The interpretation of partitioned frame semantics

The advocate of modal logic or relevant logic has traditionally argued that her preferred system offers the best regimentation of the theory of entailment. Essential to the projects of modal and relevant logic is the importation of non-truth-functional expressive resources into the object language on which the logic is defined. The most elegant technique for giving the semantics of such languages is that of frame semantics , a variation on which features the device of partitioned frames that divide ‘points of evaluation’ into two types: normal points and abnormal points. This essay attempts to provide satisfying philosophical interpretations of the partitioned frame semantics for some weak modal logics and relevant logics. According to the current best interpretation, partitioned frame semantics are a kind of possible worlds semantics that appeal to ‘logically impossible worlds’ where the laws of logic may fail to hold. This is undermined by the fact that all of the target logics fail to satisfy criteria of expressibility needed to support the thesis that laws of logic fail to hold at abnormal points. This suggests that we should not look for a monistic interpretive paradigm for partitioned frame semantics. In response, new interpretations of the weak modal logic S0.5 and the ‘basic’ relevant logic B are constructed. It is argued that S0.5 is a logic of alethic modality according to a conventionalist conception of necessity, and that B is a logic of meaning inclusion according to a dialetheist conception of the space of possibilities. The given interpretations help to shed light on relations between systems of modal logic and relevant logic, and explain some of the most interesting features of such logics, like failure of the ride of necessitation and failure of contraction principles.