文档介绍：ARTICLE IN PRESS
S0004-3702(01)00164-3/FLA AID:1868 Vol.•••(•••) (1-40)
ELSGML 2001/10/15 Prn:16/10/2001; 15:09 AIJ1868 by:. p. 1
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Artificial Intelligence •••(••••) •••–•••
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9 Quantum logic, Hilbert space, revision theory 9
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11 a,∗ b 11
12 Kurt Engesser , Dov M. Gabbay 12
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13 Birkenweg 3, 78573 Wurmlingen, Germany 13
b Department puter Science, King’s College London, Strand, London WC2R 2LS, UK
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15 Received 24 May 2000; received in revised form 19 September 2001 15
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18 Abstract 18
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Our starting point is the observation that with a given Hilbert space H we may, in a way to be made
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precise, associate a class of non-monotonic consequence relations in such a way that there exists a
21 one-to-one correspondence between the rays of H and these consequence relations. The projectors 21
22 in Hilbert space may then be viewed as a sort of revision operators. The lattice of closed subspaces 22
23 appears as a natural generalisation of the concept of a Lindenbaum algebra in classical logic. The 23
24 logics presentable by Hilbert spaces are investigated and characterised. Moreover, the individual 24
25 consequence relations are studied. A key concept in this context is that of a consequence relation 25
26 having a pointer to itself. It is proved that such consequence relations have certain remarkable 26
27 properties in that they reflect their metatheory at the object level to a surprising extent. The tools 27
28 used in the investigation stem from two different areas of research, namely from the disciplines 28
29 of non-monotonic logic on the one hand and from Hilbert space theory on the other. There exist 29
surprising connections between these two fields of research the investigation of which constitutes the
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purpose of this paper. 2001 Published by Elsevier Science .
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32 Keywords: Quantum logic; Hilbert space; Revision theory; Conseque