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### Fundamental Properties of Fuzzy Implications

2018 Formalized Mathematics
We develop a framework of Mizar attributes allowing us for a smooth proving of basic properties of these fuzzy connectives [9].  ...  In the article we continue in the Mizar system [8], [2] the formalization of fuzzy implications according to the monograph of Baczyński and Jayaram "Fuzzy Implications" [1].  ...  On the other hand, in the final section I prove the formal characterization of fundamental fuzzy implications in terms of four elementary properties [12] expressed in Table 1 I LK + + + + I GD + +  ...

### Fuzzy Implication Classes Satisfying a Boolean-Like Law [chapter]

Anderson Cruz, Benjamín Bedregal, Regivan Santiago
2013 Advances in Intelligent Systems and Computing
We focus the investigation on the following classes of fuzzy implications: (S,N)-, R-, QL-, D-, (N,T)-, f -, gand h-implications.  ...  Properties that are always true in the classical theory (Boolean laws) have been extended to fuzzy theory and so-called Boolean-like laws.  ...  In fuzzy logics, the lack of a consensus on Boolean implication meaning entails non-equivalent acceptable definitions of fuzzy implications (see [15, 16, 27, 28, 34] as examples) and classes of fuzzy  ...

### On EQ-Fuzzy Logics with Delta Connective

Martin Dyba, Vilem Novak
2011 Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)
The former is a new kind of many-valued logic which based on EQ-algebra of truth values, i.e. the algebra in which fuzzy equality is the fundamental operation and implication is derived from it.  ...  First, we extend the EQ-algebra by the ∆ operation and then introduce axioms and inference rules of EQ ∆ -logic. We also prove the deduction theorem formulated using fuzzy equalities.  ...  For all a, b ∈ E we put a → b = (a ∧ b) ∼ a (1) and call this operation an implication. The following theorem demonstrates the fundamental properties of the fuzzy equality and implication.  ...

### On the minimization of incompletely specified fuzzy functions

Abraham Kandel
1974 Information and Control
and with the mathematical properties involved in fuzzy logic.  ...  It is our hope that this new method will stimulate more researchers to look into the algebraic properties of fuzzy structures.  ...  Compute the fuzzy prime implicants of F, @ F¢ ; 2. Remove the essential fuzzy prime implicants; 3. Find a minimal complexity cover of the remainder.  ...

### Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems

Ulrich Bodenhofer, Martine De Cock‡, Etienne E. Kerre
2003 International Journal of General Systems
Firstly, a general concept of closedness of fuzzy sets under fuzzy preorderings is proposed and investigated along with the corresponding opening and closure operators.  ...  The purpose of this paper is two-fold.  ...  In order to provide the reader with the basic properties of residual implications, we will now briefly recall them.  ...

### Fuzzy logic controllers are universal approximators

J.L. Castro
1995 IEEE Transactions on Systems, Man and Cybernetics
We try to answer this fundamental question by proving that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, and for each fixed type of membership function belonging to a wide class  ...  In this paper, we consider a fundamental theoretical question, Why does fuzzy control have such good performance for a wide variety of practical problems?.  ...  is 3c' = P by: which in the case of a t-norm implication is the general expression d) The combination of all fuzzy inferences is made by means of T*: iv) The defuzzification method verifies the property  ...

### On the Paradoxical Success of Mamdani's Minimum-Based Inference [chapter]

Marcin Detyniecki, Benjamin Moubêche, Bernadette Bouchon-Meunier
2012 Studies in Fuzziness and Soft Computing
introducing a simple variant of the Generalized Modus Ponens (GMP) that uses standard fuzzy implications.  ...  Yet, it is well known that it uses a min-based implication that does not verify classical boolean logic requirements.  ...  standard fuzzy implications.  ...

### Page 8069 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews
the set of all infinitely A-distributive implications on L in detail and point out that three classes of fundamental fuzzy implications and Yager’s implication on [0,1] are all infinitely A-distributive  ...  A series of properties of subal- gebras and generalized tautologies of product logical systems are given.  ...

### Page 1506 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews
Incidentally, in the diagram on page 281, the implication-does- not-hold arrows ( # ) should point up instead of down. S. E.  ...  The author investigates the question of when a space with a property # has a countably compact extension with property ?.  ...

### A short note on fuzzy relational inference systems

Martin Štěpnička, Balasubramaniam Jayaram, Yong Su
2018 Fuzzy sets and systems (Print)
fuzzy relational models of fuzzy rule bases, namely, the Mamdani-Asilian and the implicative models.  ...  This paper is a short note contribution to the topic of fuzzy relational inference (FRI) systems and the preservation of their desirable properties.  ...  Introduction The notion of fuzzy inference systems (or fuzzy rule based systems) denotes systems incorporating the fundamental idea of Zadeh to imitate the human way of thinking based on fuzzy rules, see  ...

### Formalization of Human Categorization Process Using Interpolative Boolean Algebra

2015 Mathematical Problems in Engineering
Such relations are directly constructed from a Boolean consistent fuzzy partial order relation, which is treated by Boolean implication.  ...  The introduced formalism secures the principles of categorization showing that Boolean laws are fundamental in the categorization process.  ...  Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.  ...

### Rough Approximations under Level Fuzzy Sets [chapter]

W. -N. Liu, JingTao Yao, Yiyu Yao
2004 Lecture Notes in Computer Science
The combination of fuzzy set and rough set theories lead to various models. Functional and set approaches are two categories based on different fuzzy representations.  ...  In this paper, we study rough approximations based on the notion of level fuzzy sets. Two rough approximation models, namely α-level rough set and β-level rough set, are proposed.  ...  The present study examines some of the fundamental issues of the combination from the perspective of level fuzzy sets.  ...

### Formal Introduction to Fuzzy Implications

2017 Formalized Mathematics
In the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets.  ...  of fuzzy logic (together with t-norms and t-conorms, formalized previously).  ...  fuzzy implication has also properties of a classical implication.  ...

### Properties of Fuzzy Implications obtained via the Interval Constructor

B.R.C. Bedregal, R.H. Santiago, R.H.S. Reiser, G.P. Dimuro
2007 TEMA
This work considers an interval extension of fuzzy implication based on the best interval representation of continuous t-norms.  ...  Some related properties can be naturally extended and that extension preserves the behaviors of the implications in the interval endpoints.  ...  Este trabalho considera a extensão intervalar da implicação fuzzy baseada no conceito de melhor representação intervalar de t-normas contínuas, previamente introduzido por Bedregal e Takahashi.  ...

### Page 4447 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews
This paper reveals a fundamental property of the mu- calculus that can be viewed as an explanation of the determinacy result in purely algebraic terms; it also gives a new proof of this result which does  ...  Burhan (3-TRNT-I; Toronto, ON); Kreinovich, Vladik (1-TXEP-C; El Paso, TX); Yager, Ronald R. (1-IONA-M; New Rochelle, NY) A new class of fuzzy implications. Axioms of fuzzy implication revisited.  ...
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