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  1. (1 other version)Die Grundlagen der Arithmetik §§82-83.Richard Heck & George Boolos - 1998 - In Matthias Schirn (ed.), The Philosophy of mathematics today. New York: Clarendon Press.
    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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  • Frege meets dedekind: A neologicist treatment of real analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...)
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  • Is Hume's principle analytic?Crispin Wright - 1999 - Notre Dame Journal of Formal Logic 40 (1):307-333.
    This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...)
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  • The standard of equality of numbers.George Boolos - 1990 - In Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge and New York: Cambridge University Press. pp. 261--77.
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  • (1 other version)Nominalist platonism.George Boolos - 1998 - In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. pp. 73-87.
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  • Foundations without foundationalism: a case for second-order logic.Stewart Shapiro - 1991 - New York: Oxford University Press.
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...)
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  • Frege's conception of numbers as objects.Crispin Wright - 1983 - [Aberdeen]: Aberdeen University Press.
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  • Nominalism through de-nominalization.Agustin Rayo & Stephen Yablo - 2001 - Noûs 35 (1):74–92.
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  • The development of arithmetic in Frege's Grundgesetze der Arithmetik.Richard Heck - 1993 - Journal of Symbolic Logic 58 (2):579-601.
    Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...)
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  • New V, ZF and Abstraction.Stewart Shapiro & Alan Weir - 1999 - Philosophia Mathematica 7 (3):293-321.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...)
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  • Well- and non-well-founded Fregean extensions.Ignacio Jané & Gabriel Uzquiano - 2004 - Journal of Philosophical Logic 33 (5):437-465.
    George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...)
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  • (1 other version)To be is to be a value of a variable (or to be some values of some variables).George Boolos - 1984 - Journal of Philosophy 81 (8):430-449.
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  • Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
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  • Powers of regular cardinals.William B. Easton - 1970 - Annals of Mathematical Logic 1 (2):139.
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  • Die Grundlagen der Arithmetik, §§ 82-3. [REVIEW]William Demopoulos - 1998 - Bulletin of Symbolic Logic 6 (4):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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  • (1 other version)Is Hume's principle analytic?G. Boolos - 1998 - Logic, Logic, and Logic:301--314.
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  • Abstraction and set theory.Bob Hale - 2000 - Notre Dame Journal of Formal Logic 41 (4):379--398.
    The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties confronting the (...)
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  • The limits of abstraction.Kit Fine - 2002 - New York: Oxford University Press. Edited by Matthias Schirn.
    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.
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  • (1 other version)Nominalist platonism.George Boolos - 1985 - Philosophical Review 94 (3):327-344.
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  • 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): ∀P∀Q[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 (...)
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  • (1 other version)Grundlagen der Arithmetik: Studienausgabe mit dem Text der Centenarausgabe.Gottlob Frege - 1988 - Meiner, F.
    Die Grundlagen gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen. Studienausgabe aufgrund der textkritisch herausgegebenen Jubiläumsausgabe (Centenarausgabe). Mit Einleitung, Anmerkungen, Literaturverzeichnis und Namenregister.
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  • 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 ofGs 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 for (...)
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  • (1 other version)Philosophy of Mathematics.Stewart Shapiro - 2003 - In Peter Clark & Katherine Hawley (eds.), Philosophy of science today. New York: Oxford University Press.
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
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  • Beyond Plurals.Agust\’in Rayo - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute generality. New York: Oxford University Press. pp. 220--54.
    I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higherorder quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of (...)
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  • ‘Neo-logicist‘ logic is not epistemically innocent.Stewart Shapiro & Alan Weir - 2000 - Philosophia Mathematica 8 (2):160--189.
    The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...)
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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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  • On the philosophical significance of Frege's theorem.Crispin Wright - 1997 - In Richard G. Heck (ed.), Language, thought, and logic: essays in honour of Michael Dummett. New York: Oxford University Press. pp. 201--44.
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  • Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.
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  • Iteration one more time.Roy T. 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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  • Crispin Wright, Frege's Conception of Numbers as Objects. [REVIEW]Boguslaw Wolniewicz - 1986 - Studia Logica 45 (3):330-330.
    The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.
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  • Implicit definition and the a priori.Bob Hale & Crispin Wright - 2000 - In Paul Artin Boghossian & Christopher Peacocke (eds.), New Essays on the A Priori. Oxford, GB: Oxford University Press. pp. 286--319.
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  • (2 other versions)The Limits of Abstraction.Kit Fine - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993. Oxford, England: Clarendon Press.
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  • (2 other versions)The Limits of Abstraction.Kit Fine - 2005 - Philosophical Studies 122 (3):367-395.
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  • (2 other versions)The Limits of Abstraction.Kit Fine - 2004 - Bulletin of Symbolic Logic 10 (4):554-557.
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