Connectives in Quantum and other Cumulative Logics

The nonmonotonic logics definable by definability-preserving choice functions that satisfy Coherence have been studied in [7]. Larger families correspond to weakenings of this property. The cumulative and loop-cumulative relations of [6] are characterized by such models and, as a consequence, one may study the natural connectives for those logics. The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of models, not only definable sets and no definability-preservation property is required in the soundness part. For those logics, proper conjunction and negation may be defined, but no proper disjunction, contrary to the situation studied in [7]. Quantum Logics, as defined by [3] are such Logics but the orthogonal complement does not provide a proper negation. ∗This work was partially supported by the Jean and Helene Alfassa fund for research in Artificial Intelligence.