A fuzzy logic system based on Schweizer-Sklar t-norm

Abstract

Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL * is established, and its soundness theorem and completeness theorem are proved. The following facts are pointed out: the well-known formal system SBL~ is a semantic extension of UL * ; the fuzzy logic system IMTL Δ is a special case of UL * when two negations in UL * coincide. Moreover, the connections between the system UL * and some fuzzy logic formal systems are investigated. Finally, starting from the concepts of "the strength of an 'AND' operator" by R.R. Yager and "the strength of fuzzy rule interaction" by T. Whalen, the essential meaning of a parameter p in UL * is explained and the use of fuzzy logic system UL * in approximate reasoning is presented.In recent years, the basic research of fuzzy logic and fuzzy reasoning is growing actively day by day, such as the basic logic system BL proposed by Hajek [1] ; fuzzy logic system MTL proposed by Esteva and Godo [2] ; fuzzy reasoning, implication operators and fuzzy quantifiers were deeply investigated by Ying [3,4] . Moreover, Wang [5][6][7] combined fuzzy logic with fuzzy reasoning and built up the formal deductive system L * , created fuzzy reasoning triple I method which is superior to CRI method, and so on.In fuzzy logic, how to choose implication operators has direct influence on the result of the fuzzy reasoning. Because the t-norm reflects the property of the logic connective "AND" to a large extent, most of the implication operators chosen in the fuzzy logic systems are related to certain t-norms. However, in order to describe the flexibility in human's logical reasoning, choosing some parametric t-norms may be more reasonable. In fact, Klement and Navara [8] had studied the fuzzy logic system based on the parametric t-norm-Frank t-norm. Wu [9] and Wang [10] studied the parametric Kleene system and H α system respectively. Moreover, the residual implications induced by the Schweizer-Sklar

Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Schweizer–Sklar Muirhead Mean Aggregation Operators

“…To promote the classical triangular inequality, Karl Menger [12] proposed the concept of trigonometric function that is the prototype of t-norm and t-conorm. Schweizer and Skar [13] gave a detailed definition of t-norm and t-conorm, and SS t-norm and t-conorm (SSTT) are one of the forms of t-norm and t-conorm [14][15][16]. Schweizer-Sklaroperation is an instance of ATT, but SS operation contains an alterable parameter; therefore, they are more agile and superior, and can better reflect the property of "logical and" and "logical or", respectively.…”

MADM Based on Generalized Interval Neutrosophic Schweizer-Sklar Prioritized Aggregation Operators

“…Sh-Sk operations [39] are the particular cases from Atn and Atcn, and they are with a changeable parameter, so they are additionally supple and better than the former operations. Still, the majority of research concerning Sh-Sk mostly determined the elementary theory and types of Sh-Sk TN (Sh-Sktn) and TCN (Sh-Sktcn) [40,41]. Recently, Liu et al [42] and Zhang [43] merged Sh-Sk operations with interval-valued IFS (IVIFS) and IFS and anticipated power WA/geometric AGOs and weighted power WA/geometric AGOs for IVIFSs and IFSs, respectively.…”

First-order logic system $\mathbf{IMTL_{Q}^{*}}$ and triple I method in fuzzy reasoning with linguistic quantifiers

“…As a theoretical basis for approximate reasoning, tnorm based fuzzy logics are developed in the recent years (see [4,5,9,13,19,20]). For t-norm based first-order fuzzy logic, many interesting researches are published, for example, the first-order system K * is constructed and its complete theorem is proved in [8], and the present state of predicate fuzzy logics is surveyed in [1].…”