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23 pages, 1382 KiB

Open AccessArticle

Three-Way Fuzzy Sets and Their Applications (III)

by Qingqing Hu and Xiaohong Zhang

Axioms 2023, 12(1), 57; https://doi.org/10.3390/axioms12010057 - 3 Jan 2023

Cited by 3 | Viewed by 2268

Three-way fuzzy inference is the theoretical basis of three-way fuzzy control. The proposed TCRI method is based on a Mamdani three-way fuzzy implication operator and uses one inference and simple composition operation. In order to effectively improve the TCRI method, this paper proposes a full implication triple I algorithm for three-way fuzzy inference and gives the triple I solution to the TFMP problem. The emphasis of our research is

R_{0}

and G

\ddot{o}

del triple I solution, which is related to three-way residual implication, as well as Zadeh’s and Mamdani’s triple I solution, which is based on three-way fuzzy implication operator. Then the three-way fuzzy controller is constructed by the proposed Zadeh’s and

R_{0}

triple I algorithm. Finally, the proposed triple I algorithm is applied to the three-way fuzzy control system, and its advantage is illustrated by the three-dimensional surface diagram of the control variable. Full article

(This article belongs to the Special Issue Soft Computing with Applications to Decision Making and Data Mining)

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9 pages, 298 KiB

Open AccessArticle

Third Zadeh’s Intuitionistic Fuzzy Implication

by Krassimir Atanassov

Mathematics 2021, 9(6), 619; https://doi.org/10.3390/math9060619 - 15 Mar 2021

Cited by 7 | Viewed by 3032

Abstract

George Klir and Bo Yuan named after Lotfi Zadeh the implication

p \to q = max (1 - p, min (p, q))

(also Early Zadeh implication). In a series of papers, the author introduced two intuitionistic [...] Read more.

George Klir and Bo Yuan named after Lotfi Zadeh the implication

p \to q = max (1 - p, min (p, q))

(also Early Zadeh implication). In a series of papers, the author introduced two intuitionistic fuzzy forms of Zadeh’s implication and studied their basic properties. In the present paper, a new (third) intuitionistic fuzzy form of Zadeh’s implication is proposed and some of its properties are studied. Full article

(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)

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13 pages, 572 KiB

Open AccessArticle

A Fuzzy Take on the Logical Issues of Statistical Hypothesis Testing

by Matthew Booth and Fabien Paillusson

Philosophies 2021, 6(1), 21; https://doi.org/10.3390/philosophies6010021 - 15 Mar 2021

Cited by 1 | Viewed by 2736

Abstract

Statistical Hypothesis Testing (SHT) is a class of inference methods whereby one makes use of empirical data to test a hypothesis and often emit a judgment about whether to reject it or not. In this paper, we focus on the logical aspect of this strategy, which is largely independent of the adopted school of thought, at least within the various frequentist approaches. We identify SHT as taking the form of an unsound argument from Modus Tollens in classical logic, and, in order to rescue SHT from this difficulty, we propose that it can instead be grounded in t-norm based fuzzy logics. We reformulate the frequentists’ SHT logic by making use of a fuzzy extension of Modus Tollens to develop a model of truth valuation for its premises. Importantly, we show that it is possible to preserve the soundness of Modus Tollens by exploring the various conventions involved with constructing fuzzy negations and fuzzy implications (namely, the S and R conventions). We find that under the S convention, it is possible to conduct the Modus Tollens inference argument using Zadeh’s compositional extension and any possible t-norm. Under the R convention we find that this is not necessarily the case, but that by mixing R-implication with S-negation we can salvage the product t-norm, for example. In conclusion, we have shown that fuzzy logic is a legitimate framework to discuss and address the difficulties plaguing frequentist interpretations of SHT. Full article

(This article belongs to the Special Issue Logic and Science)

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18 pages, 767 KiB

Open AccessArticle

On Extended Representable Uninorms and Their Extended Fuzzy Implications (Coimplications)

by Aifang Xie

Symmetry 2017, 9(8), 160; https://doi.org/10.3390/sym9080160 - 18 Aug 2017

Cited by 1 | Viewed by 3689

Abstract

In this work, by Zadeh’s extension principle, we extend representable uninorms and their fuzzy implications (coimplications) to type-2 fuzzy sets. Emphatically, we investigate in which algebras of fuzzy truth values the extended operations are type-2 uninorms and type-2 fuzzy implications (coimplications), respectively. Full article

(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)

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