Table Of ContentQU ANTUM LOGIC
SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Academy of Finland and Stanford University
Editors:
ROBERT S. COHEN, Boston University
DON ALD DAVIDSON, University of Chicago
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, University of Arizona
VOLUME 126
QUANTUM LOGIC
by
PETER MITTELST AEDT
University of Cologne, Germany
D. REIDEL PUBLISHING COMPANY
DORDRECHT: HOLLAND I BOSTON: U.S.A.
LONDON: ENGLAND
Ubrary of Congress Cataloging In PubHcation Data
Mittelstaedt, Peter, 1929-
Quantum logic.
(Synthese library; v. 126)
Bibliography: p.
Includes index.
1. Quantum theory. 2. Logic, Symbolic and mathe
matical. I. Title.
QCI74.17.M35M57 530.1'2 78-10433
ISBN-13: 978-94-009-9873-5 e-ISBN-13: 978-94-009-9871-1
DOl: 10.1007/978-94-009-9871-1
Published by D. Reidel Publishing Company,
P.O. Box 17, Dordrecht, Holland
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Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland
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TO MECHTHILD
T ABLE OF CONTENTS
INTRODUCTION
CHAPTER 1/ THE HILBERT SPACE FORMULATION OF
QUANTUM PHYSICS 6
1.1 The Hilbert Space 6
1.2 The Lattice of Subspaces of Hilbert Space 11
1.3 Projection Operators 16
1.4 States and Properties of a Physical System 21
CHAPTER 2/ THE LOGICAL INTERPRETATION OF THE
LATTICE Lq 27
2.1 The Quasimodular Lattice Lq 27
2.2 The Relation of Commensurability 31
2.3 The Material Quasi-implication 37
2.4 The Relation between Lattice Theory and Logic 42
CHAPTER 3 / THE MATERIAL PROPOSITJONS OF
QUANTUM PHYSICS 48
3.1 Elements of a Language of Quantum Physics 48
3.2 Argument-rules for Compound Propositions 53
3.3 Commensurability and Incommensurability 60
3.4 The Material Dialog-game 65
CHAPTER 4/ THE CALCULUS OF EFFECTIVE QUANTUM
LOGIC 72
4.1 Formally True Propositions 72
4.2 Formal Dialogs with Material Commensurabilities 76
4.3 The Formal Dialog-game 82
4.4 The Calculus Qeff of Effective Quantum Logic 88
CHAPTER 5/ THE LATTICE OF EFFECTIVE QUANTUM
LOGIC 99
5.1 The Quasi-implicative Lattice Lqi 99
5.2 Properties of the Lattice L 104
qj
viii TABLE OF CONTENTS
5.3 The Relation between Lqi and the Lattice Li 109
5.4 The Relation between Lqi and the Lattice Lq 113
CHAPTER 6/ THE CALCULUS OF FULL QUANTUM LOGIC 119
6.1 Value-definite Material Propositions 119
6.2 The Value-definiteness of Compound Propositions 124
6.3 The Extension of the Calculus Qelf 128
6.4 The Principle of Excluded Middle 134
CONCLUDING REMARKS: CLASSICAL LOGIC AND
QU ANTUM LOGIC 140
BIBLIOGRAPHY 144
INDEX 147
INTRODUCTION
In 1936, G. Birkhoff and J. v. Neumann published an article with the
title The logic of quantum mechanics'. In this paper, the authors
demonstrated that in quantum mechanics the most simple observables
which correspond to yes-no propositions about a quantum physical
system constitute an algebraic structure, the most important proper
ties of which are given by an orthocomplemented and quasimodular
lattice Lq. Furthermore, this lattice of quantum mechanical proposi
tions has, from a formal point of view, many similarities with a
Boolean lattice L8 which is known to be the lattice of classical
propositional logic. Therefore, one could conjecture that due to the
algebraic structure of quantum mechanical observables a logical
calculus Q of quantum mechanical propositions is established, which
is slightly different from the calculus L of classical propositional logic
but which is applicable to all quantum mechanical propositions (C.F.
v. Weizsacker, 1955). This calculus has sometimes been called 'quan
tum logic'.
However, the statement that propositions about quantum physical
systems are governed by the laws of quantum logic, which differ from
ordinary classical logic and which are based on the empirically
well-established quantum theory, is exposed to two serious objec
tions:
(a) Logic is a theory which deals with those relationships between
various propositions that are valid independent of the content of the
respective propositions. Thus, the validity of logical relationships is
not restricted to a special type of proposition, e.g. to propositions
about classical physical systems.
(b) The laws of logic, though valid for all statements encountered in
experience, do not derive their validity from experience. Instead,
logic must be considered as a theory the statements of which can be
justified exclusively by its inherent evidence and irrespective of all
empirical knowledge.
These arguments become particularly apparent in the framework of
the operational foundation of logic, which explains the 'truth' of a
2 INTRODUCTION
logical statement by the existence of a strategy of success in a dialog.
In this approach, the laws of logic are completely determined by the
possibilities of proving or disproving elementary and compound pro
positions within a well-defined proof procedure which can be
represented by a dialog. Hence, the logical statements are 'true'
independent of the special content of the actual propositions and for
this reason they can neither be proved nor disproved by arguments
which are based on empirical knowledge.
The controversial situation which thus arises can, however, be
clarified if the dialogic justification of ordinary propositional logic is
investigated in more detail. In this way, it turns out that in the
framework of the dialogic method an assumption is always tacitly
made which restricts the propositions considered to those of classical
physics and mathematics. Technically, this means that within the
dialogic proof procedure of compound propositions only the truth or
falsity of the respective sUb-propositions is tested but not their
mutual commensurability. Therefore, one could try to search for an
operational foundation of logic which is independent of the presup
position just mentioned. By this it is not meant that some emprical
knowledge about quantum mechanical propositions should be in
corporated into the foundation of logic. Conversely, we will rather
eliminate that empirical supposition which is still contained in the
operational foundation of ordinary logic - namely the assertion that
all propositions are mutually commensurable. Technically, this
generalization of the dialogic method can be performed by in
corporating an additional testing procedure which - apart from the
truth and falsity - examines the mutual commensurability of two
propositions. In this way, the generalized dialog-game can be equally
applied to propositions of classical physics and of quantum physics.
Starting from this generalized operational foundation of logic one
arrives at a logical calculus Qeff. which will be called the calculus of
effective quantum logic, and which differs from the well-known cal
culus of effective (intuitionistic) logic inasmuch as some of the laws
of this logic are valid only in a relaxed version. Furthermore, by a
rather weak assumption concerning the measurability of commen
surabilities, the 'tertium non datur' can be justified for all finite
compound propositions and the calculus Qeff can be extended to the
calculus Q of full quantum logic which incorporates the principle of
excluded middle as a general law.
It can then be shown that this calculus Q of full quantum logic is, in
INTRODUCTION 3
fact, a model of the orthocomplemented and quasimodular lattice Lq
which has been obtained previously from the algebraic structure of
quantum mechanical observables. In this way, one obtains the
following important results:
(a) The orthocomplemented quasimodular lattice Lq which follows
from a formal analysis of quantum mechanics can be interpreted as a
generalized propositional logic. This 'quantum logic' is universal in
the sense that the validity of its laws is not restricted to a special type
of proposition. Instead, the laws of quantum logic are equally valid
for all propositions of classical physics and of quantum physics.
(b) Since the calculus Q of full quantum logic can be justified by
theoretical reasons only and independent of any empirical knowledge,
it follows that at least those structures of quantum mechanics which
are summarized in the lattice Lq are non-empirical and may be
considered as cognitions a priori.
In order to demonstrate these results, we proceed in the following
way: In Chapter 1, we summarize some formal properties of quantum
mechanics which are of special importance for the problem of quan
tum logic. It is shown that the yes-no propositions A, B, ... about a
physical system S correspond respectively to sub-spaces MA, MH, •••
of the Hilbert space ~(S) which is associated with the system S as its
state-space. On the other hand, these subspaces of a Hilbert space
constitute a lattice, the most important properties of which are given
by an orthocomplemented and quasimodular lattice L Hence, quan
q•
tum mechanical propositions form a lattice which has many proper
ties in common with the Boolean lattice La of classical propositional
logic and the operations of which have many similarities with the
logical operations 'and', 'or' and 'not'.
The formal properties of the lattice Lq and its relations to a
Boolean lattice La are investigated in Chapter 2. Firstly, we show that
a commensurability relation can be defined in Lq such that mutually
commensurable propositions form Boolean sublattices of L
q.
Secondly, it is shown that the most important syntactical require
ments for a logical calculus which can be formulated on the basis of
La are also fulfilled by the lattice Lq. Hence, there are no formal
arguments which exclude the possibility of an interpretation of Lq as
a logical calculus. However, it is obvious that the formal similarities
of Lq and La are by no means sufficient in order to justify a logical
interpretation of the lattice L
q•
Therefore, for the present, we leave lattice theory and in Chapter 3
