Abstract

The crucial role that fuzzy implications play in many applicable areas was our motivation to revisit the topic of them. In this paper, we apply classical logic’s laws such as De Morgan’s laws and the classical law of double negation in known formulas of fuzzy implications. These applications lead to new families of fuzzy implications. Although a duality in properties of the preliminary and induced families is expected, we will prove that this does not hold, in general. Moreover, we will prove that it is not ensured that these applications lead us to fuzzy implications, in general, without restrictions. We generate and study three induced families, the so-called -implications, -implications, and -implications. Each family is the “closest” to its preliminary-“creator” family, and they both are simulating the same (or a similar) way of classical thinking.

1. Introduction

Although, in classical logic, the implication is uniquely determined, in fuzzy logic, there are not only several formulas but also families of fuzzy implications. Moreover, among fuzzy implications, there are several properties, which a fuzzy implication satisfies or violates. The necessity of this variety had been addressed by many authors [1–3], and it is the election of the proper fuzzy implication to any applied problem. More specifically, Mas et al. in [1] addressed the following:

Of course, all these expressions for implications are equivalent in any Boolean algebra and consequently in classical logic. However, in fuzzy logic, these four definitions yield to distinct classes of fuzzy implications. Thus, the following question naturally arises: why so many different models to perform this kind of operation? The main reason is because they are used to represent imprecise knowledge. Note that any “if then” rule in fuzzy systems is interpreted through one of these implication functions. So, depending on the context and on the proper rule and its behaviour, different implications can be adequate in any case.

Among the families or the construction methods of fuzzy implications, there are several types. There are fuzzy implications that are constructed by generalizations of classical tautologies, such as ()-implications, QL-implications, D-implications [1, 3–6], and ()-implications [2, 7, 8]. All these generalizations are not always an easy process. For example, in the case of ()-implications [4, 6] and ()-implications [2, 7, 8], the generalization is smooth and direct. On the contrary, in the case of QL- and D-operations [4, 6], the generalization holds under conditions, which need investigation and remain an open problem, when a QL- (respectively, D-) operation is a fuzzy implication [4]. There are generalizations from classical set theory, such as R-implications [4–6]. There are fuzzy implications that are constructed by generalizations of the aforementioned generalizations, such as ()- and RU-implications [4]. There are fuzzy implications that are constructed by function generators with specific properties [4, 9–12]. Moreover, there are generation methods of fuzzy implications from known fuzzy implications [4, 13].

In this paper, we focus on these fuzzy implications, in which the formula contains at least a t-norm or a t-conorm. Such families are ()-implications, ()-implications, QL-implications, D-implications, and R-implications.

We have to remind that, in classical logic, there are De Morgan’s laws, which are expressed by tautologies and and the classical law of double negation, which is expressed by the tautology .

So, the central idea is to apply De Morgan’s laws and, if necessary, the classical law of double negation in known formulas of fuzzy implications and investigate the results. These are the generation of new families of fuzzy implications. As we will show in the following, each new induced family does not generally have the same properties as its preliminary-“creator” family. On the contrary, some duality in the properties is remarkable and expected, but this does not hold, in general. So, any case must be studied individually from the beginning.

2. Preliminaries

Definition 1 (see [4, 5, 14, 15]). A decreasing function is called fuzzy negation if and . Moreover, a fuzzy negation is called(i)Strict if it is continuous and strictly decreasing.(ii)Strong if it is an involution, i.e.,(iii)Nonfilling if

Remark 1 (see [4]). (i)We call the classical fuzzy negation, which is a strong negation.(ii)Moreover, we will need the following fuzzy negations:

Definition 2 (see [4]). A function is called a triangular norm (shortly t-norm) if it satisfies, for all , the following conditions:Dually, a function is called a triangular conorm (shortly t-conorm) if it satisfies, for all , conditions (4), (5), and (6), and additionally,

Definition 3 (see [4]). A t-norm (respectively, a t-conorm ) is called(i)Continuous if it is continuous in both arguments(ii)Idempotent if (respectively, ), for all (iii)Strict if it is continuous and strictly monotone, i.e., whenever and (respectively, whenever and )(iv)Positive if or (respectively, or )Tables 1 and 2 list a few of the common t-norms and t-conorms, respectively (see Tables 2.1 and 2.2 of [4]).

Definition 4 (see [4]). Let be a t-conorm and be a fuzzy negation. We say that the pair satisfies the law of excluded middle if

Definition 5 (see [4]). Let be a t-norm and be a fuzzy negation. We say that the pair satisfies the law of contradiction if

Definition 6 (see [4]). A triple , where is a strong negation, is called a De Morgan triple ifMoreover, in the first case, is called N-dual of , and in the second case, is called N-dual of .

Definition 7 (see [4, 16]). By , we denote the family of all increasing bijections from to . We say that functions are -conjugate if there exists such that , where

Remark 2 (see [4]). It is easy to prove that if and is a t-norm, is a t-conorm, and is a fuzzy negation (respectively, strict and strong), then is a t-norm, is a t-conorm, and is a fuzzy negation (respectively, strict and strong).

Definition 8 (see [4, 5]). A function is called a fuzzy implication if

Remark 3. By axioms (14) and (15), we deduce the normality conditionFurthermore, by Definition 8, it is easy to prove the left and right boundary conditions [4]:Table 3 lists a few of basic fuzzy implications (see Table 1.3 and Proposition 1.1.7 of [4]).

Definition 9 (see [4]). A fuzzy implication is said to satisfy(i)The left neutrality property if(ii)The exchange principle if

Remark 4 (see [4]). It is proved that if and is a function, which satisfies axiom (13) (respectively, (14), (15), (16), and (17)), then is a function, which also satisfies axiom (13) (respectively, (14), (15), (16), and (17)). So, if is a fuzzy implication, then is also a fuzzy implication.

Proposition 1 (see [4]). If a function satisfies (13), (15), and (17), then the function is a fuzzy negation, where

Definition 10 (see [4]). Let be a fuzzy implication. The function defined by Proposition 1 is called the natural negation of .

Definition 11 (see [4]). Let be a fuzzy negation and be a fuzzy implication. A function defined byis called the -reciprocal of . When is the classical negation , then is called the reciprocal of and is denoted by .
In Table 4, we present the known families of fuzzy implications that are generalizations from the classical logic (except R-implications that are generalizations from the classical set theory via the isomorphism that exists between classical two-valued logic and classical set theory and a set theoretic identity (see page 68 of [4])). We consider that the reader knows these families, as well as QL- and D-operations.

3. ()-Implications

()-implications are mentioned as formulas by many authors [15, 17–19]. Baczyński and Jayaram in Corollary 2.5.31 of [4] related them with R-implications when the t-norm is left continuous and is a strong negation. Pradera et al. in Remark 30 of [21] mentioned the same formula, using aggregation functions, in general. Bedregal in Proposition 2.6 of [7] defined them for any t-norm and any fuzzy negation. They obtained their name in [2] and were studied by Pinheiro et al. in [2, 8] for any t-norm and any fuzzy negation.

()-implications are the “closest” to ()-implications since for strong negations, they are the same family (see p. 234 of [15], [2] Theorem 3.1 of [8]). On the contrary, for nonstrong negations, they have no common implications since unlike -implications (see Proposition 2.4.3 of [4]), -implications violate (21) (see Proposition 3.3(i) of [2]). All these results lead us to Figure 1.

Figure 1 

Intersection between - and -implications.

The aforementioned lead us to the claim that these two families simulate the same (or a similar) way of classical thinking. Firstly, we must explain the meaning of this simulation. The meaning is that there are tautologies in the classical logic, which can be proved without using truth tables or other rules, but only De Morgan’s laws and the classical law of double negation.

So, firstly, we work on classical logic’s tautologies, and secondly, we generalize them in fuzzy logic. Unfortunately, this generalization is not always an easy process, as we will show in the following.

Let us remark that ()-implications are the generalization of the tautologyand then by applying De Morgan’s laws and the classical law of double negation, we get the following tautology:which leads us to ()-implications.

Similarly, QL-operations (respectively, implications) are the generalization of the tautology

At this point, let us make clear what is the meaning of “simulate the same (or a similar) way of classical thinking” via an example.

Example 1. Someone could claim that since ()- and QL-implications simulate the same (or a similar) way of classical thinking. This is not acceptable since we cannot prove the tautologyusing only De Morgan’s laws and maybe the classical law of double negation.

4. ′-Implications

Similarly, D-operations (respectively, implications) are the generalization of the tautologyand after the application of De Morgan’s laws and the classical law of double negation, we get the following tautology:which leads us to a new family of fuzzy operations (respectively, implications) obtained by the following formula.

Definition 12. A function is called a -operation if there exist a t-norm , a t-conorm , and a fuzzy negation such thatIf is a -operation generated by the triple , then we denote it by .

Theorem 1. Let be a -operation; then, satisfies (13)–(18) and (20), and if is a strong negation, then satisfies (21). Furthermore, , where .

Proof. Let be a -operation; then, for , ifwhich means that satisfies (13).
satisfies (15) since satisfies (16) since satisfies (17) since satisfies (18) since satisfies (20) since , it is(21) is satisfied if is a strong negation since , it isLastly, we haveBy Theorem 1, it follows that a -operation is generated by a unique negation. The reason for which we use the name -operations instead of -implications is that they do not generally satisfy (14), so they are not always increasing with respect to the second variable, as it can be seen in the next example.

Example 2. Consider the triple . The obtained -operation iswhich does not satisfy (14) sinceTherefore, the first main problem is the characterization of those -operations, which satisfy (14). In this paper, we try to characterize these triples that produce -operations, which satisfy (14). In the following, we will give only partial results as partial results are known in the literature for QL- and D-operations, too [4]. Following the terminology [8, 16], only if the -operation is a fuzzy implication, we use the term -implication. So, we have the following proposition, without proof.

Proposition 2. A function is called a -implication if it is a -operation and satisfies (14).

All the above are useful, but they do not ensure that the set of -implications is nonempty. This is ensured by the next example.

Example 3. Consider the triple . The obtained -operation iswhich is a -implication.
By Example 3, someone could automatically prove that the set of -implications is nonempty. Furthermore, it is very easy to observe and prove the following lemma. The proof is omitted due to its simplicity.

Lemma 1. Let be t-norms, be t-conorms, and be a fuzzy negation.(i)If , then (ii)If , then (iii)If and , then

Although Lemma 1 is simple, it is very useful for the following. It is , for all t-norms (see Remark 2.1.4.(ix), page 43, of [4]) and , for all t-conorms (see Remark 2.2.5.(viii), page 46, of [4]). Thus, it is easy to prove that the strongest -operation, which can be obtained by the fuzzy negation , iswhich is a -implication, too. By this result, we can conclude that is not a -implication. That is, because its natural negation is and, furthermore, there are , such that . is stronger than the strongest -implication that has as its natural negation, meaning that is not a -implication.

The formula of -operations is complicated. Moreover, we have to check every time if axiom (14) is satisfied. If we use a nonfilling negation, the following theorem will be a useful tool to overcome these difficulties.

Theorem 2. Let be a -implication and be a nonfilling negation; then, it is , that is, the pair satisfies the law of contradiction (10).

Proof. If is a -implication, then it satisfies (19), that is, for all since is a nonfilling negation. So, .

Remark 5. By Theorem 2, it is obvious that if is a nonfilling negation and the pair does not satisfy the law of contradiction (10), i.e., , for some , then the obtained -operation, for any t-conorm , is not a fuzzy implication.

Theorem 3. If and is a -operation (respectively, implication), then is a -operation (respectively, implication), and moreover,

Proof. Let be a -operation (respectively, implication); then, is a -operation (respectively, implication) according to Remark 4. Moreover, for all , we deduce thatSince we have studied some general results for -operations (respectively, implications), we are going to study some more specific cases. This family has no interest if we use strong negations. This is explained by Theorem 4 after the following Proposition 3.