Analogue of the quantum total probability rule from Paraconsistent bayesian probability theory

TL;DR

This paper develops a paraconsistent Bayesian probability theory that generalizes classical probability and derives a total probability rule resembling quantum mechanics, especially in the context of SIC-POVM measurements.

Contribution

It introduces a novel probability framework based on paraconsistent logic, leading to a quantum-like total probability rule derived from logical and probabilistic principles.

Findings

Derives a total probability rule analogous to quantum mechanics.

Shows that certain probability assignments reproduce quantum total probability expressions.

Provides a logical foundation for quantum probability within a paraconsistent logic framework.

Abstract

We derive an analogue of the quantum total probability rule by constructing a probability theory based on paraconsistent logic. Bayesian probability theory is constructed upon classical logic and a desiderata, that is, a set of desired properties that the theory must obey. We construct a new probability theory following the desiderata of Bayesian probability theory but replacing the classical logic by paraconsistent logic. This class of logic has been conceived to handle eventual inconsistencies or contradictions among logical propositions without leading to the trivialisation of the theory. Within this Paraconsistent bayesian probability theory it is possible to deduce a new total probability rule which depends on the probabilities assigned to the inconsistencies. Certain assignments of values for these probabilities lead to expressions identical to those of Quantum mechanics, in particular to the quantum total probability rule obtained via symmetric informationally complete positive- operator valued measure.