Linguistic Truth-Valued Lattice Implication Algebra and Its Properties

Abstract

The subject of this work is to establish a mathematical framework that provides the basis and tool for automated reasoning and uncertainty reasoning based on linguistic information. This paper focuses on a flexible and realistic approach, i.e., the use of linguistic terms, specially, the symbolic approach acts by direct computation on linguistic terms. An algebra model with linguistic terms, which is based on a logical algebraic structure, i.e., lattice implication algebra, is constructed and applied to represent imprecise information and deal with both comparable and incomparable linguistic terms (i.e., non-ordered linguistic terms). Some properties and its substructures of this algebraic model are discussed.

α-Generalized Semantic Resolution Method in Linguistic Truth-valued Propositional Logic P(X)

“…described in natural language, usually, in a quantitative setting the information is expressed by means of numerical values, However, when we work in a qualitative setting, that is, with vague or imprecise knowledge, this cannot be estimated with an exact numerical value, Then, it may be more realistic to use linguistic truth values instead of numerical values 20,21,22,23 . In 2006, based on symbolic approaches, Xu 24 proposed linguistic truthvalued lattice implication algebra, this algebraic model can be applied for linguistic truth-valued automated reasoning and uncertainty reasoning. In 2007, Xu 25 proved the weak completeness of resolution in a linguistic truth-valued propositional logic which truth-value field is linguistic truth-valued lattice implication algebra.…”

“…20,24 Let AD n = {a 1 , a 2 , · · · , a n } be a set with n modifiers and a 1 < a 2 < · · · < a n , MT = { f ,t} be a set of meta truth values and…”

“…20,24 Denote MT ={ f ,t}, which is called as the set of meta truth values. The lattice implication algebra of defined on the set of meta truth values is called a meta linguistic truth-valued lattice implication algebra, where f < t, the operation is defined as f = t and t = f , the operation → is defined as: →:…”

Ordering based decision making – A survey

“…They [15,113] also extended the possibilistic logic by defining new combination rules to aggregate multiple-source information, which provides a coherent way to represent and reason uncertain information from different sources. Inspired from hedge algebra and by analyzing semantic heredity of linguistic hedges, based on the extensive work on lattice implication algebras and the corresponding logic systems [114], Xu, et al proposed linguistic truth-valued lattice implication algebra [115,116], as simply shown in Fig. 3, for modelling ordinal linguistic information, and discussed the corresponding logic system [116,117] and the approximate reasoning approaches based on it [118,119].…”

Determination of alpha-resolution for lattice-valued first-order logic based on lattice implication algebra

“…In [25], some opinions on fuzzy logic from the view of machine intelligence are presented, characteristics of fuzzy logic that adapted to investigate machine intelligence are analyzed, and the following opinions are proposed: (1) machine intelligence requires uncertain reasoning with linguistic expressions; (2) study uncertain reasoning based on logic is one of scientific methodologies; (3) machine intelligence demands uncertain reasoning with linguistic information under the guidance of logic; (4) machine intelligence also requires automated reasoning with linguistic information based on logic. In order to establishing algebraic structure of linguistic truth values, linguistic truth-valued lattice implication algebra [9] is established and its properties are discussed. In [11], weak completeness of α -resolution in a linguistic truth-valued propositional logic based on lattice implication algebra are discussed.…”