An Application of Quantum Logic to Experimental Behavioral Science - Louis Narens (Quantum Reports, 2021)
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 <semantics>σ</semantics>-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 <semantics>σ</semantics>-additive function from the closed subspaces of a Hilbert space onto <semantics>[0,1]</semantics>. 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.