Logic, Geometry And Probability Theory

Abstract

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.

Generalized Probabilities in Statistical Theories

“…As we have seen in previous sections, there are two versions of CP, namely, the approach of R. T. Cox [24,25] and the one of A. N. Kolmogorov [3]. The Kolmogorovian approach can be generalized in order to include non-Boolean models, as we have shown in Section 4. In what follows, we will see that Cox's method can also be generalized to non-distributive lattices, and thus the non-commutative character of QP can be captured in this framework [19,27].…”

“…This was recognized very quickly as a nonclassical feature. These peculiarities and the formal aspects of the probabilities involved in quantum theory have been extensively studied in the literature [12][13][14][15][16][17][18][19]. We refer to the probabilities related to quantum phenomena as quantum probabilities (QP).…”

Quantum Logic and Geometric Quantization

“…What we need to complete the description of the quantum picture is a notion of probability on L (H). An answer to this problem was given by [15] that introduced a probability function p : L (H) → [0, 1]. The function p is σ-additive and can be understood in the sense of [16] with (H, L (H) , p) as probability space.…”

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

“…Interestingly enough, there is also a connection between the faces of the convex set of states of a given model and its lattice of properties (in the quantum-logical sense), providing an unexpected connection between geometry, lattice theory and statistical theories [11,23,103]. F is a face if for all x satisfying:…”