Modeling with Words: Steps Towards a Fuzzy Quantum Logic

Abstract

On 1936, Birkhoff and von Newmann proposed the introduction of a “quantum logic”, as the lattice of quantum mechanical proposition which is not distributive and also not a Boolean. Seven years later, Mackey tried to provide a set of axioms for the propositional system to predict of the outcome set of experiments. He indicated that the system is an orthocomplemented partially ordered set. Physical complex systems can be modeled by using linguistic variables which are variables whose values may be expressed in terms of a specific natural or artificial language, for example $\mathbb {L}$= {very less young; less young; young; more young; very young; very very young ...}. In language of hedge algebra ($\mathbb{H}\mathbb{A}$), $\mathbb {L}$ set which is generated from $\mathbb{H}\mathbb{A}$ is the POSET (partial order set). In this paper, we introduce a quantum logic $\ell$ to assert that, let $\bot$ be the orthocomplementation map $\bot : \ell \rightarrow \ell$, all $\clubsuit , \spadesuit \in \bot$ must satisfy the following conditions:

• $(\clubsuit ^\bot )^\bot =\clubsuit$

• If $\spadesuit \le \clubsuit$ then $\clubsuit ^\bot \le \spadesuit ^\bot$

• The greatest lower bound $\clubsuit \vee \clubsuit ^\bot \in \ell$ and the least upper bound $\clubsuit \wedge \clubsuit ^\bot \in \ell$