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  1. Takeuti's proof theory in the context of the Kyoto School.Andrew Arana - 2019 - Jahrbuch Für Philosophie Das Tetsugaku-Ronso 46:1-17.
    Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...)
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  • 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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  • Neo-Logicism and Its Logic.Panu Raatikainen - 2020 - History and Philosophy of Logic 41 (1):82-95.
    The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...)
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  • Resolution and the consistency of analysis.Peter B. Andrews - 1974 - Notre Dame Journal of Formal Logic 15 (1):73-84.
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  • Self-verifying axiom systems, the incompleteness theorem and related reflection principles.Dan Willard - 2001 - Journal of Symbolic Logic 66 (2):536-596.
    We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...)
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  • Cut-Elimination for Quantified Conditional Logic.Christoph Benzmüller - 2017 - Journal of Philosophical Logic 46 (3):333-353.
    A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.
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  • (2 other versions)Non-constructive Proofs of a Metamathematical Theorem Concerning the Consistency of Analysis and its Extension.Sh^|^Ocirc Maehara, Ji, Toshio Nishimura & Setsuya Seki - 1960 - Annals of the Japan Association for Philosophy of Science 1 (5):269-288.
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  • The machinery of consistency proofs.Mariko Yasugi - 1989 - Annals of Pure and Applied Logic 44 (1-2):139-152.
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  • ALONZO: Deduktionsagenten höherer Ordnung für Mathematische Assistenzsysteme.Benzmüller Christoph - 2003
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