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  1. Logic, Logic and Logic.George Boolos - 1998 - Harvard University Press.
    This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; ...
  • Frege's Conception of Logic.Patricia A. Blanchette - 2012 - Oup Usa.
    In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic.
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  • The Limits of Abstraction.Kit Fine - 2002 - Oxford University Press.
    Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.
  • New Essays on Tarski and Philosophy.Douglas Patterson (ed.) - 2008 - Oxford University Press.
    The essays can be seen as addressing Tarski's seminal treatment of four basic questions about logical consequence. (1) How are we to understand truth, one of ...
  • 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.
  • Finitude and Hume’s Principle.Richard G. 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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  • Frege's Conception of Numbers as Objects.John P. Burgess - 1984 - Philosophical Review 93 (4):638.
  • 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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  • Ineffability Within the Limits of Abstraction Alone.Stewart Shapiro & Gabriel Uzquiano - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics. Oxford University Press.
    The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...)
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  • Conservativeness, Stability, and Abstraction.R. T. Cook - 2012 - British Journal for the Philosophy of Science 63 (3):673-696.
    One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable (...)
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  • Notions of Invariance for Abstraction Principles.G. A. Antonelli - 2010 - Philosophia Mathematica 18 (3):276-292.
    The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...)
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  • Die Grundlagen der Arithmetik, 82-3.George Boolos & Richard G. Heck - 1998 - In Matthias Schirn (ed.), Bulletin of Symbolic Logic. Clarendon Press. pp. 407-28.
    This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in (...)
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  • Logic, Logic and Logic.George Boolos & Richard C. Jeffrey - 1998 - Studia Logica 66 (3):428-432.
     
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  • Tarski's Thesis.Gila Sher - 2008 - In Douglas Patterson (ed.), New Essays on Tarski and Philosophy. Oxford University Press. pp. 300--339.
  • Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.
  • Iteration One More Time.R. Cook - 2003 - Notre Dame Journal of Formal Logic 44 (2):63--92.
    A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that (...)
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  • Abstraction and Identity.Roy T. Cook & Philip A. Ebert - 2005 - Dialectica 59 (2):121–139.
    A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
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  • The Limits of Abstraction.Kit Fine - 2004 - Bulletin of Symbolic Logic 10 (4):554-557.
     
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  • Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF.Thomas Forster - 2016 - Philosophia Mathematica 24 (1):nku005.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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  • Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF.Thomas Forster - 2016 - Philosophia Mathematica 24 (1):50-59.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted _aussonderung_ but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for "low" sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no (...)
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