Deduction theorems for weak implicational logics
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 the author looks in detail at special cases of the Gentzen rules for \(\vdash\) and shows 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 \(E_{\to}\), \(R_{\to}\), T, P-W and P-W-I.
scientific article; zbMATH DE number 3851042