The probability theory used in quantum mechanics is currently being employed by psychologists to model the impact of context on decision. Its event space consists of closed subspaces of a Hilbert space, and its probability function sometimes violate the law of the finite additivity of probabilities. Results from the literature indicate that such a “Hilbert space probability theory” cannot be extended to standard, finitely additive, probability theory by the addition of new events with specific probabilities. This chapter presents a new kind of probability theory that shares many fundamental algebraic characteristics with Hilbert space probability theory but does extend to standard probability theory by adjoining new events with specific probabilities. The new probability theory arises from considerations about how psychological experiments are related through counterfactual reasoning.
The probability theory used in quantum mechanics is currently being employed by psychologists to model the impact of context on decision. Its event space consists of closed subspaces of a Hilbert space, and its probability function sometimes violate the law of the finite additivity of probabilities. Results from the literature indicate that such a “Hilbert space probability theory” cannot be extended to standard, finitely additive, probability theory by the addition of new events with specific probabilities. This chapter presents a new kind of probability theory that shares many fundamental algebraic characteristics with Hilbert space probability theory but does extend to standard probability theory by adjoining new events with specific probabilities. The new probability theory arises from considerations about how psychological experiments are related through counterfactual reasoning.
Various proposals for generalizing event spaces for probability functions have been put forth in the mathematical, scientific, and philosophic literatures. In cognitive psychology such generalizations are used for explaining puzzling results in decision theory and for modeling the influence of context effects. This commentary discusses proposals for generalizing probability theory to event spaces that are not necessarily boolean algebras. Two prominent examples are quantum probability theory, which is based on the set of closed subspaces of a Hilbert space, and topological probability theory, which is based on the set of open sets of a topology. Both have been applied to a variety of cognitive situations. This commentary focuses on how event space properties can influence probability concepts and impact cognitive modeling.
In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.
Abstract In this paper, a modified formulation of generalised probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, is presented and the guidelines to how to extend this work to infinite dimensional Hilbert spaces are given. Moreover, this new formulation which will be called as extended operational‐probabilistic theories, applies not only to quantum systems, but also equally well to classical systems, without violating Bell's theorem, and at the same time solves the measurement problem. A new answer to the question of why our universe is quantum mechanical rather than classical will be presented. Besides, this extended probability theory shows that it is non‐determinacy, or to be more precise, the non‐deterministic description of the universe, that makes the laws of physics the way they are. In addition, this paper shows that there is still a possibility that there might be a deterministic level from which our universe emerges, which if understood correctly, may open the door wide to applications in areas such as quantum computing. In addition, this paper explains the deep reason why complex Hilbert spaces in quantum mechanics are needed.
The purpose of this article is to discuss principle ideas of quantum cognition research program, which comprise elements of the formalism of quantum mechanics (mainly Hilbert space theory and quantum probability theory) for modeling human cognition and decision processes. In the opinion of authors of this program, paradox empirical findings in psychological literature may be explained based on concepts of quantum mechanics. Formally, there is described a discrete-time random chain χ which is defined on a finite interval [0, T] and χ(t) can assume only finite number of values. The space H of such processes will be finite-dimensioned. Then some properties and applications of the quantum probability space on H are studied.
Quantum cognition, a growing topic at the interface of quantum physics and cognitive research, provides a new viewpoint on the intricacies of human thought processes. In contrast to classic Boolean algebraic models, which have long dominated cognitive psychology, quantum cognition proposes a new framework influenced by quantum mechanics' algebraic structure. This deviation derives from the understanding that natural notions, based on prototypes, are more closely related to convex sets than to the rigid rules of Boolean algebra. Drawing on the fundamental work of luminaries such as John von Neumann, quantum cognition reveals a unique probability theory based on ortho-algebras, challenging traditional probabilistic reasoning based on Boolean algebras. This change opens up solutions to complex challenges in rationality, logical thinking, and probabilistic reasoning, ushering in a new era of cognitive modelling. Several arguments in the literature support the importance of quantum cognition in explaining cognitive phenomena.
Based on a review of the achievements of quantum mechanics and the psychology of perception, some of their possibilities for expanding the classical theory of probability to the field of representing psychological events, for which a combination of outcomes may have a probability greater than the probability of single events from their common set, are considered. It is noted that a model based on a common space of elementary non-intersecting elements is insufficient for describing the psychology of behavior associated with generative processes and co-represented phenomena. To extend elementary probability theory, results from quantum theory relating to ideas about test spaces and the possibility of combining individual events in a test can be used. The proposed models can include both systems of individual tests with repetition of events in additional tests and the possibility of combining individual test events themselves using combinatorial and geometric (projection) methods. Simple examples of expanding classical probabilistic models show that the "Conjunction Fallacy" in the psychology of heuristic behavior should be considered not so much an error of subjects as a scientific illusion of researchers, when they try to fit the behavior of a more complex system into the Procrustean bed of an overly simple model.