An algebraic approach to belief contraction and nonmonotonic entailment

AbstractThe approach of Alchourrón, Gärdenfors 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. It can also be shown that ordinary entailment can be retrieved from the family of entailments with finite restrictions. Nonmonotonic closure or consequence C, defined algebraically, 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. For example, necessary and sufficient conditions are given for nonmonotonic closure to be equal to ordinary closure: C=Cn