Quantum theory as a relevant framework for the statement of probabilistic and many-valued logic

TL;DR

This paper introduces a quantum-inspired framework for probabilistic and many-valued logic, representing logical propositions as quantum states and connectives as transformations, enabling unified analysis across disciplines.

Contribution

It proposes a novel quantum-based approach to plausible logic, generalizing classical logical systems and applying quantum engineering methods for transformations.

Findings

Logical connectives as positive transformations of likelihood matrices

Reproduction and generalization of classical logical systems

Application to psychophysics and social sciences

Abstract

Based on ideas of quantum theory of open systems we propose the consistent approach to the formulation of logic of plausible propositions. To this end we associate with every plausible proposition diagonal matrix of its likelihood and examine it as density matrix of relevant quantum system. We are showing that all logical connectives between plausible propositions can be represented as special positive valued transformations of these matrices. We demonstrate also the above transformations can be realized in relevant composite quantum systems by quantum engineering methods. The approach proposed allows one not only to reproduce and generalize results of well-known logical systems (Boolean, Lukasiewicz and so on) but also to classify and analyze from unified point of view various actual problems in psychophysics and social sciences.