A Method of Generating Fuzzy Implications from n Increasing Functions and n + 1 Negations

: In this paper, we introduce a new construction method of a fuzzy implication from n increasing functions g i : [ 0, 1 ] → [ 0, ∞ ) , ( g ( 0 ) = 0 ) (i = 1, 2, . . . , n, n ∈ N ) and n + 1 fuzzy negations N i (i = 1, 2, . . . , n + 1, n ∈ N ). Imagine that there are plenty of combinations between n increasing functions g i and n + 1 fuzzy negations N i in order to produce new fuzzy implications. This method allows us to use at least two fuzzy negations N i and one increasing function g in order to generate a new fuzzy implication. Choosing the appropriate negations, we can prove that some basic properties such as the exchange principle (EP), the ordering property (OP), and the law of contraposition with respect to N are satisﬁed. The worth of generating new implications is valuable in the sciences such as artiﬁcial intelligence and robotics. In this paper, we have found a novel method of generating families of implications. Therefore, we would like to believe that we have added to the literature one more source from which we could choose the most appropriate implication concerning a speciﬁc application. It should be emphasized that this production is based on a generalization of an important form of Yager’s implications.


Introduction
Fuzzy implications are the generalization of the classical (Boolean) implication in the interval of [0, 1]. They play an important role in the area of fuzzy logic, decision theory, and fuzzy control. We can generate fuzzy implications from aggregation functions and fuzzy negations ( [1][2][3][4][5]). Other ways of generating fuzzy implications can be achieved by additive generating functions or by some initials implications ( [6][7][8][9][10][11]). Fuzzy implications are used for the application of the 'if-then' rule in fuzzy systems and inference processes, through Modus Ponens and Modus Tollens [12]. This paper is inspired by Yager's f-generated implications where f: [0,1] → [0, ∞] is a strictly decreasing and continuous function and f(1) = 0. In addition, a fuzzy implication I: [0,1] → [0,1] is defined by: I(x, y) = f xf(y) , x, y ∈ [0,1] with the understanding 0ˑ∞=0 (see [1] Definition 3.1.1). In this paper, we use functions g : [0,1] → [0, ∞), which are increasing and continuous, and also g (0) = 0. We present a new production machine of fuzzy implications. Such a type of generating fuzzy implications can be found in the literature ( [1][2][5][6][7][8]), for example, I = 1 − x + xy. The production of new fuzzy implications is accomplished with the help of any fuzzy negations and increasing functions. These generated fuzzy implications fulfill the necessary properties required to be fuzzy implications (see [1] Definition 1.1.1.). Moreover, if the negations are selected with certain properties, then the generated implications may also fulfill additional properties like the neutrality property (NP), exchange principle (EP), identity principle (IP), and some others. The worth of this production of implications could be estimated at artificial intelligence, robotics science, etc. ( [13][14][15]). This method of producing implications gives us the possibility, in a fuzzy environment, to find a large number of implications, which could help any researcher choose the most appropriate one.
The paper is organized as follows. In Section 2, we recall the basic concepts and definitions used in the paper. In Section 3, we study the new constructed method of fuzzy implications. Firstly, we present a constructed method using one increasing function g and two negations N , N , then a second method using two increasing functions g , g and three negations N , N , N . Finally, we generalize our constructed method using n functions g , g , … , g and n + 1 negations N , N , … , N , N .

Preliminaries
In order to help the reader get familiar with the theory, we recall here some of the concepts and results employed in the rest of the paper. x ≤ x then I(x , y) ≥ I(x , y), i. e., I is decreasing in the first variable (1) y ≤ y then I(x, y ) ≤ I(x, y ), i. e. , I is increasing in the second variable. (2) The set of all fuzzy implications will be denoted by . A fuzzy negation N is a generalization of the classical complement or negation ¬, whose truth N is decreasing (7)

A fuzzy negation is called strict if, in addition,
N is strictly decreasing (8) N is continuous (9)

Definition 6. (see [1] Definition 1.4.2 (ii)).
A fuzzy negation is said to be non-filling if is called the natural negation of I. Important negations that will be used throughout this paper are the standard negation = 1 − , the least or Godel, and the greatest or dual Godel fuzzy negations given respectively by ii. The exchange principle if: I x, I(y, z) = I y, I(x, z) , x, y, z ∈ [0,1] iii. The identity principle if: iv. The ordering property if: v.
The law of contraposition with respect to if: vi. The law of left contraposition with respect to if: vii. The law of right contraposition with respect to N if: If y ≤ z, then T(x, y) ≤ T(x, z) , i. e. , T(x,·) is increasing (28)

It is proved that, if φ ∈ Φ, T is a continuous t-norm, is a continuous t-conorm, is a fuzzy (strict, strong) negation, and is a fuzzy implication, then is a t-norm, is a t-conorm, is a fuzzy (strict, strong) negation, and
is a fuzzy implication.

The equation p→ q ≡ ¬p ∨ q creates a new class of fuzzy implications. A function : [0,1] ⟶ [0,1] is called an (S, N)-Implication if there exist a t-conorm S and a fuzzy negation
such that:

Definition 13. (see [1] Subsection 7.3).
The equation (p ∧ q) → r ≡ (p → (q →r)) is known as the law of importation and is a tautology in classical logic. The general form of the above equivalence is given by

Proposition 1. (see [1] Definition 7.4.2).
If I ∈ FI is such that there exist x, y ∈ (0, 1) such that x > y and I(x, y) = 1, then I does not satisfy (32) with any t-norm T.

Let I ∈ FI, a t-norm T satisfy (32), is the natural negation of I and is the natural negation of T, then
≤ , ℎ .
is called the N-reciprocal of I. When N is the classical negation , then is called the reciprocal of I and is denoted by .

The Main Results
In this section, we give definitions of new generated implications and prove some useful properties of them.

Theorem 1
If , are two fuzzy negations and : [0,1] → [0, ∞) is an increasing and continuous function with is a fuzzy implication.

Proof.
Let

If φ ∈ Φ and I is the fuzzy implication of Theorem 1, then Iφ is a fuzzy implication.
Proof.
According to Remark 1, I is a fuzzy implication. □   The graph of the above surface is plotted in Figure 3. g (x ) g (1) ≤ N (y) g (x ) g (1) and N (y) We conclude that (15)   The graph of the above surface is plotted in Figure 5.
is a fuzzy implication.

Conclusions
In this paper, a new production machine of fuzzy implications from n continuous increasing functions and n+ 1 negation are introduced. We studied certain properties of these new fuzzy implications, as the left neutrality property (14), exchange principle (15), identity principle (16), ordering property (17), law of contraposition (18), and T-Conditionality (32), where some results are obtained if the fuzzy negations are strong or the least fuzzy negations. The advance of this method relies on the fact that we can combine a lot of fuzzy negations N and increasing functions g in order to generate fuzzy implications.
Finally, we believe that this production machine needs to be investigated further. It has been observed that in order to be satisfied, certain desirable properties by the implications generated by this method must use strong fuzzy negations or the least fuzzy negation. A question that arises is the following one: Are there non-strong fuzzy negations that satisfy the left neutrality property (14) or the exchange principle (15)? In addition, in a future paper, we will study the behavior of non-continuous functions in terms of the validity of certain basic properties.