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Summary

Several natural languages contain a grammatical distinction between singular and plural expressions. The distinction also concerns quantification. Alongside singular quantifiers (‘something’, ‘everything’), we can find plural quantifiers (‘some things’, ‘all things’). Plural logic is a formal system that regiments plural quantification as a sui generis form of quantification, distinct from singular quantification. When treated as sui generis, plural quantification and plural logic have been thought to be philosophically significant and have found a number of applications especially in philosophy of mathematics and metaphysics. For the most part, these applications can be traced back to two of the virtues that plural quantification is alleged to have, i.e. ontological innocence and expressive power. On the one hand, it is assumed that plural quantifiers range in a special plural way over the entities in the range of the singular quantifiers and not over special plural entities (e.g. sets, collections, or any kind of set-like entities). Thus they do not incur ontological commitments exceeding those of the singular quantifiers. On the other hand, as shown by Boolos, plural quantification can interpret monadic second-order logic. As a result, plural quantification has been thought to provide more expressive power than singular quantification as captured by first-order logic. While the growing philosophical literature focuses primarily on the logical and foundational features of plural quantification, research in natural language semantics targets the meaning-theoretic and compositional features of plurals, often from an algebraic perspective. These two strands of research appear largely unreconciled.

Key works

Classic papers are Boolos 1984 and Boolos 1985. Yi 1999Oliver & Smiley 2001, and Rayo 2002 argue forcefully for the significance of plural quantification. The question of logicality is addressed in Linnebo 2003. Full treatments of plural logic are given in Yi 2005 and Yi 2006McKay 2006, and Oliver & Smiley 2013. Influential contributions in linguistics include Link 1983 and Link 1998, Schein 1993, Schwarzschild 1996, and Landman 2000.

Introductions  Rayo 2007 and Linnebo 2009 provide overviews of the philosophical literature. For a linguistically oriented introduction, see Schein 2006
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  1. Hanoch Ben-Yami (2013). Higher‐Level Plurals Versus Articulated Reference, and an Elaboration of Salva Veritate. Dialectica 67 (1):81-102.
    In recent literature on plurals the claim has often been made that the move from singular to plural expressions can be iterated, generating what are occasionally called higher-level plurals or superplurals, often correlated with superplural predicates. I argue that the idea that the singular-to-plural move can be iterated is questionable. I then show that the examples and arguments intended to establish that some expressions of natural language are in some sense higher-level plurals fail. Next, I argue that these and some (...)
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  2. Hanoch Ben-Yami (2012). Response to Westerstahl. Logique Et Analyse 55 (217):47-55.
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  3. Hanoch Ben-Yami (2009). Generalized Quantifiers, and Beyond. Logique Et Analyse (208):309-326.
    I show that the contemporary dominant analysis of natural language quantifiers that are one-place determiners by means of binary generalized quantifiers has failed to explain why they are, according to it, conservative. I then present an alternative, Geachean analysis, according to which common nouns in the grammatical subject position are plural logical subject-terms, and show how it does explain that fact and other features of natural language quantification.
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  4. Hanoch Ben-yami (2009). Plural Quantification Logic: A Critical Appraisal. Review of Symbolic Logic 2 (1):208-232.
    I first show that most authors who developed Plural Quantification Logic (PQL) argued it could capture various features of natural language better than can other logic systems. I then show that it fails to do so: it radically departs from natural language in two of its essential features; namely, in distinguishing plural from singular quantification and in its use of an relation. Next, I sketch a different approach that is more adequate than PQL for capturing plural aspects of natural language (...)
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  5. Alan Berger (2002). A Formal Semantics for Plural Quantification, Intersentential Binding and Anaphoric Pronouns as Rigid Designators. Noûs 36 (1):50–74.
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  6. Boccuni (2011). On the Consistency of a Plural Theory of Frege’s Grundgesetze. Studia Logica 97 (3):329-345.
    PG (Plural Grundgesetze) is a predicative monadic second-order system which is aimed to derive second-order Peano arithmetic. It exploits the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. In this paper, a model-theoretical consistency proof for the system PG is provided.
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  7. Francesca Boccuni (2013). Plural Logicism. Erkenntnis 78 (5):1051-1067.
    PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically (...)
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  8. Francesca Boccuni (2010). Plural grundgesetze. Studia Logica 96 (2):315-330.
    PG ( Plural Grundgesetze ) is a predicative monadic second-order system which exploits the notion of plural quantification and a few Fregean devices, among which a formulation of the infamous Basic Law V. It is shown that second-order Peano arithmetic can be derived in PG. I also investigate the philosophical issue of predicativism connected to PG. In particular, as predicativism about concepts seems rather un-Fregean, I analyse whether there is a way to make predicativism compatible with Frege’s logicism.
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  9. Einar Duenger Bohn (2012). Monism, Emergence, and Plural Logic. Erkenntnis 76 (2):211-223.
    In this paper I argue that we need to take irreducibly plural logic more seriously in metaphysical debates due to the fact that the verdict of many metaphysical debates hangs on it. I give two examples. The main example I focus on is the debate recently revived by Jonathan Schaffer over the fundamental cardinality of the world. I show how the three main arguments provided by Schaffer are unsound in virtue of an employment of plural logic. The second example I (...)
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  10. George Boolos (1998). Reply to Charles Parsons' ``Sets and Classes''. In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. 30-36.
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  11. George Boolos (1985). Nominalist Platonism. Philosophical Review 94 (3):327-344.
  12. George Boolos (1984). To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables). Journal of Philosophy 81 (8):430-449.
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  13. Phillip Bricker (1989). Quantified Modal Logic and the Plural De Re. Midwest Studies in Philosophy 14 (1):372-394.
    Modal sentences of the form "every F might be G" and "some F must be G" have a threefold ambiguity. in addition to the familiar readings "de dicto" and "de re", there is a third reading on which they are examples of the "plural de re": they attribute a modal property to the F's plurally in a way that cannot in general be reduced to an attribution of modal properties to the individual F's. The plural "de re" readings of modal (...)
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  14. John P. Burgess (2004). E Pluribus Unum: Plural Logic and Set Theory. Philosophia Mathematica 12 (3):193-221.
    A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.
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  15. J. R. Cameron (1999). Plural Reference. Ratio 12 (2):128–147.
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  16. Massimiliano Carrara, Alessandra Arapinis & Friederike Moltmann (eds.) (forthcoming). Unity and Plurality. Philosophy, Logic, and Semantics. Oxford University Press.
    This volume brings together new work on the logic and ontology of plurality and a range of recent articles exploring novel applications to natural language semantics. The contributions in this volume in particular investigate and extend new perspectives presented by plural logic and non-standard mereology and explore their applications to a range of natural language phenomena. Contributions by P. Aquaviva, A. Arapinis, M. Carrara, P. McKay, F. Moltmann, O. Linnebo, A. Oliver and T. Smiley, T. Scaltsas, P. Simons, and B.-Y. (...)
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  17. Massimiliano Carrara & Enrico Martino (2011). On the Infinite in Mereology with Plural Quantification. Review of Symbolic Logic 4 (1):54-62.
    In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification (...)
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  18. Massimiliano Carrara & Enrico Martino (2009). On the Ontological Commitment of Mereology. Review of Symbolic Logic 2 (1):164-174.
    In Parts of Classes (1991) and Mathematics Is Megethology (1993) David Lewis defends both the innocence of plural quantification and of mereology. However, he himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out as a whole. In the case of plural quantification . Instead, in the mereological case: (Lewis, 1991, p. 87). The aim of the paper is to (...)
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  19. Helen Morris Cartwright (1993). On Plural Reference and Elementary Set Theory. Synthese 96 (2):201 - 254.
    The view that plural reference is reference to a set is examined in light of George Boolos's treatment of second-order quantification as plural quantification in English. I argue that monadic second-order logic does not, in Boolos's treatment, reflect the behavior of plural quantifiers under negation and claim that any sentence that properly translates a second-order formula, in accordance with his treatment, has a first-order formulation. Support for this turns on the use of certain partitive constructions to assign values to variables (...)
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  20. Nino B. Cocchiarella (2005). Denoting Concepts, Reference, and the Logic of Names, Classes as Many, Groups, and Plurals? Linguistics and Philosophy 28 (2):135 - 179.
    Bertrand Russell introduced several novel ideas in his 1903 Principles of Mathematics that he later gave up and never went back to in his subsequent work. Two of these are the related notions of denoting concepts and classes as many. In this paper we reconstruct each of these notions in the framework of conceptual realism and connect them through a logic of names that encompasses both proper and common names, and among the latter, complex as well as simple common names. (...)
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  21. Salvatore Florio (forthcoming). Untyped Pluralism. Mind.
    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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  22. Enrico Franconi (1993). A Treatment of Plurals and Plural Quantifications Based on a Theory of Collections. Minds and Machines 3 (4):453-474.
    Collective entities and collective relations play an important role in natural language. In order to capture the full meaning of sentences like The Beatles sing Yesterday, a knowledge representation language should be able to express and reason about plural entities — like the Beatles — and their relationships — like sing — with any possible reading (cumulative, distributive or collective).In this paper a way of including collections and collective relations within a concept language, chosen as the formalism for representing the (...)
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  23. A. P. Hazen (1997). Relations in Lewis’s Framework Without Atoms. Analysis 57 (4):243–248.
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  24. Allen Hazen (2000). Relations in Lewis's Framework Without Atoms: A Correction. Analysis 60 (4):351–353.
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  25. Simon Hewitt (2012). The Logic of Finite Order. Notre Dame Journal of Formal Logic 53 (3):297-318.
    This paper develops a formal system, consisting of a language and semantics, called serial logic ( SL ). In rough outline, SL permits quantification over, and reference to, some finite number of things in an order , in an ordinary everyday sense of the word “order,” and superplural quantification over things thus ordered. Before we discuss SL itself, some mention should be made of an issue in philosophical logic which provides the background to the development of SL , and with (...)
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  26. K. Hossack (2000). Plurals and Complexes. British Journal for the Philosophy of Science 51 (3):411-443.
    Atomism denies that complexes exist. Common-sense metaphysics may posit masses, composite individuals and sets, but atomism says there are only simples. In a singularist logic, it is difficult to make a plausible case for atomism. But we should accept plural logic, and then atomism can paraphrase away apparent reference to complexes. The paraphrases require unfamiliar plural universals, but these are of independent interest; for example, we can identify numbers and sets with plural universals. The atomist paraphrases would fail if plurals (...)
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  27. Keith Hossack (2013). Sets and Plural Comprehension. Journal of Philosophical Logic:1-23.
    The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. (...)
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  28. Joongol Kim (2010). Yi on 2. Philosophia Mathematica 18 (3):329-336.
    Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two (in number)’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.
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  29. Kevin C. Klement, Early Russell on Types and Plurals.
    In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was short-lived; rejected before PoM even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about (...)
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  30. Juha Kontinen & Jakub Szymanik (2008). A Remark on Collective Quantification. Journal of Logic, Language and Information 17 (2):131-140.
    We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize all collective (...)
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  31. Peter Lasersohn (2011). Mass Nouns and Plurals. In Claudia Maienborn, Klaus von Heusinger & Paul Portner (eds.), Semantics: An International Handbook of Natural Language Meaning. De Gruyter Mouton. 2.
  32. Godehard Link (1998). Algebraic Semantics in Language and Philosophy. CSLI Publications.
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  33. Øystein Linnebo (2010). Pluralities and Sets. Journal of Philosophy 107 (3):144-164.
    Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?
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  34. Øystein Linnebo, Plural Quantification. Stanford Encyclopedia of Philosophy.
    Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...)
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  35. Øystein Linnebo (2007). Burgess on Plural Logic and Set Theory. Philosophia Mathematica 15 (1):79-93.
    John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led to substantial (...)
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  36. Øystein Linnebo (2003). Plural Quantification Exposed. Noûs 37 (1):71–92.
  37. Øystein Linnebo & David Nicolas (2008). Superplurals in English. Analysis 68 (299):186–197.
    where ‘aa’ is a plural term, and ‘F’ a plural predicate. Following George Boolos (1984) and others, many philosophers and logicians also think that plural expressions should be analysed as not introducing any new ontological commitments to some sort of ‘plural entities’, but rather as involving a new form of reference to objects to which we are already committed (for an overview and further details, see Linnebo 2004). For instance, the plural term ‘aa’ refers to Alice, Bob and Charlie simultaneously, (...)
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  38. Enrico Martino & Massimiliano Carrara (2010). To Be is to Be the Object of a Possible Act of Choice. Studia Logica 96 (2):289-313.
    Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite (...)
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  39. Vann McGee & Agustín Rayo (2000). A Puzzle About de Rebus Beliefs. Analysis 60 (4):297–299.
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  40. Thomas J. McKay (2008). Review of H. Laycock, Words Without Objects: Semantics, Ontology, and Logic for Non-Singularity. [REVIEW] Canadian Journal of Philosophy 38 (2):pp. 301-323.
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  41. 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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  42. Friederike Moltmann (forthcoming). Plural Reference and Reference to a Plurality. A Reassessment of the Linguistic Facts. In Massimiliano Carrara, Alessandra Arapinis & Friederike Moltmann (eds.), Unity and Plurality. Logic, Philosophy, and Semantics. Oxford University Press.
    This paper defends 'plural reference', the view that definite plurals refer to several individuals at once, and it explores how the view can account for a range of phenomena that have been discussed in the linguistic literature.
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  43. Adam Morton (1975). Complex Individuals and Multigrade Relations. Noûs 9 (3):309-318.
    I relate plural quantification, and predicate logic where predicates do not need a fixed number of argument places, to the part-whole relation. For more on these themes see later work by Boolos, Lewis, and Oliver & Smiley.
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  44. David Nicolas, Can Mereological Sums Serve as the Semantic Values of Plurals?
    Abstract: Friends of plural logic—like Oliver & Smiley (2001), Rayo (2002), Yi (2005), and McKay (2006)—have argued that a semantics of plurals based on mereological sums would be too weak, and they have adduced several examples in favor of their claim. However, they have not considered various possible counter-arguments. So how convincing are their own arguments? We show that several of them are easily answered, while some others are more problematic. Overall, the case against mereological singularism—the idea that mereological sums (...)
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  45. David Nicolas (2008). Mass Nouns and Plural Logic. Linguistics and Philosophy 31 (2):211 - 244.
    A dilemma put forward by Schein (1993, Plurals and events. Cambridge: MIT Press) and Rayo (2002, Nous, 36, 436-464) suggests that, in order to characterize the semantics of plurals, we should not use predicate logic, but plural logic, a formal language whose terms may refer to several things at once. We show that a similar dilemma applies to mass nouns. If we use predicate logic and sets when characterizing their semantics, we arrive at a Russellian paradox. And if we use (...)
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  46. Hiroki Nomoto (forthcoming). Proceedings of the Poster Session of the 29th Annual West Coast Conference on Formal Linguistics (WCCFL 29). In Proceedings of the Poster Session of the 29th Anual West Coast Conference on Formal Linguistics (WCCFL 29). University of Arizona Linguistics Circle.
    Dayal's (2004) theory of kind terms accounts for the definiteness and number marking patterns in kind terms in many languages. Brazilian Portuguese has been claimed to be a counter-example to her theory as it seems to allow bare ``singular'' kind terms, which are predicted to be impossible according to her theory. However, the empirical status of the relevant data has not been clear so far. This paper presents a new data point from Singlish and confirms the existence of bare ``singular'' (...)
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  47. Alex Oliver & Timothy Smiley (2013). Plural Logic. Oxford University Press.
    Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.
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  48. Alex Oliver & Timothy Smiley (2009). Sharvy's Theory of Descriptions: A Paradigm Subverted. Analysis 69 (3):412-421.
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  49. Alex Oliver & Timothy Smiley (2008). Is Plural Denotation Collective? Analysis 68 (297):22–34.
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  50. Alex Oliver & Timothy Smiley (2007). Erata: What Are Sets and What Are They For? Noûs 41 (2):354 -.
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