Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems

TL;DR

This paper introduces a quantum-inspired matrix model for multi-valued and fuzzy logic systems, utilizing Hilbert space observables to unify logical and quantum concepts, with applications to quantum cognition.

Contribution

It develops a novel quantum-based framework for multi-valued and fuzzy logic, connecting logical observables with quantum angular momentum and providing explicit models for three-valued logic.

Findings

Logical observables correspond to quantum operators in Hilbert space.

Fuzzy membership functions are derived from quantum mean values.

Explicit models for three-valued logic connectives are constructed.

Abstract

We propose a matrix model for two- and many-valued logic using families of observables in Hilbert space, the eigenvalues give the truth values of logical propositions where the atomic input proposition cases are represented by the respective eigenvectors. For binary logic using the truth values {0,1} logical observables are pairwise commuting projectors. For the truth values {+1,-1} the operator system is formally equivalent to that of a composite spin 1/2 system, the logical observables being isometries belonging to the Pauli group. Also in this approach fuzzy logic arises naturally when considering non-eigenvectors. The fuzzy membership function is obtained by the quantum mean value of the logical projector observable and turns out to be a probability measure in agreement with recent quantum cognition models. The analogy of many-valued logic with quantum angular momentum is then established. Logical observables for three-value logic are formulated as functions of the Lz observable of the orbital angular momentum l=1. The representative 3-valued 2-argument logical observables for the Min and Max connectives are explicitly obtained.