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Graham Leach-Krouse
Kansas State University
  1.  66
    Yablifying the Rosser Sentence.Graham Leach-Krouse - 2014 - Journal of Philosophical Logic 43 (5):827-834.
    In a recent paper , Urbaniak and Cieśliński describe an analogue of the Yablo Paradox, in the domain of formal provability. Just as the infinite sequence of Yablo sentences inherit the paradoxical behavior of the liar sentence, an infinite sequence of sentences can be constructed that inherit the distinctive behavior of the Gödel sentence. This phenomenon—the transfer of the properties of self-referential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—suggests a number of interesting logical questions. The (...)
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  2.  61
    What Russell Should Have Said to Burali–Forti.Salvatore Florio & Graham Leach-Krouse - 2017 - Review of Symbolic Logic 10 (4):682-718.
    The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some (...)
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  3. Structural-Abstraction Principles.Graham Leach-Krouse - 2015 - Philosophia Mathematica:nkv033.
    In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, (...)
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  4. Provability, Mechanism, and the Diagonal Problem.Graham Leach-Krouse - 2016 - In Leon Horsten & Philip Welch (eds.), Gödel's Disjunction: the Scope and Limits of Mathematical Knowledge. Oxford, UK: pp. 211-240.
  5. Carnap: An Open Framework for Formal Reasoning in the Browser.Graham Leach-Krouse - 2018 - Electronic Proceedings in Theoretical Computer Science 267:70-88.
    This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap’s implementation, along (...)
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  6.  38
    Burali-Forti as a Purely Logical Paradox.Graham Leach-Krouse - 2019 - Journal of Philosophical Logic 48 (5):885-908.
    Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, (...)
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  7.  29
    Generalizing Boolos’ Theorem.Graham Leach-Krouse - 2017 - Review of Symbolic Logic 10 (1):80-91.
    It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory  HP2, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of HP2 in PA2 . Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of PA2 to be recovered from some model of HP2. So the space (...)
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  8. 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 in a (...)
     
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