Fuzzy Reasoning Based on Truth-Value Progression: A Control-Theoretic Design Approach

Fuzzy logic, especially fuzzy reasoning, has been widely used in many real-world applications for its tremendous practical value. This paper mainly focuses on a new theoretical principle of fuzzy logic, fuzzy reasoning, and its applications. It starts with enunciating a novel foundation: the Fundamental Axiom of Reasoning. We then propose a practical framework, which enables the Fundamental Axiom of Reasoning to be applied to various applications. This framework is divided into discrete case and continuous case. In the discrete case, we first solve the sorites paradox. We then proceed to a pattern recognition problem. So far, these are all passive approach in that the Fundamental Axiom of Reasoning is used as is to measure if the reasoning is valid. In the continuous case, we proceed to actively designing the reasoning so that the Fundamental Axiom of Reasoning is always guaranteed through Lyapunov stability approaches. This is formulated as a control-theoretic task. The problem of reasoning is converted to a control design one. This is then applied to a vehicle following problem. Through theoretical analysis and practical demonstrations, we show how the Fundamental Axiom of Reasoning can be conceptualized, articulated, and formalized in fuzzy reasoning.