Research On Lattice Implication Algebra And Uncertain Reasoning Based On Linguistic Truth Value

In objectivity and the process of cognizing objectivity by human, there exist many kinds of information with uncertainty. Hence, human always have to deal with various uncertain information. In dealing with uncertain information by human, uncertain information is expressed or processed by natural language in many cases. So, theory and methods of uncertain information processing based on natural language need to be studied. Hence, non-classical logical algebra, non-classical logic system and corresponding with linguistic truth value logical algebra and algebraic logic system of truth which is expressed by natural language need to be studied, all of these are very important meaningful for intelligent information processing. Based on these researching background and academic ideas, lattice implication algebra and uncertain inference of linguistic truth value are studied in the Ph.D thesis. Our main study works are focused on as following:1. Obtaining several equivalence definitions of lattice implication algebra and constructing method of lattice implication algebra on a non-empty set. Constructing methods of implication operators "→", order-reversing involutions "′" and concrete forms of lattice implication algebra are given on chain and Cartesian product of two finite chains, meanwhile, it is proved that infinite numbers of lattice implicationalgebra on [0,1] could be constructed. According to equivalence definition of latticeimplication algebra, some special properties of lattice implication algebra are discussed. Operations "∧" and "∨" of implication operators "→" of lattice implication algebra are defined. It is proved that operations "∧" and "∨" are closed.2. Consistency of component and wider than on modular lattice is proved. It is obtained that sufficient and necessary condition of deciding component and concrete forms of component in lattice implication algebra. It is proved that finite "∧" and finite "∨" are closed in lattice implication algebra.3. The properties of implication algebra on partial ordered set and associated implication algebra are deeper discussed, several sufficient and necessary conditions, by which implication algebra becomes associated implication algebra, are obtained. Filter and generating filter of implication algebra on partial ordered set are defined, and structure of generating filter is obtained. The sufficient and necessary conditions, by which residuated lattice becomes lattice implication algebra and R₀ - algebra, areobtained, respectively. A simply method to decide whether or not lattice implication algebra becomes R₀ - algebra is proved. Moreover, the properties and includedrelationship among MTL—algebra,IMTL-algebra,BL-algebra,R₀—algebra and lattice implication algebra are discussed.4. Based on hedge operator and basic linguistic, Cartesian product of hedge operator and basic linguistic which are always used in natural language are obtained by hedge algebra. Based on the linguistics properties of hedge, lattice implication algebra of linguistic truth value is obtained, from logic inference point of view, propositional calculus with linguistic truth value are discussed, especially, linguistic truth value three-I inference and its properties are studied.