On the weak intuitionistic fuzzy implication based on  operation

: In this paper, a new weak intuitionistic fuzzy implication is introduced. Fulfillment of some axioms and properties, together with the Modus Ponens and Modus Tollens inference rules are investigated. Negation induced by the newly proposed implication is presented.


Introduction
In 1983, Atanassov presented in [1] a concept of a kind of vague sets, that were named Intuitionistic Fuzzy Sets (IFSs). The concept was directly inspired by the concept of Fuzzy Sets (FSs) introduced by Zadeh in 1965. IFS, however, differs from FS in the two independently defined functions that assign the membership degree and the non-membership degree of a given element x to a given set A. While in FSs the degree of non-membership of the element x to the FS A is equal to 1 A(x), where A(x) is the membership degree, Atanassov introduced separate values A(x) and A(x) of membership and non-membership of x to the IFS A.
In the so developed Intuitionistic Fuzzy Logic (IFL) a variable x has its truth-value presented by the ordered pair a, b, for which it holds that a, b, a + b  [0, 1]. Such a pair is called an Intuitionistic Fuzzy Pair (IFP), presented in details in [5]. The numbers a and b are interpreted as the degrees of validity and non-validity of x, respectively. We denote the truth-value of x by V(x).
The variable with truth-value Truth, as in the classical logic, will be denoted by 1 and the variable Falsity will be denoted by 0. Therefore, for these variables it holds that V(1) = 1, 0 and V(0) = 0, 1. In addition, the variable having the truth-value 0, 0 is also used, symbolically presented as V(FI) = 0, 0. It is called in the literature Full Ignorance (FI) or Uncertainty. Notably, such a variable does not exist in either the classical or the fuzzy logic.
We call the variable x an Intuitionistic Fuzzy Tautology (IFT), if and only if (shortly: iff) it holds for V(x) = a, b that a b and, similarly, we call x an Intuitionistic Fuzzy co-Tautology (IFcT), if it holds that a b. For every x we can define the value of negation of x in the typical form V(x) = b, a.
For the IF pairs different operations can be defined. One of them is introduced in [12] and later considered in [13]. Definition 1. ([12, p. 24]) For two IFPs a, b and c, d, where a + b + c + d > 0, the operation  is defined as follows: 5. For the following considerations we introduce first some ordering relation for the intuitionistic truth-values. For V(x) = a, b and V(y) = c, d, where a, b, c, d, a One of the important logical connectives in the IFL is the Intuitionistic Fuzzy Implication (IFI). In this paper, we will omit the formal difference between an implication as a logical connective and an implicator as a binary operator, although for some considerations, this difference can be important.
The general conditions for the IFI were given first by Cornelis and Deschrijver [15], Cornelis, Deschrijver and Kerre [17,18], Cornelis, Deschrijver, Cock and Kerre [16], and later by Liu and Wang [24], and Zhou, Wu and Zhang [26]. These conditions are grounded on the conditions formulated for the classical fuzzy implication (see e.g. [14], Def. 1.1.1., p. 2). We can see that the condition (IFI 6) can be omitted. The (IFI 6) condition can be obtained as a corollary from the (IFI 5) and (IFI 2) conditions.
In the existing literature, there is the definition of the intuitionistic fuzzy implicator (implication) without the conditions (IFI 1) and (IFI 2) (see e.g. [25,Def. 10,p. 3]). It is, however, an isolated case, and, moreover, neglecting the monotonicity conditions (IFI 1) and (IFI 2), it is inappropriate as it allows too much freedom in defining the 'implicator' or 'implication'.
In the literature on the subject, almost 200 different intuitionistic fuzzy implications have been introduced (see e.g. [2][3][4]). One of them is presented by Atanassova in [6]. Such kind of implication is called by Dworniczak in [19] a Weak Intuitionistic Fuzzy Implication (WIFI). The WIFIs are studied in [711, 2022].
The most important kind of the IFIs is called an (S, N)-implication (or an S-implication). These are implications with the truth value , where S is some s-norm and N is some IF negation operator. In this case the s-norm S must be an intuitionistic counterpart of the classical s-norm (see e.g. [17]).

Main results
We introduce now a new weak intuitionistic fuzzy implication . The given below implication is a result of using of the operation  playing the role of the s-norm in the (S, N)-implication. Owing to this fact, we will call this implication the implication based on  operation. The negation is in this case the classical negation . Symbolically, we write: We can formulate the following Theorem 1.
Proof. We start with a preliminary note.
The connective  fulfills the conditions (WIFI 1)(WIFI 5) because of the following reasoning: , then a1  a2 and b1  b2 and a1b2  a2b1. Therefore: In the same manner, we can check the inequality Therefore,  y), and the proof of (WIFI 1) is completed.
is equivalent for the inequality c + 1  d, which holds for c, d  [0, 1], therefore, 0  y is an IFT. In the literature 1 on fuzzy implications (not necessarily intuitionistic fuzzy implications), in addition to (WIFI 1)(WIFI 5) or (IFI 1)(IFI 6), the following axioms are further postulated: f and   a, b, c, d, e, f, a + b, c + d, e + f  [0, 1], and  is an implication.
As we can see, therefore, the implication  is not a generalization of the classical implication.
There exist two basic rules of inference: Modus Ponens and Modus Tollens. These are the tautologies, given in the two-valued logic in the form (p  (p  q))  q and ((p  q)  N(q))  N(p), respectively. We assume that the Modus Ponens in the IFL-case is as follows: if x is an IFT and x  y is an IFT, then y is an IFT. Similarly, we assume the Modus Tollens rule in the IFL-case as follows: if x  y is an IFT and y is an IFcT then x is an IFcT.  So, 0  ab and ab  cd. It follows that 0  cd. Hence c  d and y is an IFT. b) Let x  y be an IFT and y be an IFcT. Then, Hence, a  b and x is an IFcT. This completes the proof.
One of the fundamental tautologies of classical logic is the relationship between the implication and negation. This relationship says that the truth-value of negation of the variable x is equal to the value of the logical implications of the antecedent x and the consequent False. Symbolically, this tautology is written in the classical logic in the form of N(x)  (x  0). Using this relationship, we can, for every intuitionistic fuzzy implication, designate a corresponding negation, called a generated (induced) negation. Theorem 4. Let V(x) = a, b. The negation N generated by the implication  is expressed by formula: Proof. It follows from the definition of the  implication. The first equality in the remark R6 shows that the above negation does not fulfill the basic property of negations in form V(N(0)) = V (1), however N(0) is an IFT. The property presented in remark R7 should not be satisfied because the negation of an IFT should be an IFcT and the negation of an IFcT should be an IFT. For this reason, the negation N should be carefully used in different applications.

Conclusion
In the paper the new fuzzy intuitionistic implication based on the operation  is presented together with its basic properties. The implication may be the subject of further research, both in terms of its properties or with regards to comparisons with other intuitionistic fuzzy implications, and possible applications. Possible applications, for example, can be related to fuzzy control, reasoning with incomplete or uncertain information, or multiple criteria decision making, especially with varying degrees of criteria importance.