Comparison of Relevant Logic R and ER

Abstract

The formalization of human deductive reasoning is a main issue in artificial intelligence. Although classical logic (CL) is one of the most useful ways for the formalization, the implication of CL has some fallacies. For example, in Cl, A→B can be inferred form B for and arbitrary formula A. This inference is incorrect from the viewpoint of the meaning of implication which human has. In human deductive reasoning, when A⇾B is inferred, A and B should be related. Relevant logic has been studied for removal of implication fallacies in CL. The system R is a typical logical system from which fallacies of relevance and validity are removed. ER is a relevant typical logical system from which fallacies of relevance and validity are removed. ER is a relevant logical system from which these fallacies are removed and this system is not weaker than R. Especially, it is known that disjunctive syllogism holds in ER but does not hold in R. This inference rule is considered to be natural in human reasoning. In this paper, we prove that ER is properly stronger than R. This means that, for the formalization of human deductive reasoning, ER is more suitable than other relevant logical systems. The proof consists of the following steps: First, the natural deduction systems FR and FR' are introduced. FR is a natural deduction system equivalent to R. It is proved that FR' is stronger than FR. Next, we show that he normalization theorem holds in FR' and that there is a proof of ER corresponding to each normal proof of FR'. In addition, We show the fact that there is theorem of ER which can not be inferred in R. It follows that ER is properly stronger than R.