Results for 'Frege's abstraction scheme'

968 found
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  1.  30
    Fregean Extensions of First‐Order Theories.John L. Bell - 1994 - Mathematical Logic Quarterly 40 (1):27-30.
    It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.
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  2.  69
    Frege’s Attack on “Abstraction” and his Defense of the “Applicability” of Arithmetic.Daniël F. M. Strauss - 2003 - South African Journal of Philosophy 22 (1):63-80.
    The traditional understanding of abstraction operates on the basis of the assumption that only entities are subject to thought processes in which particulars are disregarded and commonalities are lifted out (the so-called method of genus proximum and differentia specifica). On this basis Frege criticized the notion of abstraction and convincingly argued that (this kind of) “entitary- directed” abstraction can never provide us with any numbers. However, Frege did not consider the alternative of “property- abstraction.” In this (...)
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  3.  94
    Grundlagen, Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context.Paolo Mancosu - 2015 - History and Philosophy of Logic 36 (1):62-89.
    I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a (...)
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  4. Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory.Edward N. Zalta - 1999 - Journal of Philosophical Logic 28 (6):619-660.
    In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal (...)
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  5.  36
    Frege and the Fundamental Abstraction.Jim Hutchinson - forthcoming - Canadian Journal of Philosophy.
    According to Charles Travis, Frege’s principle “always to sharply separate the psychological from the logical, the subjective from the objective” involves a move called “the fundamental abstraction.” I try to explain what this abstraction is and why it is interesting. I then raise a problem for it, and describe what I think is a better way to understand Frege’s principle.
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  6.  47
    Frege on the introduction of real and complex numbers by abstraction and cross-sortal identity claims.Matthias Schirn - 2023 - Synthese 201 (6):1-18.
    In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers in (...)
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  7. 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 (...)
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  8. Frege's Result: Frege's Theorem and Related Matters.Hirotoshi Tabata - 2012 - Frontiers of Philosophy in China 7 (3):351-366.
    One of the remarkable results of Frege’s Logicism is Frege’s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume’s Principle (HP) without using Frege’s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles—the abstraction principles—incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.
     
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  9.  31
    Abstraction and Infinity.Paolo Mancosu - 2016 - Oxford, England: Oxford University Press.
    Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at (...)
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  10.  31
    Selected Letters [review of Nicholas Griffin, ed., The Selected Letters of Bertrand Russell, Vol. 1: The Private Years, 1884-1914 ]. [REVIEW]Katharine Tait - 1992 - Russell: The Journal of Bertrand Russell Studies 12 (2):211-222.
    In lieu of an abstract, here is a brief excerpt of the content:'kvieuJs SELECTED· LETTERS KATHARINE TAIT Carn Voel Porthcurno,- Cornwall TRI9 6LN, England Nicholas Griffin. The Selected Letters ofBertrand Russel~ Vol. I: The Private Years, I884-I9I4. London: Allen Lane the Penguin PreSs, 1992. Pp. xxi, 553.£25.00; C$47.99; US$35·00. Nicholas Griffin has done an admirable job of selecting and explaining the letters in this first volume. It is amazingly to his credit that he 'manages to be so well acquainted with (...)
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  11. 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 (...)
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  12. 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 (...)
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  13.  47
    Crispin Wright. On the philosophical significance of Frege's theorem. Language, thought, and logic, Essays in honour of Michael Dummett, edited by Richard G. HeckJnr., Oxford University Press, Oxford and New York 1998 , pp. 201–244. - George Boolos. Is Hume's principle analytic? Language, thought, and logic, Essays in honour of Michael Dummett, edited by Richard G. HeckJnr., Oxford University Press, Oxford and New York 1998 , pp. 245–261. - Charles Parsons. Wright on abstraction and set theory. Language, thought, and logic, Essays in honour of Michael Dummett, edited by Richard G. HeckJnr., Oxford University Press, Oxford and New York 1998 , pp. 263–271. - Richard G. HeckJnr. The Julius Caesar objection. Language, thought, and logic, Essays in honour of Michael Dummett, edited by Richard G. HeckJnr., Oxford University Press, Oxford and New York 1998 , pp. 273–308. [REVIEW]William Demopoulos - 1998 - Journal of Symbolic Logic 63 (4):1598-1602.
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  14.  89
    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 (...)
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  15. Abstraction and grounding.Louis deRosset & Øystein Linnebo - 2023 - Philosophy and Phenomenological Research 109 (1):357-390.
    The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop this (...)
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  16. 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 (...)
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  17.  76
    Analysis, abstraction principles, and slingshot arguments.James Levine - 2006 - Ratio 19 (1):43–63.
    Frege's views regarding analysis and synomymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege's criterion of synonymy, according to which two sentence express the same thought if one cannot understand them without regarding them as having the same truth–value. In a recent article in this journal, Drai (...)
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  18.  9
    Frege’s platonism and mathematical creation: some new perspectives.Matthias Schirn - 2024 - Synthese 205 (1):1-62.
    In this three-part essay, I investigate Frege’s platonist and anti-creationist position in Grundgesetze der Arithmetik and to some extent also in Die Grundlagen der Arithmetik. In Sect. 1.1, I analyze his arithmetical and logical platonism in Grundgesetze. I argue that the reference-fixing strategy for value-range names—and indirectly also for numerical singular terms—that Frege pursues in Grundgesetze I gives rise to a conflict with the supposed mind- and language-independent existence of numbers and logical objects in general. In Sect. 1.2 and 1.3, (...)
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  19. Frege’s Logicism and the Neo-Fregean Project.Matthias Schirn - 2014 - Axiomathes 24 (2):207-243.
    Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...)
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  20. Frege's Elucidatory Holism.Clinton Tolley - 2011 - Inquiry: An Interdisciplinary Journal of Philosophy 54 (3):226-251.
    Abstract I argue against the two most influential readings of Frege's methodology in the philosophy of logic. Dummett's ?semanticist? reading sees Frege as taking notions associated with semantical content?and in particular, the semantical notion of truth?as primitive and as intelligible independently of their connection to the activity of judgment, inference, and assertion. Against this, the ?pragmaticist? reading proposed by Brandom and Ricketts sees Frege as beginning instead from the independent and intuitive grasp that we allegedly have on the latter (...)
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  21. Neologicism, Frege's Constraint, and the Frege‐Heck Condition.Eric Snyder, Richard Samuels & Stewart Shapiro - 2018 - Noûs 54 (1):54-77.
    One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...)
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  22. Coalgebra And Abstraction.Graham Leach-Krouse - 2021 - Notre Dame Journal of Formal Logic 62 (1):33-66.
    Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra (...)
     
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  23. Frege's proof of referentiality.Øystein Linnebo - 2004 - Notre Dame Journal of Formal Logic 45 (2):73-98.
    I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which (...)
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  24. Frege's context principle: An interpretation.Joongol Kim - 2011 - Pacific Philosophical Quarterly 92 (2):193-213.
    This paper presents a new interpretation of Frege's context principle on which it applies primarily to singular terms for abstract objects but not necessarily to singular terms for ordinary objects.
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  25. Frege’s Cardinals as Concept-correlates.Gregory Landini - 2006 - Erkenntnis 65 (2):207-243.
    In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation (...)
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  26.  20
    Frege's Argument for Platonism.Ivan Kasa - 2011 - In Michael Bruce & Steven Barbone, Just the Arguments. Chichester, West Sussex, U.K.: Wiley‐Blackwell. pp. 370–372.
  27.  94
    Frege's Commitment to an Infinite Hierarchy of Senses.Daniel R. Boisvert & Christopher M. Lubbers - 2003 - Philosophical Papers 32 (1):31-64.
    Abstract Though it has been claimed that Frege's commitment to expressions in indirect contexts not having their customary senses commits him to an infinite number of semantic primitives, Terrence Parsons has argued that Frege's explicit commitments are compatible with a two-level theory of senses. In this paper, we argue Frege is committed to some principles Parsons has overlooked, and, from these and other principles to which Frege is committed, give a proof that he is indeed committed to an (...)
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  28.  96
    Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint.Crispin Wright - 2000 - Notre Dame Journal of Formal Logic 41 (4):317--334.
    We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term (...) Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)
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  29. 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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  30. Frege's Paradise and the Paradoxes.Sten Lindström - 2003 - In Frederick Stoutland, Krister Segerberg & Rysiek Śliwiński, A philosophical smorgasbord: essays on action, truth, and other things in honour of Frederick Stoutland. Uppsala: Uppsala Universitet.
    The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the so-called Russell-Myhill antinomy, an intensional analogue of Russell’s more well-known paradox for extensions. Frege’s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth and (...)
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  31.  85
    Frege’s Puzzle on the Santa Monica Beach De Jure Co-reference and the Logical Appraisal of Rational Agents.Emiliano Boccardi - 2018 - Manuscrito 41 (1):1-31.
    ABSTRACT In this paper, I argue that a number of influential Millian responses to Frege’s puzzle, which consist in denying that Frege’s data apply to natural languages, are not viable if logic is to play its role in legitimizing the logical appraisal of rational subjects. A notion of validity which does justice to the normativity of logic must make room for a distinction between valid inferences and enthymemes. I discuss the prospects of formal, relevant and manifest validity as candidates for (...)
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  32.  76
    Reflections on Frege’s Theory of Real Numbers†.Peter Roeper - 2020 - Philosophia Mathematica 28 (2):236-257.
    ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.
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  33.  79
    Abstraction without exceptions.Luca Zanetti - 2021 - Philosophical Studies 178 (10):3197-3216.
    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 (...)
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  34. Abstraction, Properties, and Immanent Realism.E. Jonathan Lowe - 1999 - The Proceedings of the Twentieth World Congress of Philosophy 2:195-205.
    Objects which philosophers have traditionally categorized as abstract are standardly referred to by complex noun phrases of certain canonical forms, such as ‘the set of Fs’, ‘the number of Fs’, ‘the proposition that P’, and ‘the property of being F’. It is no accident that such noun phrases are well-suited to appear in ‘Fregean’ identity-criteria, or ‘abstraction’ principles, for which Frege’s criterion of identity for cardinal numbers provides the paradigm. Notoriously, such principlesare apt to create paradoxes, and the most (...)
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  35. 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 (...)
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  36.  64
    Frege’s Foundations and Intuitionistic Logic.G. Kreisel - 1984 - The Monist 67 (1):72-91.
    Summary. This article develops two principal points. First, the so-called rivals of logical foundations, associated with Zermelo, Hilbert, and Brouwer, are here regarded as variants; specifically: to simplify, refine, resp. extend Frege’s scheme. Each of the variations is seen as a special case of a familiar strategy in the pursuit of knowledge. In particular, the extension provided by Brouwer’s intuitionistic logic concerns the class of propositions considered: about incompletely defined objects such as choice sequences. In contrast, Frege or, for (...)
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  37.  50
    Frege's horizontal and the liar-paradox.Dirk Greimann - 2003 - Manuscrito 26 (2):359-387.
    According to Peter Aczel, the inconsistency of Frege’s system in Grundgesetze is due, not to the introduction of sets, as is usually thought, but to the introduction of the Horizontal. His argument is that the principles governing sets are intuitively correct and therefore consistent, while the scheme introducing the Horizontal amounts to an internal definition of truth conflicting with Tarski’s classic result on the undefinability of truth in the object language. The aim of this paper is to show that (...)
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  38.  52
    The semantics of Frege's Grundgesetze.John N. Martin - 1984 - History and Philosophy of Logic 5 (2):143-176.
    Quantifiers in Frege's Grundgesetze like are not well-defined because the part Fx & Gx stands for a concept but the yoking conjunction is horizontalised and must stand for a truth-value. This standard interpretation is rejected in favor of a substitutional reading that, it is argued, both conforms better to the text and is well-defined. The theory of the horizontal is investigated in detail and the composite reading of Frege's connectives as made up of horizontals is rejected. The sense (...)
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  39.  61
    Abstract singular reference: A dilemma for Dummett.Alexander Miller - 1991 - Southern Journal of Philosophy 29 (2):257-269.
    Michael Dummett has attempted to give an account of the semantics of abstract singular terms which steers a middle course between reductionism and full-blown Platonism concerning their references: according to this middle position, reference, in the case of abstract singular terms, becomes "a matter wholly internal to the language." My main aim in this paper is to show that Dummett's arguments are in some considerable tension with more general features of his interpretation of Frege's philosophical semantics, so that given (...)
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  40. 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 (...)
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  41.  78
    Singular Propositions, Abstract Constituents, and Propositional Attitudes.Edward N. Zalta - 1989 - In Joseph Almog, John Perry & Howard Wettstein, Themes From Kaplan. New York: Oxford University Press. pp. 455--78.
    The author resolves a conflict between Frege's view that the cognitive significance of coreferential names may be distinct and Kaplan's view that since coreferential names have the same "character", they have the same cognitive significance. A distinction is drawn between an expression's "character" and its "cognitive character". The former yields the denotation of an expression relative to a context (and individual); the latter yields the abstract sense of an expression relative to a context (and individual). Though coreferential names have (...)
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  42.  70
    A Reassessment of Cantorian Abstraction based on the ε-operator.Nicola Bonatti - forthcoming - Synthese.
    Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...)
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  43. What is Frege's Julius caesar problem?Dirk Greimann - 2003 - Dialectica 57 (3):261-278.
    This paper aims to determine what kind of problem Frege's famous “Julius Caesar problem” is. whether it is to be understood as the metaphysical problem of determining what kind of things abstract objects like numbers or value‐courses are, or as the epistemological problem of providing a means of recognizing these objects as the same again, or as the logical problem of providing abstract sortal concepts with a sharp delimitation in order to fulfill the law of excluded middle, or as (...)
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  44.  41
    Frege's Realism.Gregory Currie - 1978 - Inquiry: An Interdisciplinary Journal of Philosophy 21 (1-4):218-221.
    In this note the claim is defended that Frege was a realist in the sense that he attributed causal efficacy to certain abstract objects. The arguments of Dummett and Sluga (cf. Inquiry, Vols. 18, 19, and 20 [1975–77]) to the contrary are criticized.
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    (1 other version)Russell's Arguments against Frege's Sense-Reference Distinction.Paweł Turnau - 1991 - Russell: The Journal of Bertrand Russell Studies 11 (1):52-66.
    In lieu of an abstract, here is a brief excerpt of the content:RUSSELLS ARGUMENT AGAINST FREGE'S SENSE-REFERENCE DISTINCTION PAWEL TURNAu Philosophy I Jagiellonian University Cracow, Poland I n "On Denoting"l Russell argued that Frege's theory of sense and reference was an "inextricable tangle", but, ironically, many readers found the argument even more knotry. In an effort to make sense of it, commentators were often driven to attribute to Russell quite obvious and simple fallacies. A different approach was taken (...)
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  46. Metaphysical separatism and epistemological autonomy in Frege’s philosophy and beyond.Jim Hutchinson - 2022 - British Journal for the History of Philosophy 30 (6):1096-1120.
    Commentators regularly attribute to Frege realist, idealist, and quietist responses to metaphysical questions concerning the abstract objects he calls ‘thoughts’. But despite decades of effort, the evidence offered on behalf of these attributions remains unconvincing. I argue that Frege deliberately avoids commitment to any of these positions, as part of a metaphysical separatist policy motivated by the fact that logic is epistemologically autonomous from metaphysics. Frege’s views and arguments prove relevant to current attempts to argue for epistemological autonomy, particularly that (...)
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    Spiritus Asper versus Lambda: On the Nature of Functional Abstraction.Ansten Klev - 2023 - Notre Dame Journal of Formal Logic 64 (2):205-223.
    The spiritus asper as used by Frege in a letter to Russell from 1904 bears resemblance to Church’s lambda. It is natural to ask how they relate to each other. An alternative approach to functional abstraction developed by Per Martin-Löf some thirty years ago allows us to describe the relationship precisely. Frege’s spiritus asper provides a way of restructuring a unary function name in Frege’s sense such that the argument place indicator occurs all the way to the right. Martin-Löf’s (...)
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  48. A Phenomenology of Race in Frege's Logic.Joshua M. Hall - forthcoming - Humanities Bulletin.
    This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April 9) supports (...)
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  49. Relative categoricity and abstraction principles.Sean Walsh & Sean Ebels-Duggan - 2015 - Review of Symbolic Logic 8 (3):572-606.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, (...)
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    Frege’s Class Theory and the Logic of Sets.Neil Tennant - 2024 - In Thomas Piecha & Kai F. Wehmeier, Peter Schroeder-Heister on Proof-Theoretic Semantics. Springer. pp. 85-134.
    We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, (...)
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