16 found
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  1.  5
    Arithmetical Completeness Theorem for Modal Logic $$Mathsf{}$$.Taishi Kurahashi - 2018 - Studia Logica 106 (2):219-235.
    We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \ provability predicate of T whose provability logic is precisely the modal logic \. For this purpose, we introduce a new bimodal logic \, and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \.
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  2.  16
    Henkin Sentences and Local Reflection Principles for Rosser Provability.Taishi Kurahashi - 2016 - Annals of Pure and Applied Logic 167 (2):73-94.
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  3.  9
    Generalizations of Gödel’s Incompleteness Theorems for ∑N-Definable Theories of Arithmetic.Makoto Kikuchi & Taishi Kurahashi - 2017 - Review of Symbolic Logic 10 (4):603-616.
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  4.  6
    Arithmetical Soundness and Completeness for $$Varvec{Sigma }_{Varvec{2}}$$ Numerations.Taishi Kurahashi - 2018 - Studia Logica 106 (6):1181-1196.
    We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \, there exists a \ numeration \\) of T such that the provability logic of the provability predicate \\) naturally constructed from \\) is exactly \ \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.
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  5.  40
    On Proofs of the Incompleteness Theorems Based on Berry's Paradox by Vopěnka, Chaitin, and Boolos.Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai - 2012 - Mathematical Logic Quarterly 58 (4-5):307-316.
    By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...)
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  6.  22
    Rosser-Type Undecidable Sentences Based on Yablo’s Paradox.Taishi Kurahashi - 2014 - Journal of Philosophical Logic 43 (5):999-1017.
    It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type formalizations of (...)
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  7.  9
    Universal Rosser Predicates.Makoto Kikuchi & Taishi Kurahashi - 2017 - Journal of Symbolic Logic 82 (1):292-302.
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  8.  6
    Rosser Provability and Normal Modal Logics.Taishi Kurahashi - forthcoming - Studia Logica:1-21.
    In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic \. Secondly, we introduce a new normal modal logic \ which is a proper extension of \, and prove that there exists a Rosser provability predicate whose provability logic includes \.
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  9.  6
    Three Short Stories Around Gödel's Incompleteness Theorems.Makoto Kikuchi & Taishi Kurahashi - 2011 - Journal of the Japan Association for Philosophy of Science 38 (2):75-80.
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  10.  12
    Arithmetical Interpretations and Kripke Frames of Predicate Modal Logic of Provability.Taishi Kurahashi - 2013 - Review of Symbolic Logic 6 (1):1-18.
    Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical interpretation. Montagna’s results (...)
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  11.  32
    Liar-Type Paradoxes and the Incompleteness Phenomena.Makoto Kikuchi & Taishi Kurahashi - 2016 - Journal of Philosophical Logic 45 (4):381-398.
    We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s (...)
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  12.  3
    On Arithmetical Completeness of the Logic of Proofs.Sohei Iwata & Taishi Kurahashi - 2019 - Annals of Pure and Applied Logic 170 (2):163-179.
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  13.  6
    On Partial Disjunction Properties of Theories Containing Peano Arithmetic.Taishi Kurahashi - 2018 - Archive for Mathematical Logic 57 (7-8):953-980.
    Let \ be a class of formulas. We say that a theory T in classical logic has the \-disjunction property if for any \ sentences \ and \, either \ or \ whenever \. First, we characterize the \-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of (...)
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  14.  16
    On Predicate Provability Logics and Binumerations of Fragments of Peano Arithmetic.Taishi Kurahashi - 2013 - Archive for Mathematical Logic 52 (7-8):871-880.
    Solovay proved (Israel J Math 25(3–4):287–304, 1976) that the propositional provability logic of any ∑2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179–189, 1984) that predicate provability logics of Peano arithmetic and Bernays–Gödel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289–1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a binumeration of the (...)
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  15.  2
    Provability Logics Relative to a Fixed Extension of Peano Arithmetic.Taishi Kurahashi - 2018 - Journal of Symbolic Logic 83 (3):1229-1246.
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  16.  5
    Illusory Models of Peano Arithmetic.Makoto Kikuchi & Taishi Kurahashi - 2016 - Journal of Symbolic Logic 81 (3):1163-1175.
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