In making decisions involving probabilistic phenomena, people's behavior often violate economic and philosophic principles of rationality. Various theories in economics and psychology have been developed to account for these violations, Prospect Theory of Kahneman and Tversky [4] being currently the most influential. Almost always the accounts assumed an underlying boolean algebra of events. The deviations from SEU are modeled by changing or generalizing characteristics of a finitely additive probability function. Relatively recently, a different approach has been taken: Change the event space to accommodate the violations of economic and philosophic rationality. Topological event spaces of the previous section are one example of such an approach. More commonly in the literature are modeling techniques inspired by von Neumann's approach to quantum mechanics, for example, Busemeyer and Bruza [5].

In his classic Mathematische Grundlagen der Quantenmechanik, von Neumann [6] modeled probabilistic quantum phenomena using closed subspaces of a Hilbert space as events. The seminal article by Birkhoff and von Neumann [1], “The Logic of Quantum Mechanics,” isolated the algebraic properties of the event spaces that von Neumann thought underly the probability theory inherent in quantum phenomena. The logic consisted of the following:

For lattice algebras, the modular law is a generalization of the distributive law, B ⋓ (A ⋒ C) = (B ⋓ A) ⋒ (B ⋓ C). Thus, the von Birkhoff-von Neumann logic is a generalization of a boolean lattice algebra, that is, of an orthocomplemented lattice algebra satisfying the above distributive law. It applies to the lattice algebra of all subspaces of a finite dimensional Hilbert space. However, as Husimi [7] pointed out, the lattice algebra of closed subspaces of an infinite dimensional Hilbert space does not satisfy the modular law. He suggested replacing the modular law with the following consequence of it that he called the orthomodular law: For all A, B, C, if B⊆A then A=B⋓(B⊥⋒A). Today, Husimi's suggestion won out and the term quantum logic applies to lattice event algebras with orthocomplementation satisfying the orthomodular law. In this article, lattice terminology is used instead, and such lattices are called orthomodular lattices.

In psychology, ideas derived from quantum mechanics have been implemented in various ways, from borrowing methods that assume some physics, to using only Hilbert space probability theory, to using only orthomodular lattices. All of these have foundational issues: Why should methods based on physical laws, e.g., methods based on the conservation of energy, apply to psychology? How does one derive the geometrical properties of Hilbert space used in quantum probability from psychological considerations? What does orthomodularity have to do with how experiments are designed and conducted? To my knowledge, the first two questions has not been adequately addressed in the literature. This article makes some progress on the third.

The behavioral modeling described in this section concerns a simplified experimental situation. It differs from a similar model presented in Narens [8] in that minor errors and ambiguities in the construction of that model are eliminated and the material is presented in a more clear manner. The version presented here is also more general.

The assumed simplified situation makes for easier mathematical modeling and philosophical analysis, which are the principal goals of this article. The cost for this is a loss of realism and a design that may require much larger numbers of subjects than is practical for usual psychological experimentation.

The experimental situation under consideration has a large population of subjects, where each is put into exactly one of a finite number of experiments. In psychology, this is called a between-subject paradigm.

Each experiment has a finite, nonempty set of choices—called outcomes—and each of an experiment's subjects must choose exactly one of the experiment's outcomes. Different experiments are assumed to have different outcomes. Thus, each outcome occurs only in one experiment.

To simplify the presentation, only a specific case involving two experiments is considered throughout most of this article. The definitions, concepts, and methods of proof developed for this specific case are formulated in manners so that they generalize to the case of finitely many experiments. Such a generalization is briefly discussed in Section 3.5.

The (experimental) paradigm (P) has two experiments, (A) and (B). Experiment (A) has a set of 3 outcomes, OA={a,b,c}, and experiment (B) has a set of 3 outcomes, OB={d,e,f}. (P), which spans (A) and (B), has the set six outcomes, O={a,b,c,d,e,f}. The set of (P)'s subjects, S, is randomly divided in half, with one of the halves participating in (A) and the other in (B). In each experiment, the identity of each subject is recorded along with the outcome she chose. Thus, the number of subjects, N, and the number N_x of subjects who chose outcome x, x = a, b, c, d, e, f, are known. This is the collected data of (P).

Paradigm (P) also has a theory that connects its experiments (A) and (B). This connection is described counterfactually, for example,

Such counterfactuals exist only in theory, not in data: For a subject s who chose some outcome of E in ℘(OA) and E≠OA in experiment (A), it is not possible to determine from (P)'s data alone whether or not s's choice would have been in F∈℘(OB), where in experiment (B), F≠OB and F ≠ ∅. Such a determination must be a consequence of the theory posited by paradigm (P).

Theoretical assumptions of (P). The following three theoretical assumptions are made about (P):

(T1) Each subject in S paradigmatically choses exactly one outcome from each of (P)'s experiments; and each of (P)'s outcomes is paradigmatically chosen by some subject in S.

(T1) is a general theoretical assumption that extends to paradigms having finitely many experiments. (T2) and (T3) are theoretical assumptions that are specific to properties of (P). Section 3.5 describes modified versions of them that apply more widely to paradigms having finitely many experiments.

By assumption, OA∩OB=∅. However, there are situations where outcomes in OA and OB are needed to be identified. This accomplished through the use of counterfactual statements. Assumption (T2) above is an example of this: For the purposes of analysis and drawing conclusions about (P), it counterfactually identifies c and d as being the same outcome.

The following notation and concepts are useful.

The following definition provides a method for identifying events across experiments.

Notation For each H∈℘(O), let <-H>=<O-H> and -<H>=<O>-<H>.

The following lemma is a simple consequence of Definition 5.

Lemma 1. Let H be the proposition. Then < −H > = − < H >.

Proof. Each subject p makes one unique actual choice. If this choice is in −H then p is in < −H > and therefore p must be in − < H >, i.e., < −H > ⊆ − < H >. If p is in − < H >, then her actual choice is in −H, i.e., − < H > ⊆ < −H >.

Elements of P are described later in Figure 1.

It follows from P's theory and data that ∅ is a proposition and O is a proposition.

It will be shown that the proposition a is {a, e, f}.

{a} ⊆ {a, e, f}. Because each subject paradigmatically selects exactly one outcome in OA, it follows that b ∉ a and c ∉ a, and thus by assumption (T2), d ∉ a. By assumption (T1) each subject who paradigmatically chooses some element of a must also paradigmatically choose some element in OB. This element cannot be d. Therefore, it is either e or f. If it is e, then another subject who paradigmatically chose a must have chosen f, for otherwise a ⊆ e, contradicting assumption (T3). Similarly if a subject who chose an element of a paradigmatically chose f, then another subject who paradigmatically chose and element of a must have paradigmatically chosen e. Thus, it has been shown that a = {a, e, f}. Similarly b = {b, e, f}. Note that

However {e, f} is not a proposition. Note that the proposition that is the ⊆-greatest-lower bound of a and b is the empty set, ∅.

Note by Lemma 1 and the meanings of “< >” and “∪” that for all E and F in ℘(O),

Lemma 2. Let C, D, and E, respectively be, respectively, propositions associated with C, D, and E. Then C^⊥ is a proposition, C =C^⊥⊥, and D ⊆ E iff E^⊥ ⊆ D^⊥.

Proof. C^⊥ = σ(C) is a proposition by Definition 7.

By Equation (3),

Because

and

it follows that D ⊆ E iff E^⊥ ⊆ D^⊥.

Lemma 3. ⋓ is the ⊆-least upper bound operation on P.

Proof. Let F and G, respectively, be propositions associated with F and G. Then, because

it follows that

and therefore F ⋓ G is a proposition. Then

and

making F ⋓ G an upper bound of F and G. Suppose H is the proposition associated with H and is such that F ⊆ H and G ⊆ H. Then

Thus

and therefore,

showing that F ⋓ G is the least upper bound of F and G.

Lemma 4. ⋒ is the ⊆-greatest lower bound operation on P.

Proof. Let F and G, respectively, be propositions associated with F and G. Then, because < F > ∩ < G > = < F ∩ G >, it follows that

and therefore F ⋒ G is a proposition. Thus,

and

making F ⋒ G a lower bound of F and G. Suppose the proposition H associated with H is such that

Then < F > ⊇ < H > and < G > ⊇ < H >, and thus

and therefore,

showing that F ⋒ G is the greatest lower bound of F and G.

Lemma 5. 𝕻=〈P,⋓,⋒,⊥,O,∅〉 is a complemented lattice event algebra.

Proof. O is clearly the ⊆-largest element of P and ∅ is clearly the ⊆-smallest element of P.

Because, by Lemmas 3 and Lemma 4, ⋓ and ⋒ are, respectively, the ⊆-least upper bound and ⊆-greatest lower bound operators on P, 𝕻 is a lattice event algebra. The following shows that ^⊥ is a complementation operation on 𝕻.

Let E and F, respectively, be the propositions associated with E and F. Then, by E^⊥ = σ(−E) and Equation (3),

and

Lemma 6. The complemented lattice event algebra 𝕻=〈P,⋓,⋒,⊥,O,∅〉 satisfies DeMorgan's Laws.

Proof. It is a well-known result of lattice theory (e.g., Theorem 2.14 of [2]) that DeMorgan's Laws for 𝕻 are equivalent to the following: For all E and F in P,

Equation (4) follows from Lemma 2.

The above lemmas show that the description of the experimental situation gives rise to an orthocomplemented lattice. Aerts and Gabora [9] have a similar result for a different psychological paradigm: They show that their empirical data is representable as an orthocomplemented lattice that they imbed in a Hilbert space.

Theorem 3 given later shows that 𝕻=〈P,⋓,⋒,⊥,O,∅〉 also satisfies the Orthomodular Law. The proof, which generalizes to a wide class of paradigms with finitely many experiments, uses a probability function that is defined on the set of 𝕻's propositions. The probability theory for this function is developed in the following two sections.

The following is the intended interpretation of ℙ: For each proposition E in P, ℙ(E) is the probability that a randomly chosen paradigm subject actually chose some outcome e in E. If the subjects in < E > are known through data and theory, then the value of ℙ(E) completely computable from data.

Propositions E that span experiments are necessarily partially based on counterfactuals. Because of this, they are theoretical in nature. Nevertheless, as discussed at the end of Section 3, for paradigm (P), ℙ's value for a proposition F is estimable from data to a good approximation. As discussed at the end of Section 3 this is not generally true of other paradigms. However, for the special case where a proposition comes from one of a paradigm's experiments it is generally true by an analog of the following argument given for (P).

Because of the large numbers of subjects participating in (P)'s experiments and the way they were randomly assigned in equal numbers to each of (P)'s experiments, it follows that for each E in ℘(OA),

where ≈ stands for “approximately” and | < E^⋆ > | for “the number of subjects in (B) that counterfactually chose some outcome in E.” Thus, for E in ℘(OA),

which is computable since | < E > | and |<O>| are known from data.

Thus for each proposition E in ℘(OA) and similarly for each proposition F in ℘(OB), ℙ(E) and ℙ(F) are estimable to a good approximation from data. For G∈℘(O) where G spans experiments can be more complicated. For such spanning propositions, theoretical assumptions as well as data are needed to calculate ℙ's probabilities. As discussed at the end of Section 3, this is possible for paradigm (P) but may not be possible for other paradigms where the the theory may not complete enough to estimate all spanning propositions.

Figure 1 is a Hasse diagram of the lattice 𝕻=〈P,⋓,⋒,O,∅〉. The set-theoretic boolean algebra generated by O has 2⁶ = 64 elements. The elements at the bottom of Figure 1 but above ∅ are called atoms. They are lattice elements E such that there there does not exist a lattice element F such that ∅ ⊂ F ⊂ E. Figure 1 has 5 atoms, a, b, k, e, f. The set-theoretic boolean algebra generated by these atoms has 2⁵ = 32 elements. P has 12 elements—a substantial reduction from 64 or 32.

In Figure 1, the lattice-theoretic intersection ⋒ of atoms, e.g., a ⋒ f, is the proposition ∅. This is a consequence of assumption (T3).

The following concepts are useful for the understanding of the structure of orthomodular lattices.

Lemma 7. Suppose 𝖃=〈X,⋓,⋒,⊥,X,∅〉 is an ortholattice that has no O₆ subalgebra. Then 𝖃 is orthomodular.

Proof. Theorem 2 (pp. 22–23) of Kalmbach [10]. (Also Theorem 2.25 of [2].)

Lemma 8. ℙ is a ⊂ -monotonic orthoprobability function on 𝕻=〈P,⋓,⋒,⊥,O,∅〉.

Proof. Suppose F and G are arbitrary elements of P such that G ⊆ F^⊥. We first show

It is immediate that < F > ⊆ < F ⋓ G > and < G > ⊆ < F ⋓ G >. Thus,

Suppose s is in

Then is is in σ(F) or s is in σ(G). Without loss of generality, suppose s is in σ(F) = F. Then

and Equation (5) follows from Equations (6) and (7).

By the definitions of “proposition” and “^⊥” and Equation (3), < F^⊥ > = < − F > = − < F >. Thus, < F > ∩ < F^⊥ > = ∅, and therefore, because G ⊆ F^⊥, < F > ∩ < G > = ∅. Thus,

Therefore,

showing ortho-additivity.

To show monotonicity suppose F and H are arbitrary elements of P such that F ⊂ H. Then < F > ⊂ < H >. Then, by the definition of ℙ, ℙ(F) < ℙ(H).

Theorem 3. 𝕻=〈P,⋓,⋒,⊥,O,∅〉 is an orthomodular lattice and ℙ is an orthoprobability function on 𝕻.

Proof. By Lemma 8, ℙ is a monotonic orthoprobability function on 𝕻. Suppose 𝕻 is not an orthomodular lattice. A contradiction will be shown. Then by Lemma 7 there exists a sublattice of 𝕻 that has a Hasse diagram of the form displayed in Figure 2. By the monotonicity of ℙ,

and by the ortho-additivity of ℙ,

Equations (8) and (9) contradict one another, because, by the monotonicity of ℙ, ℙ(G^⊥) < ℙ(F^⊥).

The literature has studied orthomodular lattices as generalizations of the logic underlying quantum mechanics. Unfortunately, not all orthomodular lattices admit orthoprobability functions [11]. This in itself is a clue that for science something more than general orthomodular lattices are needed. For 𝕻, the probability function ℙ was derived directly from (P)'s theory and empirical considerations.

Thus far, our analysis has focussed on the paradigm (P) and the probabilistic structure 𝕻. Although the analysis sometimes used special features of them, care was taken to present, when possible, concepts and methods of proof that generalized to a wider class experimental situations and a wider class of probabilistic structures. There are, however, some conditions special to 𝕻 and ℙ that do not apply to all between-subject paradigms involving finitely many experiments. These are concerned with the use of 𝕻's atoms.

The boolean algebra 𝕭=〈℘(O),∪,∩,-,O,∅〉 spans (P)'s experiments. Its set of atoms is O={a,b,c,d,e,f}, which is the set of outcomes of (P)'s experiments. (P)'s data consist of records of the choices in O made by its subjects. In shifting the analysis from 𝕭 to 𝕻, it is desirable to keep the data intact. (P)'s theoretical axioms and concepts does this by making the propositions a, b, k, e, and f the atoms of 𝕻. (k results from theoretical assumption (T2) that requires the identification of c and d.) This allows the collected data about O to be transferred to a, b, k, e, f. Concepts and theorems can exploit this transfer. For example, this transfer is needed to implement the important concept of “actually determined” in obtaining consequences of (P)'s theory and data.

Many of the previous results about 𝕻 generalize to a paradigm (Q) involving finitely many experiments, (A₁), …, (A_n), where (Q) has disjoint experimental outcomes, disjoint subject populations, and where for 1 ≤ i, j ≤ n, A_i's subject population is randomly sampled from (Q)'s subject pool and is the same size as A_j's subject population. In particular, with appropriate generalizations of (P)'s theory, Lemmas 2 to 8 and Theorem 3 generalize to (Q)'s lattice of propositions using the methods of proof similar to the those presented for 𝕻 and ℙ.

(Q)'s theory consists of a set of statements describing relationships among subjects' responses across experiments. Among these are statements that generalize (T1), (T2), and (T3) of (P)'s theory in the following manner:

(T*1) Each of (Q)'s subjects paradigmatically choses an outcome from each of (Q)'s experiments; and each of (Q)'s outcomes is paradigmatically chosen by at least one of (Q)'s subjects.

“< >” and ℙ have analogous definitions and results for (Q) to those for (P).

Many researchers of the formal foundations of quantum mechanics have speculated that the underlying probability theory for quantum mechanics is not interpretable in a physically acceptable manner into a boolean probability theory (e.g., [1, 12–14]). Others have disagreed (e.g., [15]), producing a long-running controversy that continues to the present (e.g., 16).

Von Neumann was well aware of foundational difficulties presented in his seminal 1932 book, Mathematische Grundlagen der Quantenmechanik. It appears to me that such difficulties are sharply increased and compounded by the importation of formalisms involving probability from quantum mechanics to cope with the difficult contextual issues presented in the behavioral sciences.

Rédei [17] writes the following about the evolution of von Neumann's position about the nature of probability in quantum mechanics.

Von Neumann's concerns about probability theory in quantum mechanics do not hold for the multi-experiment behavioral paradigms presented in this article. The paradigms' orthomodular lattice event structures follows directly from their experimental designs and theories linking experiments. This produces an orthoprobability probability function ℚ for a paradigm's lattice of propositions, 𝕼. Because for propositions, actual probabilities coincide paradigmatic probabilities, ℚ can be estimated through a relative-frequency process for events for which the underlying theory and collected data specify to a good approximation which subjects paradigmatically chose outcomes for those events. Paradigm (P) is an example where such a relative frequency approach applies to all of its events: Its event lattice has twelve elements. Of these, the probabilities of two, O and ∅, are determined by definition. Five others, a, b, k, e, f, are atoms and their actual probabilities are estimable by collected data and thus, as described earlier, their paradigmatic probabilities are estimable. The remaining five are complements of the five atoms and these have as probabilities 1 minus the probability of its atom, and thus they too are estimable. Now consider the general case 𝕼 where F and G are lattice disjoint propositions where it is known which subjects chose an element of F and which chose an element of G. If it is the case that F ∪ G ⊂ F ⋓ G then more information is required to estimate the number of subjects who are in F ⋓ G. The additional information has to come from the paradigm's theory. For (P), its theory tells us that a ⋓ f = {a, b, e, f}, which is the complement of k and thus has number of subjects |O|-|k|. This number is known because |O| and |k| are known.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.