A 3-Valued Logical Foundation for Evidential Reasoning

The famous Cox’s Theorem provides a two-valued logical justification of probability theory, which was crowned with the Logic of Science by Edwin Jaynes. In this paper, we similarly study a three-valued logical foundation for belief functions in this paper by focusing on the connection of Dempster’s rule of combination and common knowledge. Dempster’s rule allows one to combine belief functions from different sources and common knowledge is a mutual knowledge shared by an interacting group of agents. First, by interpreting belief functions as probabilities on knowledge, we prove that the combination of belief functions according to Dempster’s rule is probabilities of common knowledge. Next, we explore a three-valued logical foundation of evidential reasoning. Belief (mass) functions are derived from any probability function over the set of all three-valued truth assignments. Moreover, we show that, in the Carnapian universe consisting of all atoms where each propositional letter or its negation appears only once, each three-valued assignment actually induces an S5 knowledge operator. Common knowledge is interpreted naturally as the combination of consistent three-valued assignments. Through these derived knowledge operators, we establish a natural connection between Dempster’s rule of combination and common knowledge by reasoning in the three-valued logic.