André Fuhrmann CONDITIONAL LOGICS AND CUMULATIVE LOGICS

Introduction Inference ivolving (so-called counterfactual) conditionals is a well-understood variety of nonmonotonic modes of reasoning. There is, accordingly , some prospect that by establishing links between logics of con-ditionals and varieties of nonmonotonic reasoning our understanding of the latter will improve. The purpose of this paper is to exhibit a close relationship between familiar conditional logics and a kind of nonmono-tonic reasoning recently studied under the name of cumulative inference by Kraus, Lehmann and Magidor (henceforward KLM). I refer the reader to KLM [1] for information concerning the origins and motivation of systems of cumulative inference. KLM claim (in [1, 170]) that the link between their work and previous work on conditional logic " is mainly at the level of the formal systems and not at the semantic level ". And indeed, the semantics of KLM does not seem to fit easily into the range of semantics offered for conditionals (see [2] or [3] for a survey). But while cumulative inference can be modeled in a way that contrasts with familiar semantics for conditionals, it need not be thus modeled. The link between cumulative and conditional logic is just as strong at the level of semantics as it is at the level of proof theory. Cumulative logics are based on a sentential language L with ∧ (conjunction) and ¬ (negation) chosen as primitive sentential connectives. There are two relations of inference between formulae: the straight turnstile, , denoting a familiar notion of deducibility, and a curly turnstile |∼, used for a nonmonotonic relation inference (cumulative inference). The following system CB emerges as a basis for all logics discussed in the context of cumulative inference; it is based on the classical sentential calculus and is closed under the rule of Modus Ponens.