Algebraic hierarchy of logics unifying fuzzy logic and quantum logic

TL;DR

This paper introduces a unified algebraic framework for various logics, including fuzzy and quantum logics, by defining a rigorous concept of fuzzy negation and exploring their hierarchical relationships.

Contribution

It provides a novel algebraic definition of fuzzy negation and unifies multiple logic systems within a hierarchical lattice framework.

Findings

Fuzzy negation satisfying specific algebraic conditions is rigorously defined.

Most general logics include fuzzy, quantum, intuitionistic, and Boolean logics as special cases.

New examples of non-contradictory and non-orthomodular fuzzy logics are presented.

Abstract

In this paper, a short survey about the concepts underlying general logics is given. In particular, a novel rigorous definition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented. According to this definition, a fuzzy negation satisfies the weak double negation condition, requiring double negation to be expansive, the antitony condition, being equivalent to the disjunctive De Morgan law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other. From this perspective, the most general logics are fuzzy logics, containing as special cases paraconsistent (quantum) logics, quantum logics, intuitionistic logics, and Boolean logics, each of which given by its own algebraic restrictions. New examples of a non-contradictory logic violating the conjunctive De Morgan law, and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given.