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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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135 found
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1 — 50 / 135
  1. We Belong Together? A Plea for Modesty in Modal Plural Logic.Simon Hewitt - manuscript
    It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.
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  2. Frege's Theorem in Plural Logic.Simon Hewitt - manuscript
    A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.
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  3. Properties, Plurals and Paradox.Philippe Schlenker - manuscript
    It has been argued that an objectual semantics for plurals falls victim to Russell’s paradox, and that a nominalistic semantics should therefore be preferred (Boolos 1984); similar considerations have sometimes been extended to other types of abstract reference, in particular to property talk. We suggest that this line of argument is mistaken: deeply entrenched features of ordinary language guarantee that property and plural talk do give rise to paradoxes. In the case of properties, the grammar of English is untyped, which (...)
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  4. Rigid and Flexible Quantification in Plural Predicate Logic.Lucas Champollion, Justin Bledin & Haoze Li - forthcoming - Semantics and Linguistic Theory 27.
    Noun phrases with overt determiners, such as <i>some apples</i> or <i>a quantity of milk</i>, differ from bare noun phrases like <i>apples</i> or <i>milk</i> in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable fixed (...)
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  5. Groups as Pluralities.John Horden & Dan López de Sa - forthcoming - Synthese:1-35.
    We say that each social group is identical to its members. The group just is them; they just are the group. This view of groups as pluralities has tended to be swiftly rejected by social metaphysicians, if considered at all, mainly on the basis of two objections. First, it is argued that groups can change in membership, while pluralities cannot. Second, it is argued that different groups can have exactly the same members, while different pluralities cannot. We rebut these objections, (...)
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  6. Plural Reference and Syntactic Three-Dimensionality (Book Proposal, Under Contract).Friederike Moltmann - forthcoming - Oxford University Press.
    The book argues for Plural Reference for the semantics of natural language and makes the connection between Plural Reference and Alternative Semantics for the purpose of the interpretation of three-dimensional syntactic structures of coordinate sentences (in the sense of my 1992 MIT Ph D thesis).
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  7. Proceedings of the Poster Session of the 29th Annual West Coast Conference on Formal Linguistics (WCCFL 29).Hiroki Nomoto - forthcoming - 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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  8. Composition as Identity, Now with All the Pluralities You Could Want.Jonathan D. Payton - forthcoming - Synthese:1-22.
    According to ‘composition as identity’ (CAI), a composite object is identical to all its parts taken together. Thus, a plurality of composite objects is identical to the plurality of those objects’ parts. This has the consequence that, e.g., the bricks which compose a brick wall are identical to the atoms which compose those bricks, and hence that the plurality of bricks must include each of those atoms. This consequence of CAI is in direct conflict with the standard analysis of plural (...)
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  9. How to Identify Wholes with Their Parts.Jonathan D. Payton - forthcoming - Synthese:1-23.
    I claim that a whole is identical to its parts. Many find this claim incredible: it seems that a whole and its parts must be distinct, for the whole is one thing while its parts are many things. Byeong-uk Yi has developed a version of this argument which exploits the resources of plural logic. Yi provides logical analyses of the predicates ‘one’ and ‘many’ which seem to show that nothing can satisfy them both. But there are two senses of the (...)
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  10. Plural Logic.Michael Rieppel - forthcoming - Analysis:anv010.
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  11. Indefinite Extensibility and the Principle of Sufficient Reason.Geoffrey Hall - 2021 - Philosophical Studies 178 (2):471-492.
    The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefinite extensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefinite extensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefinite extensibility of the concept set, plausible principles permit the derivation of modal collapse.
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  12. Critical Plural Logic.Salvatore Florio & Øystein Linnebo - 2020 - Philosophia Mathematica 28 (2):172-203.
    What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.
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  13. Baxter and Cotnoir on Composition as Identity.Joongol Kim - 2019 - 철학사상 [CHUL HAK SA SANG: Journal of Philosophical Ideas] 73:105-125.
    This paper provides a critical examination of three related attempts to defend Composition as Identity (CI), namely the thesis that if some things compose something, then they are it. First, it will be argued against Donald Baxter’s view of composition as ‘loose identity’ that by construing composition as strictly a many-many relation, the view trivializes CI, and cannot be an option for the advocate of CI who takes composition as a genuine many-one relation. Second, it is argued against Baxter’s modified (...)
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  14. Modal Structuralism and Reflection.Sam Roberts - 2019 - Review of Symbolic Logic 12 (4):823-860.
    Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...)
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  15. Logic and Plurals.Salvatore Florio & Øystein Linnebo - 2018 - In Kirk Ludwig & Marija Jankovic (eds.), The Routledge Handbook of Collective Intentionality. Routledge. pp. 451-463.
    This chapter provides an overview of the philosophical and linguistic debate about the logic of plurals. We present the most prominent singularizing analyses of plurals as well as the main criticisms that such analyses have received. We then introduce an alternative approach to plurals known as plural logic, focusing on the question whether plural logic can count as pure logic.
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  16. Mereological Composition and Plural Quantifier Semantics.Manuel Lechthaler & Ceth Lightfield - 2018 - Philosophia 46 (4):943-958.
    Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity (...)
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  17. Composition as Identity and Plural Cantor's Theorem.Einar Duenger Bohn - 2016 - Logic and Logical Philosophy 25 (3).
    I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.
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  18. Unity and Plurality. Philosophy, Logic, and Semantics.Massimiliano Carrara, Alessandra Arapinis & Friederike Moltmann - 2016 - 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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  19. On the Innocence and Determinacy of Plural Quantification.Salvatore Florio & Øystein Linnebo - 2016 - Noûs 50 (3):565–583.
    Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic (...)
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  20. Plurals and Modals.Øystein Linnebo - 2016 - Canadian Journal of Philosophy 46 (4-5):654-676.
    Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.
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  21. Plural Reference and Reference to a Plurality. Linguistic Facts and Semantic Analyses.Friederike Moltmann - 2016 - In Massimiliano Carrara, Alexandra Arapinis & Friederike Moltmann (eds.), Unity and Plurality. Logic, Philosophy, and Semantics. Oxford: Oxford University Press. pp. 93-120.
    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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  22. Plural Logic: Second Edition, Revised and Enlarged.Alex Oliver & Timothy Smiley - 2016 - 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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  23. Book Review: Alex Oliver and Timothy Smiley, Plural Logic. [REVIEW]Thomas Brouwer & Casper Storm Hansen - 2015 - Studia Logica 103 (5):1095-1100.
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  24. Grounding Megethology on Plural Reference.Massimiliano Carrara & Enrico Martino - 2015 - Studia Logica 103 (4):697-711.
    In Mathematics is megethology Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis’work is very attractive. However, the alleged (...)
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  25. Plural Logic and Sensitivity to Order.Salvatore Florio & David Nicolas - 2015 - Australasian Journal of Philosophy 93 (3):444-464.
    Sentences that exhibit sensitivity to order (e.g. 'John and Mary arrived at school in that order' and 'Mary and John arrived at school in that order') present a challenge for the standard formulation of plural logic. In response, some authors have advocated new versions of plural logic based on fine-grained notions of plural reference, such as serial reference (Hewitt 2012) and articulated reference (Ben-Yami 2013). The aim of this article is to show that sensitivity to order should be accounted for (...)
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  26. When Do Some Things Form a Set?Simon Hewitt - 2015 - Philosophia Mathematica 23 (3):311-337.
    This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...)
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  27. Plural Logic, by Alex Oliver and Timothy Smiley: Oxford: Oxford University Press, 2013, Pp. Xiv + 336, £40. [REVIEW]Lloyd Humberstone - 2015 - Australasian Journal of Philosophy 93 (1):192-195.
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  28. Pluralities and Plural Logic.Michael Rieppel - 2015 - Analysis 75 (3):504-514.
  29. Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic.Marcus Rossberg - 2015 - Journal of Philosophical Logic 44 (3):341-350.
    Boolos has suggested a plural interpretation of second-order logic for two purposes: to escape Quine’s allegation that second-order logic is set theory in disguise, and to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and Yablo’s interpretation does not achieve the (...)
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  30. 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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  31. 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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  32. 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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  33. 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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  34. Semantics and the Plural Conception of Reality.Salvatore Florio - 2014 - Philosophers' Imprint 14:1-20.
    According to the singular conception of reality, there are objects and there are singular properties, i.e. properties that are instantiated by objects separately. It has been argued that semantic considerations about plurals give us reasons to embrace a plural conception of reality. This is the view that, in addition to singular properties, there are plural properties, i.e. properties that are instantiated jointly by many objects. In this article, I propose and defend a novel semantic account of plurals which dispenses with (...)
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  35. Sets and Plural Comprehension.Keith Hossack - 2014 - Journal of Philosophical Logic 43 (2-3):517-539.
    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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  36. Early Russell on Types and Plurals.Kevin Klement - 2014 - Journal for the History of Analytical Philosophy 2 (6):1-21.
    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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  37. Review of Oliver & Smiley, Plural Logic, 2013. [REVIEW]David Nicolas - 2014 - Notre Dame Philosophical Reviews 2.
  38. Consequences of Collapse.Theodore Sider - 2014 - In Donald Baxter & Aaron Cotnoir (eds.), Composition as Identity. Oxford University Press. pp. 211-221.
    "Composition as identity" is the radical claim that the whole is identical to the parts - radical because it implies that a single object can be identical to many objects. Composition as identity, together with auxiliary assumptions, implies the principle of "collapse": an object is one of some things if and only it is part of the fusion of those things. Collapse has important implications: the comprehension principle of plural logic must be restricted, plural definite descriptions such as "the Cheerios (...)
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  39. Dependent Plural Pronouns with Skolemized Choice Functions.Yasutada Sudo - 2014 - Natural Language Semantics 22 (3):265-297.
    The present paper discusses two interesting phenomena concerning phi-features on plural pronouns: plural pronouns that denote atomic individuals, and plural pronouns with more than one binder. A novel account of these two phenomena is proposed, according to which all occurrences of phi-features are both semantically and morphologically relevant. For such a ‘uniformly semantic account’ of phi-features, dependent plural pronouns constitute a theoretical challenge, while partial binding is more or less straightforwardly accounted for. In order to make sense of the semantic (...)
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  40. Plural Quantifiers: A Modal Interpretation.Rafal Urbaniak - 2014 - Synthese 191 (7):1-22.
    One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and set-theoretic—only the latter can provide the language of plurals with the desired expressive power (given that (...)
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  41. Higher‐Level Plurals Versus Articulated Reference, and an Elaboration of Salva Veritate.Hanoch Ben-Yami - 2013 - 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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  42. Plural Logicism.Francesca Boccuni - 2013 - 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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  43. Plural Logic.Alex Oliver & Timothy Smiley - 2013 - Oxford University Press UK.
    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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  44. Response to Westerstahl.Hanoch Ben-Yami - 2012 - Logique Et Analyse 55 (217):47-55.
  45. Monism, Emergence, and Plural Logic.Einar Duenger Bohn - 2012 - 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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  46. The Logic of Finite Order.Simon Hewitt - 2012 - 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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  47. Modalising Plurals.Simon Thomas Hewitt - 2012 - Journal of Philosophical Logic 41 (5):853-875.
    There has been very little discussion of the appropriate principles to govern a modal logic of plurals. What debate there has been has accepted a principle I call (Necinc); informally if this is one of those then, necessarily: this is one of those. On this basis Williamson has criticised the Boolosian plural interpretation of monadic second-order logic. I argue against (Necinc), noting that it isn't a theorem of any logic resulting from adding modal axioms to the plural logic PFO+, and (...)
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  48. 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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  49. On the Consistency of a Plural Theory of Frege’s Grundgesetze.Francesca Boccuni - 2011 - 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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  50. On the Infinite in Mereology with Plural Quantification.Massimiliano Carrara & Enrico Martino - 2011 - 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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