Deduction theorems for weak implicational logics

Studia Logica 41 (2-3):95 - 108 (1982)
The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I.
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DOI 10.1007/BF00370338
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Haskell B. Curry (1952). The System LD. Journal of Symbolic Logic 17 (1):35-42.
Alan Ross Anderson & Nuel D. Belnap (1977). Entailment. Vol. 1. Canadian Journal of Philosophy 7 (2):405-411.

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