Some Observations on the Systems LFI1 and LFI1

One of the well-known systems of paraconsistent logic called LFI1 is designed to be a base system in constructing evolutionary databases. This system LFI1 is proved to be a 3-valued logic and also maximal relative to classical logic enriched with inconsistency operator in an obvious manner. The present paper aims to examine the system LFI1 from the viewpoint of Belnap\´s 4-valued logic. More concretely, we develop the Belnapian 4-valued system of Logics of Formal Inconsistency(LFIs) which can be seen as a natural generalization of LFI1. As a consequence, from the viewpoint of the Belnapian logic, we obtain a formalization which contains the notion of "normality" instead of the constant ⊥. On the other hand, from the viewpoint of LFIs, we lose the maximality but might be able to cope with more data in constructing databases. This is because the fourth value of the Belnapian matrix corresponds to incomplete data which cannot be dealt with in LFI1. Our results contain an axiomatization of the Belnapian LFI, a characterization of "normality" in the system, and a translation result between the existing Belnapian system and the system we introduce.