28 found
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  1.  19
    Realizability and recursive set theory.Charles McCarty - 1986 - Annals of Pure and Applied Logic 32:153-183.
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  2.  74
    Completeness and incompleteness for intuitionistic logic.Charles Mccarty - 2008 - Journal of Symbolic Logic 73 (4):1315-1327.
    We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic (...)
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  3.  31
    Intuitionistic sets and numbers: small set theory and Heyting arithmetic.Stewart Shapiro, Charles McCarty & Michael Rathjen - forthcoming - Archive for Mathematical Logic.
    It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. (...)
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  4.  19
    Variations on a thesis: intuitionism and computability.Charles McCarty - 1987 - Notre Dame Journal of Formal Logic 28 (4):536-580.
  5.  36
    Skolem's paradox and constructivism.Charles McCarty & Neil Tennant - 1987 - Journal of Philosophical Logic 16 (2):165 - 202.
  6.  32
    Constructive validity is nonarithmetic.Charles McCarty - 1988 - Journal of Symbolic Logic 53 (4):1036-1041.
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  7. The Axiom of Choice is False Intuitionistically (in Most Contexts).Charles Mccarty, Stewart Shapiro & Ansten Klev - 2023 - Bulletin of Symbolic Logic 29 (1):71-96.
    There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some (...)
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  8.  20
    Markov's principle, isols and Dedekind finite sets.Charles McCarty - 1988 - Journal of Symbolic Logic 53 (4):1042-1069.
  9.  24
    Antirealism and constructivism: Brouwer’s weak counterexamples: Antirealism and constructivism: Brouwer’s weak counterexamples.Charles Mccarty - 2013 - Review of Symbolic Logic 6 (1):147-159.
    Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to _Test_, the recognition condition, and the _BHK_ interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the _BHK_, then every relevant instance of the classical principle _Test_ is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is (...)
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  10.  90
    Paradox and Potential Infinity.Charles McCarty - 2013 - Journal of Philosophical Logic 42 (1):195-219.
    We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.
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  11. Carnap and Quine on Intuitionism.Charles McCarty - 2007 - Soochow Journal of Philosophical Studies 16:93 - 109.
  12.  13
    Subcountability under realizability.Charles McCarty - 1986 - Notre Dame Journal of Formal Logic 27 (2):210-220.
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  13.  11
    Arithmetic, Convention, Reality.Charles McCarty - 2014 - In Arithmetic, Convention, Reality. pp. 83-96.
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  14.  11
    At the Heart of Analysis: Intuitionism and Philosophy.Charles McCarty - 2006 - Philosophia Scientiae:81-94.
    One’s first impression is that Brouwer’s Continuity Theorem of intuitionistic analysis, that every total, real-valued function of a real variable is continuous, stands in straightforward contradiction to a simple theorem of conventional real analysis, that there are discontinuous, real-valued func­tions. Here we argue that, despite philosophical views to the contrary, first impressions are not misleading; the Brouwer Theorem, together with its proof, presents mathematicians and philosophers of mathematics with an antimony, one that can only be resolved by a close, foundational (...)
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  15.  4
    At the Heart of Analysis: Intuitionism and Philosophy.Charles McCarty - 2006 - Philosophia Scientiae:81-94.
    One’s first impression is that Brouwer’s Continuity Theorem of intuitionistic analysis, that every total, real-valued function of a real variable is continuous, stands in straightforward contradiction to a simple theorem of conventional real analysis, that there are discontinuous, real-valued func­tions. Here we argue that, despite philosophical views to the contrary, first impressions are not misleading; the Brouwer Theorem, together with its proof, presents mathematicians and philosophers of mathematics with an antimony, one that can only be resolved by a close, foundational (...)
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  16.  19
    Brouwer’s weak counterexamples and testability: Further remarks: Brouwer’s weak counterexamples and testability: Further remarks.Charles Mccarty - 2013 - Review of Symbolic Logic 6 (3):513-523.
    Straightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. (...)
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  17.  59
    Intuitionism and logical syntax.Charles McCarty - 2008 - Philosophia Mathematica 16 (1):56-77.
    , Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...)
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  18.  26
    Logical rules and the determinacy of meaning.Charles McCarty - 2018 - Studies in Logic, Grammar and Rhetoric 54 (1):89-98.
    The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove that, given (...)
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  19.  28
    Philosophy of Mathematics in the Twentieth Century: Selected Essays.Charles McCarty - 2016 - Philosophical Review Recent Issues 125 (2):298-302.
  20.  21
    Reconstructing a Logic from Tractatus: Wittgenstein’s Variables and Formulae.Charles McCarty & David Fisher - 2016 - In Sorin Costreie (ed.), Early Analytic Philosophy – New Perspectives on the Tradition. Cham, Switzerland: Springer Verlag.
    It is and has been widely assumed, e.g., in Hintikka and Hintikka, that the logical theory available from Wittgenstein’s Tractatus Logico-Philosophicus affords a foundation for the conventional logic represented in standard formulations of classical propositional, first-order predicate, and perhaps higher-order formal systems. The present article is a detailed attempt at a mathematical demonstration, or as much demonstration as the sources will allow, that this assumption is false by contemporary lights and according to a preferred account of argument validity. When Wittgenstein’s (...)
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  21.  54
    Structuralism and Isomorphism.Charles McCarty - 2013 - Philosophia Mathematica (1):nkt024.
    If structuralism is a true view of mathematics on which the statements of mathematicians are taken ‘at face value’, then there are both structures on which classical second-order arithmetic is a correct report, and structures on which intuitionistic second-order arithmetic is correct. An argument due to Dedekind then proves that structures and structures are isomorphic. Consequently, first- and second-order statements true in structures must hold in , and conversely. Since instances of the general law of the excluded third fail in (...)
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  22.  25
    Satisfiability is False Intuitionistically: A Question from Dana Scott.Charles McCarty - 2020 - Studia Logica 108 (4):803-813.
    Satisfiability or Sat\ is the metatheoretic statementEvery formally intuitionistically consistent set of first-order sentences has a model.The models in question are the Tarskian relational structures familiar from standard first-order model theory, but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat\ to be false outright. Following the lead of Carter :75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic. These metatheorems (...)
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  23.  3
    Thesis and Variations.Charles McCarty - 2006 - In Adam Olszewski, Jan Wolenski & Robert Janusz (eds.), Church's Thesis After 70 Years. Ontos Verlag. pp. 281-303.
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  24.  59
    The coherence of antirealism.Charles McCarty - 2006 - Mind 115 (460):947-956.
    The project of antirealism is to construct an assertibility semantics on which (1) the truth of statements obeys a recognition condition so that (2) counterexamples are forthcoming to the law of the excluded third and (3) intuitionistic formal predicate logic is provably sound and complete with respect to the associated notion of validity. Using principles of intuitionistic mathematics and employing only intuitionistically correct inferences, we show that prima facie reasonable formulations of (1), (2), and (3) are inconsistent. Therefore, it should (...)
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  25.  15
    Two questions from Dana Scott: Intuitionistic topologies and continuous functions.Charles McCarty - 2009 - Journal of Symbolic Logic 74 (2):689-692.
  26.  44
    Intuitionism: An introduction to a seminar. [REVIEW]Charles McCarty - 1983 - Journal of Philosophical Logic 12 (2):105 - 149.
  27.  40
    Wittgenstein on the Foundations of Mathematics. [REVIEW]Charles McCarty - 1981 - Grazer Philosophische Studien 14 (1):165-175.
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  28.  9
    Wittgenstein on the Foundations of Mathematics. [REVIEW]Charles McCarty - 1981 - Grazer Philosophische Studien 14 (1):165-175.
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