Academic literature on the topic 'Fuzzy logic. Fuzzy logic Fuzzy mathematics. Logic, Symbolic and mathematical'

What is the nature of symbols? This word is used for traffic signs, for mathematical notations, and motivationally loaded cultural objects, which may inspire war and piece. This chapter explains relationships among symbols, cognition, and language. Symbols are explained as processes in the mind involving cognition and language. Relationships between cognition and language were a mystery until recently. Linguists often considered language as relationships among words and other linguistic entities, separately from its relationships to the world. Mechanisms of language in the mind and brain were considered separate and different from thinking and cognition. Neural mechanisms integrating language and cognition are unknown. Yet, language and cognition are intertwined in evolution, ontogenesis, learning, and in everyday usage, therefore a unified understanding of working of the mind is essential. A mathematical description of such unifying mechanisms is the subject of this paper. We discuss relationships among computational intelligence, known mechanisms of the mind, semiotics, computational linguistics, and describe a process integrating language and cognition. Mathematical mechanisms of concepts, emotions, and instincts are described as a part of information processing in the mind and related to perception and cognition processes in which an event is understood as a concept. Development of such mathematical theories in the past often encountered difficulties of fundamental nature manifested as combinatorial complexity. Here, combinatorial complexity is related to logic underlying algorithms and a new type of logic is introduced, dynamic fuzzy logic, which overcomes past limitations. This new type of logic is related to emotional signals in the brain and combines mechanisms of emotions and concepts. The mathematical mechanism of dynamic logic is applicable to both language and cognition, unifying these two abilities and playing an important role in language acquisition as well as cognitive ontogenesis. The mathematical description of thought processes is related to semiotic notions of signs and symbols.

“Incorporate nuance, because of fuzzy logic” offers a basic introduction to fuzzy logic, a multi-valued logical system that accommodates a range of absolute and partial truths. Fuzzy logic differs from traditional logic in which statements are considered either absolutely true or absolutely false. Fuzzy logic offers a method for programming computers with intelligent instructions that emulate human thought and decision making. The discussion is supplemented with numerous hand-drawn sketches and explanations of real-life applications of fuzzy logic in electric trains, washing machines, digital cameras, rice cookers, facial recognition software, drones, and medical devices. Mathematics students and enthusiasts are encouraged to fuzzify their mathematical and life pursuits involving uncertain circumstances in which an absolute “yes” or an absolute “no” may not be the best decision. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

This paper establishes close relationships between fundamental problems in the philosophical and mathematical theories of mind. It reviews the mathematical concepts of intelligence, including pattern recognition algorithms, neural networks and rule systems. Mathematical difficulties manifest as combinatorial complexity of algorithms are related to the roles of a priori knowledge and adaptive learning, the same issues that have shaped the two-thousand year old debate on the origins of the universal concepts of mind. Combining philosophical and mathematical analyses enables tracing current mathematical difficulties to the contradiction between Aristotelian logic and Aristotelian theory of mind (Forms). Aristotelian logic is shown to be the culprit for the current mathematical difficulties. I will also discuss connections to Gödel’s theorems. The conclusion is that fuzzy logic is a fundamental requirement for combining adaptivity and apriority. Relating the mathematical and philosophical helps clarifying both and helps analyzing future research directions of the mathematics of intelligence.

Abstract Interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters. The basic mathematical operations are defined in the interval algebra. Neural networks is an approach leading to engineering expert systems that are capable of learning, self adapting to particular engineering applications and handling fuzzy and interval input information. In traditional machine learning, symbolic representations, such as first order predicate calculus, are used to represent knowledge. The resulting algorithms are specific to the selected representation and presum