An algebraic approach to belief contraction and nonmonotonic entailment

The approach of Alchourron, Gardenfors and Makinson to belief contraction is treated algebraically. This is then used to give an algebraic treatment of nonmonotonic entailment in the context of a belief set. The algebra used is a preboolean algebra whose elements are sets of sentences and whose order relation is restricted entailment. Under plausible assumptions restricted entailment is computable; so I have proposed elsewhere [4] that restricted entailment be taken as the deductive process of an agent. It can also be shown (not here) that ordinary entailment can be retrieved from the family of entailments with finite restrictions. Nonmonotonic closure satisfies inclusion, supraclassicality and distribution, but satisfaction of idempotency and cumulativity depend on certain conditions being fulfilled. Casting the notions of belief contraction and nonmonotonic entailment in algebraic formalism facilitates the understanding and analysis of these ideas.