The subsystem S of Parry's AI [10] (obtained by omitting modus ponens for the material conditional) is axiomatized and shown to be strongly complete for a class of three valued Kripke style models. It is proved that S is weakly complete for the class of consistent models, and therefore that Ackermann's rule is admissible in S. It also happens that S is decidable and contains the Lewis system S4 on translation — though these results are not presented here. S is (...) arguably the most relevant relevant logic known at this time to be decidable. (shrink)

Logic is easily misunderstood, especially relevant logic. Thus the philosopher Peter Geach deplores the current preoccupation with "inconsistent" logic that are non-trivial and yet inconsistent). But what use are these, anyway? And, as for entailment , the concept is well-developed and well-established. Why meddle with it? ;In this work, we do not meddle with logical consequence. Instead, we develop a group of well-behaved logics that are conforming; i.e. that conform to the Boolean Order of Things, according to which a contradiction (...) is the very worst, whereas a logical truth is the very best. These systems are relevant, however, in that they reject modus ponens. Each admits Ackermann's rule ). (shrink)