TY - GEN
T1 - Connectives in cumulative logics
AU - Lehmann, Daniel
PY - 2008
Y1 - 2008
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=49949083741&partnerID=8YFLogxK
U2 - 10.1007/978-3-540-78127-1_22
DO - 10.1007/978-3-540-78127-1_22
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AN - SCOPUS:49949083741
SN - 3540781269
SN - 9783540781264
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 424
EP - 440
BT - Pillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday
A2 - Avron, Arnon
A2 - Dershowitz, Nachum
A2 - Rabinovich, Alexander
ER -