The proofs of α → α in P – W

Abstract

AbstractThe syntactic structure of the system of pure implicational relevant logic P − W is investigated. This system is defined by the axioms B = (b → c) → (a → b) → a → c, B′ = (a → b) → (b → c)→ a → c, I = a → a, and the rules of substitution and modus ponens. A class of λ-terms, the closed hereditary right-maximal linearλ-terms, and a translation of such λ-terms M to BB′ I-combinators M+ is introduced. It is shown that a formula a is provable in P − W if and only if α is a type of some λ-term in this class. Hence these λ-terms represent proof figures in the Natural Deduction version of P − W.Errol Martin (1982) proved that no formula with form α → α is provable in P − W without using the axiom I. We show that a β-normal form λ-term M in the class is η reducible to λx.x if the translated BB′ I-combinator M+ contains I. Using this theorem and Martin's result, we prove that a λ-term in the class is βη-reducible to λx.x if the λ-term has a type α → α. Hence the structure of proofs of α → α in P − W is determined.