### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 1841-1849 |

Number of pages | 9 |

Journal | Journal of Symbolic Logic |

Volume | 65 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Philosophy
- Logic

### Cite this

*Journal of Symbolic Logic*,

*65*(4), 1841-1849. https://doi.org/10.2307/2695080

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

Research output: Contribution to journal › Article

*Journal of Symbolic Logic*, vol. 65, no. 4, pp. 1841-1849. https://doi.org/10.2307/2695080

}

TY - JOUR

T1 - A lambda proof of the P-W theorem

AU - Hirokawa, Sachio

AU - Komori, Yuichi

AU - Nagayama, Misao

PY - 2000/1/1

Y1 - 2000/1/1

N2 - 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.

AB - 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.

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U2 - 10.2307/2695080

DO - 10.2307/2695080

M3 - Article

VL - 65

SP - 1841

EP - 1849

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

SN - 0022-4812

IS - 4

ER -