Classic Probability Revisited (II): Algebraic Operations of the Extended Probability Theory

Part II of this paper presents a set of comprehensive algebraic operators on the extended mathematical structures of the general probability theory. It is recognized that the classic probability theory is cyclically defined among a small set of highly coupled operations. In order to solve this fundamental problem, a reductive framework of the general probability theory is introduced. It is found that conditional probability operation on consecutive events is the key to independently manipulate other probability operations. This leads to a revisited framework of rigorous manipulations on general probabilities. It also provides a proof for a revisited Bayes’ law fitting in more general contexts of variant sample spaces and complex event relations in fundamental probability theories. The revisited probability theory enables a rigorous treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, and brain informatics.