The proofs of α→α in P - W

Journal of Symbolic Logic 61 (1):195-211 (1996)
Abstract
The 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 linear λ-terms, and a translation of such λ-terms M to BB'I-combinators M + is introduced. It is shown that a formula α 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 λ x.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 λ x.x if the λ-term has a type α → α. Hence the structure of proofs of α → α in P - W is determined
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