文档介绍：arXiv:quant-ph/0101028 v2 6 Jan 2001
Maria
QUANTUM
Luisa
Dalla
Decem
Chiara
b
er
12,
and
LOGICS
2001
Rob
erto
Giun
tini
Contents
1 Introduction 3
2 Orthomodular quantum logic and orthologic 11
3 The implication problem 22
4 Metalogical properties and anomalies 28
5 A modal interpretation of OL and OQL 32
6 An axiomatization of OL and OQL 35
7 The intractability of orthomodularity 40
8 Hilbert quantum logic and the orthomodular law 45
9 First-order quantum logic 50
10 Quantum set theories and theories of quasisets 56
11 The unsharp approaches 59
12 Effect structures 61
13 Paraconsistent quantum logic 70
14 The Brouwer-Zadeh logics 74
15 Partial quantum logics 86
16 Luk asiewicz quantum logic 92
17 Conclusion 96
List of Figures
1 Failure of bivalence in QT . . . . . . . . . . . . . . . . . . . . 8
2 Quasi-model for γ. . . . . . . . . . . . . . . . . . . . . . . . 31
3 The Greechie diagram of . . . . . . . . . . . . . . . . . . 46
G12
4 The orthomodular lattice . . . . . . . . . . . . . . . . . . 47
G12
1
5 The Greechie diagram of . . . . . . . . . . . . . . . . . . 47
B30
6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
G14
7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
M4
8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
M3
2
1 Introduction
The official birth of quantum logic is represented by a famous article of
Birkhoff and von Neumann “The logic of quantum mechanics”(Birkhoff
and von Neumann 1936). At the very beginning of their paper, Birkhoff
and von Neumann observe:
One of the aspects of quantum theory which has attracted the
most general attention, is the novelty of the logical notions which
it presupposes .... The object of the present paper is to discover
what logical structures one may hope to find in physical theories
which, like quantum mechanics, do not conform to classical logic.
In order to understa