The Discrete-Continuous Logic and its possible quantum realizations

TL;DR

The paper introduces a novel discrete-continuous probabilistic logic (DCL) with matrix representations, enabling both classical and continuous logical operations, and explores its potential applications in quantum physics and cognitive sciences.

Contribution

It proposes a new generalized probabilistic logic (DCL) with matrix-based propositions, integrating discrete and continuous logical operations, and links classical propositions to quantum-like structures.

Findings

DCL represents propositions as 2x2 positive matrices with unit trace.

Logical operations include negation, disjunction, and rotations.

Classical propositions are special cases of DCL, derived from disjunctions of identical GP.

Abstract

We propose a new version of generalized probabilistic propositional logic, namely, discrete-continuous logic (DCL) in which every generalized proposition (GP) is represented as 2x2 nondiagonal positive matrix with unit trace. We demonstrate that on the set of propositions of this kind one can define both the discrete logical operations (connectives) such as negation and strong logical disjunction and in addition one parameter group of continuous operations (logical rotations). We prove that an arbitrary classical proposition (which in this logic is represented by the purely diagonal matrix) can be considered as the result of strong disjunction of two identical GP. This fact gives one a good reason to presume the DCL as a prime logical substructure underlying to ordinary propositional logic, which is recorded by our consciousness. We believe that proposed version of DCL will find many applications both in physics (quantum logic) and also in cognitive sciences (mental imagery) for better understanding of the pecular nature of mental brain operations.