## Abstract

Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.

## 1. Introduction

Recently, ideas from quantum physics have filtered into cognitive psychology to account for the troublesome influence context has in human cognition and decision-making. Some investigators, following Busemeyer and co-workers (see [1] for a summary) use physical quantum theory's concepts and methods, which are based on non-Boolean event spaces, to describe paradoxical empirical findings in human decision-making. Others (e.g. Dzhafarov & Kujala [2] and Narens [3]) model similar findings using classical Boolean probability theory. In particular, as discussed in §4, Narens formulates his modelling in terms of a non-Boolean, quantum-like, probability space that is imbedded in a classical Boolean probability space. It is controversial in the foundations of quantum mechanics as to what extent a similar kind of imbedding is possible for its probability theory.

Section 4's modelling is based on a theory about the kind of reasoning involved in the linking of theory and data across experiments. Many psychological paradigms require their experiments to have different instructions, stimuli or outcomes, with each of their subjects participating in only a single experiment. In order to formulate hypotheses or draw conclusions across such experiments, the researcher needs to be able to theorize about their linkage. A natural way is through counterfactual reasoning: e.g. providing answers to questions like, ‘If a subject choose *a* in Experiment 1, what would she have chosen if she had been put instead in Experiment 2?’ Such reasoning allows the experimenter to formulate and test theories across experiments. As shown in §4, such theories combined with collected data produce probabilistic event spaces analogous to those seen in quantum mechanics.

This counterfactually based process is very different from the powerful methods involving transformations and invariance that quantum mechanics employs to achieve similar goals. Because of this, I believe that the counterfactual reasoning methodology presented in this article is unlikely to provide much insight or interesting new modelling for quantum phenomena. However, it nevertheless has the potential to provide new insights and modelling methods for psychology, and more generally for the behavioural, social and economic sciences.

Another form of non-Boolean probability theory is based on event spaces that form a topology instead of a Boolean algebra. It too has appeared recently in the psychological literature to model decision phenomena (see ch. 10 of [3]). Theorem 3.1 shows that it can be imbedded in classical Boolean probability theory. This result is used in §3 to provide a ‘Dutch Book Argument’ showing that this non-Boolean probability theory yields a rational assignments of probabilities.

The development of the ideas of this article require a number of mathematical concepts. The following section presents some of these, with a bit of their history.

## 2. Probability theories

The following conventions are used throughout this article: *X* refers to a non-empty set, to the empty set, ℘(*X*) to the power-set of *X*, and − to the operation of set-theoretic complementation, ∪ to the operation of set-theoretic union and ∩ to the operation of set-theoretic intersection.

denotes a function on ℘(*X*) such that for all *A* and *B* in ℘(*X*),

— , , ,

—

*finite additivity*: if , then and— ⊂-

*monotonicity*: if*A*⊂*B*, then .

is called a ⊂-*monotonic probability function.*

An *event space* is a subset of ℘(*X*) such that *X* and are in . Throughout this article, will denote an event space. is said to be a *belief function* on if and only if is a function on such that if *A*, *B*, *A*∪*B* and *A*∩*B* are in then , and ,

—

*distributive finite additivity*: and— ⊂-

*monotonicity*: if*A*⊂*B*, then .

Suppose is a belief function on . Then is said to be a *probabilistic frame* for if and only if . A major part of this article is concerned with intrinsic characterizations of so that it has a belief function with a probabilistic frame. By results discussed in §3, this establishes the rationality of the probabilistic assignments to elements of by one common test for probabilistic rationality.

Throughout this article, *classical probability theory* will refer to Kolmogorov's 1933 [4] axiomatization of probability as a *σ*-measurable function on a Boolean algebra of events. About the same time as Kolmogorov's axiomatization, von Neumann [5] formulated his foundation for quantum mechanics which employed a different kind of probability function, one whose event algebra was non-Boolean. His event space consisted of closed linear subspaces of a Hilbert space, and his event algebra consisted of the following:
where is the set of topologically closed subspaces of a Hilbert space with domain *H*, and for *A* and *B* in , (i) *A*⊔*B* is the smallest subspace in containing both *A* and *B*, and (ii) *A*^{⊥} is the orthogonal complement of *A*, that is, the set of vectors in *H* that is orthogonal to *A*. It easily follows that is a lattice (i.e. ⊔ and ∩ are, respectively, ⊆-least upper bound and ⊆-greatest lower bound operators), ^{⊥} is a complentation operator on (i.e. and *A*⊔*A*^{⊥}=*H*), and ∩, ⊔ and ^{⊥} satisfies DeMogan's laws,
It is well known that is also *modular*, that is it satisfies the following modular law: for all *A*, *B* and *C* in ,
The Hilbert space modular law is obviously a generalization of the distributive law of a Boolean algebra of events,

Birkhoff & von Neumann [6] considered the major change in the logical structure of quantum mechanics from that of classical mechanics to be the replacing of the distributive law by the Hilbert space modular law. Husumi [7], however, showed that the closed subspaces of an infinite Hilbert space did not, in general, satisfy the Hilbert space modular law, but instead satisfied the following orthomodular law, that is a generalization of the Hilbert space modular law. Husumi concluded that the logical structure of quantum mechanics is better captured with an orthomodular law than with a modular law. His view won out, and today the logical structure of quantum mechanics—‘quantum logic’—is generally formulated using a version of an orthomodular law.

## 3. Belief for distributed algebras of events

A *distributive algebra of events* is a structure of the form , where ∪ and ∩ are binary operations on . If, in addition, set-theoretic complementation, −, is also an operation on , then it becomes a Boolean algebra of events. It is distributive, because the set-theoretic operations of ∪ and ∩ satisfy the distributive law, *A*∩(*B*∪*C*)=(*A*∩*B*)∪(*A*∩*C*). A topological space is an example of a distributive algebra of events that is not necessarily a Boolean algebra of events.

Topological spaces has a logical structure that generalizes Boolean logic. Like the Boolean case, ∪ is interpreted as ‘or’ and ∩ as ‘and’. However, ‘not’ and ‘implies’ have different interpretations: ‘not *A*’ is interpreted as the interior of −*A*, and ‘*A* implies *B*’ is interpreted as the interior of −*A*∪*B*. With these interpretations, a topology becomes an algebraic structure known as a ‘Heyting algebra of events’, or in the philosophy of mathematics, as an ‘intutionistic logic’. All finite distributive lattices are topologies and therefore Heyting algebras of events and all Heyting algebras of events are distributive algebras of events.

Heyting algebras have a variety of interpretations. In 1932, Kolmogorov [8] published an article showing that mathematical constructions as a logical notion was a Heyting algebra. Gödel in 1933 showed S4 modal logics were Heyting algebras. Today, topologies and Heyting algebras are widely used in artificial intelligence to capture reasoning, in computer science and engineering (where some programming languages are based on them) for searching large datasets, and in philosophical logic for ‘proof theory’. Recently, they have been used as event spaces in psychology to model probabilistically context effects, ambiguity and vagueness, and in game theory to model personal knowledge.

Let be a distributive algebra of events. Because need not have a rich set of disjoint events, the concept of ‘⊂-monotonic probability function’ has to be generalized if it is to be applied to , because may have too few disjoint events for finite additivity to be a useful concept. For example, consider the case where , where , , *B*∩*C*=*A* and *B*∪*C*=*X*. Then each function *Q* on such that and has the following property: for all *D* and *E* in , if *D*∩*E*=?, then *Q*(*D*∪*E*)=*Q*(*D*)+*Q*(*E*), i.e. each such function has the numerical properties of a ⊂-monotonic probability function. Thus for this kind of situation, finite additivity is not a useful probabilistic constraint. For a Boolean algebra of events, , the following two statements are equivalent, where *P* satisfies all of the conditions of a ⊂-monotonic probability function except possibly for finite additivity:

(1)

*finite additivity:*for all*A*and*B*in , if , then*P*(*A*∪*B*)=*P*(*A*)+*P*(*B*);(2)

*distributive finite additivity:*for all*A*and*B*in ,*P*(*A*∪*B*)=*P*(*A*)+*P*(*B*)−*P*(*A*∩*B*).

For distributive lattices, Statement (2) is a proper generalization of Statement (1): Statement (2) obviously implies Statement (1), and for with , , and *P*(*X*)=1,
Statement (1) is satisfied but not Statement (2). Because of these considerations, the following definition of ‘belief function’ is adopted for a distributive algebra of events: is said to be a *belief function* on a distributive algebra of events if and only if for all *A* and *B* in , , , ,

—

*distributive finite additivity*: and— ⊂-

*monotonicity*: if*A*⊂*B*, then .

For infinite , ⊂-monotonicity can conflict with being a distributive finitely additive function into the closed interval of real numbers [0,1]. A way to achieve greater generality while sparing the main ideas presented in this section is to go outside the real number system to one that includes infinitesimal positive elements. Narens [3] takes this approach. It is not done here. Instead, this section focuses on finite . This simplifies the presentation while avoiding the use of infinitesimals.

The following theorem is this section's main mathematical result.

### Theorem 3.1

*Let* *be a distributive algebra of events, X be finite and* *be a belief function on* *. Then there exists a ⊂-monotonic probability function* *on ℘(X) such that* .

### Proof.

The theorem follows from Lemma 4.1 (p. 116) and Theorem 4.19 (p. 126) of ref. [3] by substituting ‘belief function’ for ‘L-probability function’ in the statement of the lemma and the theorem. ▪

Many in science and philosophy consider classical probability theory or its generalization using finite additivity in place of *σ*-additivity as the only rational approaches to quantifying uncertainty. Various justifications for this have been proposed. One of the most prominent is the *Dutch Book Argument.*

It assumes a Boolean algebra of events . For each event *A* in , the *Bettor* assigns a price $*A* to *A*, $0≤$*A*≤$1. He will buy or sell as many tickets of the form *T*_{A}, each at the price $*A*. The purchaser of a ticket will receive $1 if *A* occurs and $0 if *A* does not occur. In the near future, the state of the world in *Y* will be determined. Another participant called the *Arbitrageur* is said to *make a Dutch Book on Bettor* if she can buy gambles from Bettor at Bettor's prices and sell gambles to Bettor at Bettor's prices so these transactions produce for Arbitrageur a net monetary gain, no matter which state in *Y* obtains. The Bettor prices are said to be *coherent* if and only if the Arbitrageur cannot make a Dutch Book against him. For each *A* in , if $*A* = the price of *A* is *x* dollars, let . Then the *Dutch Book Theorem* says the following for this situation: *the Bettor prices are coherent if and only if is a ⊂-monotonic probability function on .* The Dutch Book Theorem is a mathematical theorem, not a philosophical thesis. The argument that the Dutch Book Theorem implies that rationality, in the guise of coherence, is necessary and sufficient for the existence of a ⊂-monotonic probability function on a Boolean algebra of events (i.e. the Dutch Book Argument) is a philosophical position, not a mathematical theorem.

Let us apply the Dutch Book reasoning to the case of a distributive algebra of events with a belief function . Assume that a bettor buying and selling tickets *T*_{A}, , at price . Does this mean that the Bettor is rational by Dutch Book reasoning?

Obviously, a distributive algebra of events can have fewer events than the Boolean algebra of events generated by it, resulting in the possible elimination of some useful buying and selling opportunities by an arbitrageur, e.g. buy *A* and sell *A*−*B*. However, theorem 3.1 says that can be extended to a ⊂-monotonic probability function on . By the Dutch Book Argument, is a rational assignment. Therefore, choices for probabilities on are rational. Thus, by the Dutch Book Theorem, no Dutch Book can be made against these choices. So the Bettor is rational by Dutch Book reasoning.

As noted in [9], this suggests the following rationality principle: *If a person's probability assignments on a subset of a Boolean algebra of events can be extended to a rational probability function on , then her assignments on are rational.* The above discussion shows that acceptance of this principle and the Dutch Book Argument implies that a belief function on a distributive algebra of events is rational.

However, unlike the distributive case, there is not a necessary connection between Dutch Book coherence and a non-distributive event algebra having functions with the algebraic form of belief functions. The following example shows this. Let be the algebra with ⊔ and ⊓ being, respectively, the ⊆-least upper and the ⊆-greatest lower bound operators on , where

— , where , , ,

— for

*α*=*A*,*B*,*C*,*X*⊔*α*=*X*, and— and

*X*=*A*⊔*B*=*A*⊔*C*=*B*⊔*C*.

Let
3.1
Then has the algebraic form of a belief function in the sense that for and ,
3.2
A Dutch Book can be made against by selling each of *A*, *B* and *C* at and buying *X* at $1. Having a pricing function on that is free from Dutch Books does not guarantee that it satisfies equation (3.2): for example, let

The above results concerning the Dutch Book Argument and distributive algebras of events show that a distributive algebra of events has a natural, intrinsic belief function that extends to a probabilistic frame. The probabilistic frame provides additional ways of understanding features of the distributive algebra that are not totally apparent by just considering the distributive algebra, Dutch Book rationality being an example. There are several interesting algebras of events between a distributive algebra of events and a Boolean algebra of events, the most important being the above-discussed Heyting algebra of open sets from a topology. By theorem 3.1, all of these algebras have probabilistic frames.

## 4. Experimental psychological paradigms

This section applies a formal approach to psychological experiments that is in the spirit similar to the probabilistic modelling employed by Birkhoff and von Neumann for quantum phenomena. The goal is to describe a ‘logic’ that connects experiments about a common psychological phenomenon. The main result is that when such experiments are combined and formally described, they form a ‘quantum logic’.

### (a) Characterization of a psychological paradigm

The following definitions provide a formal characterization of an experimental psychological paradigm.

A *paradigm* consists of a non-empty set of pairwise disjoint sets. The sets in are called *outcome sets* (of ). Each outcome set has at least two distinct elements. Let . *X* is called the *domain* of . Elements of *X* are called *outcomes of* . To simplify the presentation, *it is assumed throughout this section that X is finite.*

The idea for an experimental psychological paradigm is that instructions involving the outcomes in an outcome set *Y* is given to a subject who is asked to choose one of *Y* 's outcomes. The identity of the subject and the chosen outcome are recorded as data. Other subjects are given the same instructions and the same outcome set *Y* and their identities and outcome choices are recorded as data. For convenience, this experimental process is called an *experiment* and is identified with its outcome set. Thus, in this case, the experiment is also called *Y* . By linking together the data from the various experiments in the paradigm with known and hypothesized theory, the psychological scientist is able to draw conclusions about the subjects’ psychological behaviour.

's experimental outcome sets are pairwise disjoint. This is to reflect the fact that they appear in different experiments. A main idea behind the treatments of paradigms presented in this section is that experiments act like ‘contexts’, and thus are able to change the meaning of an outcome to a subject by instructions, by other choices in its outcome set, etc. Of course, there are situations where one wants to identify certain outcomes across experiments, and a mechanism for doing this is presented later.

Throughout this section, is a non-empty set. It is called the *set of subjects for *. A *between-subject paradigm based on with the set of subjects * is where each subject *s* in participates in exactly one experiment *Y* in . In that experiment, the subject is required to pick one and only one of the outcomes in the experiment's outcome set *Y* . A *within-subject paradigm* is defined later.

Note that *in a between-subject experiment, each subject has a (theoretical) choice for an outcome in each of 's experiments, but she participates—that is, makes an actual choice—in only one of 's experiments.*

The following defines the *frame probability function * for a *between-subject experiment.* Let
where ℘(*X*) is the power-set of *X*. The following notation is useful. For each *A* in ℘(*X*), let 〈〈*A*〉〉=the set of all subjects *s* of who would choose some element *a* in *A* if *s* were put in the unique experiment *Y* of such that *a* was an outcome of *Y* . Let
denote the number of subjects in 〈〈*A*〉〉. is defined on ℘(*X*) as follows: for each *A* in ℘(*X*),
Because 〈〈*A*〉〉 is counterfactually defined, i.e. ‘if *s* *were put* into the unique experiment …’, it is necessarily theoretical in nature. But that alone does not necessarily mean that it cannot be computed from data.

### (b) Random approximation of based on data

In applications there are often several ways of specifying 's probabilities. In psychology, it is usually done by taking sufficiently large, same sized random samples of subjects for each experiment in . For example, suppose has *n* experiments, *Y* _{1},…,*Y* _{n}, each with *k* participating subjects, and such that each subject in was randomly assigned to one of the experiments. Suppose the number *N*=*kn* of subjects in is sufficiently large. For each *x* in *X*, let

#

*s*(*x*) be the number of subjects*s*inthat participated in the unique experiment in which had*S**x*as an outcome and chose*x*.

Let be the following function on ℘(*X*): for each *F* in ℘(*X*),
4.1
After the paradigm has been run, the probability that some element *a* of *F* *was observed* to be selected is . is the probability that a randomly selected subject *s* in , *would select some outcome of F if she were to have participated in F.* and are related as follows: is a good approximation for , in symbols,
4.2
This is the case, because by the method of random selection, for each

*a*in

*F*, and thus Equation (4.2) relates 's theory to its data. As an example, suppose that a set of theoretical constraints about are formulated in terms linear equations or inequalities involving . Then these can be tested against 's data by the substitution of for (equation (4.2)). Because and are in a constant ratio, the same constraints will approximately hold for if in reality they held for . However, the constraints for are now observable and are available for empirical testing.

The ideas concerning experimental paradigms presented in this article are easily modified so that they apply beyond between-subject paradigms. For example, for describing a *within-subject behaviour across the experiments of ,* is taken to be the set of cognitive states a subject enters into while participating in 's experiments. It is assumed that for any given experiment *Y* in , there is a probability distribution on the subject's cognitive states in such that for each outcome *a* of *Y* , is the probability that the subject will choose outcome *a* if put in experiment *Y* . Then for this paradigm situation, becomes the unique probability function on ℘(*X*) such that for each *b* in *X*,
where *Z* is the unique experiment in that has *b* as an outcome. How is determined or estimated for a within-subject paradigm does not play a role in this article. This within-subject use of has the same kind of mathematical structure as the previously discussed between-subject use with the cognitive states of a subject corresponding to the subjects participating in a between-subject experiment.

For the remainder of this article, it will be assumed that is a between-subject paradigm.

### (c) Determinable events and propositions

The outcomes of the experiments in are distinct, that is, *Y* _{i}≠*Y* _{j} for *i*≠*j*. In some behavioural paradigms, a ‘same outcome’ is interpreted as occurring in different experiments, e.g. in a psychophysical experiment, physically identical stimuli occurring as choices in more than one paradigm experiment. When *a*∈*Y* _{i} and *b*∈*Y* _{j}, *i*≠*j*, this article's interpretation of ‘*a* and *b* are the same outcome’ is that the underlying theory implies that the set of subjects *s* who would choose *a* if put in an experiment with *a* as an outcome is the same as the set of subjects *s* who would choose *b* if put in an experiment with *b* as an outcome. A similar concept holds more generally for events.

Events *F* and *G* in ℘(*X*) are said to be *equivalent*, in symbols, * F≡G,* if and only if the subset of subjects

*s*of who would choose some element

*a*of

*F*if

*s*were put in 's experiment in which

*a*is an element is the same as the subset of subjects

*s*of who would choose some element

*b*of

*G*if

*s*were put in 's experiment in which

*b*is an element. Note that previous usage of ‘

*a*and

*b*are the same outcome, when

*a*≠

*b*,’ is the same as saying, ‘{

*a*}≡{

*b*} when

*a*≠

*b*.’

It easily follows that ≡ is an equivalence relation.

The equivalence of distinct events is established as a consequence of the paradigm's theoretical assumptions. In some cases, such an equivalence is suggested by the paradigm's empirical data. For example, if the subjects are appropriately randomly sampled and empirically all the subjects in experiment *Y* chose one alternative *a* and all the subjects in experiment *Z*≠*Y* chose one alternative *b*, then one might want to use this data to make the theoretical assumption that {*a*}≡{*b*}. Distinct events in the same experiment cannot be equivalent, because it is assumed that each subject chooses one and only one outcome from each experiment.

The following concept roughly corresponds to the concept of ‘observable’ of quantum physics.

An event *F* in ℘(*X*) is said to be *determinable* if and only if for each subject *s* in , either (i) it can be determined from the paradigm's theory and data that *s* is in 〈〈*F*〉〉, or (ii) it can be determined from the paradigm's theory and data that *s* is in 〈〈*X*−*F*〉〉. Let **Det** stand for the set of determinable events.

**Det** does not have the right kind of algebraic structure in terms of the ⊆ relation to be a good candidate for a probability space. Instead, a subset of it, called the ‘set of propositions’, has a rich algebraic structure and captures the relevant probabilistic information contained in **Det**.

The equivalence relation ≡ partitions **Det** into equivalence classes. Each equivalence class has a ⊆-largest element, . Such largest elements are called *propositions.* Each proposition is used to identify its equivalence class, and, throughout the remainder of this section, propositions will be used to denote equivalence classes.

**P** denotes the set of paradigm 's propositions. The following theorem shows that 〈**P**,⊆〉 has a rich algebraic structure. In mathematics, this structure is known as a ‘orthomodular lattice’.

### Theorem 4.1

*The following five statements hold*:

(i)

*〈***P***,⊆〉 is a lattice, that is, is an algebraic structure such that*—

*X∈***P***, and for all A∈***P**,*and*—

*for all F and G in***P***the ⊆-least upper bound of F and G, F⊔G, and the ⊆-greatest lower bound of F and G, F⊓G, are in***P**.

(ii)

*is complemented, where − is the operation of set-theoretic complementation restricted***P***, that is, for all F in***P**,—

*−F is in***P**,—

*F⊔−F=X and*— .

(iii)

*is an ortholattice, that is, it satisfies DeMorgan's laws: for all F and G in***P**,(iv)

*restricted to***P***is an orthoprobability function on**, that is, for all F and G in***P**,—

*and*,—

*if F⊂G, then**and*—

*if G ⊆ −F, then*.

(v)

*The lattice**is orthomodular, that is it satisfies the following condition for all F and G in***P**:

A detailed proof of theorem 4.1 is given in §6. Statements (i)–(iii) of theorem 4.1 follow, with some calculation, from the definitions of ‘determinable’ and ‘proposition’. Statement (iv) follows from the facts that is a probabilistic frame on ℘(*X*) and **P**⊆℘(*X*). Statement (v) is a consequence of Statements (i)–(iv) and theorems of lattice theory that characterize the orthomodular law.

The following notation is useful. Let . M is called the *between-subject orthomodular lattice.* As discussed previously, in terms of current terminology, M has the algebraic structure of a ‘quantum logic’, which is an algebraic generalization of the ‘logic’ described by Birkhoff & von Neumann [6] for quantum mechanics.

The approach to modelling experimental phenomena developed in this article is expressed in theorem 4.1 and the fact that the lattice M differs from that of Birkhoff & von Neumann in the following manners:

— this article's approach is designed for behavioural modelling while Birkhoff's & von Neumann's approach was designed for modelling quantum mechanics;

— the lattice M has set-theoretic complementation as its complementation operation, whereas the Hilbert space modelling uses an orthocomplementation operation that is not set-theoretic complementation, except for the degenerate case of a one-dimensional space;

— for ortholattices, Birkhoff & von Neumann generalized the

*distributive law,*to the*modular law,*Lattice M generalizes further by generalizing, for ortholattices, the modular law to the orthomodular law. (It is a well-known theorem of lattice theory that for an ortholattice, the distributive law implies the modular law which implies the orthomodular law.); and—

*most importantly, lattice M arose directly out of considerations about how between-subjects experiments are conceptualized, run and analysed.*Birkhoff & von Neumann's orthocomplemented modular event space arose out of a mathematical formulation of quantum mechanics that assumed the existence of a Hilbert space as part of the formulation.

### (d) Examples from physics and voting

Although paradigms in quantum mechanics and behavioural science are about radically different subject matters, they can generate event spaces that share much algebraic structure. The following physical situation described by Foulis & Randall is an example.
Suppose we have a device that, from time to time, emits a particle and projects it along a linear scale. We consider two experiments,

*E*_{1} and *E*_{2} defined as follows: in *E*_{1}, we look to see if there is a particle present. If there is not, we record the outcome of *E*_{1} as the symbol *n*. If there is, we measure its position coordinate *x*. If *x*≥1, we record the outcome of *E*_{1} as the symbol *a*, while if *x*<1, we record the outcome of *E*_{1} as the symbol *b*. Thus, the set of outcomes of *E*_{1} is *O*_{1}={*n*,*a*,*b*}. In *E*_{2}, we look to see if there is a particle present. If there is not, we record the outcome of *E*_{2} as the symbol *n*. If there is, we measure the *x*-component *p*_{x} of its momentum, recording the symbol *c* as the outcome of *E*_{2} if *p*_{x}≥1 and the symbol *d* as the outcome of *E*_{2} if *p*_{x}<1. Thus, the set of outcomes for *E*_{2} is *O*_{2}={*n*,*c*,*d*}. (The reason for our identification of the outcome *n* of *E*_{1} with the outcome *n* of *E*_{2} should be clear to the reader.) [10], p. 1674

Note that there are 12 events in figure 1. In the Boolean algebra of events generated by *I*=*O*={*a*,*b*,*n*,*c*,*d*}, there are 32 events.

As a behavioural example that produces the same event space of propositions, suppose there are two committees, 1 and 2, with no member in common. These committees are to choose among candidates *x*, *y* and *z* to lead a company. Each committee's choice will be done by voting with each committee member having a single vote for her preferred candidate. Candidates *x* and *y* have similar visions for the company, while candidate *z* has a radically different vision. Using this section's terminology, let be the paradigm with its set subjects, , be the union of the two committees, its experiments *Y* _{i}, *i*=1,2, be having the members of committee *i* make ballot choices for *x*, *y* and *z*. These ballot choices are denoted by *x*_{i}, *y*_{i} and *z*_{i} to emphasize that *x*_{i} denotes the candidate *x* on the ballot for Committee *i*. Let the following be the paradigm's theoretical assumptions: for each *s* in , if *s* voted for *z*_{1}, that is, if *s* chose *z*_{1} in *Y* _{1}, then she would have voted for *z*_{2} if she were allowed to participate in Committee 2, and if she voted for *z*_{2}, then she would have voted for *z*_{1} if she were allowed to participate in Committee 1. Under these assumptions, produces an event space of propositions that is isomorphic to the lattice in figure 1.

## 5. Conclusion

Classical probability theory has dominated almost all applied research involving uncertainty and error, except for phenomena related to quantum mechanics. This article considered two of its generalizations, one involving a distributive algebra of events and a belief function, and the other involving an orthomodular lattice of events and an orthoprobability function. Both generalized theories provided richer sets of probabilistic concepts than the classical theory. There are scientific situations where the classical theory appears inadequate, e.g. when too few events are realizable or empirically observable to form a Boolean algebra of events, or when context influences observed probabilities. For some situations, the generalizations can be used to mitigate such inadequacies.

The material presented on a distributive algebra of events show that a belief function on a distributive algebra of events satisfies one of the main arguments for the rationality of classical probability: the Dutch Book Argument. This result holds because a theorem establishes that can be extended to a ⊂-monotonic probability function on ℘(*X*). The standard Dutch Book Theorem for a Boolean algebra of events then implies that the restriction of to is a rational assignment of probabilities, that is is rational. This is an example of how probability on can be viewed in two different but related ways: as a belief function that captures the probabilistic intrinsic structure inherent in , and as part of a ⊂-monotonic probability function on ℘(*X*) that allows and to be examined in terms of results from ⊂-monotonic probability theory.

Orthoprobability functions on an orthomodular event space yield a generalization of classical probability theory. Two kinds of event spaces for such functions were compared. One is based on closed subspaces of a Hilbert space and is almost universally used throughout quantum mechanics. The other is constructed out of considerations about relationships among the experiments of a psychological paradigm. It is controversial in the foundations of quantum mechanics whether its Hilbert space probability theory is extendable in rational or useful manners to a classical Boolean probability theory (see [11] for a discussion). This is not an issue for the psychological experimental approach of §4, because its orthoprobability function extends in a completely natural manner to a classical probability function.

## 6. Proof of theorem thm4.1

### Theorem 6.1

*The following five statements hold*:

(i)

*〈***P***,⊆〉 is a lattice, that is, is an algebraic structure such that*—

*and X∈***P***, and*—

*for all F and G in***P***the ⊆-least upper bound of F and G, F⊔G, and the ⊆-greatest lower bound of F and G, F⊓G, are in***P**.

(ii)

*is complemented, where − is the operation of set-theoretic complementation restricted***P***, that is, for all F in***P**,—

*−F is in***P**,—

*F⊔−F=X and*— .

(iii)

*is an ortholattice, that is, it satisfies DeMorgan's laws: for all F and G in***P**,(iv)

*restricted to***P***is an orthoprobability function on**, that is, for all F and G in***P**,—

*and*,—

*if F⊂G, then**and*—

*if G ⊆ −F, then*.

(v)

*The lattice**is orthomodular, that is, it satisfies the following condition for all F and G in***P***:*

Following [3], the proof of theorem 4.1 is divided into seven parts, stated here as lemmas. Throughout the lemmas, the notation 〈〈*A*〉〉 stands for the number of 's subjects *s* that would choose some outcome *a* in *A* if *s* were put into the unique experiment *Y* of having *a* as an outcome.

### Lemma 6.2

*X* *and* *are in* **P**.

### Proof.

Immediate from definition of **P**. ▪

### Lemma 6.3

*Suppose* *A*∈**P**. *Then* −*A*∈**P** *and*
6.1

### Proof.

It follows from the definitions of ‘proposition’ and ‘determinable’ that −*A* is determinable. It will be shown by contradiction that
6.2
Suppose *s*∈(〈〈*A*〉〉∩〈〈−*A*〉〉). Because *s*∈〈〈−*A*〉〉, let *a* in −*A* be such that *s* would choose *a*. Because *A* is determinable, each of its outcomes that would be chosen by each subject in 〈〈*A*〉〉 is in *A*. Therefore, because by hypothesis *s* is also in 〈〈*A*〉〉, it follows that *a* is in *A*, which is impossible since *a*∈−*A*.

It follows from the definition of ‘determinable’ that
6.3
It follows from equations (6.2) and (6.3) that for each proposition *A*, 〈〈*A*〉〉 and 〈〈−*A*〉〉 are set-theoretic complements with respect to of each other. Therefore,
6.4
Thus, in order to complete the proof of lemma 6.3, it needs to only be shown that −*A* is a proposition. This is done by contradiction. Suppose −*A* were not a proposition. Let *B* be a proposition such that −*A*⊂*B* and 〈〈−*A*〉〉=〈〈*B*〉〉. Then let *b* be an element of *B*−(−*A*)=*B*∩*A*, and, because *B* is a proposition, let *s*′ be an element of 〈〈*B*〉〉 such that *s*′ would choose *b* if put in the unique experiment of that *b* is an outcome. By hypothesis, 〈〈−*A*〉〉=〈〈*B*〉〉. Thus
6.5
Because *b*∈(*A*∩*B*), *b*∈*A*. Because *b*∈*A* and, by hypothesis, *s*′ would choose *b*, it follows that
6.6
Thus by equations (6.5) and (6.6), *s*′∈(〈〈−*A*〉〉∩〈〈*A*〉〉), contradicting equation (6.2).

The following notation is used in lemmas 6.4–6.7. For each determinable event *D*, let
and let
Note that for each proposition *E*,
6.7
▪

### Lemma 6.4

*and* *X*∈**P**, *and for all* *F* *and* *G* *in* **P** *the* ⊆-*least upper bound of* *F* *and* *G*, *F*⊔*G*, *exists and the* ⊆-*greatest lower bound of* *F* *and* *G*, *F*⊓*G*, *exists, and both are in* **P**.

### Proof.

*X* and are in **P** by lemma 6.2. It is immediate that they are, respectively, the ⊆-largest and the ⊆-smallest elements of 〈**P**,⊆〉. Let *A* and *B* be arbitrary elements of **P**. Then from the paradigm's theory and data it is known for each subject *s* whether or not *s* is in 〈〈*A*〉〉 and whether or not *s* is in 〈〈*B*〉〉. Thus it is known, for each subject *s*, (i) whether or not *s* is in 〈〈*A*〉〉∪〈〈*B*〉〉, and (ii) whether or not *s* is in 〈〈*A*〉〉∩〈〈*B*〉〉. Let
6.8
Then, by the definition of *σ*, *A*⊔*B* and *A*⊓*B* are in **P**. The following shows that *A*⊔*B* is the ⊆-least upper bound in **P** of *A* and *B*. Suppose *C* in **P** is such that
Then
and, therefore,
A similar argument shows that *A*⊓*B* is the ⊆-greatest lower bound in **P** of *A* and *B*. ▪

### Lemma 6.5

*The following two statements hold*:

(

*i*)*is a complemented lattice, where − is set-theoretic complementation*.(

*ii*)*Let**A**and**B**be arbitrary elements of***P***such that**B*⊆ −*A*.*Then*6.9

### Proof.

(i) By lemma 6.4, is a lattice. The following shows that −, the operation of set-theoretic complementation, is a complementation operation of . Using lemma 6.3, and

(ii) Because 〈〈*A*〉〉 ⊆ 〈〈*A*∪*B*〉〉 and 〈〈*B*〉〉 ⊆ 〈〈*A*∪*B*〉〉,
Thus, to show equation (6.9), it needs only be shown that
6.10
Assume *s* is an arbitrary subject in 〈〈*A*⊔*B*〉〉. Because − is the operation of set-theoretic complementation and *A*∈**P** and *s* must choose some element of *X* for each experiment in in which she were to participate, it follows that either *s*∈〈〈*A*〉〉 or *s*∈−〈〈*A*〉〉. If *s*∈〈〈*A*〉〉, then *s*∈(〈〈*A*〉〉∪〈〈*B*〉〉). If *s*∈−〈〈*A*〉〉, then *s* would not choose any element *a* of *A* if she were to participate in the unique experiment of **P** that had *a* as an outcome. Because *s*∈〈〈*A*∪*B*〉〉, *s* would then have chosen some element *b* of *B* if she were to participate in the unique experiment of **P** that *b* were an outcome, that is, *s*∈〈〈*B*〉〉, and thus *s*∈(〈〈*A*〉〉∪〈〈*B*〉〉). This shows equation (6.10). ▪

### Lemma 6.6

*is an ortholattice, where − is set-theoretic complementation.*

### Proof.

By lemma 6.5, only DeMorgan's laws needs to be shown. Let *A* and *B* be arbitrary propositions. Then, by lemma 6.4, *A*⊔*B* and *A*⊓*B* are in **P**, and, by lemma 6.3, −(*A*⊔*B*) and −(*A*⊓*B*) are in **P**.

By equation (6.1), −(〈〈*A*⊔*B*〉〉)=〈〈−(*A*⊔*B*)〉〉. Thus by equations (6.1) and (6.8),
6.11
and
6.12
By equations (6.12) and (6.11),
6.13
Similarly, by equation (6.1), −(〈〈*A*⊓*B*〉〉)=〈〈−(*A*⊓*B*)〉〉. Thus by equations (6.1) and (6.8),
6.14
and
6.15
By equations (6.15) and (6.14),
6.16
Equations (6.16) and (6.13) show De Morgan's laws. ▪

### Lemma 6.7

*is an orthoprobability function on* .

### Proof.

By lemma 6.6, is an ortholattice. Let *A* and *B* be arbitrary elements of **P** such that *B*⊆−*A*. Because 〈〈*B*〉〉⊆−〈〈*A*〉〉, it follows that . For each proposition *E*, let
Then, it follows from equation (6.9) and the definition of that
and thus that is an orthoprobability function on . ▪

### Lemma 6.8

*satisfies the orthomodular law. By lemma* 6.6, *is an ortholattice. Then, by Theorem 2* (*pp*. 22–23) *of Kalmbach* [12], *it follows that to show the orthomodular law one needs to only show that the following Statement H holds*:
*Statement H:* *there does not exist propositions* *A* *and* *B* *in* **P** *such that the following three conditions hold*:

(

*i*)(

*ii*)*A*⊔−*B*=*B*⊔−*A*=*X*.(

*iii*)

This is represented in figure 2. (In lattice theory Statement H says that there does not exist an *O*_{6} subalgebra of .) Suppose Statement H were false, a contradiction will be shown. Let *A* and *B* in be such that condition (i) holds. By lemma 6.7, is an orthoprobability function on . Then by Statement H,
6.17
and
6.18
and equations (6.17) and (6.18) contradict one another.

## Competing interests

I declare I have no competing interests.

## Funding

The research for this article was supported by grant no. FA9550-13-1-0012 from AFOSR and grant no. SMA-1416907 from NSF.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

- Accepted August 14, 2015.

- © 2015 The Author(s)