Representing rules as random sets, II: Iterated rules

I. R. Goodman, H. T. Nguyen, and others have proposed the theory of random sets as a unifying paradigm for knowledge‐based systems. The more types of ambiguous evidence that can be brought under the random‐set umbrella, the more potentially useful the theory becomes as a systematic methodology for comparing and fusing seemingly incongruent kinds of information. Conditional event algebra provides a general framework for dealing with rule‐based evidence in a manner consistent with probability theory. This and a previous companion article bring conditional event logic—and through it, rules and iterated rules—under the random set umbrella. In this article, we derive a new conditional event algebra, denoted GNW2, for rules which are contingent on other rules. We show that this logic is the only one which extends the GNW first‐degree logic, admits a simple embedding into random sets, and has GNW‐like behavior. We show that GNW events are contained in a Boolean subalgebra GNW* of GNW2. Finally, we show how GNW2 can be used to recursively define a GNW‐like Boolean logic for iterated rules of any degree. © 1996 John Wiley & Sons, Inc.