D.J. Foulis, Dept. of Mathematics & Statistics University of Massachusetts, 'A Half Century of Quantum Logic--What Have we Learned?

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.

Index

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].

Index

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 identities as 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 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.

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].

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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.

Index

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.

Students of Quantonics note that Foulis' mandate of 'mutual exclusion'
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.

Index

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.

Index

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.

Index

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.

Index

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 Attributes To 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 Attributes An 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!

Index

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.

Index

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 := UwÎF 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.

Index

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.

Index

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.

Index

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].

Index

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!

Index

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.

Index

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."

Index

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8. Bennett, M.K. and Foulis, D.J., Tensor products of orthoalgebras, Order 10, No. 3 (1993), 271-282

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14. D'Andrea, A.B. and De Lucia, P. The Brooks-Jewett theorem on an orthomodular lattice, J. Math. Anal. Appl. 154 (1991.) 507-552

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19. Dalla Chiara, M.L., and Giuntini, R. "Paraconsistent quantum logics, Found. Physics 19, No. 7 (1989) 891-904

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22. Dvurecenskij, A., Quantum logics and completeness criteria of inner product spaces, International J. Theor. Phys. 31, (1992) 1899-1907

23. Dvurecenskij, A., Gleason's Theorem and its Applications, Kluwer, Dordrecht/Boston/London 1993

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28. Foulis, D., Coupled physical systems, Found. Phys. 7 (1989) 905-922

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Index

Footnotes

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."
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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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Index

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Transposed: 18-22Jan99 PDR

Revised: 25Feb2003 PDR

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(26Jan2000 rev by Doug: Add ref. link 10, and add link to June99QQA on 'thelogos.')
(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' Conclusion: Firefly Example 1. Answer in red text & dated.)
(25Feb2003 rev - Add Note 11 comments in red text. Add 'mutual exclusion' note to Sec. 4.)