Modal Probability Logic

The combination of modal logic and probabilistic logic can trace back to Carnap.On the one hand, he constructs the semantic frame of modal logic by the conception of“state-description”and so on. On the other hand, he constructs the modal inductive logic by introducing a probability method in the frame. For Carnap is the first who made the semantic explanation of modal logic, there was not a mature semantics of modal logic when he started to construct the modal inductive logic. By the hard work of Carnap and some other logicians, the semantics of modal logic has become mature. The subsequent logicians mainly adopt the same method as Carnap, namely the introduction of probabilistic method in the frame of modal logic. This paper intends to construct a probabilistic system on the basis of the classical relation semantics of modal logic. The system should fulfill the following intuitive thoughts:(A) Every event of some probability to occur in a certain world would necessarily occur in a certain or some possible worlds accessible to the current world.(B) The modal probabilistic logic is elaborate modal logic. The elaboration is represented in two sides:(B1) Distinguishing between the different degree of the accessibility from the current world to other worlds.(B2) Distinguishing between the event of necessity and the event of probability 1. A event of necessity must be a event of probability 1,and not conversely.The task of this paper is as following:(A) Every explanation of probability has some relative meaning by the exposition of different explanations of probability theories, probabilistic logic and probability.(B) Simply introduces K system of modal logic and its modal, that is, the conceptions of necessity and probability in modal logic have relative meaning.(C) Constructs the modal of unary probabilistic modal logic and proves some common valid formulas while gives its axiom and proves its soundness.(D) Constructs the modal of binary probabilistic modal logic and proves some common valid formulas while gives its axiom. The conditional logic can be explained by binary probabilistic logic.(E) For the modal of modal probabilistic logic is just a extension of modal logic,analogous as modal logic, we can translate the expressions in modal probabilistic logic into expressions in first order logic by using modal.(F) We can apply modal probabilistic logic to dynamic game logic for the modal probabilistic logic does not change the modal of modal logic and only attaches a probabilistic explanation. That is a natural extension to the game logic which is basically constructed on modal logic.