Quantum logic, Hilbert space, revision theory

• Published in: Artificial intelligence 2002-03, Vol.136 (1), p.61-100
• Main Authors: Engesser, Kurt, Gabbay, Dov M.
• Format: Article
• Language: English
• Subjects: Artificial intelligence; Consequence relation; Hilbert space; Non-monotonic logic; Quantum logic; Revision theory
• ISSN: 0004-3702; 1872-7921
• DOI: 10.1016/S0004-3702(01)00164-3

Our starting point is the observation that with a given Hilbert space H we may, in a way to be made precise, associate a class of non-monotonic consequence relations in such a way that there exists a one-to-one correspondence between the rays of H and these consequence relations. The projectors in Hilbert space may then be viewed as a sort of revision operators. The lattice of closed subspaces appears as a natural generalisation of the concept of a Lindenbaum algebra in classical logic. The logics presentable by Hilbert spaces are investigated and characterised. Moreover, the individual consequence relations are studied. A key concept in this context is that of a consequence relation having a pointer to itself. It is proved that such consequence relations have certain remarkable properties in that they reflect their metatheory at the object level to a surprising extent. The tools used in the investigation stem from two different areas of research, namely from the disciplines of non-monotonic logic on the one hand and from Hilbert space theory on the other. There exist surprising connections between these two fields of research the investigation of which constitutes the purpose of this paper.