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A lambda proof of the P-W theorem

Published online by Cambridge University Press:  12 March 2014

Sachio Hirokawa ,
Yuichi Komori  and
Misao Nagayama
Sachio Hirokawa
Affiliation:
Computer Center, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-8581, Japan, E-mail:[email protected]
Yuichi Komori
Affiliation:
Department of Mathematics, Faculty of Science, Chiba University, Yayoi-Cho 1-33, Inage-Ku. CHIBA 263-8522, Japan, E-mail:[email protected]
Misao Nagayama
Affiliation:
Department of Mathematics, Tokyo Woman's Christian University, Zempuku-JI 2-6-1, Suginami, Tokyo 167-8585, Japan, E-mail:[email protected]
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Abstract

The logical system P-W is an implicational non-commutative intuitionistic logic denned by axiom schemes B − (b → c) → (a → b) → a → c. B′ = (a → b) → (b → c) → a → c. I - a → a with the rules of modus ponens and substitution. The P-W problem is a problem asking whether α - β holds if α → β and β → α are both provable in P-W. The answer is affirmative. The first to prove this was E. P. Martin by a semantical method.

In this paper, we give the first proof of Martin's theorem based on the theory of simply typed λ-calculus. This proof is obtained as a corollary to the main theorem of this paper, shown without using Martin's Theorem, that any closed hereditary right-maximal linear (HRML) λ-term of type α → α is βη-reducible to λxx. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1841 - 1849
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Anderson, A. R. and Belnap, N. D. Jr., Entailment, vol. 1, Princeton University Press, Princeton. N.J., 1975.Google Scholar
[2]Barendregt, H., The lambda calculus, second ed., North-Holland, Amsterdam, 1984.Google Scholar
[3]Dezani-Ciancaglini, M., Characterization of normal forms possessing inverse in the λ-β-η calculus, Theoretical Computer Science, vol. 2 (1976), pp. 323–337.CrossRefGoogle Scholar
[4]Fine, K. and Martin, E. P., Progressive logic — the conjunction case, manuscript. October 13, 1994.Google Scholar
[5]Hirokawa, S., The proof of α → a in P-W, this Journal, vol. 61 (1996), pp. 195–221.Google Scholar
[6]Komori, Y., Syntactic investigations into BI logic and BB′I Logic, Studia Logica, vol. 53 (1994), pp. 397–416.CrossRefGoogle Scholar
[7]Kron, A., A constructive proof of a theorem in relevant logic, Zeitschrift fÜr Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 423–430.CrossRefGoogle Scholar
[8]Martin, E. P. and Meyer, R. K., Solution to the P-W problem, this Journal, vol. 47 (1982), pp. 867–887.Google Scholar
[9]Nagayama, M., Normalization theorem for P-W, Bulletin of the Section of Logic, vol. 28 (1999), pp. 83–85.Google Scholar
[10]Powers, L., On P-W, The Relevance Logic Newsletter, vol. 1 (1976), pp. 131–142.Google Scholar
[11]Trigg, P., Hindley, J. R., and Bunder, M. W., Combinatory abstraction using B, B′ and friends, Theoretical Computer Science, vol. 135 (1994), pp. 405–422.CrossRefGoogle Scholar