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  1. Abstraction Without Exceptions.Luca Zanetti - forthcoming - Philosophical Studies:1-20.
    Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down the foundations for (...)
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  • Russell’s Paradox and Free Zig Zag Solutions.Ludovica Conti - forthcoming - Foundations of Science:1-19.
    I present the traditional debate about the so called explanation of Russell’s paradox and propose a new way to solve the contradiction that arises in Frege’s system. I briefly examine two alternative explanatory proposals—the Predicativist explanation and the Cantorian one—presupposed by almost all the proposed solutions of Russell’s Paradox. From the discussion about these proposals a controversial conclusion emerges. Then, I examine some particular zig zag solutions and I propose a third explanation, presupposed by them, in which I emphasise the (...)
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  • Consistency, Models, and Soundness.Matthias Schirn - 2010 - Axiomathes 20 (2-3):153-207.
    This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...)
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  • Introduction.Øystein Linnebo - 2009 - Synthese 170 (3):321-329.
    Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable ones. This is the "bad company problem." In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.
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  • Bad Company Tamed.Øystein Linnebo - 2009 - Synthese 170 (3):371 - 391.
    The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts (...)
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  • Amending Frege’s Grundgesetze der Arithmetik.Fernando Ferreira - 2005 - Synthese 147 (1):3-19.
    Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this (...)
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  • Frege, Boolos, and Logical Objects.David J. Anderson & Edward N. Zalta - 2004 - Journal of Philosophical Logic 33 (1):1-26.
    In this paper, the authors discuss Frege's theory of "logical objects" and the recent attempts to rehabilitate it. We show that the 'eta' relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the 'eta' relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for (...)
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  • First-Order Frege Theory is Undecidable.Warren Goldfarb - 2001 - Journal of Philosophical Logic 30 (6):613-616.
    The system whose only predicate is identity, whose only nonlogical vocabulary is the abstraction operator, and whose axioms are all first-order instances of Frege's Axiom V is shown to be undecidable.
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  • On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze.Fernando Ferreira & Kai F. Wehmeier - 2002 - Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...)
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  • Frege's Principle.Richard Heck - 1995 - In J. Hintikka (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Kluwer Academic Publishers.
    This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.
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  • 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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  • The Finitistic Consistency of Heck’s Predicative Fregean System.Luís Cruz-Filipe & Fernando Ferreira - 2015 - Notre Dame Journal of Formal Logic 56 (1):61-79.
    Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system.
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  • Neo-Fregeanism: An Embarrassment of Riches.Alan Weir - 2003 - Notre Dame Journal of Formal Logic 44 (1):13-48.
    Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume's Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of (...)
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  • Grundgesetze der Arithmetik I §§29‒32.Richard G. Heck - 1997 - Notre Dame Journal of Formal Logic 38 (3):437-474.
    Frege's intention in section 31 of Grundgesetze is to show that every well-formed expression in his formal system denotes. But it has been obscure why he wants to do this and how he intends to do it. It is argued here that, in large part, Frege's purpose is to show that the smooth breathing, from which names of value-ranges are formed, denotes; that his proof that his other primitive expressions denote is sound and anticipates Tarski's theory of truth; and that (...)
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  • Book Review: Kit Fine. The Limits of Abstraction. [REVIEW]John P. Burgess - 2003 - Notre Dame Journal of Formal Logic 44 (4):227-251.
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  • 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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  • Impredicativity and Paradox.Gabriel Uzquiano - 2019 - Thought: A Journal of Philosophy 8 (3):209-221.
    Thought: A Journal of Philosophy, EarlyView.
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  • Is Frege's Definition of the Ancestral Adequate?Richard Heck - 2016 - Philosophia Mathematica 24 (1):91-116.
    Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...)
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  • Abstraction Reconceived.J. P. Studd - 2016 - British Journal for the Philosophy of Science 67 (2):579-615.
    Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...)
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  • Predicative Fragments of Frege Arithmetic.Øystein Linnebo - 2004 - Bulletin of Symbolic Logic 10 (2):153-174.
    Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...)
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  • The Logical Significance of Assertion: Frege on the Essence of Logic.Walter B. Pedriali - 2017 - Journal for the History of Analytical Philosophy 5 (8).
    Assertion plays a crucial dual role in Frege's conception of logic, a formal and a transcendental one. A recurrent complaint is that Frege's inclusion of the judgement-stroke in the Begriffsschrift is either in tension with his anti-psychologism or wholly superfluous. Assertion, the objection goes, is at best of merely psychological significance. In this paper, I defend Frege against the objection by giving reasons for recognising the central logical significance of assertion in both its formal and its transcendental role.
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  • Structural-Abstraction Principles.Graham Leach-Krouse - 2015 - Philosophia Mathematica:nkv033.
    In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, (...)
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  • A Logic for Frege's Theorem.Richard Heck - 2011 - In Frege’s Theorem: An Introduction. Oxford University Press.
    It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...)
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  • The Convenience of the Typesetter; Notation and Typography in Frege’s Grundgesetze der Arithmetik.Jim J. Green, Marcus Rossberg & A. Ebert Philip - 2015 - Bulletin of Symbolic Logic 21 (1):15-30.
    We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik.
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  • Sir Michael Anthony Eardley Dummett, 1925-2011.R. G. Heck - 2013 - Philosophia Mathematica 21 (1):1-8.
    A remembrance of Dummett's work on philosophy of mathematcis.
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  • Grundgesetze der Arithmetic I §10.Richard Heck - 1999 - Philosophia Mathematica 7 (3):258-292.
    In section 10 of Grundgesetze, Frege confronts an indeterm inacy left by his stipulations regarding his ‘smooth breathing’, from which names of valueranges are formed. Though there has been much discussion of his arguments, it remains unclear what this indeterminacy is; why it bothers Frege; and how he proposes to respond to it. The present paper attempts to answer these questions by reading section 10 as preparatory for the (fallacious) proof, given in section 31, that every expression of Frege's formal (...)
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  • Frege’s Permutation Argument Revisited.Kai Frederick Wehmeier & Peter Schroeder-Heister - 2005 - Synthese 147 (1):43-61.
    In Section 10 of Grundgesetze, Volume I, Frege advances a mathematical argument (known as the permutation argument), by means of which he intends to show that an arbitrary value-range may be identified with the True, and any other one with the False, without contradicting any stipulations previously introduced (we shall call this claim the identifiability thesis, following Schroeder-Heister (1987)). As far as we are aware, there is no consensus in the literature as to (i) the proper interpretation of the permutation (...)
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  • Fragments of Frege’s Grundgesetze and Gödel’s Constructible Universe.Sean Walsh - 2016 - Journal of Symbolic Logic 81 (2):605-628.
    Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...)
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  • Julius Caesar and Basic Law V.Richard G. Heck - 2005 - Dialectica 59 (2):161–178.
    This paper dates from about 1994: I rediscovered it on my hard drive in the spring of 2002. It represents an early attempt to explore the connections between the Julius Caesar problem and Frege's attitude towards Basic Law V. Most of the issues discussed here are ones treated rather differently in my more recent papers "The Julius Caesar Objection" and "Grundgesetze der Arithmetik I 10". But the treatment here is more accessible, in many ways, providing more context and a better (...)
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  • Did Frege Commit a Cardinal Sin?A. C. Paseau - 2015 - Analysis 75 (3):379-386.
    Frege’s _Basic Law V_ is inconsistent. The reason often given is that it posits the existence of an injection from the larger collection of first-order concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis.
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  • Russell's Paradox in Consistent Fragments of Frege's Grundgesetze der Arithmetik.Kai F. Wehmeier - 2004 - In Godehard Link (ed.), One Hundred Years of Russell’s Paradox. de Gruyter.
    We provide an overview of consistent fragments of the theory of Frege’s Grundgesetze der Arithmetik that arise by restricting the second-order comprehension schema. We discuss how such theories avoid inconsistency and show how the reasoning underlying Russell’s paradox can be put to use in an investigation of these fragments.
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  • What is Neologicism?Bernard Linsky & Edward N. Zalta - 2006 - Bulletin of Symbolic Logic 12 (1):60-99.
    In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...)
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  • Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic.Sean Walsh - 2016 - Journal of Philosophical Logic 45 (3):277-326.
    This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...)
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  • Mending the Master: JOHN P. BURGESS, Fixing Frege. Princeton, N. J.: Princeton University Press, 2005. ISBN 0-691-12231-8. Pp. Xii + 257. [REVIEW]O. Linnebo - 2006 - Philosophia Mathematica 14 (3):338-400.
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  • The Strength of Abstraction with Predicative Comprehension.Sean Walsh - 2016 - Bulletin of Symbolic Logic 22 (1):105–120.
    Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...)
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  • On the Varieties of Abstract Objects.James E. Davies - 2019 - Australasian Journal of Philosophy 97 (4):809-823.
    I reconcile the spatiotemporal location of repeatable artworks and impure sets with the non-location of natural numbers despite all three being varieties of abstract objects. This is possible because, while the identity conditions for all three can be given by abstraction principles, in the former two cases spatiotemporal location is a congruence for the equivalence relation featuring in the relevant principle, whereas in the latter it is not. I then generalize this to other ‘physical’ properties like shape, mass, and causal (...)
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  • The Groundedness Approach to Class Theory.Jönne Kriener - 2014 - Inquiry: An Interdisciplinary Journal of Philosophy 57 (2):244-273.
    Kripke showed how to restrict Tarski’s schema to grounded sentences. I examine the prospects for an analogous approach to the paradoxes of naive class comprehension. I present new methods to obtain theories of grounded classes and test them against antecedently motivated desiderata. My findings cast doubt on whether a theory of grounded classes can accommodate both the extensionality of classes and allow for class definition in terms of identity.
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  • Introduction to Special Issue: Reconsidering Frege's Conception of Number Dedicated to the Memory of Aldo Antonelli.Erich H. Reck & Roy T. Cook - 2016 - Philosophia Mathematica 24 (1):1-8.
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  • 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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  • Plural Grundgesetze.Francesca Boccuni - 2010 - 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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  • Review of Kazuyuki Nomoto "Frege Tetsugaku No Zenbou ". [REVIEW]Hidenori Kurokawa - 2014 - Journal of the Japan Association for Philosophy of Science 42 (1):39-54.
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  • Term Models for Abstraction Principles.Leon Horsten & Øystein Linnebo - 2016 - Journal of Philosophical Logic 45 (1):1-23.
    Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is (...)
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  • Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
    This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper (...)
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  • What Russell Should Have Said to Burali–Forti.Salvatore Florio & Graham Leach-Krouse - 2017 - Review of Symbolic Logic 10 (4):682-718.
    The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...)
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  • Modality and Paradox.Gabriel Uzquiano - 2015 - Philosophy Compass 10 (4):284-300.
    Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. (...)
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