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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. Eduardo Alejandro Barrio (2007). Modelos, Autoaplicación Y Máxima Generalidad (Models, Self-Application and Absolute Generality). 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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  2. Arvid Båve (2011). How To Precisify Quantifiers. 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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  3. George S. Boolos (1975). On Second-Order Logic. Journal of Philosophy 72 (16):509-527.
  4. Tim Button (2010). Dadaism: Restrictivism as Militant Quietism. 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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  5. Iris Einheuser (2010). The Model-Theoretic Argument Against Quantifying Over Everything. 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.
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  6. Kit Fine (2006). Relatively Unrestricted Quantification. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 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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  7. Salvatore Florio (2014). Unrestricted Quantification. 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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  8. S. Gandon (2013). Variable, Structure, and Restricted Generality. 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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  9. Michael Glanzberg (2006). Context and Unrestricted Quantification. In A. Rayo & G. Uzquiano (eds.), Absolute Generality. Oxford University Press. 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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  10. Michael Glanzberg (2006). Quantifiers. In Ernest Lepore & Barry Smith (eds.), The Oxford Handbook of Philosophy of Language. Oxford University Press. 794--821.
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  11. Michael Glanzberg (2004). A Contextual-Hierarchical Approach to Truth and the Liar Paradox. 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 (...)
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  12. Michael Glanzberg (2004). Quantification and Realism. 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 (...)
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  13. Michael Glanzberg (2001). The Liar in Context. 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 (...)
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  14. Santos Gonçalo (forthcoming). Numbers and Everything. Philosophia Mathematica.
    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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  15. Shaughan Lavine (2006). Something About Everything: Universal Quantification in the Universal Sense of Universal Quantification. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 98--148.
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  16. Øystein Linnebo (2006). Sets, Properties, and Unrestricted Quantification. 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 (...)
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  17. Øystein Linnebo & Agustín Rayo (2012). Hierarchies Ontological and Ideological. 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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  18. Laureano Luna (2013). Indefinite Extensibility in Natural Language. 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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  19. Andrew McCarthy & Ian Phillips (2006). No New Argument Against the Existence Requirement. Analysis 66 (289):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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  20. Vann McGee (2006). There's a Rule for Everything. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 179--202.
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  21. Thomas J. McKay (2006). Plural Predication. 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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  22. T. Parent, Modal Realism and the Meaning of 'Exist'.
    Here I first raise an argument purporting to show that Lewis’ Modal Realism ends up being completely trivial. But although I reject this line, the argument reveals how difficult it is to interpret Lewis’ thesis that possibilia “exist.” Four natural interpretations are considered, yet upon reflection, none appear entirely adequate. In particular, under the three different “concretist” interpretations of ‘exist’, Modal Realism looks insufficient for genuine ontological commitment. Whereas under the “multiverse” interpretation, Modal Realism ends up being a theory of (...)
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  23. Charles Parsons (2006). The Problem of Absolute Universality. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 203--19.
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  24. Graham Priest (2007). Review of Agustn Rayo, Gabriel Uzquiano (Eds.), Absolute Generality. [REVIEW] Notre Dame Philosophical Reviews 2007 (9).
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  25. Agustín Rayo (2012). 4. Absolute Generality Reconsidered. Oxford Studies in Metaphysics 7:93.
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  26. Agustin Rayo (2006). Beyond Plurals. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 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 (...)
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  27. Agustín Rayo (2003). When Does ‘Everything’ Mean Everything? Analysis 63 (278):100–106.
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  28. Agustin Rayo (1999). Toward a Theory of Second-Order Consequence. 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.
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  29. Agustín Rayo & Gabriel Uzquiano (eds.) (2006). Absolute Generality. 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.
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  30. Agustin Rayo & Gabriel Uzquiano (2006). Introduction. 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.
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  31. Agustín Rayo & Timothy Williamson (2003). A Completeness Theorem for Unrestricted First-Order Languages. 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 (...)
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  32. Gonçalo Santos (2013). Numbers and Everything. 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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  33. Stewart Shapiro (2003). All Sets Great and Small: And I Do Mean ALL. 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 (...)
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  34. Peter Smith (2008). Review of A. Rayo and G. Uzquiano (Eds.), Absolute Generality. [REVIEW] Bulletin of Symbolic Logic 14 (3).
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  35. Paul Vincent Spade (1976). A Note on Truth and Security for Modal and Quantificational Paradoxes. Philosophical Studies 29 (3):211 - 214.
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  36. Gabriel Uzquiano (forthcoming). Varieties of Indefinite Extensibility. Notre Dame Journal of Formal Logic.
    We look at two recent accounts of the indefinite extensibility of set, and compare them with a linguistic model of the indefinite extensibility. I suggest the linguistic model has much to recommend over extant accounts of the indefinite extensibility of set, and we defend it against three prima facie objections.
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  37. Gabriel Uzquiano (2006). The Price of Universality. Philosophical Studies 129 (1):137 - 169.
    I present a puzzle for absolutely unrestricted quantification. One important advantage of absolutely unrestricted quantification is that it allows us to entertain perfectly general theories. Whereas most of our theories restrict attention to one or another parcel of reality, other theories are genuinely comprehensive taking absolutely all objects into their domain. The puzzle arises when we notice that absolutely unrestricted theories sometimes impose incompatible constraints on the size of the universe.
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  38. Gabriel Uzquiano (2006). Unrestricted Unrestricted Quantification: The Cardinal Problem of Absolute Generality. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 305--32.
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  39. Gabriel Uzquiano & Agustin Rayo (eds.) (2006). Absolute Generality. 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.
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  40. Alan Weir (2006). Is It Too Much to Ask, to Ask for Everything. In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Oxford University Press. 333--68.
    Most of the time our quantifications generalise over a restricted domain. Thus in the last sentence, ‘most of the time’ is arguably not a generalisation over all times in the history of the universe but is restricted to a sub-group of times, those at which humans exist and utter quantified phrases and sentences, say. Indeed the example illustrates the point that quantificational phrases often carry an explicit restriction with them: ‘some people’, ‘all dogs’. Even then, context usually restricts to a (...)
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  41. Timothy Williamson (2007). Absolute Identity and Absolute Generality. In Gabriel Uzquiano & Agustin Rayo (eds.), Absolute Generality. OUP. 369--89.
    In conversations between native speakers, words such as ‘same’ and ‘identical’ do not usually cause much difficulty. We take it for granted that others use them with the same sense as we do. If it is unclear whether numerical or qualitative identity is intended, a brief gloss such as ‘one thing not two’ for the former or ‘exactly alike’ for the latter removes the unclarity. In this paper, numerical identity is intended. A particularly conscientious and logically aware speaker might explain (...)
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  42. Takashi Yagisawa (2012). Unrestricted Quantification and Reality: Reply to Kim. [REVIEW] 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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