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**1 - 4**of**4**### Fuzzy logics as the logics of chains

"... The paper proposes a formal delimitation of the class of ‘fuzzy logics ’ and answers some objections that can be raised against the definition; the focus is put on informal and motivational aspects of the problem. Key words: Non-classical logics, formal fuzzy logic, weakly implicative fuzzy logics 1 ..."

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The paper proposes a formal delimitation of the class of ‘fuzzy logics ’ and answers some objections that can be raised against the definition; the focus is put on informal and motivational aspects of the problem. Key words: Non-classical logics, formal fuzzy logic, weakly implicative fuzzy logics 1991 MSC: 03B52, 03A05, 03B22 1

### Editors

"... About this Series The series “Studies in Fuzziness and Soft Computing ” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fiel ..."

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About this Series The series “Studies in Fuzziness and Soft Computing ” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing ” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results.

### Complex Fuzzy Sets and Complex Fuzzy Logic an Overview of Theory and Applications

"... Abstract Fuzzy Logic, introduced by Zadeh along with his introduction of fuzzy sets, is a continuous multi-valued logic system. Hence, it is a generalization of the classical logic and the classical discrete multi-valued logic (e.g. Łukasiewicz’ three/many-valued logic). Throughout the years Zadeh a ..."

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Abstract Fuzzy Logic, introduced by Zadeh along with his introduction of fuzzy sets, is a continuous multi-valued logic system. Hence, it is a generalization of the classical logic and the classical discrete multi-valued logic (e.g. Łukasiewicz’ three/many-valued logic). Throughout the years Zadeh and other researches have introduced extensions to the theory of fuzzy setts and fuzzy logic. Notable exten-sions include linguistic variables, type-2 fuzzy sets, complex fuzzy numbers, and Z-numbers. Another important extension to the theory, namely the concepts of complex fuzzy logic and complex fuzzy sets, has been investigated by Kandel et al. This extension provides the basis for control and inference systems relating to complex phenomena that cannot be readily formalized via type-1 or type-2 fuzzy sets. Hence, in recent years, several researchers have used the new formalism, often in the context of hybrid neuro-fuzzy systems, to develop advanced complex fuzzy logic-based inference applications. In this chapter we reintroduce the concept of complex fuzzy sets and complex fuzzy logic and survey the current state of com-plex fuzzy logic, complex fuzzy sets theory, and related applications.

### Fuzzy Class Theory

"... The paper introduces a simple, yet powerful axiomatization of Zadeh’s notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order th ..."

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The paper introduces a simple, yet powerful axiomatization of Zadeh’s notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order theory over fuzzy logic, which is generalized to simple type theory. We show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism. In this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed; however, we also demonstrate how to add structure to the universe of discourse within our framework. Key words: Formal fuzzy logic, fuzzy set, foundations of fuzzy mathematics, LΠ logic, higher-order fuzzy logic, fuzzy type theory, multi-sorted fuzzy logic 1991 MSC: 03B52, 03E70, 03E72 1