Lattice-Valued Logic: An Alternative Approach to Treat Fuzziness and Incomparability

I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.