On a semantics for non-classical logics
Satoru Nagai · Proceedings of the Japan Academy Series A Mathematical Sciences · 1973
In [2] and [3], Ono showed some incompleteness results on two types of semantics for the intermediate predicate logics, that is, the algebraic and the Kripke-type.More precisely, he proved that there exist many intermediate predicate logics without characteristic sets of algebraic models, and that there exist those without characteristic Kripke models.This situation is more serious in the case of the modal predicate logics.In fact he proved the existence of a modal predicate logic having neither characteristic sets of algebraic models nor char- acteristic Kripke models.Thus the existing semantical methods proved incomplete in the above sense.Therefore, some new type of semantics is required since the semantical method is indispensable for the study of logics.This note proposes one of such semantics that contains the algebraic semantics as well as Kripke-type one as special cases.Our new semantics is obtained by combining these two types of semantics quite naturally.