Connectives in Cumulative Logics

Cumulative logics are studied in an abstract setting, i.e., without connectives, very much in the spirit of Makinson’s [11] early work. A powerful representation theorem characterizes those logics by choice functions that satisfy a weakening of Sen’s property α, in the spirit of the author’s [9]. The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of worlds, not only definable sets and no definability-preservation property is required in the soundness part. For abstract cumulative logics, proper conjunction and negation may be defined. Contrary to the situation studied in [9] no proper disjunction seems to be definable in general. The cumulative relations of [8] that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by [3] are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation.