About this topic
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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  1. 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 (...)
  2. 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 (...)
  3. On Second-Order Logic.George S. Boolos - 1975 - Journal of Philosophy 72 (16):509-527.
  4. 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’.
  5. Absolute Generality. [REVIEW]P. Cobreros - 2008 - Teorema: International Journal of Philosophy 27 (2).
  6. Review: Agustin Rayo and Gabriel Uzquiano (Eds): Absolute Generality. [REVIEW]P. Dieveney - 2008 - Mind 117 (467):719-722.
  7. Art and the Absolute.Martin Donougho - 1987 - Review of Metaphysics 40 (3):567-569.
  8. Quantifier Variance and the Collapse Theorems.Cian Dorr - 2014 - The Monist 97 (4):503-570.
  9. The Model-Theoretic Argument Against Quantifying Over Everything.Iris Einheuser - 2010 - Dialectica 64 (2):237-246.
    A variant of Hilary Putnam's model-theoretic argument against metaphysical realism appears to show that our quantifiers do not determinately range over absolutely everything. This paper argues that some recent attempts to respond to the quantificational skeptic are unsuccessful and offers an alternative response: the key to answering the skeptic is not to refute her argument but to realize that the argument's setup prevents it from being convincing to those it is directed at.
  10. 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 (...)
  11. 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 (...)
  12. 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 (...)
  13. What Russell Should Have Said to Burali-Forti.Salvatore Florio & Graham Leach-Krouse - forthcoming - Review of Symbolic Logic.
    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 (...)
  14. 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 (...)
  15. 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 (...)
  16. Quantifiers.Michael Glanzberg - 2006 - In Ernest Lepore & Barry Smith (eds.), The Oxford Handbook of 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 (...)
  17. 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.
  18. Quantification and Realism.Michael Glanzberg - 2004 - Philosophy and Phenomenological Research 69 (3):541–572.
    This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...)
  19. A Contextual-Hierarchical Approach to Truth and the Liar Paradox.Michael Glanzberg - 2004 - Journal of Philosophical Logic 33 (1):27-88.
    This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad (...)
  20. The Liar in Context.Michael Glanzberg - 2001 - Philosophical Studies 103 (3):217 - 251.
    About twenty-five years ago, Charles Parsons published a paper that began by asking why we still discuss the Liar Paradox. Today, the question seems all the more apt. In the ensuing years we have seen not only Parsons’ work (1974), but seminal work of Saul Kripke (1975), and a huge number of other important papers. Too many to list. Surely, one of them must have solved it! In a way, most of them have. Most papers on the Liar Paradox offer (...)
  21. God as Absolute, Yet Related to All.Charles Hartshorne - 1947 - Review of Metaphysics 1 (1):24 - 51.
  22. 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 (...)
  23. 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 (...)
  24. Quest for the Absolute.Joseph W. Koterski - 1994 - Review of Metaphysics 48 (1):153-154.
  25. Everything, and Then Some.Stephan Krämer - 2017 - Mind 126 (502):499-528.
    On its intended interpretation, logical, mathematical and metaphysical discourse sometimes seems to involve absolutely unrestricted quantification. Yet our standard semantic theories do not allow for interpretations of a language as expressing absolute generality. A prominent strategy for defending absolute generality, influentially proposed by Timothy Williamson in his paper ‘Everything’, avails itself of a hierarchy of quantifiers of ever increasing orders to develop non-standard semantic theories that do provide for such interpretations. However, as emphasized by Øystein Linnebo and Agustín Rayo, there (...)
  26. 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.
  27. Sets, Properties, and Unrestricted Quantification.Øystein Linnebo - 2006 - In Gabriel Uzquiano & Agustin Rayo (eds.), Absolute Generality. Oxford University Press.
    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 (...)
  28. 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.
  29. 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 (...)
  30. 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 (...)
  31. There's a Rule for Everything.Vann McGee - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 179--202.
  32. Plural Predication.Thomas J. McKay - 2006 - 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 (...)
  33. 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 (...)
  34. The Problem of Absolute Universality.Charles Parsons - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 203--19.
  35. Review of Absolute Generality. [REVIEW]G. Priest - forthcoming - Notre Dame Philosophical Reviews.
  36. Review of Agustn Rayo, Gabriel Uzquiano (Eds.), Absolute Generality[REVIEW]Graham Priest - 2007 - Notre Dame Philosophical Reviews 2007 (9).
  37. The Absolute Being. [REVIEW]D. G. R. - 1959 - Review of Metaphysics 12 (4):667-667.
  38. 4. Absolute Generality Reconsidered.Agustín Rayo - 2012 - Oxford Studies in Metaphysics 7:93.
  39. Beyond Plurals.Agustin Rayo - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. pp. 220--54.
    I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higherorder quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of (...)
  40. When Does ‘Everything’ Mean Everything?Agustín Rayo - 2003 - Analysis 63 (2):100–106.
  41. Absolute Generality.Agustín Rayo & Gabriel Uzquiano (eds.) - 2006 - Oxford University Press.
    The problem of absolute generality has attracted much attention in recent philosophy. Agustin Rayo and Gabriel Uzquiano have assembled a distinguished team of contributors to write new essays on the topic. They investigate the question of whether it is possible to attain absolute generality in thought and language and the ramifications of this question in the philosophy of logic and mathematics.
  42. Introduction.Agustin Rayo & Gabriel Uzquiano - 2006 - In Agustin Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press.
    Whether or not we achieve absolute generality in philosophical inquiry, most philosophers would agree that ordinary inquiry is rarely, if ever, absolutely general. Even if the quantifiers involved in an ordinary assertion are not explicitly restricted, we generally take the assertion’s domain of discourse to be implicitly restricted by context.1 Suppose someone asserts (2) while waiting for a plane to take off.
  43. Toward a Theory of Second-Order Consequence.Agustin Rayo & Gabriel Uzquiano - 1999 - Notre Dame Journal of Formal Logic 40 (3):315-325.
    There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.
  44. A Completeness Theorem for Unrestricted First-Order Languages.Agustín Rayo & Timothy Williamson - 2003 - In Jc Beall (ed.), Liars and Heaps. Oxford University Press.
    Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as (...)
  45. Indefinite Divisibility.Jeffrey Sanford Russell - 2015 - Inquiry : An Interdisciplinary Journal of Philosophy (3):1-25.
    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 (...)
  46. 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.
  47. All Sets Great and Small: And I Do Mean ALL.Stewart Shapiro - 2003 - Philosophical Perspectives 17 (1):467–490.
    A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...)
  48. Review of A. Rayo and G. Uzquiano (Eds.), Absolute Generality[REVIEW]Peter Smith - 2008 - Bulletin of Symbolic Logic 14 (3).
  49. A Note on Truth and Security for Modal and Quantificational Paradoxes.Paul Vincent Spade - 1976 - Philosophical Studies 29 (3):211 - 214.
  50. 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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