Formalizing a Deductively Open Belief Space

A knowledge representation and reasoning system must be able to deal with contradictions and revise beliefs. There has been much research in belief revision in the last decade, but this research tends to be either in the Coherence camp (AGM) or the Foundations (TMS) camp with little crossover. Most theoretical postulates on belief revision and belief contraction assume a deductively closed belief space something that is computationally hard (or impossible) to produce in an implementation. This makes it difficult to analyze implemented belief revision systems using the theoretical postulates. This paper offers a formalism that describes a deductively open belief space (DOBS). It then uses this formalism to alter the AGM integrity constraints for a DOBS. A DOBS uses a base set of hypotheses, but only deduces beliefs from that base as the result of specific queries. Thus, it can grow over time even if the base remains static, and can never be referred to as consistent only either inconsistent or "not known to be inconsistent." This work and future alterations to the traditional postulate formalisms will better enable system/postulate comparisons.