文档介绍：Artificial Intelligence 136 (2002) 61–100
Quantum logic, Hilbert space, revision theory
Kurt Engesser a,∗, Dov M. Gabbay b
a Birkenweg 3, 78573 Wurmlingen, Germany
b Department puter Science, King’s College London, Strand, London WC2R 2LS, UK
Received 24 May 2000; received in revised form 19 September 2001
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. 2001 Published by Elsevier Science .
Keywords: Quantum logic; Hilbert space; Revision theory; Consequence relation; Non-monotonic logic
1. Introduction
In this paper we establish connections between two seemingly unrelated areas of logical
research, namely that of quantum logic on the one hand and that of non-monotonic logic
on the other. In particular, we look at the connection between Hilbert space and logic in
a new way from the po