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  1.  21
    On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects.Yuta Takahashi - 2018 - Annals of the Japan Association for Philosophy of Science 27:1-26.
    Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...)
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  2.  15
    Completeness of Second-Order Intuitionistic Propositional Logic with Respect to Phase Semantics for Proof-Terms.Yuta Takahashi & Ryo Takemura - 2019 - Journal of Philosophical Logic 48 (3):553-570.
    Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. In this paper, (...)
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  3.  3
    Reading Gentzen's Three Consistency Proofs Uniformly.Ryota Akiyoshi & Yuta Takahashi - 2013 - Journal of the Japan Association for Philosophy of Science 41 (1):1-22.
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  4. Gentzen’s 1935 Consistency Proof and the Interpretation of its Implication.Yuta Takahashi - 2018 - Proceedings of the XXIII World Congress of Philosophy 55:73-78.
    In this paper, I will argue from a historical perspective that Gentzen’s 1935 consistency proof of 1st order Peano Arithmetic PA principally aimed to give a finitist interpretation of implication and this aspect of the 1935 proof emerged as the attempt to cope with the non-finiteness in BHK-interpretation of implication. My argument consists of two parts. First, I will explain that the fundamental idea of the 1935 proof is to show the soundness of PA on some finitist interpretation and Gentzen (...)
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