A Note on the Rational Closure of Knowledge Bases with Both Positive and Negative Knowledge

The notion of the rational closure of a positive knowledge base K of conditional assertions θi |∼ φi (standing for if θi then normally φi) was first introduced by Lehmann in [2] and developed by Lehmann and Magidor in [3]. Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice however one might expect a knowledge base to consist not just of positive conditional assertions, θi |∼ φi, but also negative conditional assertions, θi 6 |∼ φi (standing for not {if θi then normally φi}). Restricting ourselves to a finite language we show that the rational closure still exists for satisfiable knowledge bases containing both positive and negative conditional assertions and has similar properties to those exhibited in [3]. In particular an algorithm in [3] which constructs the rational closure can be adapted to this case and yields, in turn, completeness theorems for the conditional assertions entailed by such a mixed knowledge base. Introduction, Notation and Preliminaries The results presented in this paper can best be motivated, or justified, by first considering the following problem. Let us suppose that we have somehow learnt that an intelligent, rational, agent believes some finite set, K say, of so called conditional, or defeasible, assertions, If θi then normally (or usually) φi i = 1, . . . ,m where θi, φi are sentences from some finite propositional language L. (To fix a little notation here let SL be the set of sentences of L, built up using the standard connectives ∧,∨,→,¬ from the propositional variables p1, p2, . . . , pn.) In this case what other assertions of this form can we, or should we, conclude that the agent believes? Much of the knowledge we possess and, more importantly communicate, is in the form of just such assertions and our ability to use them to correctly draw further conclusions is seen as a manifestation of our intelligence. In consequence this question has received considerable attention in the AI community with many different approaches and solutions being proposed (see, for example [4],[5],[6],[7],[8], [12]). The key distinguishing feature of such assertions, as opposed to classical notions of logical consequence, is their possible non-montonicity. For it is now certainly possible for and agent to believe