Metamathematics of Fuzzy Logic

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Metamathematics of Fuzzy Logic

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Metamathematics of Fuzzy Logic

31 Aug 1998-

TL;DR: This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.

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Abstract: Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks. References. Index.

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Journal Article•DOI•

Is there a need for fuzzy logic

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Lotfi A. Zadeh¹•Institutions (1)

University of California, Berkeley¹

01 Jul 2008-Information Sciences

TL;DR: In this paper, fuzzy logic is viewed in a nonstandard perspective and the cornerstones of fuzzy logic-and its principal distinguishing features-are: graduation, granulation, precisiation and the concept of a generalized constraint.

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1,253 citations

Cites background from "Metamathematics of Fuzzy Logic"

...The agenda of FLl is similar in spirit to the agenda of classical logic [24,27,56,47,19,49]....
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...In a narrow sense, fuzzy logic is a logical system which is a generalization of multivalued logic [27]....
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Journal Article•DOI•

Monoidal t-norm based logic : towards a logic for left-continuous t-norms

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Francesc Esteva¹, Lluís Godo¹•Institutions (1)

Intel¹

16 Dec 2001-Fuzzy Sets and Systems

TL;DR: This paper investigates a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua, and completeness of MTL with respect to linearly ordered MTL-algebras is proved.

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900 citations

Book Chapter•DOI•

Aggregation operators: properties, classes and construction methods

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Tomasa Calvo¹, Anna Kolesárová, Magda Komorníková, Radko Mesiar•Institutions (1)

University of the Balearic Islands¹

01 Jan 2002

TL;DR: In this article, the authors restrict their considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞].

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Abstract: Aggregation (fusion) of several input values into a single output value is an indispensable tool not only of mathematics or physics, but of majority of engineering, economical, social and other sciences. The problems of aggregation are very broad and heterogeneous, in general. Therefore we restrict ourselves in this contribution to the specific topic of the aggregation of finite number of real inputs only. Closely related topics of aggregating infinitely many real inputs [23,109,64,52,43,42,44,99], of aggregating inputs from some ordinal scales [41,50], of aggregating complex inputs (such as probability distributions [107,114], fuzzy sets [143]), etc., are treated, among others, in the quoted papers, and we will not deal with them. In this spirit, if the number of input values is fixed, say n, an aggregation operator is a real function of n variables. This is still a too general topic. Therefore we restrict our considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞]. It is a matter of rescaling to fix I = [0,1].

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599 citations

Journal Article•DOI•

Fuzzy transforms: Theory and applications

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Irina Perfilieva¹•Institutions (1)

University of Ostrava¹

01 Apr 2006-Fuzzy Sets and Systems

TL;DR: A method of image compression and reconstruction on the basis of the F-transform, which is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granules etc.), is presented.

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548 citations

Cites background from "Metamathematics of Fuzzy Logic"

...An example of a uniform fuzzy partition of [1, 4] by sinusoidal membership functions....
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...(1) Our first assumption reduces a residuated lattice on [0, 1] with which we worked before, to a more specific structure known as a BL-algebra [4], by adding the identity: x ∗ (x → y) = x ∧ y....
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...An example of a fuzzy partition of [1, 4] by triangular membership functions....
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...(1) Our first assumption reduces a residuated lattice on [0, 1] with which we worked before, to a more specific structure known as a BL-algebra [4], by adding the identity: x ∗ (x → y) = x ∧ y....
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...Note that LŁ is also a special BL-algebra on [0, 1]....
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