Re: quantum logic

[Peter McBurney <p.j.mcburney@csc.liv.ac.uk>]

John --

Although not a categorical treatment, a recent paper by Kurt Engesser
and Dov Gabbay in <Artificial Intelligence> discusses a connection
between Quantum Logic and Hilbert spaces.   (The reason the work
appeared in the leading AI journal is that there are applications to
nonmonotonic reasoning, which is a major area of research in AI.)

Citation details and abstract below.

-- Peter


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Artificial Intelligence
Volume 136, Issue 1 , March 2002 , Pages 61-100

"Quantum logic, Hilbert space, revision theory"

Kurt Engesser and Dov M. Gabbay

a Birkenweg 3, 78573 Wurmlingen, Germany
b Department of Computer Science, King's College London, Strand, London
WC2R 2LS, UK


Abstract

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.

Author Keywords: Quantum logic; Hilbert space; Revision theory;
Consequence relation; Non-monotonic logic

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