# A lambda proof of the P-W theorem

Sachio Hirokawa, Yuichi Komori, Misao Nagayama

### 抄録

The logical system P-W is an implicational non-commutative intuitionistic logic defined 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 λx.x. Here the HRML λ-terms correspond, via the Curry-Howard isomorphism, to the P-W proofs in natural deduction style.

元の言語 英語 1841-1849 9 Journal of Symbolic Logic 65 4 https://doi.org/10.2307/2695080 出版済み - 1 1 2000

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Theorem
Natural Deduction
P Systems
Intuitionistic Logic
Term
Axiom
Substitution
Isomorphism
Corollary
Calculus
Closed
Style
Calculi
Logic
Modus Ponens

• Philosophy
• Logic

### これを引用

A lambda proof of the P-W theorem. / Hirokawa, Sachio; Komori, Yuichi; Nagayama, Misao.

：: Journal of Symbolic Logic, 巻 65, 番号 4, 01.01.2000, p. 1841-1849.

Hirokawa, S, Komori, Y & Nagayama, M 2000, 'A lambda proof of the P-W theorem', Journal of Symbolic Logic, 巻. 65, 番号 4, pp. 1841-1849. https://doi.org/10.2307/2695080
Hirokawa, Sachio ; Komori, Yuichi ; Nagayama, Misao. / A lambda proof of the P-W theorem. ：: Journal of Symbolic Logic. 2000 ; 巻 65, 番号 4. pp. 1841-1849.
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