**—***What Have we Learned?*

Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003, USA Foulis presented this paper at the Einstein Meets Magritte Conference in Brussels, Belgium during May-June 1995. Note that Pirsig presented his SODV paper on 1Jun1995 at this same conference. Reproduced with permission. Notify us of any typos at **:**

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1-317-THOUGHT

Doug - 5Feb2011.

A List of Foulis' Publications

Also see our ongoing critical review of Max Jammer's

Chapter 8, 'Quantum Logic'*The Philosophy of Quantum Mechanics*

__Index__:- 1. Introduction
- 2. Quantum Logic
**:**What Good is it? - 3. A Brief History of Quantum Logic
- 4. The Firefly Box and its Event Logic
- Figure 1
- The Firefly Box

Figure 2 - The Logic of Events - 5. The Logic of Experimental Propositions
- Figure 3 - The Logic of Experimental Propositions
- 6. The Logic of Attributes
- Figure 4 - Attribute Logic
- 7. States and Irreducible Attributes
- 8. Probability Models
- Figure 5 - Probability Model
- 9. Testing and Inference
- 10. The Quantum Firefly Box
- 11. Conclusion
**:**What Have We Learned? - 11.1 - Logical Connectives
- 11.2 - A Hierarchy of Logical Systems
- 11.3 - Events vs. Experimental Propositions
- 11.4 - Experimental Propositions vs. Attributes
- 11.5 - States and Probability Models
- References
- Footnotes

**1** __Introduction__ This expository
paper comprises my *personal* *response *to the question
in the title. Before giving my answers to this question, I discuss
the utility of quantum logic in Section **2**, offer a succinct
review of the history of quantum logic in Section **3**, and
present in Section **4** two simple thought experiments involving
a firefly in a box. The two thought experiments are pursued in
Sections **5** through **9**, where they give
rise to natural and (I hope) compelling illustrations of
the basic ideas of quantum logic. In Section **10**, I replace
the firefly by a "quantum firefly," and in Section **11**,
I summarize the lessons that we have (or should have) learned.

This paper is written for so-called laypersons (although some
of the ideas presented here have yet to be fully appreciated even
by some expert quantum logicians). Thus, in using the firefly
box to motivate and exemplify the fundamental notions of quantum
logic, I need only the simplest mathematical tools, i.e., sets
and functions. This does not necessarily imply that a casual reading
of the narrative will guarantee an adequate understanding of the
basic principles**—**a certain
amount of attentiveness to detail is still required.

**2** __Quantum Logic__**:**__
What Good is it?__ Before proceeding, I should address a question
germane to any meaningful discussion of what we have learned,
namely the related question *what good is quantum* *logic?
*Until now, quantum logic has had little or no impact on mainstream
physics; indeed some physicists go out of their way to express
a contempt for the subject (note 1). Whether
or not the insights achieved by quantum logicians contribute directly
to an achievement of whatever the Holy Grail of contemporary or
future physicists happens to be (note 2),
quantum logic has already made significant contributions to the
philosophy of science and to both mathematical
and philosophical logic.

Prior to Galileo's celebrated declaration that the Great Book of Nature is written in mathematical symbols, what we now call physical science was commonly referred to as natural philosophy. Quantum logic offers the possibility of reestablishing some of the close bonds between physics and philosophy that existed before the exploitation of powerful techniques of mathematical analysis changed not only the methods of physical scientists, but their collective mindset as well. The hope is that quantum logic will enable the mathematics of Descartes, Newton, Leibniz, Euler, Laplace, Lagrange, Gauss, Riemann, Hamilton, Levi-Civita, Hilbert, Banach, Borel, and Cartan, augmented by the mathematical logic of Boole, Tarski, Church, Post, Heyting, and Lukasiewicz to achieve a new and fertile physics/philosophy connection.

The mathematics of quantum mechanics involves operators on infinite dimensional vector spaces. Quantum logic enables the construction of finite, small, easily comprehended mathematical systems that reflect many of the features of the infinite dimensional structures, thus considerably enhancing our understanding of the latter. For instance, a finite system of propositions relating to spin-one particles constructed by Kochen and Specker [50] settled once and for all an aspect of a long standing problem relating to the existence of so-called hidden variables [5]. Another example is afforded by the work of M. Kläy in which a finite model casts considerable light on the celebrated paradox of Einstein, Podolsky, and Rosen [47].

One facet of quantum logic, yet to be exploited, is its potential
as an instrument of pedagogy. In
introductory quantum physics classes (especially in the United
States), students are informed *ex cathedra *that the state
of a physical system is represented by a complex-valued wavefunction
y, that observables correspond
to self-adjoint operators, that the temporal evolution of the
system is governed by a Schrödinger equation, and so on.
Students are expected to accept all this uncritically, as their
professors probably did before them. Any question of *why is*
dismissed with an appeal to authority and an injunction to wait
and see how well it all works. Those students whose curiosity
precludes blind compliance with the gospel according to Dirac
and von Neumann are told that they have no feeling for physics
and that they would be better off studying mathematics or philosophy.
A happy alternative to teaching by dogma is provided by basic
quantum logic, which furnishes a sound and intellectually satisfying
background for the introduction of the standard notions of elementary
quantum mechanics.

Quantum logic is a recognized, autonomous, and rapidly developing field of mathematics (note 3) and it has engendered related research in a number of fields such as measure theory [11,14,15,20,23,26,34,36,45,70,71,74,76] and functional analysis [4,11,16, 22,23,25,35,72]. The recently discovered connection between quantum logic and the theory of partially ordered abelian groups [29,33] promises a rich cross fertilization between the two fields. Also, quantum logic is an indispensable constituent of current research on quantum computation and quantum information theory [24].

**3** __A Brief History of Quantum
Logic__ In 1666, G.W. Leibniz envisaged a universal scientific
language, the *characteristica* *universalis, *together
with a symbolic calculus, the *calculus* *ratiocinator,
*for formal logical deduction within this language. Leibniz
soon turned his attention to other matters, including the creation
of the calculus of infinitesimals, and only partially developed
his logical calculus. Nearly two centuries later, in *Mathematical
Analysis of Logic *(1847) and *Laws of Thought *(1854),
G. Boole took the first decisive steps toward the realization
of Leibniz's projected calculus of scientific reasoning (note
4).

From 1847 to the 1930's, Boolean algebra, which may be considered as a classical precursor of quantum logic, underwent further development in the hands of De Morgan, Jevons, Peirce, Schröder, et al, and received its modern axiomatic form thanks to the work of Huntington, Birkhoff, Stone, et al. Nowadays, Boolean algebras are studied either as special kinds of lattices [9], or equivalently as special kinds of rings (note 5). In 1933, Kolmogorov, building upon an original idea of Fréchet, established the modern theory of probability using Boolean sigma-algebras of sets as a foundation [51].

The genesis of quantum logic is Section
5, Chapter 3 of J. von Neumann's 1932 book on the mathematical
foundations of quantum mechanics [59]. Here
von Neumann argued that certain linear operators, the *projections
*defined on a *Hilbert space *(note
6), could be regarded as representing experimental propositions
affiliated with the properties of a quantum mechanical system.
He wrote,

"...the relation between the properties of a physical system on the one hand, and the projections on the other, makes possible a sort of logical calculus with these."

In 1936, von Neumann, now in collaboration with G. Birkhoff
published a definitive article on the logic of quantum mechanics
[10]. Birkhoff and von Neumann proposed that
the specific quantum logic of projection operators on a Hilbert
space should be replaced by a general class of quantum logics
governed by a set of axioms, much in the same way that Boolean
algebras had already been characterized axiomatically. They observed
that, for propositions P, Q, R pertaining to a classical mechanical
system, the *distributive law*

P & (Q or R) = (P & Q) or (P & R)

holds, they gave an example to show that this law can fail for propositions affiliated with a quantum mechanical system, and they concluded that,

"...whereas logicians have usually assumed that properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the

distributive identitiesas the weakest link in the algebra of logic."

Birkhoff and von Neumann went on to argue
that a quantum logic ought to satisfy only a weakened version
of the distributive law called the *modular law note* (note 7);* *however, they pointed out that
projection operators on a Hilbert space can fail to satisfy even
this attenuated version of distributivity. Much of von Neumann's
subsequent work on *continuous geometries *[60]
and *rings of operators *[61] was motivated
by his desire to construct logical calculi satisfying the modular
law. In 1937, K. Husimi [41] discovered that
projection operators on a Hilbert space satisfy a weakened version
of the modular law, now called the *orthomodular identity*
(note 8)*.*

From 1937 until 1955, all research on
quantum logic ceased as scientists turned their attention to military
applications of physics. In 1955, L. Loomis [53]
and S. Maeda [57] independently rediscovered
Husimi's orthomodular identity in connection with their efforts
to extend von Neumann's dimension theory for rings of operators
to more general structures. The structures studied by Husimi,
Loomis, and Maeda are now called *orthomodular* *lattices*
(note 9).

In 1957, G. Mackey wrote an expository article on quantum mechanics
[55] based on lectures he was giving at Harvard.
In 1963, he published an expanded version of these lectures in
the form of an influential monograph [56]
in which he referred to propositions affiliated with a physical
system as *questions. *Under fairly reasonable hypotheses,
it is easy to show that Mackey's questions form an orthomodular
lattice.

The simplicity and elegance of Mackey's formulation and the
natural and compelling way in which it gave rise to a system of
experimental propositions inspired a renewed interest in the study
of quantum logic, now identified with the study of orthomodular
lattices. Could it be that these lattices provide a basis for
Leibniz's long awaited calculus ratiocinator? Thus motivated,
a small but devoted group of researchers**—**Catlin,
Finch, Foulis, Greechie, Gudder, Holland, Janowitz, Jauch, Kotas,
MacLaren, Maeda, Piron, Pool, Ramsay, Randall, Schreiner, Suppes,
Varadarajan, et al.,**—**began
in the early 1960's the task of working out a general mathematical
theory of orthomodular lattices. A comprehensive account of the
resulting theory and an extensive bibliography up to about 1983
can be found in [44].

In 1964, C. Piron introduced an alternative to Mackey's approach
in which questions again band together to form an orthomodular
lattice, but this time possessing more of the special features
of the lattice of projection operators on a Hilbert space [62]. In fact, Piron was able to show that his
questions could be represented as actual projection operators
on a so-called *generalized Hilbert space. *Piron's work
raised the issue of how to characterize the standard Hilbert spaces
among the class of generalized Hilbert spaces.

A list of more or less "natural conditions" on generalized Hilbert spaces was soon proposed in the hopes of singling out the "true" Hilbert spaces. In 1980, H. Keller dashed these hopes by constructing an example of a generalized Hilbert space satisfying all of the proposed natural conditions, but that is not a standard Hilbert space [46]. In 1995 M. Solèr showed that Keller's counter example could be bypassed by adding just one more natural condition to the previous list [72]. Thanks to Solèr's work, we are now in possession of a satisfactory axiomatic approach to Hilbert-space based quantum mechanics [40].

As early as 1962 [27], it was noticed by
some of the aforementioned researchers that, even without the
imposition of additional hypotheses, Mackey's questions form an
intriguing structure called an *orthomodular poset. *For
this reason, orthomodular posets were also considered as possible
candidates for quantum logics and were studied in parallel with
orthomodular lattices, especially by S. Gudder [34,35] and his students. A comprehensive account
of orthomodular lattices and posets as models for quantum logics
can be found in [65].

In orthodox quantum
mechanics, when systems are combined or coupled to form composite
systems [3,28,43],
the combined system is represented mathematically by a so-called
*tensor** product of*
Hilbert spaces. Even in the early 1960's, researchers realized
that the entire quantum logic program would falter unless a suitable
version of **tensor** product
could be found for the more general logical structures then under
consideration.

After many unsuccessful attempts to formulate a suitable **tensor **[See **abelian**, just below. Doug - 5Aug2009. Doug's
same comments apply, here, to Hermitian conjugate operations whose
purpose is to remove 'imaginary' content from quantum~wave functions
in order to make them classically 'true.' Those 'operations' are
invalid in quantum reality since they require commutation. Commutation
is invalid in quantum~reality! Doug - 5Feb2011.]
product for orthomodular lattices and posets, it was discovered
in 1979 that **all such attempts were doomed
to failure** owing to the fact that the category of orthomodular
posets is too small to admit a **tensor**
product [67]. C. Randall and D. Foulis showed
that, to accommodate the construction of **tensor**
products, a larger category of mathematical structures called
*orthoalgebras *has to be employed [8,31,49,66,68]. For a while, it seemed that orthoalgebras
were the true quantum logics [8,20,26,30,36,71].

Composite physical systems were studied from the perspective
of quantum logic in an important and influential sequence of papers
by D. Aerts [1,2,3].
In these studies, Aerts introduced the crucial notion of an entity
which, roughly speaking, consists of a quantum logic of questions
or propositions affiliated with a physical system together with
a related system of *properties, *or *attributes, *of
the system. Among other things, Aerts showed convincingly that
a proper representation of a composite system requires consideration
of the way in which properties of a total system depend on the
properties of its constituents.

In parallel with the development of
quantum logic, and starting as early as 1970 [16,38,39], Davies, Lewis, Holevo,
Ludwig, Prugovecki, Ali, Busch, Lahti, Mittelstaedt, Schroeck,
'Bujagski, Beltrametti, et al worked out a theory of quantum statistics
and quantum measurement based on so-called effect *operators
*(note 10)* *on a Hilbert space
[54]. Every projection operator is an effect
operator, but not conversely, and the effect operators do not
even form a lattice, let alone an orthomodular lattice, or even
an orthoalgebra..

In 1989, R. Guintini and H. Greuling introduced axioms for
a generalized orthoalgebra and argued that effect operators on
a Hilbert space form such a structure [32].
The generalized orthoalgebras of Guintini and Greuling, which
turned out to be mathematically equivalent to the D-posets of
Kôpka and Chovanec [52], have come to
be called *effect algebras *[29]. It
can be argued that fuzzy or unsharp propositions are properly
represented as elements of an effect algebra [13,17,18,19,52].

In 1994, Bennett and Foulis [29] discovered
a connection between effect algebras and partially ordered **abelian** groups. In subsequent papers,
they went on to show that virtually every structure previously
proposed for a quantum logic, and indeed every Boolean algebra,
can be represented as an interval in a such a group. An interval
in [a]
partially ordered **abelian**
groups organized in a natural way into an effect algebra, is called
an *interval effect algebra. *As a mathematical theory, quantum
logic is thus subsumed by the theory of partially ordered **abelian** groups. [Doug's
violet bold highlights on **abelian**. Readers please fathom that one of **abelian** group
theory's core axioms is commutation. As we emphasize elsewhere, classical axiomatic
commutation is bogus in quantum~logic described by Quantonics'
coquecigrues. Doug - 5Aug2009. This
applies to Hermitian operators too! Doug - 5Feb2011.]

Because there was no serious work on quantum logic *per se*
during the years 1938-1957, I consider that quantum logic has
been under development for roughly half a century. The history
of quantum logic has been a story of more and more general mathematical
structures**—**Boolean algebras,
orthomodular lattices, orthomodular posets, orthoalgebras, and
effect algebras**—**being proposed
as basic models for the logics affiliated with physical systems.
Whether effect algebras are the end of the line remains to be
seen.

Those wishing to read more about quantum logic and its connections with quantum physics are encouraged to consult the following standard references [6,12,21,35,42,43,54,55,56,59,62, 65].

**4** __The Firefly Box and its Event
Logic__ Now I invite you to contemplate with me some "thought
experiments" involving a firefly in a box (Figure 1). The
box is to have two translucent (but not transparent) windows,
one on the front and one on the side. The remaining four sides
of the box are opaque. At any given moment, the firefly might
or might not have its light on. If the light is on, it can be
seen as a blip by looking at either of the two windows.

Looking directly at the front window of the box when the light
is on, one can tell by the position of the blip whether the firefly
is in the left (l)* *or right (r) half of the box. Likewise,
looking directly at the side window when the light is on, one
can tell whether the firefly is in the front (f) or back (b) half
of the box. Because the windows are not transparent, one cannot
rely on depth perception to determine from the front window whether
the firefly is in the front or back half of the box, nor from
the side window, whether the firefly is in the left or right half
of the box.

Now consider two experimental procedures
F and S. Procedure F is conducted by looking directly at the front
window and recording l, r, or nf according to whether the blip
is on the left, on the right, or there is no blip, respectively.
Procedure S is conducted by looking directly at the side window
and recording f, b, or ns according to whether the blip is in
the front, the back, or there is no blip, respectively. One cannot
conduct both procedures F and S at the same time because of the
necessity of looking *directly *at one window or the other.
Indeed, if one stands in a position to see both windows, parallax
could spoil the accuracy of the observation (note
11).

Imagine that we plan to conduct an experimental study of the firefly's habits using the only means available to us, namely the two experimental procedures F and S. Our work will be guided by an emerging "firefly box theory" (FBT) that may have to be amended as we collect more and more experimental data or change our mind about what is going on inside the box.

To begin with, let us provisionally incorporate into our FBT
the simplifying assumption that there are no baffles within the
box behind which the firefly might hide from either window. If
this is so, then a blip would be seen on the front window if and
only if it would be seen on the side window. (Note the use of
the subjunctive here**—**we can
not meaningfully perform both F and S simultaneously!) This assumption
is implemented simply by *identifying *outcome nf of F with
outcome ns of S. Thus, we set n **:**= nf = ns. (The notation
**:**= means equals by definition.)

Let Ef **:**= {l, r, n} and Es **:**= {f, b, n} be the
**mutually exclusive** and exhaustive
outcome sets for the experimental procedures F and S, respectively.
Execution of F will yield one and only one outcome l, r, or n;
execution of S will yield one and only one outcome f, b, or n.

of Ef and Es is a classical-logic mandate which denies quantum reality's included-middle.

By an event for F we will mean a subset A of Ef. Including
the empty set and Ef itself, there are 8 such events, namely Æ, {l}, {r}, {n}, {l,r},
{l,n}, {r,n}, {l,r,n}. If F is executed and the resulting outcome
is e E Î Ef, say that an event
A Í Ef *occurs *if e Î A and that it *nonoccurs *if
e Ï A. Likewise, an event for
S is understood to be one of the 8 subsets Æ,
{f}, {b}, {n}, {f,b}, {f,n}, {b,n}, {f,b,n} of Es. If S is executed,
then an event A for *S occurs or nonoccurs *according to
whether the outcome belongs or does not belong to A, respectively.

An event A can only occur or nonoccur when *tested* by
the execution of either F (if A Í
Ef) or S (if A Í Es). The null
event Æ, the event
{n}, and only these two events are tested by both F and S. Let

e:= {Æ, {l}, {r}, {l,r}, {l,n}, {r,n}, {l,r,n},{f}, {b}, {n}, {f,b}, {f,n}, {b,n}, {f,b,n} }

be the collection of all events and let

E

:= Ef È Es = {l, r, n, f, b)

be the set of all outcomes of the available experimental procedures.
There are 32 subsets of E, but only 14 events in **e**.
For example, {r,f) is a subset of E, but it is not an event since
there is no conclusive way to test it. For instance, if S is executed
and b is the outcome, we would hardly say that {r,f} failed to
occur since F was not executed and it is meaningless to ask whether
or not the outcome r was secured.

The set **e
**of all events can be organized into a rudimentary logical
structure as follows**:** Let A,B,C Î
**e**. Say
that A and B are *compatible *iff they are simultaneously
testable in the sense that A,B Í
Ef or A,B Í Es. (We abbreviate
if and only if as iff,) Call A and B *orthogonal *iff they
are compatible and disjoint. (Two sets are disjoint iff they have
no elements in common.) Say that A and C are *local complements
*iff they are orthogonal and their union is either Ef or Es.
If A and B share a common local complement C, say that A and B
are perspective *with axis C.*

For instance, the events {l,r} and {r,n} are compatible, since they are both tested by F, but they are not orthogonal since they have a common outcome r. If two events are orthogonal, they can be tested simultaneously and, when so tested, at most one of them can occur. The events {l,r} and {n} are local complements, since they are disjoint and their union is Ef. When tested by F, one and only one of them will occur. The events {f,b} and {n} are also local complements, and both are tested by S. Therefore, {l,r} and {f,b} are perspective with {n} as an axis.

If A Î **e**, then A has at least
one local complement C in **e**.
Indeed, if A Í Ef, then the
complement C **:**= Ef\A of A in Ef is a local complement of
A. Likewise, if A Í Es, then
D = Es\A is a local complement of A. Therefore, every event A
is perspective to itself (with any local complement of A as an
axis). Note that the two events Ef and Es are perspective, with
Æ as an axis.

There is an obvious sense in which perspective events are "logically
equivalent." For instance {l,r} occurs iff the firefly's
light is on, and likewise for {f,b}. Also, Ef = {l,r,n} *always
occurs *(when tested, of course), and so does Es = {f,b,n}.
The collection **e**
is partially ordered by set containment Í.
If A and B are events and A Í
B, there is an obvious sense in which A "implies" B.
For instance, {n} Í {r,n} and,
if {n} occurs, the firefly's light is out, and presumably {r,n}
would have occurred too, *had it been tested. *(Again, note
the use of the subjunctive. Indeed, if S is executed and {n} occurs,
then {r,n} was not tested, so it neither occurred nor nonoccurred.)

Figure 2 shows a diagram (called a *Hasse diagram) *of
the *event logic *for the firefly box. The 14 events in **e **are shown as
nodes in this diagram, and event A is a subset of an event B iff
either A = B or it is possible to go upward from A to B along
a sequence of connecting line segments. The perspective events
{l,r} and {f,b}, as well as {l,r,n} and {f,b,n} are enclosed in
shaded ellipses on the diagram.

**5** __The Logic of Experimental Propositions__
As we have seen, in the event logic **e
**of the firefly box, two perspective events are, in
some sense, logically equivalent. The temptation to identify logically
equivalent events is irresistible, and we do so now by collapsing
the diagram in Figure 2 as indicated by the shaded ellipses. Call
the elements of the collapsed diagram *experimental propositions,
*denote the experimental proposition corresponding to an event
A Î **e
**by p(A), and
let P be the set of all
p(A) as A runs through
**e**. For
simplicity, if A = {e} is an event with only one outcome, we write
p(e) rather than p({e}).

Denote the proposition p(Æ) by 0 **:***=
*p(Æ)
Î P.
Define the proposition 1 Î P by 1 **:**= p({l,r,n})
= p({f,b,n}). The proposition
p({l,r}) = p({f,b}),
which can be regarded as asserting that the firefly's light is
on, is denoted by p(n').
Likewise, the proposition p({l,n})
can be regarded as asserting, "it is false that the firefly
is in the right half of the box with its light on," so we
denote it by p(r'), and
so on. The resulting Hasse diagram for the *logic *P *of experimental* *propositions
*is shown in Figure 3.

The partial order on P
depicted in Figure 3, is denoted by __<__ and called
*implication, or entailment. *Note that, if A,B Î
**e**, then
p(A) __<__ p(B) iff there is an event
Bl such that A Í Bl and Bl is
perspective to B. Evidently, 0 £
p(A)* *__<__
1 for all p(A)
Î P.

To test an experimental proposition p(A) Î
P, we select any event
Al (including A itself) such that p(A)
= p(Al), we choose a test
for Al, and we carry out the test. If Al occurs, we say that the
proposition p(A) is *confirmed,
*otherwise, we say that it *is refuted. *Thus, confirmation
and refutation of experimental propositions is linked to occurrence
and nonoccurrence of events. For instance, to test whether p(n') is confirmed (i.e., whether
the light is on), we can execute either F or S and conclude that
the light is on iff the outcome n is not secured. In reporting
that a proposition p(A) is confirmed or refuted *it* *is
not* *necessary, to specify which test was executed.*

There is a natural notion of "logical negation" for
the experimental propositions in the logic P.
Indeed, if A Î **e**, define p(A)'
**:**= p(C), where
C Î **e
**is any local complement of A. The proposition p(A)', which is easily seen
to be well defined, is regarded as a *logical negation, or denial,
of *p(A)*. *Evidently,
0' = 1 and 1' = 0. In Figure 3, the logical negations of each
proposition in the first row above 0 are located directly above
that proposition in the first row below 1. For instance, p(l)' =
p(l'),

The partially ordered set P
depicted in Figure 3 is actually a lattice; that is, any
pair of propositions p(A)
and p(B) have a *least
upper bound, or join, *p(A)
Ú p(B)
and a *greatest lower* *bound, or meet, *p(A)
Ù p(B)
with respect to the implication relation £.
For instance, p(l) Ú p(r)
= p(n') and p(r)
Ù p(f)
= 0.

The mapping p(A) ® p(A)'
is an *orthocomplementation on *the lattice P
in that it has the following properties for all experimental
propositions p,q Î P**:**

- (i) p Ù p' = 0
- (ii) p Ú p' = 1
- (iii) p'' = p

- (iv) p £ q Þ q' £ p'

As a consequence, P
satisfies the De Morgan Laws**:**

- (v) (p Ù q)' = p' Ú q' and
- (vi) (p Ú q)' = p' Ù q'

Furthermore, the following *orthomodular identity *holds
in n**:**

- (vii) p
<q Þ q = p Ú (q Ù p' ).

Therefore, P forms
a so-called *orthomodular *lattice [27,44].

Say that propositions p(A)
and p(B) in n are *compatible
iff* there are compatible events Al and Bl with p(A)
= p(Al) and p(B)
= p(Bl). Note that a common
test for the events Al and Bl is then a common test for the propositions
p(A) and p(B).
For instance, p(l) and
p(n) are compatible, p(n) and p(b)
are compatible, but p(l)
and p(b) are incompatible.

Say that propositions p(A)
and p(C) in P
are *orthogonal iff* there are orthogonal events Al and Bl
with p(A) = p(Al)
and p(B) = p(Bl).
Note that orthogonal propositions are necessarily compatible and
that p(A) is orthogonal
to p(C) iff p(A)
£ p(C)'.

If p(A) £ p(B) and if p(A) is confirmed, it is understood that p(B) is also confirmed and that every proposition p(C) that is orthogonal to p(A) is refuted. Note that p(A) is confirmed iff p(A)' is refuted. Evidently, 1 is always confirmed, and 0 is always refuted.

In classical (Boolean) logic, the meet p Ù
q of two propositions p and q is effective as their *logical
conjunction* p & q. This is certainly not the case in the
logic P; for instance,
p(l) Ù
p(f) = 0, whereas p(l) & p(f)
would be the proposition asserting that the firefly is in the
left front quadrant of the box with its light on. What is happening
here is perfectly clear**—**the conjunction p(l)
& p(f) is not in the logic P because
*there is no way to test *it!

How does one account for the nonclassical nature of the firefly
box logic P, given that
its source, a box with windows and a firefly, is utterly classical?
The answer, as we shall see in Section **10** below, is that there are pairs of nonclassical quantum-mechanical
experiments that yield the same event logic **e**, and therefore
the same experimental logic P, as the
firefly box. The event logic **e**
does not "know" the difference between the firefly box
and the quantum-mechanical system, so it produces an experimental
logic P compatible with
both.

**6** __The Logic of Attributes__
Usually there are certain *properties,* or attributes, associated
with a physical system **j**,
such as "**j **is
green," or "**j**
carries an electric charge of 1.60217733 X 10^-19 coulomb,"
or "**j*** *has
a spin component +1/2 in the z direction." An attribute can
be either actual or potential. Those attributes that are always
actual, such as the color of a raven or the charge of an electron,
are said to be *intrinsic. *Attributes that can be either
actual or potential, such as the color of a chameleon or the spin
component of an electron, are called *accidental.*

An actual attribute a of a physical
system **j **can manifest
itself experimentally only in terms of outcomes of experimental
procedures. In fact, a induces a division
of the set E of all outcomes of experimental procedures into two
disjoint parts**:** P **:**= those outcomes that are *possible
*when a is actual, and E\P = those
outcomes that are *impossible *when a
is actual. a

What are the attributes associated with our firefly box? Attributes such as "the temperature in the box is 18° C" or "the box weighs 15 kg" do not concern us here since they are unrelated to the only experimental procedures at our disposal, namely F and S. However, consider the attribute a = "either the firefly's light is off, or else it is on and the firefly is in the left front quadrant of the box." If a is actual, outcomes r and b are impossible and the set of possible outcomes is P = {l,f,n}. Conversely, given that the possible outcomes when a is actual are l, f, and n, one can easily identify the original attribute a.

More generally, those attributes a of the firefly box that
can manifest themselves by way of the experimental procedures
F and S can always be recaptured as soon as we know the subset
P of E consisting of outcomes that are possible when the attribute
is actual. Therefore, the set P Í
E provides a perspicuous mathematical representation of the attribute
a, and in what follows, we shall simply
*identify* a *with *P.

Suppose that P Í E is an
attribute of our firefly box and let A Î
**e **be
an event. If A Ç *P *=
Æ, then A consists
entirely of outcomes that are impossible when P is actual. Thus,
if A Ç *P *= Æ
and P is actual, then A is *impossible in* the sense that
it must nonoccur when tested. Recall that the two events {l,r}
and {f,b} are logically equivalent; hence if {l,r} is impossible,
so is {f,b} and vice versa. Consequently,

(1) P Ç {l,r} = Æ Û P Ç {f,b} = Æ.

Of the 32 subsets of E = {l,r,n,f,b},
only 20 satisfy condition (1), and each of these can be interpreted
as a meaningful attribute of the firefly box. Even the empty set
Æ can be interpreted
as an attribute, albeit one that is *never* actual. On the
other hand, the set E of all outcomes satisfies (1) and represents
an attribute that is *always actual* (note
l2). Let us denote by Ã
the collection of all attributes P Í
E, so that P Î Ã
iff P Í E and P satisfies (1).

The set Ã is partially ordered
by the relation 9 of set containment. Furthermore, if P,Q Î Ã,*
*then P Í Q iff Q is actual
whenever P is actual. Therefore, Í
can be regarded as a kind of *implication relation *on Ã and, in this sense, Ã
becomes a logical system called the attribute *logic *of
the firefly box. It is easy to see that, if P,Q Í
E and both P and Q satisfy condition (1), then so does P È Q. Therefore, Ã
is closed under the formation of unions. Thus, if P,Q Î
A, then P and Q have a join P Ú
Q = P È Q in Ã.
Also, if P,Q Î Ã,
then P and Q have a meet P Ù
Q in Ã; in fact P Ù
Q is the union of all attributes in Ã
that are contained in both P and Q.

Although Ã is closed under
unions, it is not closed under intersections. For instance, P
**:**= {l,f,n} Î Ã
and Q **:**= {l,b,n} Î Ã, but P Ç
Q = {l,n}Ï Ã.
In fact, P Ù Q = {n} ¹
P Ç Q. In general, if P,Q Î Ã,
then the attribute P Ù Q is
a subset of the set P Ç
Q, but P Ù Q ¹
P Ç Q unless it happens that P Ç Q Î Ã.

Suppose P Î Ã
and P is actual. If the experimental procedure F is executed,
one of the outcomes l, r, or n in Ef must be secured and, since
the outcomes in Ef\P are impossible, it follows that one of the
outcomes in P Ç Ef must be secured.
In other words, the event P Ç
Ef *necessarily occurs *when P is actual. Likewise, if P
is actual, the event P Ç
Es necessarily occurs when tested. Furthermore, if P is actual,
A Î **e**,
and one of the conditions P Ç
Ef Í A or P
Ç Es Í
A holds, then A necessarily occurs when tested. Let us say that
P *guarantees *A iff one of the conditions
P Ç Ef Í
A or P Ç Es Í
A holds. The fact that the empty attribute Æ
guarantees every event is harmless since the empty attribute is
never actual.

If A Î **e**, let [A] denote the union
of all attributes P Î Ã
that guarantee A. In other words, [A] is *the largest attribute*
*that, when actual, necessitates the occurrence of *A*
when tested. *It is easy to check that, if A,B Î
**e**, then

(2) p(A) £ p(B) Û [A] Í [B],

so we can and do define [p(A)]
**:**= [A] for all events A Î
**e**.

For simplicity, if e Î E, we write [e] rather than [{e}] and we write [e'] rather than [p(e)'].

An attribute of the form [A] is called a *principal* attribute.
In view of (2), the mapping p(A)
® [A] embeds the experimental logic
P in the attribute logic
Ã, whence 12 of the 20 attributes
in Ã are principal, and 8 are
nonprincipal. Although the embedding p(A)
® [A] preserves joins, it fails
to preserve meets. For instance, p(l)
Ù p(f)
= 0 in P, but [l] = {l,f,b),
[f] = {l,,r,f) and [l] Ù [f]
= {l,f} ¹ Æ
in Ã.

Each of the 8 nonprincipal attributes in Ã can be written as a meet of principal attributes; for instance {l,f,n} is a nonprincipal attribute in Ã and {l,f,n) = [r'] Ù [b']. For simplicity, we write [r'b'] rather than [r'] Ù [b']. Similarly, we write [lf] rather than [l] Ù [f], and so on. With this notation, the Hasse diagram for the attribute logic is shown in Figure 4.

**7** __States and Irreducible Attributes__
Nearly every scientific theory utilizes, explicitly or implicitly,
the notion of the *state *of a physical system. The usual
understanding is that, at any given moment, the system is *in
*a particular state y. All information
about outcomes of experimental procedures executed on the system
in state y are supposed to be encoded
into y. The state of the system can
change in time under a deterministic or stochastic dynamical law,
it can change because an experimental procedure is executed, or
it can change spontaneously.

Until now, our firefly box theory (FBT) has recognized only
one explicit principle, namely n = nf = ns. (However, one could
argue that much of the discussion in Section **6** regarding
the attributes of the firefly box constitutes a further evolution
of the FBT). Now we have to face the issue of incorporating into
our FBT a suitable mathematical representation for the set y of all possible states of the firefly
box.

We cannot see inside the box, but we are formulating our FBT under the supposition that the blips of light on the windows are caused by a firefly. The firefly could be located in any one of the four quadrants of the box, and its light could be on or off, so there seem to be eight different possible states of the firefly box. However, when the light is off our available experimental procedures F and S provide no information about the location of the firefly. In view of this experimental limitation, it seems more reasonable to restrict our state space Y to five possible states, namely (in Dirac's "ket" notation)

Y = {|lf>, |lb>, |rf>, |rb>, |n>}.

The first four states correspond to the location of the firefly
in the left-front, left-back, right-front, and right-back quadrant
with its light on. In the fifth state |n>, the light is off*.*

Notice in Figure 4 that there are exactly five minimal nonempty
attributes in Ã, namely [lf],
[lb], [r,f], [r,b], and [n]. These are the attributes that are
*irreducible *in the sense that they cannot be decomposed
into more elementary attributes, and they are in obvious one-to-one
correspondence with the five states in Y.
Two new principles suggest themselves, and we now incorporate
them into our FBT**:**

Principle of Irreducible AttributesTo each state y Î Y there corresponds a uniquely determined irreducible attribute Py, Î Ã.The attribute Py, is actual iff the system is in state y.

Principle of Actuality for AttributesAn attribute P Î Ã is actual iff the system is in a state y for which Py Í P.

Thanks to the principle of irreducible attributes, one and only one of the irreducible attributes is actual at any given moment. As a consequence of both principles, this unique irreducible attribute is the meet of all the attributes that are actual at that moment.

Suppose P,Q Î Ã.
In spite of the fact that P Ù
Q is not necessarily P Ç Q,
it turns out (and is not difficult to verify) that P Ù
Q is actual iff both P and Q are actual. Thus, P Ù
Q is effective as a true *logical conjunction *of P and Q
in the attribute logic Ã. That
the join P Ú Q = P È
Q of two attributes is not necessarily a logical disjunction of
P and Q is a profound observation first made by D. Aerts [1]. For instance, a glance at Figure 4 shows that
[lf] Ú [rb] = [n']; yet the
attribute [n'] can be actual (i.e., the light is on) in a state
(e.g., |rf>) for which neither [lf] nor [rb] is actual.

The fact that a join of attributes need not be a logical disjunction
of the attributes can and should be regarded as the true basis
for the notion of "superposition of states." Say that
a state y Î
Y is a *proper superposition *of
states a,b Î
Y iff Py
Í Pa Ú
Pb but y
¹ a,b. For instance, |rf> is a proper superposition
of |lf> and |rb>.

Since the publication in 1930 of Dirac's seminal monograph
on the mathematical foundations of quantum mechanics [21],
it has been an article of faith among physicists that a fundamental
distinction**—**if not *the*
fundamental distinction**—**between
quantum and classical mechanics is that there are proper superpositions
of states in the former, but not in the latter. If this is so
(and I am not entirely convinced that it is [7]),
then our firefly box is already exhibiting quantal behavior!

**8** __Probability Models__ The
system of real numbers is denoted by the symbol Â,
and the closed interval of real numbers between 0 and 1 is written
as [0,1]. By a *probability model *for the firefly box, we
mean a function w**:**E ®
[0,1] Í Â
mapping each outcome e Î E into
a real number w(e) between 0 and 1
in such a way that

(1) w(l) + w(r) + w(n) = 1 and w(f) + w(b) + w(n) = 1

If e Î E, then w(e)
is to be interpreted as the *probability,* according to the
model w, that the outcome e will be secured when an experimental
procedure (F or S) is conducted for which e is a possible outcome.
Denote by W the set of
all probability models w for the firefly
box.

If A Î **e **is an element of the
event logic and w Î
W is a probability model,
we define

- (2) w(A)
**:**= S w (e), - e Î A

and interpret w(A) as the probability,
according to the model w, that the
event A will occur *if tested. *In this way, probability
models w Î
W can be "lifted"
to the logic **e **of
events. If A,B Î **e **with A Í
B, it is clear that w(A) £
w(B). If A,C Î
**e **and
A is orthogonal to C, then w(A È C) = w*(A)
*+ w(C).
Therefore, (1) implies that w(A) +
w(C) = 1 for local complements A,C
Î **e**.

If w Î
W and A and B are perspective
events with axis C, then w(A) + w(C) = 1 = w(B)
+ w(C), and it follows that w(A)
= w(B). Hence, for an experimental
proposition w(A) we can and do define
w(p(A)) **:**= w(A).
In this way, probability models w Î W
can be lifted to the logic P
of experimental propositions. Naturally, w(p(A)) is interpreted as the
probability, according to the model w,
that p(A) will be confirmed
*if tested.*

If w Î
W, then, regarded as a
function w**:**P
® [0,1] Í
Â, w
is a *probability measure *in the sense that w(l)
= 1 and, for orthogonal propositions p(A)
and p(C), w(p(A) Ú
p(C)) = w(p(A)) + w(p(C)). For the firefly box,
W provides a so-called
*full*, or *order determining, *set of probability measures
in the sense that, if w(p(A))
£ w(p(B)) for all w
Î W,,
then p(A) £
p(B).

If w1,w2,¼ wn Î W,
and tl,t2,...tn are positive real numbers such that t1 + t2 +¼ + tn = 1, the function w**:**E
® Â
defined for all e Î E by

(3) w(e) **:**= tlwl(e)
+ t2w2(e) +¼
+ tnwn(e) is called a *convex combination,
or mixture, *of wl, w2,...
wn with *mixing coefficients *tl,t2,...,tn.
It is not difficult to see that such a mixture takes on values
between 0 and 1 and satisfies (1), so it is again a probability
model w Î
W. In other words, W is a *convex set, *i.e.,
it is closed under the formation of convex combinations.

Each w Î W is completely determined by the three real numbers

(4) x **:**= w(l), y **:**=
w(f), and z **:**= w(n).

Indeed, as a consequence of (1),

(5) w(r) = 1 - x - z and w(b) = 1 - y - z.

Of course, the numbers x,y,z are subject to the conditions

(6) 0 £ x,y,z £ 1, x + z £ 1, and y + z £ 1.

The set of all points (x,y,z) in coordinate 3-space Â^3
that satisfy (6) is a pyramid with a square base (Figure 5). A
point (x,y,z) in the pyramid may be identified with the corresponding
state w by (4) and (5), so the pyramid
provides a geometric representation of the space W
of probability models for the firefly box. The five vertices
w|lf> **:**= (1,1,0), w|lb>
**:**= (1,0,0), w|rf> **:**=
(0,1,0), w|rb> **:**= (0,0,0),
and w|n> **:**= (0,0,1) of the
pyramid correspond in an obvious way to the five states in Y. For instance, x = w|lf>(l) = 1, y = w|lf>(f)
= 1, and z = w|lf>(n) = 0 for the
probability model w|lf>.

In the geometric representation of W
as a pyramid (Figure 5), the five vertices correspond to
*extreme points *of the convex set W,
that is, points w that cannot be written
in the form (3) unless w1 = w2
=...*= *w. For a *polytope*,
such as W, there are only
finitely many extreme points, and every point is a convex combination
of extreme points.

If w Î
W, we define the *support
of *w, in symbols supp(w)
by

(7) supp(w) **:**= {e Î
E | 0 < w(e)},

noting that supp(w) is the set of
all outcomes e Î E that are *possible
*according to the model w. In view
of our discussion in Section 7, it should come as no surprise
(and it is easy to check) that supp(w)
Î Ã.
Furthermore, the supports of extreme points produce irreducible
attributes corresponding to states, just as one would expect.
For instance, supp(w|lf>) = {l,f}
= [lf] which corresponds to the state |lf>. Note that the support
of a convex combination (3) is the union of the supports of w1, w2,..., wn Î W from which it is formed.

The five vertices, eight edges, four triangular faces, and
the square base of the pyramid in Figure 5 are called *faces
*of W. In addition
it is convenient to include the empty set Æ
and W itself as (improper)
faces, making a total of twenty faces in all. The four triangular
faces and the square base**—**that
is, the maximal proper faces**—**are
called facets. The five vertices are the minimal proper faces.
The intersection of two faces is again a face, and, given any
two faces there is a unique smallest face containing both. Therefore,
partially ordered by inclusion Í
the faces form a lattice, called the face lattice of W,
and denoted by Á.

The fact that both the attribute logic Ã
and the face lattice Á have
twenty elements is no accident. In fact there is a natural one-to-one
correspondence P « F
between attributes P Î
Ã and faces F
Î Á given by F **:**= {w
Î W
| supp(w) Í
P} and P **:**= **U**wÎF supp(w). Furthermore,
the correspondence P « F is a lattice *isomorphism *in that
it preserves meets and joins. Thus, the face lattice Á
of W provides an alternative
representation for the attributes of the firefly box.

**9** __Testing and Inference__ For
the firefly box we now have three related logical structures,
namely **e**,
P, and Ã,
as well as the state space Y,
the convex set W of probability
models, and the face Á of W, which is isomorphic to Ã.* *For Ã
Î **e
**we have three truth values**:** occur, *nonoccur,
and not tested. *For p(A)
Î P
we again have three truth values**:** *confirmed,
refuted, *and not *tested. *By their very definitions,
these truth values can be determined by executing an appropriate
experimental procedure, either F or S.

For an attribute P Î Ã we have two truth values**:**
actual and potential. Unlike the truth values for events and experimental
propositions, it might not be possible to determine the truth
value of an attribute P by conducting a single experiment. If
P = [p(A)] is a principal
attribute, we can test P by testing p(A).
If p(A) is refuted, then
P cannot have been actual since its actuality guarantees p(A). If p(A)
is confirmed, we have evidence that P might have been actual,
but it may not be conclusive. P Î
Ã is not principal, it can be
written as a conjunction of principal attributes, one of which
can be selected and tested, again supplying (usually inconclusive)
evidence that P was either actual or only potential [63].

Apparently, inferences about which properties of a physical system are actual and which are only potential will have to depend on evidence gathered from repeated testing, either under circumstances in which one has reason to believe that the state of the system remains unchanged, or on a sequence of replicas of the underlying system all of which are presumed to be in the same state. As of now, however, there seems to have been no serious attempt to develop a mathematical theory of formal scientific inference regarding the attributes of a physical system.

For each state y Î
Y, we have two truth values,
in and *not in. *To test the state y,
we can test the corresponding irreducible attribute Py
as indicated above. But then state testing will be as inconclusive
as attribute testing and it will be hindered by the same lack
of a theory of inference. For certain physical systems (if not
for our firefly) it is possible to *prepare *a preassigned
state y, that is, to bring the system
into the state y by carrying out suitable
procedures. When state preparation is possible, it may render
moot the question of how to test states (and perhaps attributes
as well).

Testing probability models is quite another matter**—**indeed this is what *statistical
inference *is all about! The usual idea is that there exists
a "true probability model" w*
Î W
representing the habits of the firefly. Although we might
not know which probability model is w*,
we might be able to make some (perhaps tentative) conclusions
about w* by repeatedly executing our
procedures F and S and processing the experimental data thus obtained.
It is often assumed that the repeated trials of F and S are "independent"
in the sense that the firefly's habits are unaffected by our experiments.
It is easy to challenge this assumption, but not so easy to design
reliable strategies of statistical inference to take into account
observation-induced changes in the firefly's behavior patterns.

Two useful mathematical tools conventionally employed in statistical
investigations are *statistical hypotheses *and *parameters.
*By a *statistical hypothesis *is meant a subset L of W,
usually subject to a condition that it be *measurable *in
some appropriate sense (e.g., Borel or Lebesgue measurable). Let
**j **denote the set of
all statistical hypotheses. By a *statistical* *parameter
*is meant a real valued function l**:**W ®
Â satisfying the condition that,
for every interval I Í Â, the set

l^-1 (I) **:**= {w
e W | l(w) Î I}
is a statistical hypothesis in **j**.*
*A statistical hypothesis of the form l^-1
(I) is called a *parametric hypothesis.*

A statistical hypothesis L Î **j**
is understood to represent the proposition w*
Î L
asserting that the true probability model belongs to L.
Partially ordered by Í,* ***j*** *forms a logical system
called the *inductive logic, *and in **j***
*the meet L Ù
G = L Ç G and
join L Ú
G *= *L
È G are
effective as the conjunction and disjunction, respectively, of
statistical propositions L,G
Î **j***.*
Under these operations, **j**
forms a Boolean algebra. The branch of statistics known as *hypothesis
testing *is concerned with the problem of deciding whether
to (tentatively) accept or reject a statistical hypotheses in
the face of experimental data, or to hold it in abeyance. Thus,
statistical hypotheses acquire three truth values**:** accepted,
*rejected, *and *held in abeyance.*

The "true value" of a statistical parameter l is of course l*
**:**= l(w*)
and *parameter estimation *is the branch of statistics devoted
to the problem of estimating l* on
the basis of experimental data. A *point estimation *of l* produces a real number l^ that one has reason to believe is a good approximation
to l*. An *interval estimation *of
l* yields a *confidence interval*
I Í Â
with the understanding that the statistical hypothesis l^-1(I)
is to be accepted. For our firefly box, the components x, y, z
of the geometric point (x,y,z) representing the probability model
w Î
W as in Figure 5 form
a *complete set *of statistical parameters in the sense that
knowledge of x*, y*, and z* would determine w*.

Suppose the experimental procedure F is executed Tf times and that the outcomes l, r, and n are secured N(l), N(r), and Nf(n) times, respectively, during these trials. Likewise, suppose S is executed Ts times and that the outcomes f, b, and n are secured N(f), N(b), and Ns(n) times, respectively, during these trials. Thus,

Tf = N(l) + N(r) + Nf(n) and Ts = N(f) + N(b) + Ns(n)

for a total of T **:**= Tf + Ts trials. If we assume that
the habits of the firefly are unaffected by our observations,
then the sequential order in which F and S are executed is presumably
irrelevant. We could carry out the Tf trials of F first, then
perform the Ts trials of S**—**or
vice versa. We could alternate trials of F and S. We could even
flip a coin after each trial to see whether to perform F or S
on the next trial. In any case, all pertinent information derived
from the T = Tf + Ts trials is encoded in the *observed frequency
vector*

h**:**= (N(l), N(r), Nf(n), Ns(n),
N(f), (N(b)).

If l is a statistical parameter,
an *estimator *for l is a function
l^(h)
that provides a numerical estimate l*
» l^(h) of l*
based on the experimentally observed frequencies.

Statisticians have developed several techniques and conditions
to assess and compare various proposed estimators. For instance,
an estimator l^
is said to be *unbiased *iff, whenever the observed frequency
vector h conforms exactly to a probability
model w in the sense that N(l) = w(l)Tf, N(r) = w(r)Tf,
Nf(n) = w(n)Tf, Ns(n) = w(n)Ts,
N(f) = w(f)Ts, and N(b) = w(b)Ts, then
l^(h)
= l(w).
A weaker, and perhaps more realistic condition is that the estimator
be *asymptotically unbiased *in the sense that l^(w) approaches l(w) as a limit
when Tf and Ts become larger and larger.

**9.1** __Example__ Let N(l,r) **:**= N(l) + N(r),
N(f,b) **:**= N(f) + N(b), Nn **:**= Nf(n) + Ns(n), and
T **:**= Tf + Tn. Then the *maximum* *likelihood estimators
*[48] for the statistical parameters x,
y, z are given by

x^ = (N(l)/N(l,r))*(1 - N(n)/ T)

y^ = (N(f)/N(f,b))*(1 - N(n)/ T)

z^ = N/T

As is easily checked, the estimators in Example 9.1 are unbiased.

**10** __The Quantum Firefly Box__
Associated with a quantum-mechanical system **j **is a vector-like quantity called spin.
However, it turns out that measurements of the spin component
in a fixed direction can produce only finitely many different
numerical outcomes rather than the continuum of possible components
that would be expected for an ordinary vector quantity. In other
words, the spin components of **j***
*in a given direction are "quantized."

The behavior of **j **in
regard to its spin is characterized by a number j which can be
0, a positive integer, or half of a positive integer. The spin
component of a spin-j system **j***,*
measured in a fixed spatial direction d, can take on only 2j +
1 different values**:** -j, -j + 1, ¼,
j* - *1, or j. For instance, the spin component of a spin-1/2
system, measured in a given direction d, can only be -1/2 or 1/2.
For a spin-1/2 system, the outcomes -1/2 and 1/2 are called *spin
down and spin up *in the direction d. An electron, for example,
is a spin-1/2 particle.

Suppose **j*** *is
a spin-1/2 system and we have a spin detecting apparatus that
will measure the spin component of **j***
*in the direction of a unit vector d = (d1,d2,d3), d1^2 + d2^2
+ d3^2 = 1. It turns out that the possible states of the system
**j** are represented
by vectors y = (yl,
y2, y3)
with y1^2 + y2^2
+ y3^2 £
1. Therefore, the state space Y of
**j **can be visualized
as a solid sphere of radius 1. According to the rules of quantum
mechanics, the probability of spin-up in direction d when **j** is in state y
Î Y
is given by

Proby(spin-up in direction d) =
1/2(l + y1d1 + y2d2
+ y3d3).* *For the special case
in which y1^2 + y2^2
+ y3^2 = 1 and g
is the angle between the unit vectors d and y,

Proby(spin-up in direction d) =
cos^2(g/2). Experimental apparatus
is rarely 100 percent efficient, and the probabilities given by
(1) and (2) have to be regarded as conditional probabilities,
*given that the detector actually* *produces a response.*

Imagine now that our firefly is really
a spin-1/2 system, that a spin detector is inside the box, and
that it signals spin-up or spin-down by producing a blip of light
on the front or side window. For the front window, with the same
symbols used in Section **4**, suppose the detector is set
so that spin-up in direction df produces outcome l, spin-down
in direction df produces outcome r, and outcome nf (no blip on
the front window) simply means that the detector failed to respond.
Likewise, for the side window, spin-up in direction ds produces
outcome f, spin-down in direction ds produces outcome b, and outcome
ns means that the detector failed to respond. As before, we must
look directly at one window or the other (note
l3).

Suppose the spin detector has the same detection efficiency,
say 100e%, 0 £
e £
1, in any one direction df as in any other direction ds. Then
we are (almost literally) "in the dark" about the spin
of **j** when we look
at the front window about as often as when we look at the side
window. In this case, there seems to be no harm in setting n **:**=
nf = ns as we did in Section **4**. Then, for either window,
the probability of outcome n is 1 - e.

Let us choose our coordinate system so that df = (1,0,0) and
ds = (cos a, sin a,
0) with 0 < a £
p/2. The state vector
y can then be written in terms of spherical
coordinates 0 £ r
£ 1, 0 £
q* *£*
*2p, and 0 £
Æ £
p as

y = (rcosq sin Æ, rsinq sin Æ, rcos Æ).

Then, with the same notation as in Section **8**,

(3) x = Proby(l) = 1/2e(1 + rcosq sin Æ),

(4) y = Proby(r) = 1/2e(l + rcos((a - q) sin Æ), and

(5) z = 1 - e

For the quantum firefly box, not every probability model w in the pyramid W
of Section **8** corresponds to a possible state y Î Y. For instance, the four vertices
on the bottom square are unattainable from (3), (4) and (5), even
if e = 1. In fact, for
e = 1 and 0 < a £ p/2, the set Ga
of all points in W that
correspond to possible states y of
the quantum firefly box is a convex region in the square base
of W bounded by an ellipse
with center (1/2,1/2), major axis along the line y = x, and having
semimajor and semi-minor axes of lengths (2^-1/2)cos(a/2)
and (2^-1/2)sin(a/2), respectively.
If a = p/2,
then Ga is a circular disk tangent
to the boundary of the square base of W
at the four midpoints (1/2,0), (1,1/2), (1/2,1), and (0,1/2).
For our original firefly box, Ga is
a statistical hypothesis asserting that the firefly is behaving
like a quantum firefly.

**11** __Conclusion__**—**__What
Have we Learned?__ Our thought experiments with the firefly
box have provided illustrations of some of the more important
ideas and tools employed in the scientific study of physical systems**:**
outcomes, events, experimentally testable propositions, states,
attributes, probability models, tests, statistical hypotheses,
and statistical parameters. Of course, this list is far from complete**—**what about observables, dynamics,
symmetries, invariants, conservation laws, amplitudes, coupled
systems, relativistic physics, causality, and so on? Although
most of these ideas can also be illustrated and studied in the
context of the firefly box (or firefly boxes), it is already possible
to discuss what I consider to be the main lessons of quantum logic,
and I propose to do that now.

Here then are my personal candidates for the answers to the
question posed in the title of this article**:**

**Logical Connectives** In dealing
with propositions associated with a physical system, *one must
question the meaning and even* *the existence of the basic
connectives of classical logic***—***and,*
*or, denial, and implication.*

In our firefly example**:**

(1) The event logic **e
**is not even a lattice, a fact which warns us not to
try forming the logical disjunction**—**let
alone the join**—**of event propositions
such as {l} and {f} that cannot be tested simultaneously. There
is no meaningful denial connective on the event logics. For instance,
what would be the denial of the event {n}? Is it {l,r}? Or is
it {f,b}? [Students of Quantonics, can you see what our answer
is here? Our answer is "Yes." Foulis demonstrates vividly
here that classical objective negation **fails** in real quantum-logical
semantics. Quantum negation is rather, subjective,
which explains why we answer "Yes!" Foulis shows us
there are n¤ *real* classical EOOOs!
His example also exemplifies exquisitely quantum comtrafactual
definiteness. We call it "Many truths." Too, in Quantonics,
we deny definiteness in any classical sense. How? We know quantum
reality, due its absolute Planck rate flux/animacy, is intrinsically
"uncertain." See our BAWAM.
To understand what we just said in a *better*, Quantonic
way, see our Bell
Theorem Chautauqua (on 'contrafactual definite'), our Aristotle Connection, our
Quantum Connection, our
Sophism Connection, and
our SOM Connection. Also as
a Quantonic learning exercise, review Foulis' uses of 'complement'
above. Does he apply a Bohrian 'complement,' or a Quantonic 'c¤mplement?'
23Jan2002 - Doug.]

(2) The logic P of experimental propositions is a lattice, but the meet (respectively, join) of experimental propositions is not their logical conjunction (respectively, disjunction) unless the propositions are simultaneously testable. For instance p(l) Ù p(f) = 0, which in no way corresponds to a logical conjunction p(l) & p(f) of p(l) and p(f). However, the logic P does carry a rather perspicuous logical negation p ® p' (which in fact is an orthocomplementation).

(3) The logic Ã of attributes is again a lattice, and the meet P Ù Q of attributes P and Q is effective as their logical conjunction. But, if there are irreducible attributes contained in P and Q that admit proper superpositions, then the join P Ú Q cannot be construed as a logical disjunction of P and Q. Also, the logic Ã, does not admit any reasonable denial connective. In particular, there is no orthocomplementation on Ã.

(4) The implication connective (p,q) ®
p É q is even more conspicuously
absent in quantum logic than the conjunction and disjunction connectives.
The material implication connective p É
q **:**= p' Ú q of classical
(Boolean) logic has been the subject of considerable philosophical
criticism and debate; in quantum logics modeled by orthomodular
lattices, one has to forfeit even this suspect connective and
make do, if at all, with severely attenuated versions thereof
[31]. Note, however, that all of the logical
systems **e***,
*P*, Ã,
*and **j **admit perspicuous
implication *relations, *namely their respective partial
order relations Í, £,
Í, and Í.

(5) It is only when one reaches the level of the inductive
logic **j** of statistical
hypotheses (a Boolean algebra) that one encounters a logical system
with a secure and well-understood meaning of conjunction, disjunction,
and denial as well as a (material) implication connective.

(6) Even at the level of the inductive logic, a *conditional*
*hypothesis *G |L
(i.e., G *given *L
) cannot be construed as a material implication L
É G,
and the logic **j** has
to be enlarged to a Heyting algebra **j|j** to accommodate this important
alternative notion of implication [73]. Conditional
events, conditional propositions, and conditional hypotheses are
currently under intense study by electronic engineers and computer
scientists because of the necessity of codifying conditional information
in expert systems.

**A Hierarchy of Logical Systems**
*There is a hierarchy of related* *but distinct logical
structures affiliated with a physical* *system. *These
include, but are not limited to, the event logic **e**, the logic P
of experimental propositions, the attribute logic Ã, the face lattice Á,
and the inductive logic **j**
and the logic **j|j** of conditional hypotheses.
*Propositions in the various* *logics are different in
kind, are tested in different ways, and* *have their own
distinct modalities.*

I have a coin.

(i) I can toss the coin, observe the outcome and determine whether or not the event "heads" occurred.

(ii) I can make a prediction that the coin will fall "heads" on the very next toss.

(iii) I can assert that the coin is fair and that I am willing
to bet at odds of 1**:**1 on either "heads" or "tails"
on the next toss.

(iv) I can claim that, in a sufficiently long sequence of independent tosses, the proportion of "heads" will be very close to 0.5.

It is patently obvious that the observation (i), the prediction
(ii), the assignment (iii) of betting odds, and the claim (iv)
regarding long run frequency are four propositions of completely
different characters. The folly of attempting to formulate a single
"unified logic" comprising all propositions affiliated
with a physical system is manifest. Perhaps the source of all
such unfortunate attempts is the use of [**ð**] the
definite article the in the title
of the seminal paper of Birkhoff and von Neumann [10].

**Events vs. Experimental Propositions**
*One cannot necessarily* *formulate quantum logic purely
on the basis of the logic *P
*of* *experimental propositions and higher-level
logics built upon* P.
Indeed, some of the information implicit in the observation that
a certain outcome was obtained (or that a certain event occurred)
may be lost in the passage from events to experimental propositions.

For the firefly box, the loss of information in the passage
from **e **to
P is of little concern.
Neither is it particularly serious in Hilbert-space based quantum
mechanics, provided that one is dealing with a *single isolated
observation, *e.g., a measurement of a spin-component with
a Stern-Gerlach apparatus. In such a case, one can safely use
elements of the standard quantum logic of Hilbert-space projection
operators to carry the pertinent experimental information.

However, if one has to deal with sequential or compound observations,
e.g., iterated Stern-Gerlach measurements [75],
the phase and amplitude information encoded in the complex wavefunction
becomes critical. In the passage from an orthonormal set (y_{i}) of Hilbert-space state vectors
to the corresponding projection operator P onto the closed linear
subspace spanned by these vectors, all *phase and amplitude*
*information is wiped out!*

**Experimental Propositions vs. Attributes**
*Experimentally testable* *propositions about a physical
system are one thing; attributes or* *properties of that
system are quite another.*

By universal agreement, the genesis of what is now called quantum
logic is von Neumann's *Grundlagen der Quantenmechanik*

[59]. Nowhere in the *Grundlagen *does
von Neumann refer to *propositions *about a physical system;
he refers only to *properties *(i.e., what we have been calling
*attributes) *of that system. However, four years after the
publication of the *Grundlagen, *von Neumann (in collaboration
with Birkhoff), writes only of *experimental propositions and
propositional calculi*—there is no further mention of
properties. I do not know how to account for von Neumann's abrupt
transition from a logic of properties to a logic of experimental
propositions. I do know that, since then, many (but not all [1,2,3,58,62,63,69]!) quantum logicians have routinely identified
experimental propositions about a physical system and attributes
of that system. *This is a* *mistake, and a serious one!*

For our firefly box, we have seen that the logic P of experimental propositions and the logic Ã of attributes are separate, distinct, and nonisomorphic logical systems, albeit linked by the mapping p(A) ® [A]. I know of no more compelling illustration of the error of confusing experimental propositions and attributes.

**States and Probability Models**
Whereas it may be useful to assume, as is often done, that there
is a probability model wy Î
W corresponding to each
physical state y Î
Y*, there is no*
*a priori reason that the mapping* y
® wy*
has to be either injective *[one-to-one,
AKA bijective]* or surjective *[onto]*.*

In the literature of pure mathematics, certain linear functionals, measures, or homomorphisms are referred to as states because they or their analogues do in fact represent physical states in some conventional theory of mathematical physics. Quantum logicians need to be more careful!.

Even in conventional Hilbert-spaced based quantum theory, where (pure) states are represented by vectors y in the unit sphere Y, there is a distinction between the a state y and the corresponding probability measure wy on the logic P of projection operators. Indeed, the probability measure wy, defined for P Î P by wy(P) = <Py,y>, determines y only up to a phase factor, and the identification of y with wy would wipe out all phase information incorporated in the state vector. Thus, in conventional quantum mechanics, the mapping y ® wy, is surjective (by a celebrated theorem of Gleason [23]), but not injective.

For our quantum firefly in Section **10**, the mapping y ® wy, from state vectors y
to probability models w Î
W is injective, but not
surjective. For the quantum firefly, the lack of surjectivity
of y ®
wy can be ascribed to the fact that
we can only measure the spin component in two different directions.
If we append spin component measurements in additional directions
to our list of available experimental procedures, we find that
n gets smaller and smaller until, finally, the mapping y
® wy
becomes surjective.

An assumption that that the mapping y ® wy from the state space Y of a physical system to the convex set W of all probability models for the system is injective, surjective, or both constitutes a significant physical assumption about the system and the available experimental procedures for its study.

**Hidden Variables** *Quantum logic enables the construction
of* *simple finite models that can help us understand the
so-called* *problem of hidden variables. *The question
of whether apparent quantal behavior can be explained by classical
experimental procedures that are currently unknown or unavailable
is called the problem of *hidden variables. *For instance,
our (non quantal) firefly box certainly admits hidden variables
in the sense that a third window on top of the box would remove
all apparent quantal behavior!

A mathematical proof showing that the quantal behavior of a
particular physical system cannot be accounted for by hidden variables
is called a *no go *proof. The first convincing no go proof
was given by von Neumann himself in the *Grundlagen *[59].* *An excellent survey of no go proofs
up until about 1973 can be found in [5].

**Complementarity** *Quantum logic also enables the construction
of* *simple models that can help us appreciate the so-called
principle* *of complementarity. *In the writings of a
number of philosophers and scientists, not the least of whom was
Bohr himself, Bohr's principle of complementarity has often been
burdened with confusing metaphysical embellishments. Stripped
of these encumbrances, the principle seems to affirm that there
may be different experimental procedures, each of which can reveal
aspects of a physical system necessary for a complete determination
of its state, but whose conditions of execution are mutually exclusive.
Our firefly box with the two experimental procedures F and S provides
a perfect example of this situation, and also exposes the strong
connection between complementarity and the problem of hidden variables.

At roughly the same time that scientists were confronted by a breakdown in the Newtonian mechanical view of the physical world, artists were discovering a "principle of complementarity" in their own world. I doubt that many artists had any direct understanding of the physical principle of complementarity in quantum mechanics. Nevertheless, there was a sympathetic resonance between the two worlds, which I leave it to the reader to contemplate after perusing the following words of the art critic Marco Valsecchi [64].

"The idea was to arrange the forms in a plane so that an object or figure could be recognized not through perspective illusion, but through an analysis of its form, and also so that it could be seen from several points of view. These multiple analyses of total vision were put into a single image, thus giving an immediate unity to what has been seen, deduced and imagined... to bring together all the multiple aspects of an object and to reduce them to the plane of the painting, like a summation all at the same time of all the different instances of poetic and rational perception."

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1. Witness the following ill-natured remark, arrogantly inserted
in the review *(Mathematical Reviews *of the American Mathematical
Society, Nov.-Dec., 1991, 91k 81010) of a paper written by a well-known
Polish physicist who (in the opinion of the reviewer, a "mainline
physicist" of some repute) had the temerity to concern himself
with matters pertaining to the foundations of physics**:**
"...a small but persistent core of diehards who find fault
with quantum mechanics is still active today. The journal *Foundations
of Physics *serves to give them somewhere to publish."

Return to text

2. Just now, the desired consummation
of theoretical physics seems to be a so called "theory of
everything," i.e., a master theory encompassing both quantum
mechanics and the general theory of relativity.

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3. Papers on quantum logic are reviewed
in Sections 03G and 81P of the American Mathematical Society's
*Mathematical Reviews.*

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4. One of Boole's primary motivations
was to construct a mathematical foundation for a theory of probability.
Indeed, the full title of his 1854 masterpiece is, *An Investigation
Into the* *Laws of Thought, On Which are Founded the Mathematical
Theories* *of Logic and Probabilities.*

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5. A Boolean algebra can be defined either
as a complemented distributive lattice or as a ring with unit
in which every element is idempotent.

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6. A Hilbert space is a vector space (over
the reals, the complexes, or the quaternions) equipped with an
inner product, and complete with respect to the metric arising
from the inner product.

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7. A lattice L satisfies the modular law
iff, for p,q,r Î L, p £
r implies that p Ú (q Ù r) = (p Ú
q) Ù r.

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8. A lattice L with an orthocomplementation
p ® p' satisfies the orthomodular
identity iff, for p,q Î L, p
£ q implies that q = p Ú (q Ù
p').

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9. The terminology "orthomodular
lattice" was suggested by I. Kaplansky because, in such a
lattice, orthogonal pairs are modular pairs.

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10. An effect operator is a self-adjoint
operator A such that __0__ £
A £ __1__.

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11. The situation is quite analogous to the fact that attempts to make simultaneous measurements of noncommuting quantum-mechanical observables lead to interference effects that spoil the accuracy of the measurements. This does not mean that such simultaneous measurements cannot or should not be made. It simply means that, when they are made, one has to deal with a certain amount of fuzziness or unsharpness.

Students of Quantonics should notice
that issues here go beyond simple parallax. First we should realize
that notions of system frameworks which have *real* 'zero
momentum' are impossible. In quantum reality there is no such
classical concept of 'zero momentum.' Then there is Foulis' fly.
It too may be in box-relative motion. And said fly's biophoton
emissions themselves are animate and polytemporal. Each of these,
and many other quantum __affects__, introduce compound sources
of quantum uncertainties.

As Henri Louis Bergson has taught us
so well, where classical reasoning (and classical mathematics
and physics) assumes reality is stable and objects in reality
are independent, quantum reasoning demands that we *physially*
view *reality* as absolutely variable-up-to-Planck-rate-animate
with quantum-objects sharing probability distributed 'codependencies.'
See our reviews of Bergson's *Introduction
to Metaphysics*, *Creative
Evolution*, and his *Time and Free Will*. See especially Bergson's remarks on Negation
is Subjective.

Also take a look at our recent work with
Zeno's Paradice which shows our view that Zeno presciently anticipated
macroscopic quantum uncertainty.

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12. All of the intrinsic attributes are
thus identified with E.

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13. Although there is no Heisenberg principle
of uncertainty for spin measurements in different directions,
the Hilbert-space operators for such measurements fail to commute.

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(26Jun2000 rev - Correct a subscript under Events vs. Experimental Prop's; 1 -> i.)

(26Jun2000 rev - Added brief def's., in red, to both 'injective' and 'surjective' under States & Probability Models.)

(25Dec2000 rev - Minor title reformat.)

(20Dec2001 rev - Add top of page frame-breaker.)

(23Jan2002 rev - See our answer to Foulis'

(25Feb2003 rev - Add Note 11 comments in red text. Add 'mutual exclusion' note to Sec. 4.)

(3Sep2004 rev - Add page top link to our critical review of Jammer's

(29Jun2007 rev - Reformat.)

(5Apr2009 rev - Add Harvard link to this page near page top. Reformat some old contact info.)

(5Aug2009 rev - Update section 3, A Brief History of Quantum Logic, with commentary on bogosity of tensors and abelian groups in quantum~reality.)

(5Feb2011 rev - Similar comments here as 5Aug2009, regarding Hermitian operators.)