Ordered structure-based semantics of linguistic terms of linguistic variables and approximate reasoning

Abstract

The paper will present an overview of an algebraic approach to approximate reasoning problems. It is shown that there exists a natural semantic ordering relation on domains of linguistic variables, and this relation makes each linguistic domain a complete distributive lattice. For linguistic variables having a unique positive primary term and negative one, their domains have sufficiently rich algebraic-logic properties for investigating fuzzy logic and approximate reasoning. The algebraic structures which model linguistic domains are called hedge algebras, because their axioms formulate directly semantics of linguistic hedges. As an example of the application of this theory, we introduce a method in linguistic reasoning, which allows us to handle directly linguistic terms. We consider this approach as having qualitative characteristics. Since, quantitative characteristics also play an important role in approximate reasoning methods, we introduce a mapping which transforms each linguistic domain into a real space. In connection with this we define notions of measure function, fuzziness measure and fuzziness degree of hedges. Then we can apply interpolation methods to solve multiple conditional fuzzy reasoning problems in a natural way, and they give more accurate results than that given by some fuzzy set-based methods. This approach has been developed in two stages. Firstly, hedge algebras were defined and investigated. Secondly, they were extended to the so-called refined hedge algebras. In this paper we shall give a presentation unifying these theories and obtain a common theory called the theory of hedge algebras, for simplicity.