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Summary An abstraction principle (AP) allows one to introduce new singular terms by providing appropriate identity conditions. For instance, the most celebrated abstraction principle, called Hume's Principle (HP), introduces numerical terms by saying: "The number of Fs is the same as the number of Gs if and only if Fs and Gs are equinumerous (the relation of equinumerosity is definable in a second-order language without reference to numbers)." The first (and unsuccessful, because inconsistent) attempt at using APs in foundations of mathematics is due to Frege. Neo-Fregeans try to salvage Frege's project. One of the tasks is to show how various mathematical theories can be derived from appropriate APs. Another task is to develop a well-motivated acceptability criterion for APs (given that Frege's Basic Law V leads to contradiction and HP doesn't). The Bad Company objection (according to which there are separately consistent but mutually inconsistent abstraction principles) indicates that mere consistency of an AP is not enough for its acceptability. Finally neo-Fregeans have to develop a philosophically acceptable story explaining why APs can play an important role in the platonist epistemology of mathematics and what role exactly it is. 
Key works Wright 1983 is a seminal book on the topic. The consistency of arithmetic based on Hume's Principle has been proven by Boolos 1987Fine 2002 is a good survey of technical aspects of neologicism. A nice anthology of papers related to the Bad Company problem is vol. 70 no 3 of Synthese edited by Linnebo 2009. A good collection of essays related to neologicism is Wright & Hale 2001.
Introductions A good place to start is Zalta 2008 and more focused Zalta 2008 and Tennant 2013. A good introductory paper focused on philosophical motivations is  Cook 2009. A nice introduction to worries surrounding the acceptability criteria of APs is Linnebo 2009.
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  1. added 2020-05-04
    Hale and Wright on the Metaontology of Neo-Fregeanism.Matti Eklund - 2016 - In Marcus Rossberg & Philip A. Ebert (eds.), Abstractionism.
  2. added 2020-05-01
    Formal Arithmetic Before Grundgesetze.Richard Kimberly Heck - 2019 - In Philip A. Ebert & Marcus Rossberg (eds.), Essays on Frege's Basic Laws of Arithmetic. Oxford: Oxford University Press. pp. 497-537.
    A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism.
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  3. added 2020-05-01
    The Basic Laws of Cardinal Number.Richard Kimberly Heck - 2019 - In Philip A. Ebert & Marcus Rossberg (eds.), Essays on Frege's Basic Laws of Arithmetic. Oxford: Oxford University Press. pp. 1-30.
    An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.
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  4. added 2020-05-01
    Logicism, Ontology, and the Epistemology of Second-Order Logic.Richard Kimberly Heck - 2018 - In Ivette Fred-Rivera & Jessica Leach (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale. Oxford: Oxford University Press. pp. 140-169.
    In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms (...)
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  5. added 2020-04-20
    Thin Objects: An Abstractionist Account, by Øystein Linnebo.J. P. Studd - 2020 - Mind 129 (514):646-656.
    Thin Objects: Anionist Account, by LinneboØystein. Oxford: Oxford University Press, 2018. Pp. xviii + 238.
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  6. added 2020-03-25
    Frege’s Theorem: AN INTRODUCTION.Richard Heck Jr - 2003 - Manuscrito 26 (2):471-503.
    Frege's work was largely devoted to an attempt to argue that the'basic laws of arithmetic' are truths of logic. That attempt had both philosophical and formal aspects. The present note offers an introduction to both of these, so that readers will be able to appreciate contemporary discussions of the philosophical significance of 'Frege's Theorem'.
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  7. added 2020-02-12
    The Limits of Abstraction.Bob Hale - 2006 - Philosophy and Phenomenological Research 72 (1):223-232.
    Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects simpler, (...)
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  8. added 2020-02-11
    Frege's Philosophy of Mathematics. [REVIEW]Bob Hale - 1999 - Philosophical Quarterly 49 (194):92-104.
  9. added 2019-11-16
    The Problem of Fregean Equivalents.Joongol Kim - 2019 - Dialectica 73 (3):367-394.
    It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...)
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  10. added 2019-09-16
    The Julio César Problem.Fraser MacBride - 2005 - Dialectica 59 (2):223-236.
    One version of the Julius Caesar problem arises when we demand assurance that expressions drawn from different theories or stretches of discourse refer to different things. The counter‐Caesar problem arises when assurance is demanded that expressions drawn from different theories. refer to the same thing. The Julio César problem generalises from the counter‐Caesar problem. It arises when we seek reassurance that expressions drawn from different languages refer to the same kind of things. If the Julio César problem is not resolved (...)
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  11. added 2019-09-10
    Thin Reference, Metaontological Minimalism and Abstraction Principles: The Prospects for Tolerant Reductionism.Andrea Sereni - 2018 - In Annalisa Coliva, Paolo Leonardi & Sebastiano Moruzzi (eds.), Eva Picardi on Language, Analysis and History. pp. 161-181.
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  12. added 2019-07-31
    The Generous Ontology of Thin Objects: Øystein Linnebo: Thin Objects: An Abstractionist Account. New York: Oxford University Press, Xvii + 231 Pp, $50.00 HB. [REVIEW]Nathaniel Gan - 2019 - Metascience 28 (1):167-169.
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  13. added 2019-06-06
    Facets and Levels of Mathematical Abstraction.Hourya Sinaceur - 2014 - Philosophia Scientiae 18 (1):81-112.
    L'abstraction mathématique consiste en la considération et la ma­nipulation d'opérations, règles et concepts indépendamment du contenu dont les nantissent des applications particulières et du rapport qu'ils peuvent avoir avec les phénomènes et les circonstances du monde réel. L'abstraction ma­thématique emprunte diverses voies. Le terme « abstraction » ne désigne pas une procédure unique, mais un processus général où s'entrecroisent divers pro­cédés employés successivement ou simultanément. En particulier, l'abstrac­tion mathématique ne se réduit pas à la subsomption logique. Je vais étudier comparativement (...)
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  14. added 2019-06-06
    Logicism and the Problem of Infinity: The Number of Numbers: Articles.Gregory Landini - 2011 - Philosophia Mathematica 19 (2):167-212.
    Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity is (...)
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  15. added 2019-06-06
    Abstraction, Structure, and Substitution: Lambda and its Philosophical Significance.Peter Simons - 2007 - Polish Journal of Philosophy 1 (1):81-100.
    λ-calculi are of interest to logicians and computer scientists but have largely escaped philosophical commentary, perhaps because they appear narrowly technical or uncontroversial or both. I argue that even within logic λ-expressions need to be understood correctly, as functors signifying functions in intension within a categorical or typed language. λ-expressions are not names but pure viable binders generating functors, and as such they are of use in giving explicit definitions. But λ is applicable outside logic and computer science, anywhere where (...)
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  16. added 2019-06-06
    The Ins and Outs of Frege's Way Out.Gregory Landini - 2006 - Philosophia Mathematica 14 (1):1-25.
    Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his _Grundgesetze der Arithmetik_. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined (...)
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  17. added 2019-06-06
    Kit Fine, The Limits of Abstraction Oxford, Clarendon Press, 2002, Cloth £18.99\Textfractionsolidus{}US &Dollar;25.00 ISBN: 0-19-924618-1. [REVIEW]Philip A. Ebert - 2004 - British Journal for the Philosophy of Science 55 (4):791-800.
    Critical Notice of The Limits of abstraction by Kit Fine, Oxford: Clarendon Press, 2002, pp.216. ISBN 9780191567261.
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  18. added 2019-06-06
    Prolegomenon To Any Future Neo‐logicist Set Theory: Abstraction And Indefinite Extensibility.Stewart Shapiro - 2003 - British Journal for the Philosophy of Science 54 (1):59-91.
    The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...)
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  19. added 2019-06-06
    Frege’s Theorem.Richard Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  20. added 2019-06-06
    Frege’s Theorem: An Introduction.Richard Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  21. added 2019-06-06
    Cantorian Abstraction: A Reconstruction and Defense.Kit Fine - 1998 - Journal of Philosophy 95 (12):599-634.
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  22. added 2019-06-06
    Frege’s Attack on Husserl and Cantor.Claire Oritz Hill - 1994 - The Monist 77 (3):345-357.
    One hundred years ago Gottlob Frege published a damaging, abusive review of Edmund Husserl’s Philosophy of Arithmetic. Although rather a lot has now been written abound Frege’s review and the role it might have played in the development of Husserl’s thought, much still remains to be rectified regarding Frege’s assessment of the book and the credence his review has been accorded. Philosophers have generally been all too willing to trust Frege’s judgment, and so all too ready to dismiss Husserl’s book (...)
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  23. added 2019-06-06
    Science and Abstraction.L. J. Russell - 1930 - Journal of Philosophical Studies 5 (17):84-93.
    It is not only in science that abstraction is found as a method of dealing with the world, and we may profitably begin by showing its wide use in ordinary practical life.
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  24. added 2019-06-05
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
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  25. added 2019-06-05
    Caesar From Frege's Perspective.Gary Kemp - 2005 - Dialectica 59 (2):179-199.
    I attempt to explain Frege's handling of the Julius Caesar issue in terms of his more general philosophical commitments. These only became fully explicit in his middle-period writings, but his earlier moves are best explained, I suggest, if we suppose them to be implicit in his earlier thinking. These commitments conditionally justify Frege in rejecting Hume's Principle as either a definition or axiom but in accepting Axiom V. However, the general epistemological picture they constitute has serious problems in accounting for (...)
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  26. added 2019-05-28
    Review of Richard G. Heck, Jr: Reading Frege’s Grundgesetze. Oxford: Oxford University Press, 2012. [REVIEW]Marcus Rossberg - 2014 - Notre Dame Philosophical Review 11.
  27. added 2019-05-08
    Kit Fine. The Limits of Abstraction. Clarendon Press, Oxford, 2002, X + 203 Pp. [REVIEW]Alan Weir - 2004 - Bulletin of Symbolic Logic 10 (4):554-557.
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  28. added 2019-03-26
    Thin Objects.Øystein Linnebo - 2018 - Oxford: Oxford University Press.
    Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the (...)
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  29. added 2019-03-22
    Frege's Theorem.Richard G. Heck - 2011 - Clarendon Press.
    The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.
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  30. added 2019-03-22
    Finitude and Hume’s Principle.Richard Heck - 1997 - Journal of Philosophical Logic 26 (6):589-617.
    The paper formulates and proves a strengthening of 'Frege's Theorem', which states that axioms for second-order arithmetic are derivable in second-order logic from Hume's Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. 'Finite Hume's Principle' also suffices (...)
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  31. added 2019-02-24
    The Limits of Reconstructive Neologicist Epistemology.Eileen S. Nutting - 2018 - Philosophical Quarterly 68 (273):717-738.
    Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...)
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  32. added 2019-02-15
    Frege's Other Program.Aldo Antonelli & Robert May - 2005 - Notre Dame Journal of Formal Logic 46 (1):1-17.
    Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of (...)
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  33. added 2019-02-04
    Content Recarving as Subject Matter Restriction.Vincenzo Ciccarelli - forthcoming - Manuscrito: Revista Internacional de Filosofía 42 (1).
    In this article I offer an explicating interpretation of the procedure of content recarving as described by Frege in §64 of the Foundations of Arithmetic. I argue that the procedure of content recarving may be interpreted as an operation that while restricting the subject matter of a sentence, performs a generalization on what the sentence says about its subject matter. The characterization of the recarving operation is given in the setting of Yablo’s theory of subject matter and it is based (...)
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  34. added 2019-02-01
    ‘Just is’-Statements as Generalized Identities.Øystein Linnebo - 2014 - Inquiry: An Interdisciplinary Journal of Philosophy 57 (4):466-482.
    Identity is ordinarily taken to be a relation defined on all and only objects. This consensus is challenged by Agustín Rayo, who seeks to develop an analogue of the identity sign that can be flanked by sentences. This paper is a critical exploration of the attempted generalization. First the desired generalization is clarified and analyzed. Then it is argued that there is no notion of content that does the desired philosophical job, namely ensure that necessarily equivalent sentences coincide in this (...)
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  35. added 2019-02-01
    N Eo-F Regeanism and Q Uantifier V Ariance.Katherine Hawley - 2007 - Aristotelian Society Supplementary Volume 81 (1):233-249.
    In his paper in the same volume, Sider argues that, of maximalism and quantifier variance, the latter promises to let us make better sense of neo-Fregeanism. I argue that neo-Fregeans should, and seemingly do, reject quantifier variance. If they must choose between these two options, they should choose maximalism.
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  36. added 2018-09-29
    Cardinality and Acceptable Abstraction.Roy T. Cook & Øystein Linnebo - 2018 - Notre Dame Journal of Formal Logic 59 (1):61-74.
    It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.
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  37. added 2018-09-29
    To Be Is to Be an F.Øystein Linnebo - 2005 - Dialectica 59 (2):201-222.
    I defend the view that our ontology divides into categories, each with its own canonical way of identifying and distinguishing the objects it encompasses. For instance, I argue that natural numbers are identified and distinguished by their positions in the number sequence, and physical bodies, by facts having to do with spatiotemporal continuity. I also argue that objects belonging to different categories are ipso facto distinct. My arguments are based on an analysis of reference, which ascribes to reference a richer (...)
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  38. added 2018-08-03
    Paradox Without Basic Law V: A Problem with Frege’s Ontology.Adam Rieger - 2002 - Analysis 62 (4):327-330.
  39. added 2018-08-01
    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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  40. added 2018-04-30
    Tuples All the Way Down?Simon Thomas Hewitt - 2018 - Thought: A Journal of Philosophy 7 (3):161-169.
  41. added 2018-03-05
    Composition as Abstraction.Jeffrey Sanford Russell - 2017 - Journal of Philosophy 114 (9):453-470.
    The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.
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  42. added 2018-03-01
    Priority, Platonism, and the Metaontology of Abstraction.Michele Lubrano - 2016 - Dissertation, University of Turin
    In this dissertation I examine the NeoFregean metaontology of mathematics. I try to clarify the relationship between what is sometimes called Priority Thesis and Platonism about mathematical entities. I then present three coherent ways in which one might endorse both these stances, also answering some possible objections. Finally I try to show which of these three ways is the most promising.
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  43. added 2018-03-01
    Ontologie Neofreghiane.Michele Lubrano - 2016 - Philosophy Kitchen 3 (4):113-125.
    In the present contribution I would like to examine some theories of the ontology of abstract entities that take inspiration from the deep insights of Gottlob Frege. These theories develop in full details some ideas explicitly or implicitly articulated in Frege’s works and try to defend a sophisticated version of Platonism about abstract entities. The review of such theories should allow us to cast light on their merits and their possible flaws and, moreover, to determine which of them is the (...)
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  44. added 2018-02-17
    Neo-Logicism and Russell’s Logicism.Kevin C. Klement - 2012 - Russell: The Journal of Bertrand Russell Studies 32 (2):159.
    Most advocates of the so-called “neologicist” movement in the philosophy of mathematics identify themselves as “Neo-Fregeans” (e.g., Hale and Wright): presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature, and when it is, often dismissed as not really logicism at all (in lights of its assumption of axioms of infinity, reducibiity and so on). In this paper I have three aims: firstly, to identify more clearly the primary metaontological (...)
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  45. added 2018-01-17
    Tuples All the Way Down?Simon Hewitt - manuscript
    We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the difficulty, (...)
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  46. added 2017-09-06
    Frege's Recipe.Roy T. Cook & Philip A. Ebert - 2016 - Journal of Philosophy 113 (7):309-345.
    In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and (...)
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  47. added 2017-05-12
    Categories for the Neologicist.Shay Allen Logan - 2017 - Philosophia Mathematica 25 (1):26-44.
    Abstraction principles provide implicit definitions of mathematical objects. In this paper, an abstraction principle defining categories is proposed. It is unsatisfiable and inconsistent in the expected ways. Two restricted versions of the principle which are consistent are presented.
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  48. added 2017-03-24
    Necessity, Necessitism, and Numbers.Roy T. Cook - 2016 - Philosophical Forum 47 (3-4):385-414.
    Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s arguments either (...)
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  49. added 2017-03-20
    Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  50. added 2017-03-14
    A Generic Russellian Elimination of Abstract Objects.Kevin C. Klement - 2015 - Philosophia Mathematica:nkv031.
    In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...)
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1 — 50 / 369