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

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 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 $L_{z}$ observable of the orbital angular momentum $\ell =1$. The representative 3-valued 2-argument logical observables for the $\mathrm {Min}$ and $\mathrm {Max}$ connectives are explicitly obtained.