A Study Of Normal Multimodal Logics

Multimodal logic,which contains two or more unreducible modal operators,is an important part of modal logic.This dissertation is a study of the basic theory of Normal Multimodal Logics,which are modal systems satisfying RN and K.The study of normal multimodal logics is the main research object of the research of multimodal logics.Using heterogeneous systems and homogeneous systems as criteria,this dissertation reviews the research on multimodal logics,and attempts to classify the numerous literature of multimodal logics.We find that most of existing multimodal logic systems are normal multimodal logic systems.Normal multimodal logic is in a dominant position of multimodal logic.Studying the normal multimodal logics has more legitimacy.Finally,analyzing these specific multimodal systems,in some way,provides motivation and possible applications of normal multimodal logic.The problem of the combination of modalities in modal logic systems is the starting point and the main motivation of our work.In the study of multimodal logics,we emphasize the importance of interaction principles,with which the interrelations between the modal concepts can be described.So the interaction axioms of multimodal logic systems is an important perspective for studying multimodal logics.In this dissertation,we study the general normal multimodal logic systems from the perspective of interaction axioms.We study three axiom schemata,such as G(a,b,c,d),G(a,b,?)and Sahlqvist axiom schema,and the last one contains the first two.According to Sahlqvist correspondence theorem,as Sahlqvist axiom schema's special cases,G(a,b,c,d),G(a.b,?)axiom schema all correspond to first-order formulas on multi-frames.Meanwhile,according to the theory of binary relations,the properties of the multi-frames,that G(a,b,c,d)and(a,b,?)schema's secondary axiom schema and the inverse of secondary axiom schema corresponded can be characterized using relational equations.In the course of doing so,we can get the determination theorem of G(a,b,?)axiom schema and the proof.On this basis,we define a new interaction axiom schema G(a,b,^)",which is a special case of Sahlqvist axiom schema and also is the extension of G(a,b,c,d),G(a,b,?,)axiom schema.G(a,b,^)n axiom schema corresponds to first-order formulas on frames,the properties of the frames can be characterized using Rai?…?Ran(?)F?(Rb1-1,…,Rbn-1)In the course of doing so,we present the soundness,the completeness and the determination theorem of G(a,b,^)n axiom schema and the proof.Furthermore,based on the determination theorem of normal multimodal logic systems,we study the separation standards of multimodal systems' axiomatization.Using the method of filtration,we study the decidability of normal multimodal logic systems.We get some decidability theorem of normal multimodal logic systems,in particular some systems which do not contain interaction axioms.However,this method has some limitations.Beside this,in this dissertation,we give expectation to using reduction method in handling decidability of normal multimodal logic systems.The general normal multimodal logic systems are abstraction and generalization of many multimodal systems proposed in the literature,which have guidance for constructing some specific multimodal systems,such as deontic logic,epistemic logic,temporal logic and so on.Finally,considering the problems in multimodal logics,we explore the direction of the study of multimodal logic in the future.