Some notes on cumulative reasoning

Abstract

We introduce a model-theoretic definition of non-monotonic inference relation and study a particular model called C^*. Gentzen-style counterpart of C^* can be regarded as the nonmonotonic infinitistic version of the system C of Kraus, Magidor, Lehmann with the consistency-preservation, property proposed by Gabbay and used in belief revision. A notion of selection function is introduced. This notion is similar to that of expectations given by Gärdenfors and Makinson and derived from partial meet contractions functions of belief revision, given by Alchourron, Gärdenfors and Makinson. The representation theorem for the cumulative system C, whose models use a binary relation < among sets of worlds, can be simplified by using selection functions. The finitistic (conditional) system of C^* turns out to be equivalent to Makinson's cumulative monotony operator as well as to the conditional system γ^* derived from the sistem γ of Gabbay by adding the cautious monotonicity axiom and using classical logic in the place of intuitionistic logic as underlying monotonic logic.

Work carried out within the framework of the agreement between Italian PT Administration and Ugo Bordoni Foundation