Connectives in Cumulative Logics
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 α, in the spirit of the author’s . 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  no proper disjunction seems to be definable in general. The cumulative relations of  that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by  are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation.
KeywordsClosed Subspace Choice Function Orthogonal Complement Quantum Logic Atomic Proposition
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