  Cumulative logics are studied in an abstract setting, i.e., without
connectives, very much in the spirit of Makinson's early work. A powerful
representation theorem characterizes those logics by choice functions that
satisfy a weakening of Sen's property alpha, in the spirit of the author's
"Nonmonotonic Logics and Semantics" (JLC). 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 "Nonmonotonic Logics and Semantics" no
proper disjunction seems to be definable in general. The cumulative relations
of KLM that satisfy some weakening of the consistency preservation property all
define cumulative logics with a proper negation. Quantum Logics, as defined by
Engesser and Gabbay are such cumulative logics but the negation defined by
orthogonal complement does not provide a proper negation.
