S. Hirokawa, Y. Komori and I. Takeuti: "A Reduction Rule for Peirce Formula", Studia Logica, 56, 3, pp. 419 -- 426, 1996. Home/Search Document Not in Database Summary Related Articles Check |
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A semantic view of classical proofs. - type-theoretic, categorical, .. - Ong (1996) (Correct)
....et al. 10] and relating them to classical proofs, there has been a great deal of interest in classical proofs. The following is a tentative (and incomplete) classification: ffl Algorithm extraction, control operators: Griffin [14] Murthy [22] Krivine [21] de Groote [9] Nakano [23] Hirokawa [16], Schwichtenberg and Berger [4] Coquand [6] etc. ffl Formal systems and calculi: Girard [11, 12] Parigot [24] Berardi and Barbanera [2] Danos, Joinet and Schellinx [8] etc. ffl Proofs and semantics of cut elimination: Girard [11] Hofmann [17] Coquand [5] Pfenning [25] Herbelin [15] ....
S. Hirokawa, Y. Komori, and I. Takeuti. A reduction rule for the peirce formula. Studia Logica, 1996. To appear.
A Notion of Classical Pure Type System - Barthe, Hatcliff, al. (1997) (6 citations) (Correct)
.... The most conventional formats are obtained by extending intuitionistic natural deduction with one rule for one of the three formulae: ffl excluded middle A :A; 8 Such a result was proved by model theoretic means in [91] Barthe, Hatcliff, S rensen Single Conclusioned DN [34,82] EM [24] PL [41,86] [69] Symmetric [4] Multi Conclusioned Act Pass [14,39,72] Intuitionistic Act Pass [39,66] Truly MC [95] Fig. 11. Classical lambda calculi and Natural Deduction Formats ffl double negation : A A; ffl Pierce s law ( A B) A) A. Remarkably, all these formats and derived endemic formats ....
S. Hirokawa, Y. Komori, and I. Takeuti. A reduction rule for the Peirce formula. Manuscript, 1994.
A Notion of Classical Pure Type System - Barthe, Hatcliff, Sørensen (1997) (6 citations) (Correct)
....should decide which of the above formulae will be taken as fundamental and define the syntax accordingly. In the case of simply typed calculus, the three possibilities have been used: double negation has been used in [30] excluded middle has been used in [22] and Pierce s law has been used in [42, 33]. In the case of classical pure type systems, it seems more reasonable to take an implicational formula as a primitive. In this paper, we have chosen to base ourselves on the double negation rule. The reason is that Pierce s law makes an implicit use of impredicativity and is therefore ....
S. Hirokawa, Y. Komori, and I. Takeuti. A reduction rule for the Peirce formula. Manuscript, 1994.
Case Calculus for Classical Logic - Baba, Hirokawa, Kashima, Komori.. (2000) Self-citation (Hirokawa Komori Takeuti) (Correct)
....implication and disjunction cannot be separated. The distinction of classical logic and intuitionistic logic can be clarified without negation and disjunction. The formula of Peirce ( ff fi) ff) ff is an example which distinguishes classical logic from intuitionistic logic. Hirokawa et al.[4, 5] introduce a formulation of classical implicational logic with this formula. But this system does not have subformula property. Therefore, we choose natural deduction formulation with single conclusion. It captures our intuition of logical reasoning such that we infer a conclusion from a set of ....
....implicational logic and its normalization. The normal form proof we obtain satisfies subformula property. We begin by Gentzen s natural deduction system for intuitionistic logic NJ [12] Since we consider implicational fragment, there are only two inference rules I and E. It is known, e.g. [4], that classical implicational logic is obtained by adding Peirce s formula as an axiom or by adding the inference rule P. 3 [ff] fi ff fi I . ff fi . ff fi E [ff fi] ff ff P We can consider this inference P as an implicational representation of the ....
S. Hirokawa, Y. Komori and I. Takeuti: "A Reduction Rule for Peirce Formula", Studia Logica, 56, 3, pp. 419 -- 426, 1996.
Combinatory Logic and lambda-Calculus for Classical Logic - Baba, Kameyama, Hirokawa (2000) Self-citation (Hirokawa) (Correct)
....defining calculi and studying relationship, we can consider how confluent subcalculus can 7 be obtained in different settings. Logical Reduction Rules The first class of classical reduction rules are called logical reduction rules, which originate from the first author and others P reduction [4]. It has a quite simple form, and moreover, the type assignment of the P reduction naturally induces the type ( ff fi) ff) ff, Peirce formula. The logical reduction rules are defined as follows: M :ff (C : ff ff N : ff ) N : ff M :ff (C) M ff fi (P ( ff fi) ff) ff N ....
....) reduces to . N : From these proof conversions, it is clear that the types are preserved by the reductions. Although the logical reduction rules in this style are natural, unrestricted use of (P) or (C) immediately causes non confluency. An example of collapse by (P) 8 is given in [4]. Let M be the following term. z ff :x ff ff ( u ff :u)z) P (v ff ff :vy ff ) Then we have the following reductions using (fi) and (P) M (P) fi) x(xy) M (fi) P) fi) xy To make the comparison with other calculi easier, we shift to another formulation of logical ....
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Hirokawa, S., S. Komori and I. Takeuti: "A Reduction Rule for Peirce Formula", Studia Logica 56, 3, pp. 419--426, 1996.
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