantifier meaning in terms of Evert Beth’s notion of implicit definition, and clarifies the oldest explicit formulation of quantifier variance: the one proposed by Rudolf Carnap. -/- (...)
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  22. Varieties of Indefinite Extensibility.Gabriel Uzquiano - 2015 - Notre Dame Journal of Formal Logic 56 (1):147-166.
    We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.
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  23. Quantifier Variance and the Collapse Theorems.Cian Dorr - 2014 - The Monist 97 (4):503-570.
  24. Untyped Pluralism.Salvatore Florio - 2014 - Mind 123 (490):317-337.
    In the semantic debate about plurals, pluralism is the view that a plural term denotes some things in the domain of quantification and a plural predicate denotes a plural property, i.e. a property that can be instantiated by many things jointly. According to a particular version of this view, untyped pluralism, there is no type distinction between objects and properties. In this article, I argue against untyped pluralism by showing that it is subject to a variant of a Russell-style argument (...)
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  25. Unrestricted Quantification.Salvatore Florio - 2014 - Philosophy Compass 9 (7):441-454.
    Semantic interpretations of both natural and formal languages are usually taken to involve the specification of a domain of entities with respect to which the sentences of the language are to be evaluated. A question that has received much attention of late is whether there is unrestricted quantification, quantification over a domain comprising absolutely everything there is. Is there a discourse or inquiry that has absolute generality? After framing the debate, this article provides an overview of the main arguments for (...)
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  26. Set Theory, Type Theory, and Absolute Generality.Salvatore Florio & Stewart Shapiro - 2014 - Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither (...)
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  27. Variable, Structure, and Restricted Generality.S. Gandon - 2013 - Philosophia Mathematica 21 (2):200-219.
    From 1905–1908 onward, Russell thought that his new ‘substitutional theory’ provided him with the right framework to resolve the set-theoretic paradoxes. Even if he did not finally retain this resolution, the substitutional strategy was instrumental in the development of his thought. The aim of this paper is not historical, however. It is to show that Russell's substitutional insight can shed new light on current issues in philosophy of mathematics. After having briefly expounded Russell's key notion of a ‘structured variable’, I (...)
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  28. Indefinite Extensibility in Natural Language.Laureano Luna - 2013 - The Monist 96 (2):295-308.
    The Monist’s call for papers for this issue ended: “if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible”. We use the Grelling-Nelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping (...)
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  29. Numbers and Everything.Gonçalo Santos - 2013 - Philosophia Mathematica 21 (3):297-308.
    I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.
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  30. Modal Tense and the Absolutely Unrestricted Quantifier.Seahwa Kim - 2012 - Acta Analytica 27 (1):73-76.
    In this paper, I examine Takashi Yagisawa’s response to van Inwagen’s ontic objection against David Lewis. Van Inwagen criticizes Lewis’s commitment to the absolutely unrestricted sense of ‘there is,’ and Yagisawa claims that by adopting modal tenses he avoids commitment to absolutely unrestricted quantification. I argue that Yagisawa faces a problem parallel to the one Lewis faces. Although Yagisawa officially rejects the absolutely unrestricted sense of a quantifying expression, he is still committed to the absolutely unrestricted sense of ‘is a (...)
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  31. Hierarchies Ontological and Ideological.Øystein Linnebo & Agustín Rayo - 2012 - Mind 121 (482):269 - 308.
    Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain superfluous restrictions are removed'. The aim of this paper is to develop a clearer understanding of Gödel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.
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  32. 4. Absolute Generality Reconsidered.Agustín Rayo - 2012 - Oxford Studies in Metaphysics 7:93.
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  33. All Things Must Pass Away.Joshua Spencer - 2012 - Oxford Studies in Metaphysics 7:67.
    Are there any things that are such that any things whatsoever are among them. I argue that there are not. My thesis follows from these three premises: (1) There are two or more things; (2) for any things, there is a unique thing that corresponds to those things; (3) for any two or more things, there are fewer of them than there are pluralities of them.
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  34. Unrestricted Quantification and Reality: Reply to Kim. [REVIEW]Takashi Yagisawa - 2012 - Acta Analytica 27 (1):77-79.
    In my book, Worlds and Individuals, Possible and Otherwise , I use the novel idea of modal tense to respond to a number of arguments against modal realism. Peter van Inwagen’s million-carat-diamond objection is one of them. It targets the version of modal realism by David Lewis and exploits the fact that Lewis accepts absolutely unrestricted quantification. The crux of my response is to use modal tense to neutralize absolutely unrestricted quantification. Seahwa Kim says that even when equipped with modal (...)
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  35. How To Precisify Quantifiers.Arvid Båve - 2011 - Journal of Philosophical Logic 40 (1):103-111.
    I here argue that Ted Sider's indeterminacy argument against vagueness in quantifiers fails. Sider claims that vagueness entails precisifications, but holds that precisifications of quantifiers cannot be coherently described: they will either deliver the wrong logical form to quantified sentences, or involve a presupposition that contradicts the claim that the quantifier is vague. Assuming (as does Sider) that the “connectedness” of objects can be precisely defined, I present a counter-example to Sider's contention, consisting of a partial, implicit definition of the (...)
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  36. Dadaism: Restrictivism as Militant Quietism.Tim Button - 2010 - Proceedings of the Aristotelian Society 110 (3pt3):387-398.
    Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’.
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  37. Absolute Generality. [REVIEW]P. Cobreros - 2008 - Teorema: International Journal of Philosophy 27 (2).
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  38. Review: Agustin Rayo and Gabriel Uzquiano (eds): Absolute Generality. [REVIEW]Patrick Dieveney - 2008 - Mind 117 (467):719-722.
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  39. Review of A. Rayo and G. Uzquiano (eds.), Absolute Generality[REVIEW]Peter Smith - 2008 - Bulletin of Symbolic Logic 14 (3):398-401.
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  40. Modelos, autoaplicación Y máxima generalidad (models, self-application and absolute generality).Eduardo Alejandro Barrio - 2007 - Theoria 22 (2):133-152.
    En este artículo, me propongo exponer algunas dificultades relacionadas con la posibilidad de que la Teoría de Modelos pueda constituirse en una Teoría General de la Interpretación. Específicamente la idea que sostengo es que lo que nos muestra la Paradoja de Orayen es que las interpretaciones no pueden ser ni conjuntos ni objetos. Por eso, una elucidación del concepto intuitivo de interpretación, que apele a este tipo de entidades, está condenada al fracaso. De manera secundaria, muestro que no hay algún (...)
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  41. Review of agustn Rayo, Gabriel Uzquiano (eds.), Absolute Generality[REVIEW]Graham Priest - 2007 - Notre Dame Philosophical Reviews 2007 (9).
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  42. Relatively Unrestricted Quantification.Kit Fine - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 20-44.
    There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned—one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem–Lowenheim Theorem appears to show that an understanding of quanti- fication over absolutely (...)
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  43. Context and unrestricted quantification.Michael Glanzberg - 2006 - In A. Rayo & G. Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 45--74.
    Quantification is haunted by the specter of paradoxes. Since Russell, it has been a persistent idea that the paradoxes show what might have appeared to be absolutely unrestricted quantification to be somehow restricted. In the contemporary literature, this theme is taken up by Dummett (1973, 1993) and Parsons (1974a,b). Parsons, in particular, argues that both the Liar and Russell’s paradoxes are to be resolved by construing apparently absolutely unrestricted quantifiers as appropriately restricted.
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  44. Quantifiers.Michael Glanzberg - 2006 - In Ernest Lepore & Barry Smith (eds.), The Oxford Handbook to the Philosophy of Language. Oxford University Press. pp. 794--821.
    The study of quantification in natural language has made remarkable progress. Quantification in natural language has been investigated extensively by philosophers, logicians, and linguists. The result has been an elegant and far-reaching theory. This article presents a survey of some of the important components of this theory. The first section presents the core of the theory of generalized quantifiers. This theory explores the range of expressions of generality in natural language, and studies some of their logical properties. The second section (...)
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  45. Something About Everything: Universal Quantification in the Universal Sense of Universal Quantification.Shaughan Lavine - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 98--148.
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  46. Sets, properties, and unrestricted quantification.Øystein Linnebo - 2006 - In Gabriel Uzquiano & Agustin Rayo (eds.), Absolute Generality. Oxford University Press. pp. 149--178.
    Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical things or all things relevant to some particular utterance or discourse but over absolutely everything there is. Prima facie, unrestricted quantification seems to be perfectly coherent. For such quantification appears to be involved in a variety of claims that all normal human beings are capable of understanding. For instance, some basic logical and mathematical truths appear to involve unrestricted quantification, such as the truth that (...)
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  47. No new argument against the existence requirement.Andrew McCarthy & Ian Phillips - 2006 - Analysis 66 (1):39–44.
    Yagisawa (2005) considers two old arguments against the existence requirement. Both arguments are significantly less appealing than Yagisawa suggests. In particular, the second argument, first given by Kaplan (1989: 498), simply assumes that existence is contingent (§1). Yagisawa’s ‘new’ argument shares this weakness. It also faces a dilemma. Yagisawa must either treat ‘at @’ as a sentential operator occupying the same grammatical position as ‘∼’ or as supplying an extra argument place. In the former case, Yagisawa’s argument faces precisely the (...)
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  48. There's a Rule for Everything.Vann McGee - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 179--202.
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  49. Plural predication.Thomas J. McKay - 2006 - New York: Oxford University Press.
    Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...)
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  50. The Problem of Absolute Universality.Charles Parsons - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 203--19.
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