Results for 'Grundgesetze der Arithmetik'

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  1.  36
    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 (...)
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  2. 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 (...)
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  3. The Consistency of Predicative Fragments of Frege’s Grundgesetze der Arithmetik.Richard G. Heck - 1996 - History and Philosophy of Logic 17 (1):209-220.
    As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell?s Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege?s Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, (...)
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  4.  82
    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 (...)
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  5.  21
    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 (...)
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  6.  32
    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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  7.  46
    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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  8. Definition by Induction in Frege's Grundgesetze der Arithmetik.Richard Heck - 1995 - In W. Demopoulos (ed.), Frege's Philosophy of Mathematics. Oxford University Press.
    This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.
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  9. Grundgesetze der Arithmetik. Section 56ff.Gottlob Frege - 1960 - In P. Geach & M. Black (eds.), Translations From the Philosophical Writings of Gottlob Frege. Blackwell.
     
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  10. Grundgesetze der Arithmetik Begriffsschriftlich Abgeleitet.Gottlob Frege - 1962 - G. Olms.
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  11. The Finite and the Infinite in Frege's Grundgesetze der Arithmetik.Richard Heck - 1998 - In Matthias Schirn (ed.), Philosophy of Mathematics Today. Oxford University Press.
    Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.
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  12.  3
    Die Unvollständigkeit der Fregeschen „Grundgesetze der Arithmetik“.Christian Thiel - 1977 - In Manfred Riedel & Jürgen Mittelstraß (eds.), Vernünftiges Denken: Studien Zur Praktischen Philosophie Und Wissenschaftstheorie. De Gruyter. pp. 104-106.
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  13.  9
    Review of Die Grundlagen der Arithmetik, §§82-3, by George Boolos and Richard G. Heck; The Finite and the Infinite in Frege's Grundgesetze der Arithmetik, by Richard G. Heck; On the Harmless Impredicativity of N= ('Hume's Principle'), by Crispin Wright; Neo-Fregeans: In Bad Company? By Michael Dummett; Response to Dummett, by Crispin Wright.William Demopoulos - 2000 - Bulletin of Symbolic Logic 6 (4):498-504.
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  14.  4
    Russell’s Notes on Frege’s Grundgesetze der Arithmetik, From §53.Bernard Linsky - 2006 - Russell: The Journal of Bertrand Russell Studies 26 (2).
    This paper completes a series of three devoted to the notes that Russell made on reading Gottlob Frege’s works beginning in the summer of 1902. Notes in the two previous papers were used in the preparation of Appendix A of The Principles of Mathematics, “The Logical and Arithmetical Doctrines of Frege”. The bulk of the notes published here are on the formal proofs in Grundgesetze der Arithmetik, which begin at §53 and continue through the rest of Vol. 1. (...)
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  15.  9
    Frege Against the Formalists. A Translation of Part of Grundgesetze der Arithmetik.Max Black & Gottlob Frege - 1953 - Journal of Symbolic Logic 18 (1):77-93.
  16.  65
    Gottlob Frege. Der Gedanke. Beiträge Zur Philosophie des Deutschen Idealismus, Vol. 1 No. 2 , Pp. 58–77. - Gottlob Frege. Die Verneinung. Beiträge Zur Philosophie des Deutschen Idealismus, Vol. 1 No. 3–4 , Pp. 143–157. - Max Black. Frege Against the Formalists. A Translation of Part of Grundgesetze der Arithmetik. Introductory Note. The Philosophical Review, Vol. 59 , Pp. 77–78. - Gottlob Frege. Frege Against the Formalists. E. Heine's and J. Thomae's Theories of Irrational Numbers. The Philosophical Review, Vol. 59 , Pp. 79–93, 202–220, 332–345. - Gottlob Frege. On Concept and Object. Mind, N.S. Vol. 60 , Pp. 168–180. - Daniela Gromska. L'Abbé Stanisław Kobyłecki. Studia Philosophica , Vol. 3 , Pp. 40–41. [4631-2; V 43.] - Daniela Gromska. Edward Stamm. Studia Philosophica , Vol. 3 , Pp. 43–45. [1851–12.3.] - Daniela Gromska. Stanisław Leśniewski. Studia Philosophica , Vol. 3 , Pp. 46–51. [2021-13; V 83, 84.] - Daniela Gromska. Leon Chwistek. Studia Philosophica , Vol. 3 , Pp. 51–54. [REVIEW]Alonzo Church - 1953 - Journal of Symbolic Logic 18 (1):93-94.
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  17.  16
    George Boolos and Richard G. HeckJnr. Die Grundlagen der Arithmetik, §§82–3. The Philosophy of Mathematics Today, Edited by Matthias Schirn, Clarendon Press, Oxford University, Oxford and New York 1998, Pp. 407–428. - Richard G. HeckJnr. The Finite and the Infinite in Frege's Grundgesetze der Arithmetik. The Philosophy of Mathematics Today, Edited by Matthias Schirn, Clarendon Press, Oxford University, Oxford and New York 1998 Pp. 429–466. - Crispin Wright. On the Harmless Impredicativity of N= . The Philosophy of Mathematics Today, Edited by Matthias Schirn, Clarendon Press, Oxford University, Oxford and New York 1998 Pp. 339–368. - Michael Dummett. Neo-Fregeans: In Bad Company? The Philosophy of Mathematics Today, Edited by Matthias Schirn, Clarendon Press, Oxford University, Oxford and New York 1998 Pp. 369–387. - Crispin Wright. Response to Dummett. The Philosophy of Mathematics Today, Edited by Matthias Schirn, Clarendon Press, Oxford University, Oxford and New York 1998 Pp. 389–4. [REVIEW]William Demopoulos - 2000 - Bulletin of Symbolic Logic 6 (4):498-504.
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  18. Michael Dummett.Grundgesetze der Arithmetik - 2010 - In Bernhard Weiss & Jeremy Wanderer (eds.), Reading Brandom: On Making It Explicit. Routledge.
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  19.  52
    Frege Against the Formalists (II): A Translation of Part of Grundgesetze der Arithmetik.Gottlob Frege - 1950 - Philosophical Review 59 (2):202-220.
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  20.  38
    Frege Against the Formalists. III: A Translation of Part of Grundgesetze der Arithmetik.Gottlob Frege - 1950 - Philosophical Review 59 (3):332-345.
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  21.  29
    Frege Against the Formalists. III: A Translation of Part of Grundgesetze der Arithmetik.Gottlob Frege - 1950 - Philosophical Review 59:332-345.
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  22.  8
    Frege Against the Formalists. I: A Translation of Part of Grundgesetze der Arithmetik.Gottlob Frege - 1950 - Philosophical Review 59 (1):77-93.
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  23.  26
    Die philosophische logik Gottlob freges. Ein kommentar, mit den texten Des vorworts zu grundgesetze der arithmetik und der logischen untersuchungen I–iv (review).Leila Haaparanta - 2011 - Journal of the History of Philosophy 49 (4):507-508.
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  24. 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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  25.  66
    Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects.Kai F. Wehmeier - 1999 - Synthese 121 (3):309-328.
    In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck (...)
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  26.  17
    Reading Frege's Grundgesetze.Richard Heck - 2012 - Oxford University Press.
    Richard G. Heck presents a new account of Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, which establishes it as a neglected masterpiece at the center of Frege's philosophy. He explores Frege's philosophy of logic, and argues that Frege knew that his proofs could be reconstructed so as to avoid Russell's Paradox.
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  27. 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 (...)
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  28.  9
    On the Consistency of the $\Delta_{1}^{1}$-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 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 schema would (...)
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  29.  36
    Referentiality in Frege's Grundgesetze.Martin Edward - 1982 - History and Philosophy of Logic 3 (2):151-164.
    In §§28-31 of his Grundgesetze der Arithmetik, Frege forwards a demonstration that every correctly formed name of his formal language has a reference. Examination of this demonstration, it is here argued, reveals an incompleteness in a procedure of contextual definition. At the heart of this incompleteness is a difference between Frege's criteria of referentiality and the possession of reference as it is ordinarily conceived. This difference relates to the distinction between objectual and substitutional quantification and Frege?s vacillation between (...)
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  30. Reading Frege's Grundgesetze. Heck Jr - 2012 - Oxford University Press UK.
    Richard G. Heck presents a new account of Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, which establishes it as a neglected masterpiece at the center of Frege's philosophy. He explores Frege's philosophy of logic, and argues that Frege knew that his proofs could be reconstructed so as to avoid Russell's Paradox.
     
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  31.  52
    O Principio Do Contexto Nas Grundgesetze de Frege (the Context Principle in Frege's Grundgesetze).Matthias Schirn - 1996 - Theoria 11 (3):177-201.
    Pretendo usar o exemplo dos nomes de percursos de valores como prova de que, contrariamente ao que Michael Resnik e Michael Dummett sustentam, Frege nunca abandonou o seu princípio do contexto: “Apenas no contexto de uma sentenya tem uma palavra significado”. Em particular, pretendo mostrar que a prova da completude com relação ao significado, que Frege tentou introduzir na linguagem formal das Grundgesetze der Arithmetik, baseia-se em uma aplicação do principio do contexto, e que, em consequencia, tambem nomes (...)
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  32.  27
    O principio do contexto nas Grundgesetze de Frege.Matthias Schirn - 1996 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 11 (3):177-201.
    Pretendo usar o exemplo dos nomes de percursos de valores como prova de que, contrariamente ao que Michael Resnik e Michael Dummett sustentam, Frege nunca abandonou o seu princípio do contexto: “Apenas no contexto de uma sentenya tem uma palavra significado”. Em particular, pretendo mostrar que a prova da completude com relação ao significado, que Frege tentou introduzir na linguagem formal das Grundgesetze der Arithmetik, baseia-se em uma aplicação do principio do contexto, e que, em consequencia, tambem nomes (...)
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  33.  11
    The Propositional Logic of Frege’s Grundgesetze: Semantics and Expressiveness.Eric D. Berg & Roy T. Cook - 2017 - Journal for the History of Analytical Philosophy 5 (6).
    In this paper we compare the propositional logic of Frege’s Grundgesetze der Arithmetik to modern propositional systems, and show that Frege does not have a separable propositional logic, definable in terms of primitives of Grundgesetze, that corresponds to modern formulations of the logic of “not”, “and”, “or”, and “if…then…”. Along the way we prove a number of novel results about the system of propositional logic found in Grundgesetze, and the broader system obtained by including identity. In (...)
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  34. Grundlagen der Arithmetik.Gottlob Frege - 1884 - Breslau: Wilhelm Koebner Verlag.
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  35.  53
    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 (...)
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  36.  29
    Die Grundlagen der Arithmetik.Gottlob Frege - 1988 - Felix Meiner Verlag.
    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.
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  37.  12
    Philosophie der Arithmetik.E. S. Husserl - 1892 - Philosophical Review 1 (3):327-330.
  38. Basic Laws of Arithmetic.Philip A. Ebert & Marcus Rossberg (eds.) - 2013 - Oxford University Press UK.
    This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik, with introduction and annotation. The importance of Frege's ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework.
     
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  39.  5
    Philosophie der Arithmetik: Mit Erganzenden Texten (1890-1901).Edmund Husserl & Lothar Eley - 1970 - Martinus Nijhoff.
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  40.  54
    Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl.Gottlob Frege - 1996 - Wittgenstein-Studien 3 (2):993-999.
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  41. 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 (...)
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  42.  20
    Frege's Approach to the Foundations of Analysis (1874–1903).Matthias Schirn - 2013 - History and Philosophy of Logic 34 (3):266-292.
    The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ?theory of quantity? (?Größenlehre?) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. (...)
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  43.  19
    Die Grundlagen der Arithmetik. Eine Logisch Mathematische Untersuchung Über den Begriff der Zahl.Gottlob Frege & Christian Thiel - 1988 - Journal of Symbolic Logic 53 (3):993-999.
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  44. Frege: A Philosophical Biography.Dale Jacquette - 2017 - Cambridge University Press.
    Gottlob Frege is one of the founding figures of analytic philosophy, whose contributions to logic, philosophical semantics, philosophy of language, and philosophy of mathematics set the agenda for future generations of theorists in these and related areas. Dale Jacquette's lively and incisive biography charts Frege's life from its beginnings in small-town north Germany, through his student days in Jena, to his development as an enduringly influential thinker. Along the way Jacquette considers Frege's ground-breaking Begriffschrift, in which he formulated his 'ideal (...)
     
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  45.  60
    Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic.Matthias Schirn - 2003 - Erkenntnis 59 (2):203 - 232.
    In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius (...)
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  46.  46
    Frege's Natural Numbers: Motivations and Modifications.Erich Reck - manuscript
    Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: the motivations (...)
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  47.  34
    Gottlob Frege: Basic Laws of Arithmetic.Philip A. Ebert & Marcus Rossberg (eds.) - 2013 - Oxford, UK: Oxford University Press.
    This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik (1893 and 1903), with introduction and annotation. As the culmination of his ground-breaking work in the philosophy of logic and mathematics, Frege here tried to show how the fundamental laws of arithmetic could be derived from purely logical principles.
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  48.  55
    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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  49.  79
    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. (...)
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  50.  53
    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 (...)
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