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  1. Natural deduction and Hilbert's ɛ-operator.Allen Hazen - 1987 - Journal of Philosophical Logic 16 (4):411 - 421.
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  • Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic.Vasil Penchev - 2021 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 13 (7):1-57.
    In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n (...)
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  • Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
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  • Semantics and Proof Theory of the Epsilon Calculus.Richard Zach - 2017 - In Ghosh Sujata & Prasad Sanjiva (eds.), Logic and Its Applications. ICLA 2017. Springer. pp. 27-47.
    The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and accessible presentations of its theory on the other. One significant early result for the original axiomatic proof system for the epsilon-calculus is the first epsilon theorem, for which a proof is sketched. The system itself is discussed, also relative to possible semantic interpretations. The problems facing (...)
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  • Harmonising natural deduction.Barry Hartley Slater - 2008 - Synthese 163 (2):187-198.
    Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can (...)
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