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  1. Begriffsschrift.Gottlob Frege - 1967 - In Jean Van Heijenoort (ed.), From Frege to Gödel. Cambridge,: Harvard University Press. pp. 1-83.
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  • Deductive versus Expressive Power: A Pre-Godelian Predicament.Neil Tennant - 2000 - Journal of Philosophy 97 (5):257.
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  • Mechanical procedures and mathematical experience.Wilfried Sieg - 1994 - In Alexander George (ed.), Mathematics and mind. New York: Oxford University Press. pp. 71--117.
    Wilfred Sieg. Mechanical Procedures and Mathematical Experience.
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  • Type Theory and Homotopy.Steve Awodey - 2012 - In Peter Dybjer, Sten Lindström, Erik Palmgren & Göran Sundholm (eds.), Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf. Dordrecht, Netherland: Springer. pp. 183-201.
    The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory and higher-dimensional category theory.
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  • Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (2):77-94.
    This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  • (1 other version)Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  • Completeness and categoricity: Frege, gödel and model theory.Stephen Read - 1997 - History and Philosophy of Logic 18 (2):79-93.
    Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...)
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  • A Defense of Second-Order Logic.Otávio Bueno - 2010 - Axiomathes 20 (2-3):365-383.
    Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, (...)
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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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  • (1 other version)Philosophy of logic.Willard Van Orman Quine - 1986 - Cambridge: Harvard University Press. Edited by Simon Blackburn & Keith Simmons.
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  • (2 other versions)Mathematical truth.Paul Benacerraf - 1973 - Journal of Philosophy 70 (19):661-679.
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  • Logical Pluralism.Gillian Russell - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    A survey of contemporary work on logical pluralism.
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  • Gaps between logical theory and mathematical practice.John Corcoran - 1973 - In Mario Bunge (ed.), The methodological unity of science. Boston,: Reidel. pp. 23--50.
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  • (1 other version)Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
    We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...)
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  • First-order logic, second-order logic, and completeness.Marcus Rossberg - 2004 - In Vincent F. Hendricks (ed.), First-order logic revisited. Berlin: Logos. pp. 303-321.
    This paper investigates the claim that the second-order consequence relation is intractable because of the incompleteness result for SOL. The opponents’ claim is that SOL cannot be proper logic since it does not have a complete deductive system. I argue that the lack of a completeness theorem, despite being an interesting result, cannot be held against the status of SOL as a proper logic.
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  • (2 other versions)Philosophy of Logic.W. V. Quine - 2005 - In José Medina & David Wood (eds.), Truth. Malden, MA: Blackwell.
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  • Second-order logic and foundations of mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
    We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...)
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  • (1 other version)Second-Order Languages and Mathematical Practice.Stewart Shapiro - 1989 - Journal of Symbolic Logic 54 (1):291-293.
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  • (1 other version)Second-order languages and mathematical practice.Stewart Shapiro - 1985 - Journal of Symbolic Logic 50 (3):714-742.
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  • (1 other version)Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.
    Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.
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  • Is There Completeness in Mathematics after Gödel?Jaakko Hintikka - 1989 - Philosophical Topics 17 (2):69-90.
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  • Ancestral arithmetic and Isaacson's Thesis.Peter Smith - 2008 - Analysis 68 (1):1-10.
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