Research On Epistemic Extension Of A Conditional Probability Logic

In recent decades,in order to meet the needs of the development of game theory,computer science,artificial intelligence and other fields,the research on probability logic that introduces modal probability operators in propositional logic has made great progress.With the help of Bayesian update in probability theory,this kind of probability logic can statically characterize the change of belief caused by the change of information.However,it is not yet possible to characterize the belief update caused by events with probability zero.One solution to this problem is to use conditional probability as an initial concept.M.S.C.Hernandes gave a conditional probability deductive system with conditional probability operators as its initial operators.This kind of system has two advantages: one is the ability to express the conditional probability based on events with probability zero,and the other is the ability to communicate well with dynamic operators.However,Hernandes’ s system cannot deal with the reasoning of knowledge.This paper attempts to extend Hernandes’ s conditional probability logic: to combine the conditional probability logic and epistemic logic,to construct a deductive system that can not only reason about the probabilistic belief but also agent’s knowledge and common knowledge;to study some meta-properties of these new deductive systems,and to explore the possible applications of this kind of logic in epistemic game theory.Specifically,the main work of this paper can be summarized as follows:(1)A deductive system that is an epistemic extension of Hernandes’ s conditional probability logic is given.First,we extend the language of Hernandes’ s conditional probability logic,that is,we add knowledge operators and common knowledge operators to his logic.The extended language can express both conditional probability and knowledge.On this basis,a conditional probabilistic epistemic deductive system is given.This system can describe the agent’s reasoning of conditional probability and knowledge.Finally,it is proved that the conditional probabilistic epistemic deductive system is weakly complete with respect to the conditional probabilistic epistemic model.(2)An infinitary deductive system which is capable of describing the reasoning of conditional probability and knowledge is given.First,an infinitary conditional probability language is defined,an infinitary conditional probability deductive system,which is conservative over Hernandes’ s conditional probability logic,is given,and the strong soundness and strong completeness of this system with respect to the conditional probability model are proved.Second,the knowledge operators and common knowledge operators are added to the infinitary conditional probability logic and we gain an axiomatic system of infinitary conditional probabilistic epistemic logic that can reason about the agent’s knowledge and common knowledge.(3)The formal method of applying conditional probabilistic epistemic logic to epistemic game theory is discussed.First,following the research route of Alexandru Baltag and Sonja Smets,we define a language for talking about the perfect information extensive game,and then we define the conditional probabilistic epistemic model that can describe the reasoning of the agent’s knowledge in the game.Second,bases on the concept of dynamic rationality,a definition of Bayesian dynamic rationality is given.Finally,taking the centipede game as an example,we informally show the application of conditional probabilistic epistemic logic in exploring the epistemic conditions of the solution concept in game theory.