Handbook of quantum logic and quantum structures; quantum logic

HANDBOOK OF QUANTUM LOGIC
AND QUANTUM STRUCTURES
QUANTUM LOGIC

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HANDBOOK OF QUANTUM LOGIC
AND QUANTUM STRUCTURES
QUANTUM LOGIC

Edited by

KURT ENGESSER
Universität Konstanz, Konstanz, Germany

DOV M. GABBAY
King's College London, Strand, London, UK

DANIEL LEHMANN
The Hebrew University of Jerusalem, Jerusalem, Israel

Amsterdam • Boston • Heidelberg • London • New York • Oxford
Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo
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CONTENTS
Foreword
Anatolij Dvureˇ
censkij

vii

Editorial Preface
Kurt Engesser, Dov Gabbay and Daniel Lehmann

ix

The Birkhoff–von Neumann Concept of Quantum Logic
os Redei
´
e
Mikl´

1

Is Quantum Logic a Logic?
Mladen Paviˇ

ci´
c and Norman D. Megill

23

Is Logic Empirical?
Guido Bacciagaluppi

49

Quantum Axiomatics
Diederik Aerts

79

Quantum Logic and Nonclassical Logics
127
Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto
Giuntini and Francesco Paoli
Gentzen Methods in Quantum Logic
Hirokazu Nishimura

227

Categorical Quantum Mechanics
Samson Abramsky and Bob Coecke

261

Extending Classical Logic for Reasoning about

Quantum Systems
Rohit Chadha, Paulo Mateus, Am´ılcar Sernadas and
Cristina Sernadas

325

Sol`er’s Theorem
Alexander Prestel

373

Operational Quantum Logic: A Survey and Analysis
David J. Moore and Frank Valckenborgh

389

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vi

Contents

Test Spaces
Alexander Wilce

443

Contexts in Quantum, Classical and Partition Logic
Karl Svozil

551

Nonmonotonicity and Holicity in Quantum Logic
Kurt Engesser, Dov Gabbay and Daniel Lehmann

587

A Quantum Logic of Down Below
Peter D. Bruza, Dominic Widdows and John H. Woods

625

A Completeness Theorem of Quantum Set Theory
Satoko Titani

661

Index

703

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vii

FOREWORD

More than a century ago Hilbert posed his unsolved (and now famous), 23 problems

of mathematics. When browsing through the Internet recently I found that Hilbert
termed his Sixth Problem non-mathematical. How could Hilbert call a problem of
mathematics non-mathematical? And what does this problem say?
In 1900, Hilbert, inspired by Euclid’s axiomatic system of geometry, formulated
his Sixth Problem as follows: To find a few physical axioms that, similar to the
axioms of geometry, can describe a theory for a class of physical events that is as
large as possible.
The twenties and thirties of the last century were truly exciting times. On the
one hand there emerged the new physics which we call quantum physics today.
On the other hand, in 1933, N. A. Kolmogorov presented a new axiomatic system
which provided a solid basis for modern probability theory. These milestones
marked the entrance into a new epoch in that quantum mechanics and modern
probability theory opened new gates, not just for science, but for human thinking
in general.
Heisenberg’s Uncertainty Principle showed, however, that the micro world is
governed by a new kind of probability laws which differ from the Kolmogorovian
ones. This was a great challenge to mathematicians as well as to physicists and
logicians. One of the responses to this situation was the, now famous, 1936 paper by Garret Birkhoff and John von Neumann entitled “The logic of quantum
mechanics”, in which they suggested a new logical model which was based on
the Hilbert space formalism of quantum mechanics and which we, today, call a
quantum logic. G. Mackey asked the question whether every state on the lattice
of projections of a Hilbert space could be described by a density operator; and
his young student A. Gleason gave a positive answer to this question. Although
this was not part of Gleason’s special field of interest, his theorem, now known
as Gleason’s theorem, had a profound impact and is rightfully considered one of
the most important results about quantum logics and structures. Gleason’s proof
was non-trivial. When John Bell became familiar with it, he said he would leave
this field of research unless there would be a simpler proof of Gleason’s theorem.
Fortunately, Bell did find a relatively simple proof of the partial result that there
exists no two-valued measure on a three-dimensional Hilbert space. An elementary

proof of Gleason’s theorem was presented by R. Cooke, M. Keane and W. Moran
in 1985.
In the eighties and nineties it was the American school that greatly enriched the
theory of quantum structures. For me personally Varadarajan’s paper and subsequently his book were the primary sources of inspiration for my work together with

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viii

Foreword

Gleason’s theorem. The theory of quantum logics and quantum structures inspired
many mathematicians, physicists, logicians, experts on information theory as well
as philosophers of science. I am proud that in my small country, Czechoslovakia
and now Slovakia, research on quantum structures is a thriving field of scientific
activity.
The achievements characteristic of the eighties and nineties are the fuzzy approaches which provided a new way of looking at quantum structures. A whole
hierarchy of quantum structures emerged, and many surprising connections with
other branches of mathemtics and other sciences were discovered. Today we can
relate phenomena first observed in quantum mechanics to other branches of science such as complex computer systems and investigations on the functioning of
the human brain, etc.
In the early nineties, a new organisation called International Quantum Structures Association (IQSA) was founded. IQSA gathers experts on quantum logic
and quantum structures from all over the world under its umbrella. It organises regular biannual meetings: Castiglioncello 1992, Prague 1994, Berlin 1996,
Liptovsky Mikulas 1998, Cesenatico 2001, Vienna 2002, Denver 2004, Malta 2006.
In spring 2005, Dov Gabbay, Kurt Engesser, Daniel Lehmann and Jane Spurr
had an excellent idea — to ask experts on quantum logic and quantum structures to
write long chapters for the Handbook of Quantum Logic and Quantum Structures.
It was a gigantic task to collect and coordinate these contributions by leading
experts from all over the world. We are grateful to all four for preparing this

monumental opus and to Elsevier for publishing it.
When browsing through this Handbook, in my mind I am wandering back to
Hilbert’s Sixth Problem. I am happy that this problem is in fact not a genuinely
mathematical one which, once it is solved, brings things to a close. Rather it has led
to a new development of scientific thought which deeply enriched mathematics, the
understanding of the foundations of quantum mechanics, logic and the philosophy
of science. The present Handbook is a testimony to this fact. Those who bear
witness to it are Dov, Kurt, Daniel, Jane and the numerous authors. Thanks to
everybody who helped bring it into existence.
Anatolij Dvureˇcenskij, President of IQSA
July 2006

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ix

EDITORIAL PREFACE

There is a wide spread slogan saying that Quantum Mechanics is the most
successful physical theory ever. And, in fact, there is hardly a physicist who does
not agree with this. However, there is a reverse of the medal. Not only is Quantum
mechanics unprecedently successful but it also raises fundamental problems which
are equally unprecedented not only in the history of physics but in the history of
science in general.
The most fundamental problems that Quantum Mechanics raises are conceptual
in nature. What is the proper interpretation of Quantum Mechanics? This is a
question touching on most fundamental issues, and it is, at this stage, safe to say
that there is no answer to this question yet on which physicists and philosophers
of science could agree. It is, moreover, no exaggeration to say that the problem of

the conceptual understanding of Quantum Mechanics constitutes one of the great
intellectual puzzles of our time.
The topic of the present Handbook is, though related to this gigantic issue, more
modest in nature. It can, briefly, be described as follows. Quantum Mechanics
owes is tremendous success to a mathematical formalism. It is the mathematical
and logical investigation of the various aspects of this formalism that constitutes
the topic of the present Handbook.
This formalism the core of which is the mathematical structure of a Hilbert
space received its final elegant shape in John von Neumann’s classic 1932 book
“Mathematical Foundations of Quantum Mechanics”. In 1936 John von Neumann
published, jointly with the Harvard matthematician Garret Birkhoff, a paper entitled “The logic of quantum mechanics”. In the Introduction the authors say: ”The
object of the present paper is to discover what logical structure we may hope to
find in physical theories which, like quantum mechanics, do not conform to classical logic”. The idea of the paper, which was as ingenious as it was revolutionary,
was that the Hilbert space formalism of Quantum Mechanics displayed a logical
structure that could prove useful to the understanding of Quantum Mechanics.
Birkoff and von Neumann were the first to put forward the idea that there is a
link between logic and (the formalism of) Quantum Mechanics, and their now famous paper marked the birth of a field of research which has become known as
Quantum Logic. The Birkhoff-von Neumann paper triggered, after some time of
dormancy admittedly, a rapid development of quantum logical research. Various
schools of thought emerged. Let us, in this Introduction, highlight just a few of
the milestones in this development.
In his famous essay “Is logic empirical?” Putnam put forward the view that the
role played by logic in Quantum Mechanics is similar to that played by geometry in

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x

Editorial Preface

the theory of relativity. On this view logic is as empirical as geometry. Putnam’s
revolutionary thesis triggered a discussion which was highly fruitful not only for
Quantum Mechanics but for our views on the nature of logic in general. The reader
will find a discussion of Putnam’s thesis in this Handbook.
Another school of thought was initiated by Piron’s “Axiomatique quantique”.
This school, which has become known as the Geneva school, aimed at reconstructing the formalism of Quantum Mechanics from first principles. It was Piron’s
student Diederik Aerts who continued this in Brussels. The achievements of the
Geneva-Brussels school are reflected in various chapters witten by Aerts and former students of his.
In Italy it was Enrico Beltrametti and Maria Luisa Dalla Chiara, just to mention
two names, who founded another highly influential school which is well represented
in this Handbook.
Highly sophisticated efforts resulted in linking the logic of Quantum Mechanics
to mainstream logic. Just to give a flavour of this, let us mention that Nishimura
studied Gentzen type systems in the context of Quantum Logic. Abramsky and
Coecke in Oxford and Sernadas in Lisbon as well as others established the connection with Categorial Logic and Linear Logic, and the connection with NonMonotonic Logic was made by the editors.
Prior to this, another direction of research had focused on the lattice structures
relevant to Quantum Logic. Essentially, this field of research was brought to
fruition in the USA by the pioneering work of Foulis and Greechie on orthomodular
lattices.
Moreover, we have to mention the Czech-Slovak school which was highly influential in establishing the vast field of research dealing with the various abstract
Quantum Structures which constitute the topic of a whole volume of this Handbook. Let us in this context just mention the names of Anatolij Dvurecenskij and
Sylvia Pulmannova in Bratislava and Pavel Ptak in Prague.
The editors are happy and grateful to have succeeded in bringing together the
most eminent scholars in the field of Quantum logic and Quantum Structures for
the sake of the present Handbook. We cordially thank all the authors for their
contributions and their cooperation during the preparation of this work. Most of
these authors are members of the Internatonal Quantum Structures Association
(IQSA). We would like to express our deep gratitude to IQSA and in particular
to its President, Professor Anatolij Dvurecenskij, for cooperating so closely with

us and supporting us so generously during the preparation of this Handbook.
The present Handbook is an impressive document of the intellectual achievements which have been made in the study of the logical and mathematical structures arising from Quantum Mechanics. We hope that it will turn out to be a
milestone on the path that will ultimately lead to the solution of one of the great
intellectual puzzles of our time, namely the understanding of Quantum Mechanics.
The Editors: Kurt Engesser, Dov Gabbay and Daniel Lehmann
Germany, London, and Israel
May 2008

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM LOGIC
Edited by K. Engesser, D. M. Gabbay and D. Lehmann
© 2009 Elsevier B.V. All rights reserved

1

THE BIRKHOFF–VON NEUMANN CONCEPT
OF QUANTUM LOGIC
Mikl´os R´edei

1

INTRODUCTION

Quantum logic was born with the following conclusion of Garrett Birkhoff and John
von Neumann in their joint paper (henceforth “Birkhoff-von Neumann paper”)
published in 1936:
Hence we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry.
[Birkhoff and von Neumann, 1936] (Emphasis in the original)

This was a striking conclusion in 1936 for two reasons, one ground breaking and one
conservative: ground breaking because it opened the way for the development of
algebraic logic in the direction of non-classical algebraic structures that have much
weaker properties than Boolean algebras. Conservative because an abstract projective geometry is an orthocomplemeneted, (non-distributive), modular lattice;
however, the non-Boolean algebra that seemed in 1936 to be the most natural
candidate for quantum logic was the non-modular, orthomodular lattice of all projections on an infinite dimensional complex Hilbert space. Indeed, subsequently
quantum logic was (and typically still is) taken to be an orthocomplemented, nonmodular, orthomodular lattice. Hence, the concept of quantum logic proposed by
Birkhoff and von Neumann in their seminal paper differs markedly from the notion that became later the standard view – it is more conservative than one would
expect on the basis of later developments.
There are not many historical investigations in the enormous quantum logic literature [Pavicic, 1992] that aim at scrutinizing the Birkhoff-von Neumann notion of
quantum logic, and especially the discrepancy between the standard notion and the
Birkhoff-von Neumann concept: [Bub, 1981b], [Bub, 1981a], [R´edei, 1996], Chapter 7. in [R´edei, 1998], [R´edei, 2001], [Dalla-Chiara et al., 2007] (see also [Popper,
1968] and [Scheibe, 1974]). The recent discovery and publication in [R´edei, 2005]
of von Neumann’s letters to Birkhoff during the preparatory phase (in 1935) of
their 1936 paper have made it possible to reconstruct in great detail [R´edei, 2007]
the conceptual considerations that culminated in the 1936 paper’s main conclusion
cited above. As a result of these historical studies we now understand quite well
why Birkhoff and von Neumann postulated the “quantum propositional calculus”

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Mikl´
os R´
edei

to be a modular lattice and rejected explicitly the idea that quantum propositional
calculus can be identified with the non-modular, orthomodular lattice of all closed

linear subspaces of an infinite dimensional complex Hilbert space (Hilbert lattice).
The aim of this review is to recall the Birkhoff-von Neumann concept of quantum
logic together with the pertinent mathematical notions necessary to understand
the development of their ideas. Special emphasis will be put on the analysis of
the difference between their views and the subsequent standard notion of quantum
logic. The structure of the review is the following. For reference, and in order to
place the Birkhoff-von Neumann concept in appropriate context, section 2 recalls
briefly the standard notion of quantum logic in terms of algebraic semantics. Based
on excerpts from the recently discovered and published letters by von Neumann
to Birkhoff, section 3 reconstructs the main steps of the thought process that led
Birkhoff and von Neumann to abandon Hilbert lattice as quantum logic and to
propose in their published paper an abstract projective geometry as the quantum
propositional system. Section 4 argues that von Neumann was not satisfied with
their idea after 1936: He would have liked to see quantum logic worked out in much
greater detail – he himself tried to achieve this but did not succeed. Section 4 also
attempts to discern the conceptual obstacles standing in the way of elaborating
quantum logic along the lines von Neumann envisaged. The concluding section 5
summarizes the main points and makes some further comments on the significance
of the Birkhoff-von Neumann concept.
2

QUANTUM LOGIC: LOGICIZATION OF NON-BOOLEAN ALGEBRAIC
STRUCTURES. THE STANDARD VIEW.

It is well known that both the syntactic and the semantic aspects of classical
propositional logic can be described completely in terms of Boolean algebras: The
Tarski-Lindenbaum algebra A of classical propositions is a Boolean algebra and a
deductive system formulated in a classical propositional logic can be identified with
a filter in A. The notions of syntactic consistency and completeness correspond
to the filter being proper and being prime (equivalently: maximal), respectively.

The notion of interpretation turns out to be a Boolean algebra homomorphism
from A into the two element Boolean algebra, and all the semantic notions are
defined in terms of these homomorphisms. All this is expressed metaphorically
by Halmos’ famous characterization of the (classical) logician as the dual of a
(Boolean) algebraist [Halmos, 1962, p. 22], a characterization which has been
recently “dualized” by Dunn and Hardegree: “By duality we obtain that the
algebraist is the dual of the logician.” [Dunn and Hardegree, 2001, p. 6].
The problem of quantum logic can be formulated as the question of whether the
duality alluded to above also obtains if Boolean algebras are replaced by other,
typically weaker algebraic structures arising from the mathematical formalism of
quantum mechanics. It turns out that formal logicization is possible for a large
class of non-Boolean structures. Following Hardegree [Hardegree, 1981b], [Hardegree, 1981a] the standard (sometimes called “orthodox”) concept is described

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The Birkhoff–von Neumann Concept of Quantum Logic

3

below, and it is this concept with which the Birkhoff von Neumann concept will
be contrasted.
Standard quantum logic comes in two forms: abstract (also called orthomodular) quantum logic and concrete (also called Hilbert) quantum logic. The semantics
is similar in both cases, but the latter determines a stronger logic.
Let K = {P, &, , ∼} be a zeroth order formal language with the set P of
sentence variables p, q . . ., two place connectives & (and), (or), negation sign ∼,
parentheses (,), and let F be the set of well formed formulas in K defined in the
standard way by induction from P : F is the smallest set for which the following
two conditions hold:
P

if φ, ψ ∈ F then

⊂

F

(1)

(φ&ψ), (φ

ψ), (∼ φ) ∈ F

(2)

Let (L, ∨, ∧, ⊥) be an orthomodular lattice. Orthomodularity of L means that the
following condition holds:
(3) orthomodularity: If A ≤ B and A⊥ ≤ C, then A∨(B∧C) = (A∨B)∧(A∨C)
Orthomodularity is a weakening of the modularity law:
(4) modularity:

If A ≤ B, then A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)

which itself is a weakening of the distributivity law:
(5) distributivity:

A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)

for all A,B,C

The set GQ ⊆ L in an orthomodular lattice is called a generalized filter if

⊥

if A ∈ GQ and A ∨ (A ∧ B) ∈ GQ then

I ∈ GQ
B ∈ GQ

(6)
(7)

Given a pair (L, GQ ) the map i : F → L is called an (L, GQ )-interpretation if
i(φ&ψ) = i(φ) ∧ i(ψ)
i(φ ψ) = i(φ) ∨ i(ψ)
i(∼ φ) = i(φ)⊥
Each interpretation i determines
⎧
⎨ 1 (true)
0 (f alse)
(11) vi (φ) =
⎩
undetermined

(8)
(9)
(10)

a (L, GQ )-valuation vi by
if
if
otherwise

i(φ) ∈ GQ
i(∼ φ) ∈ GQ

If V (L) denotes the set of all (L, GQ )-valuations and V is the class of valuations
determined by the class of orthomodular lattices, then φ ∈ F is called valid if
v(φ) = 1 for every v ∈ V , and a class of formulas Γ is defined to entail φ if
v(ψ) = 1 for all ψ ∈ Γ implies v(φ) = 1. One can define the quantum analog →Q
of the classical conditional by

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Mikl´
os R´
edei

(12) φ →Q ψ =∼ φ

(φ&ψ)

and one can formulate a deduction system in K using →Q such that one can prove
soundness and completeness theorems for the resulting quantum logical system
(see [Hardegree, 1981b],[Hardegree, 1981a]).
A specific class of orthomodular lattices is the category of Hilbert lattices: A
Hilbert lattice P(H) with lattice operations ∧, ∨, ⊥ is the set P(H) of all projections (equivalently: closed linear subspaces) of a complex, possibly infinite dimensional Hilbert space H, where the lattice operations ∧, ∨ and ⊥ are the set
theoretical intersection, closure of the sum and orthogonal complement, respectively. Note that the Hilbert lattice P(H) is not only non-distributive but it also
is non-modular if the dimension of the Hilbert space is infinite [R´edei, 1998].

It is important that while all the definitions and stipulations made above in
connection with orthomodular lattices are meaningful for Hilbert lattices, no completeness results are presently known for the semantics determined by Hilbert
lattices: The deduction system that works in the case of (abstract) orthomodular
lattices is not strong enough to yield all statements that are valid in Hilbert lattices: the “ortho-arguesian law”, which is valid in Hilbert lattices, does not hold
in every orthomodular lattice (see [Kalmbach, 1981], [Dalla-Chiara and Giuntini,
2002]).

3 THE BIRKHOFF-VON NEUMANN CONCEPT OF QUANTUM LOGIC
The Birkhoff-von Neumann paper can be viewed as the first paper in which the
suggestion to logicize a non-Boolean lattice appears. There are however several
types of non-Boolean lattice. Which one is supposed to be logicized?

3.1 Which non-Boolean lattice to logicize?
At the time of the birth of quantum logic the notion of an abstract orthomodular lattice did not yet exist; however, the canonical example of non-distributive,
orthomodular lattices, the Hilbert lattice P(H), was known already, and, since
this structure emerges naturally from the Hilbert space formalism of quantum
mechanics, P(H) was the most natural candidate in 1935 for the propositional
system of quantum logic. Indeed, Birkhoff and von Neumann did consider P(H)
as a possible propositional system of quantum logic; yet, this lattice was not their
choice: The first indication that P(H) may not be a suitable candidate for a quantum propositional system is in von Neumann’s letter of (January 19, 1935). Von
Neumann writes:

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The Birkhoff–von Neumann Concept of Quantum Logic

Using the operator-description,
a ∨ b, a ∧ b can be formed, if the
physically significant operators form a ring.

This, I think should be assumed anyhow,
even if one does not require that
all operators are phys.[ically] significant.

5

(← I believe this).

(← but I am rather doubting lately this.)

But we need probably not insist on this point too much.
(von Neumann to Birkhoff, January 19, 1935 ?1 ), [R´edei, 2005, p. 51]

3.2

Von Neumann algebras

A “ring of operators” von Neumann is referring to in the above quotation is known
today as a von Neumann algebra: a set N of bounded operators on a Hilbert
space H is a von Neumann algebra if it contains the unit, is closed with respect
to the adjoint operation and is closed in the strong operator topology. The latter
requirement means that if Qn is a sequence of operators from M such that for all
ξ ∈ H we have Qn ξ → Qξ for some bounded operator Q on H, then Q ∈ M (see
[Takesaki, 1979] for the theory of von Neumann algebras).
If S is any set of bounded operators on H, then its (first) commutant S is the
set of bounded operators that commute with every element in S i.e.:
S ≡ {Q : QX = XQ, for all X ∈ S}
The operation of taking the commutant can be iterated: S ≡ (S ) , and it is
clear that S is contained in the second commutant, so the second commutant S
is an extension of S. How much of an extension? The answer to this question,

von Neumann’s double commutant theorem, is the most fundamental result in the
theory of von Neumann algebras:
PROPOSITION 1 Double commutant theorem. S is strongly dense in S .
Proposition 1 implies that a ∗-algebra of bounded operators on a Hilbert space
that contains the unit is a von Neumann algebra if and only if it coincides with
its second commutant. A von Neumann algebra N is called a factor if the only
elements in N that commute with every other element in N are the constant
multiples of the identity, i.e. if
(13) N ∩ N = {λI : λ a complex number}
An immediate corollary of the double commutant theorem is the characteristic
property of von Neumann algebras that they contain many projections; in fact,
they contain enough projections for the set of projections P(M) to determine the
von Neumann algebra completely in the sense
1 Von Neumann’s letters are not always properly dated: the year is occasionally missing. If
this is the case, we put a question mark after the year – the context makes it clear that the year
of writing is 1935 in cases of all the letters quoted here.

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Mikl´
os R´
edei

(14) M = (P(M))
Moreover, we have
PROPOSITION 2. The set P(M) of projections of a von Neumann algebra M is
a complete, orthomodular lattice (called von Neumann lattice) with respect to the

lattice operations inherited from P(H).
In particular the Hilbert lattice P(H) of all projections on a Hilbert space is
a complete orthomodular lattice, since the set B(H) of all bounded operators on
H is a von Neumann algebra. Specifically the spectral projections of the set of
all (not necessarily bounded) selfadjoint operators coincides with the set of all
projections.
It is very important that while all von Neumann lattices are orthomodular, some
have the stronger property of modularity. There is a subtle connection between
the type of a von Neumann algebra (in the sense of the Murray-von Neumann
classification theory) and the modularity of its projection lattice. We shall return
to this issue later.

3.3 Non-modularity of Hilbert lattice
While in January 1935 von Neumann did not intend to insist on restricting the set
of physical quantities to a proper subset of all possible operators, by November
1935 he changed his mind:
I am somewhat scared to consider all physical quantities = bounded
self-adjoint operators as a lattice.
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 53]
The reason why he changed his mind was the realization that the Hilbert lattice
P(H) is not modular if the Hilbert space is infinite dimensional (note that “Blattice” means modular lattice in the next quotation):
In any linear space H the linear subspaces K, L, M, . . . form a B-lattice
L with the
“meet” K ∩ L: intersection of K and L
in the sense of set theory
“join” K ∪ L : linear sum of K and L,
i.e. set of all f + g, f ∈ K, g ∈ L
(Proof obvious.) But in a metric-linear space H the lattice L of all
closed-linear subspaces KLM , . . . , for which the “join” is defined as
“join” K ∨ K:

closure of the linear sum of K and L,
i.e. the set of all condensation points
of the f + g, f ∈ K, g ∈ L

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The Birkhoff–von Neumann Concept of Quantum Logic

7

while the “meet” is as above, is not necessarily a B-lattice. This is in
particular the case in Hilbert space. (K ∪ L and K ∨ L are identical
if K, L are both closed and orthogonal to each other, but not for any
two closed K, L !)
In fact, it is possible to find three closed-linear subspaces K, L, M of
Hilbert space H, for which
(15) K

M,

(K ∨ L) ∩ M

K ∨ (L ∩ M )

(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 54],
emphasis in original.
This letter contains a detailed proof that subspaces K, L, M exist that satisfy
(15) (and thereby violate modularity (4)). (The Birkhoff-von Neumann paper just

states this fact without detailed argument.) Von Neumann’s proof makes use of the
theory of unbounded selfadjoint operators, utilizing the fact that one can find two
unbounded selfadjoint operators X and Y on an infinite dimensional Hilbert space
such that the intersection of their domains is empty. Von Neumann emphasizes
this feature of his proof:
Examples could be constructed which make no use of operator theory,
but I think that this example shows more clearly “what it’s all about”:
It is the existence of “pathological” operators – like X, Y above – in
Hilbert space, which destroys the B-lattice character.
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 55]
Von Neumann regarded this pathological behavior of the set of all unbounded operators on a Hilbert space a very serious problem because it prohibits adding and
composing these operators in general, which entails that these operators do not
form an algebra. In von Neumann’s eyes this was a great obstacle to doing computations with those operators, and since the selfadjoint operators are representatives
of quantum physical quantities, it appeared unnatural to him that they behave
so irregularly that forming an algebra from them was not possible. He pointed
out this pathology several times in his published papers (see e.g. paragraph 6. of
Introduction in [Murray and von Neumann, 1936]), and the pathological character
of the set of all selfadjoint operators was one of the main reasons why he hoped
as late as in his famous talk on “Unsolved Problems in Mathematics” in 1954 (see
[von Neumann, 2001] and [R´edei, 1999]) that a restricted set of operators, and
therefore a specific von Neumann algebra, the type II1 factor (see below) would
be a more suitable mathematical framework for quantum mechanics than Hilbert
space theory.
In this situation von Neumann saw two options:

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(I) Either we define the “join” by ∪ (as a honest linear sum), then
the lattice is B, but we must admitt all (not-necessarily-closed-)
linear subspaces,
(II) or we define the “join” by ∨ (closure of the linear sum), then the
B-character is lost.
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 55]
Since P(H) is not modular and given that von Neumann wished to preserve modularity as a property of the quantum propositional system, one would expect von
Neumann to choose option (I). But this is not the case. Von Neumann thinks
through the consequences of choosing option (I) first:
Let us first consider the alternative (I). The orthogonal complement
K still has the property K ∪ L = (K ∩ L) , but K ∩ L = (K ∪ L)
and K = K are lost. We have K ∩K = 0, while K ∪K is everywhere
dense, but not necessarily I. There is a funny relationship between K
and its “closure” K . (For instance: All probabilities in the state K
are equal to those in the state K , but “meets” (K ∩ L and K ∩ L ,
even for closed L’s) may differ.)
The situation is strongly reminiscent of the “excluded middle” troubles,
although I did not yet compare all details with those of the classcalculus in “intuitionistic” logics.
After all it is so in normal logics, too, that these troubles arise as
soon as you pass to infinite systems, although I must admitt, that the
difficulties there are more “optional” then2 here.
It has to be said, finally, that even in this case (I) complements exist,
i.e., that for every K there exists K ∗ ’s for which K ∪K ∗ = I, K ∩K ∗ =
0, but one needs the Hammel-basis-construction to get them.
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 55]
So, while von Neumann evaluates alternative (I) as representing an option which

cannot be excluded on the grounds of being either algebraically or logically extremely weird (although it is clear from the above that he did not like the asymmetric failure of De Morgan’s law), he prefers option (II) in spite of its being
seemingly counterintuitive. Here is why:
Alternative (II) seems to exclude Hilbert space, if one sticks to Blattices.3 Still one may observe this:
Consider a ring R of operators in Hilbert space. The idempotents of R
form a lattice LR . One sees easily, that LR is irreducible (=no direct
sum), if and only if the center of R consists of the αI (α=complex
2 Spelling
3 Recall

error, should be “than”.
that B-lattice means modular lattice.

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The Birkhoff–von Neumann Concept of Quantum Logic

9

number) only, i.e. if R is a ring of the sort which Murray and I considered. (We called them “factors”.) LR contains 0, I and a complement
which dualises ∪ and ∩. (Now ∪ corresponds to what I called ∨, case
(II).) One may ask: When is LR a B-lattice? The answer is (this is not
difficult to prove): If and only if the ring R is finite in the classification
Murray and I gave. I.e.: R must be isomorphic:
1. either to the full matrix-ring of a finite-dimensional Euclidean
space (say n-dimensional, n = 1, 2, . . .),
2. to one of those of our rings, in which each idempotent has a “dimensionality”, the range of which consists of all real numbers
≥ 0, ≤ 1, and which is uniquely determined by its formal properties.
We called 1. “Case In ” and 2. “Case II1 ”.
Thus for operator-lattices the B-lattice axiom

a≤b

→

(a ∪ b) ∩ c = a ∪ (b ∩ c)

leads directly to the cases I1 , I2 , . . . and II1 !
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 56]

3.4

Types of von Neumann algebras

Von Neumann refers here to the Murray-von Neumann classification theory of factors, which was worked out by him in collaboration with F.J Murray precisely
at the time (1934-1935) when he was working with G. Birkhoff on quantum logic
[Murray and von Neumann, 1936]. Von Neumann (partly in collaboration with
F.J. Murray) published four major papers on the theory of von Neumann algebras
[Murray and von Neumann, 1936], [Murray and von Neumann, 1937], [von Neumann, 1940] and [Murray and von Neumann, 1943]. The first paper’s main result
was a classification theory of von Neumann algebras that are irreducible in the
sense of not containing non-trivial operators commuting with every other operator
in the algebra (i.e. “factors”). The set B(H) of all bounded operators is clearly a
factor and it turned out that there are five classes of factors, the different types
are denoted by von Neumann as In , I∞ , II1 , II∞ and III∞ . The classification
of factors was given in terms of a (relative) dimension function d defined on the
lattice of projections P(M) of a von Neumann algebra M. The map d from P(M)
into the set IR+ ∪ {∞} is a dimension function if
(i) d(A) > 0 if and only if A = 0,
(ii) d(A) = d(B) if there exists an isometry U ∈ M between ranges of the
projections A and B,
(iii) d(A) + d(B) = d(A ∨ B) + d(A ∧ B).

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The (relative) dimension function d on an arbitrary factor is a generalization of the
ordinary linear dimension of the closed linear subspace a projection projects to,
and the ordinary dimension takes on the positive integer values 0, 1, 2, . . . , n and
0, 1, 2, . . ., respectively, in the two well-known cases of the set of all bounded operators on a finite, n-dimensional (respectively infinite) dimensional Hilbert space.
In the cases II1 , II∞ and III∞ the ranges of the dimension function are, respectively, the following: the unit interval [0, 1], the set of non-negative real numbers
IR+ and the two element set {0, ∞}. (See [Takesaki, 1979] for a systematic treatment, or [R´edei, 1998], [Petz and R´edei, 1995] for a brief review of the Murray-von
Neumann classification theory). The result of the classification theory can thus be
summarized in the form of the following table:

range of d
{0, 1, 2, . . . , n}
{0, 1, 2, . . . , ∞}
[0, 1]
IR+
{0, ∞}

type of factor N
In
I∞
II1

II∞
III∞

the lattice P(M)
modular
orthomodular, non-modular
modular
orthomodular, non-modular
orthomodular, non-modular

3.5 The type II1 factor
Thus the significance of the existence of type II1 factors is that their projection
lattices are modular. (Accordingly, the set of all (not necessarily bounded) selfadjoint operators that they determine are free of the pathologies which von Neumann
considered undesirable.)
Von Neumann’s conclusion:
This makes me strongly inclined, therefore, to take the ring of all
bounded operators of Hilbert space (“Case I∞ ” in our notation) less
seriously, and Case II1 more seriously, when thinking of an ultimate
basis of quantum mechanics.
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 56]
As can be inferred from von Neumann’s letter to Birkhoff (November 6, 1935 ?),
[R´edei, 2005, p. 59-64], Birkhoff suggested another idea to save the modularity of
the lattice formed by some subspaces of a Hilbert space: by restricting the linear
subspaces to the finite dimensional ones. Von Neumann did not consider this idea
in detail, but thought that it was not an attractive one:
Many thanks for your letter. Your idea of requiring a ≤ c → a∪(b∩c) =
(a∪b)∩c in Hilbert space for the finite a, b, c only is very interesting, but
will it permit to differentiate between Hilbert-space and other Banachspaces?
(von Neumann to Birkhoff, November 13, 1935 ?), [R´edei, 2005, p. 59]

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The Birkhoff–von Neumann Concept of Quantum Logic

11

Rather than answering this rhetorical question, von Neumann makes his famous
confession (quoted in part by Birkhoff in [Birkhoff, 1961]), reaffirming that the
operator algebraic results related to classification theory of von Neumann algebras
reduce the privileged status of Hilbert space quantum mechanics:
I would like to make a confession which may seem immoral: I do not
believe absolutely in Hilbert space any more. After all Hilbert-space
(as far as quantum-mechanical things are concerned) was obtained by
generalizing Euclidean space, footing on the principle of “conserving
the validity of all formal rules”. This is very clear, if you consider
the axiomatic-geometric definition of Hilbert-space, where one simply
takes Weyl’s axioms for a unitary-Euclidean-space, drops the condition
on the existence of a finite linear basis, and replaces it by a minimum of
topological assumptions (completeness + separability). Thus Hilbertspace is the straightforward generalization of Euclidean space, if one
considers the vectors as the essential notions.
Now we4 begin to believe, that it is not the vectors which matter but
the lattice of all linear (closed) subspaces. Because:
1. The vectors ought to represent the physical states, but they do it
redundantly, up to a complex factor, only.
2. And besides the states are merely a derived notion, the primitive (phenomenologically given) notion being the qualities, which
correspond to the linear closed subspaces.
But if we wish to generalize the lattice of all linear closed subspaces
from a Euclidean space to infinitely many dimensions, then one does
not obtain Hilbert space, but that configuration, which Murray and I

called “case II1 .” (The lattice of all linear closed subspaces of Hilbertspace is our “case I∞ ”.) And this is chiefly due to the presence of the
rule
a ≤ c → a ∪ (b ∩ c) = (a ∪ b) ∩ c
This “formal rule” would be lost, by passing to Hilbert space!
(von Neumann to Birkhoff, November 13, 1935 ?), [R´edei, 2005, p. 59]

3.6

From the type II1 factor to abstract continuous geometry

So it would seem that the modular lattice of the type II1 factor von Neumann
algebra emerges as the strongest candidate for logicization, and so one would
expect this lattice to be declared in the Birkhoff-von Neumann paper to be the
propositional system of quantum logic. But this is not the case; in fact, the
published paper does not at all refer to the results of the Murray-von Neumann
classification theory of von Neumann algebras to support the modularity postulate.
4 With

F.J. Murray, von Neumann’s coauthor.

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Why? Von Neumann’s letters to Birkhoff also contain clues for the the answer

to this question, and the answer is that von Neumann’s mind moved extremely
quickly from the level of abstractness of von Neumann algebras to the level of
abstractness represented by continuous geometries — and this move was taking
place precisely during the preparation of the quantum logic paper: in his letter to
Birkhoff (November 6, 1935 ?) von Neumann writes:
Mathematically – and physically, too – it seems to be desirable, to try
to make a general theory of dimension in complemented, irreducible
B-lattices, without requiring “finite chain conditions”. I am convinced,
that by adding a moderate amount of continuity-conditions, the existence of a numerical dimensionality could be proved, which
1. is uniquely determined by its formal properties,
2. and after a suitable normalisation has either the range d = 1, 2, . . . , n
(n = 1, 2, . . . , finite!) or d ≥ 0, ≤ 1.
I have already obtained some results in this direction, which connect
the notion of dimension in a very funny way with the perspectivities
and projectivities in projective geometry.
It will perhaps amuse you if I give some details of this. Here they are:
(von Neumann to Birkhoff, November 6, 1935 ?), [R´edei, 2005, p. 56]
And there follows in the letter a three page long exposition of the theory of continuous geometries, which is not reproduced here. In his letter written a week later
(November 13, 1935 ?), [R´edei, 2005, p. 59-64], von Neumann gives an even more
detailed description of his results on continuous geometry, which confirm the two
conjectures 1. and 2. above completely: on every projective geometry there exists
a dimension function d having the properties
0 ≤ d(A)
d(A) + d(B)
=

≤1
d(A ∨ B) + d(A ∧ B)

(16)

(17)

and having discrete or continuous range.5 These results do not appear in the
Birkhoff-von Neumann paper on quantum logic, von Neumann published them
separately [von Neumann, 1936] (cf. footnote 33 in the Birkhoff-von Neumann
paper [Birkhoff and von Neumann, 1936]).
Thus by the time it came to the final version of the quantum logic paper, von
Neumann knew that the projection lattice of a type II1 von Neumann algebra is
just a special case of more general continuous geometries that admit well-behaving
probability measures, and this explains why the major postulate in the Birkhoffvon Neumann paper is formulated in the section entitled “Relation to abstract
projective geometries” and reads:
5 The discrepancy between Eq. (16) and the ranges mentioned under 2. above are due to
different normalizations.

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The Birkhoff–von Neumann Concept of Quantum Logic

13

Hence we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry.
[Birkhoff and von Neumann, 1936] (Emphasis in the original)

3.7

Probability and quantum logic

The finite dimension function on a projective geometry, in particular the dimension
function with the range [0, 1] on the continuous projective geometry, was for von

Neumann crucially important in his search for a proper quantum logic: he interpreted the dimension function as an a priori probability measure on the modular
lattice of the quantum propositional system. Thus by requiring quantum logic to
be an abstract continuous geometry with a dimension function, Birkhoff and von
Neumann created an analogy with classical logic and probability theory, where
a Boolean algebra is both a propositional system and a random event structure
on which probability measures are defined. While there is no detailed discussion
of this aspect of the dimension function in the von Neumann-Birkhoff correspondence, the Birkhoff-von Neumann paper points out that properties (16)-(17) of the
dimension function describe the formal properties of probability. Von Neumann
regarded it as another pathology of the total Hilbert lattice P(H) that there exists
no probability measure on it that satisfies conditions (16)-(17. This is because one
has the following theorem:
PROPOSITION 3. Let L be a bounded lattice. If there exists a finite dimension
function d on L (i.e. a map d from L into the set of real numbers that has the
properties (16)-(17)), then the lattice is modular.
It is very easy to see that subadditivity (property (17)) is a necessary condition
for a measure to be interpreted as probability understood as relative frequency in
the sense of von Mises [von Mises, 1919], [von Mises, 1981]:
Assume that the probability p(X) (X = A, B, A ∧ B, A ∨ B) is to be interpreted
as relative frequency in the following sense:
1. There exists a fixed ensemble E consisting of N events such that
2. for each event X one can decide unambiguously and
3. without changing the ensemble whether X is the case or not;
4. p(X) =
case.6

#(X)
N

where #(X) is the number of events in E for which X is the

Under the assumptions 1.-4. it trivially follows that (17) holds since one can
6 Strictly

#(X)

speaking one should write p(X) = limN →∞ N ; however, the limit is not important from the point of view of the present considerations, so we omit it.

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write
#(A ∪ B) #(A ∩ B)
+
N
N
#((A \ A ∩ B) ∪ (B \ A ∩ B) ∪ A ∩ B)) #(A ∩ B)
+
N
N
#(A \ A ∩ B) + #(B \ A ∩ B) + #(A ∩ B) + #(A ∩ B)
N
#(A) + #(B)
N

=
=
=

which is the subadditivity. Thus, if a map d on a lattice does not have subadditivity
(17) then the probabilities d(X) cannot be interpreted as probabilities in the sense
of relative frequency formulated above via 1.-4.; consequently, the lattice cannot
be viewed as representing a collection of random events in the sense of a relative
frequency interpreted probability theory specified by 1.-4. (with the understanding
that A ∧ B denotes the joint occurrence of events A and B). Since von Neumann
embraced the frequency interpretation of probability in the years 1927-1935, this
makes understandable why he considered the subadditivity (17) a key feature of
probability and, consequently, modularity an important condition to require.
Thus it would seem that within the mathematical framework of continuous
geometry, especially within the theory of the type II1 von Neumann algebras, the
Birkhoff and von Neumann concept of quantum logic could restore the harmonious
classical picture: random events can be identified with the propositions stating that
the event happens, and probabilities can be viewed as relative frequencies of the
occurrences of the events. But this restored harmony is illusory for the following
reason: von Neumann and Murray showed that a dimension function d on the
projection lattice P(N ) of a type II1 algebra N can be extended to a trace τ on
N . The defining property of a trace τ is
(18) τ (XY ) = τ (Y X)

for all X, Y ∈ N .

That is to say, the trace is exactly the functional which is insensitive (in the sense
of (18)) for the non-commutativity of the algebra. On the other hand, it can be
shown that a normal state φ on a von Neumann lattice satisfies the additivity (17)
if (and only if) it is a trace [Petz and Zemanek, 1988]. Thus, the only quantum

probability measures that mesh with the relative frequency interpretation via 1.-4.
are the ones given by the trace.
Behind the mathematical fact that only traces satisfy subadditivity lies the
conceptual difficulty that assumptions 2. and 3. of the frequency interpretation
of probability cannot be upheld in interpreting the elements of a von Neumann
lattice as random quantum events and the lattice operation A ∧ B as the joint
occurrence of A and B: assumption 3. fails if “deciding” means “measuring”,
since measuring disturbs the measured system, hence also the ensemble; therefore,
there is no single, fixed, well-defined ensemble in which to compute as relative
frequencies the probabilities of all projections representing quantum attributes.

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