In [1], we established Gentzenizations for a good range of relevant logics with distribution, but, in the process, we added inversion rules, which involved extra structural connectives, and also added the sentential constant t. Instead of eliminating them, we used conservative extension results to relate them back to the original logics. In [4], we eliminated the inversion rules and t and established a much simpler Gentzenization for the weak sentential relevant logic DW, and also for its quantificational extension DWQ, but a restriction to normal formulae (defined below) was required to enable these results to be proved. This method was quite general and hope was expressed about extending it to other relevant logics.In this paper, we develop an innovative method, which makes essential use of this restriction to normality, to establish two simple Gentzenizations for the normal formulae of the slightly weaker logic B, and then extend the method to other sentential contraction-less logics. To obtain the first of these Gentzenizations, for the logics B and DW, we remove the two branching rules (F&) and (T∨), together with the structural connective ‘,’, to simplify the elimination of the inversion rules and t. We then eliminate the rules (T&) and (F∨), thus reducing the Gentzen system to one containing only ˜ and → and their four associated rules, and reduce the remaining types of structures to four simple finite types. Subsequently, we re-introduce (T&) and (F∨), and also (F&) and (T∨), to obtain the second Gentzenization, which contains ‘,’ but no structural rules.
Logic of Violations: A Gentzen System for Reasoningwith Contrary-To-Duty Obligations