An Application of Classical Logic’s Laws in Formulas of Fuzzy Implications

,e 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.,ese 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 D′-implications, QL′-implications, and R′-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.


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. e necessity of this variety had been addressed by many authors [1][2][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. us, the following question naturally arises: why so many different models to perform this kind of operation? e 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. ere are fuzzy implications that are constructed by generalizations of classical tautologies, such as (S, N)-implications, QL-implications, D-implications [1,[3][4][5][6], and (T, N)-implications [2,7,8]. All these generalizations are not always an easy process. For example, in the case of (S, N)-implications [4,6] and (T, N)-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]. ere are generalizations from classical set theory, such as R-implications [4][5][6]. ere are fuzzy implications that are constructed by generalizations of the aforementioned generalizations, such as (U, N)and RU-implications [4].
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 (S, N)-implications, (T, N)-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 (p ∧ q) ′ ≡ p ′ ∨ q ′ and (p ∨ q) ′ ≡ p ′ ∧ q ′ and the classical law of double negation, which is expressed by the tautology (p ′ ) ′ ≡ p.
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.
ese 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.

and 2.2 of [4]).
Definition 4 (see [4]). Let S be a t-conorm and N be a fuzzy negation. We say that the pair (S, N) satisfies the law of excluded middle if Definition 5 (see [4]). Let T be a t-norm and N be a fuzzy negation. We say that the pair (T, N) satisfies the law of contradiction if Definition 6 (see [4] Moreover, in the first case, T is called N-dual of S, and in the second case, S is called N-dual of T. Definition 7 (see [4,16]). By Φ, we denote the family of all increasing bijections from [0, 1] to [0, 1]. We say that functions f, g: Remark 2 (see [4]). It is easy to prove that if ϕ ∈ Φ and T is a t-norm, S is a t-conorm, and N is a fuzzy negation (respectively, strict and strong), then T ϕ is a t-norm, S ϕ is a tconorm, and N ϕ is a fuzzy negation (respectively, strict and strong).
Proposition 1 (see [4]). If a function I: [0, 1] 2 ⟶ [0, 1] satisfies (13), (15), and (17), then the function Definition 10 (see [4] is called the N-reciprocal of I. When N is the classical negation N C , then I N is called the reciprocal of I and is denoted by I ′ . Fodor

Dubois and Prade
Rescher 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.

(T, N)-Implications
(T, N)-implications are mentioned as formulas by many authors [15,[17][18][19]. Baczyński and Jayaram in Corollary 2.5.31 of [4] related them with R-implications when the t-norm T is left continuous and N 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. ey obtained their name in [2] and were studied by Pinheiro et al. in [2,8] for any t-norm and any fuzzy negation.
(T, N)-implications are the "closest" to (S, N)-implications since for strong negations, they are the same family (see p. 234 of [15], [2]  e 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. e 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 (S, N)-implications are the generalization of the tautology and then by applying De Morgan's laws and the classical law of double negation, we get the following tautology: which leads us to (T, N)-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
(S, N)and QL-implications simulate the same (or a similar) way of classical thinking. is is not acceptable since we cannot prove the tautology using only De Morgan's laws and maybe the classical law of double negation.

D9-Implications
Similarly, D-operations (respectively, implications) are the generalization of the tautology and 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.   [1,[3][4][5][6] I S,N (x, y) � S(N(x), y) (T, N) [2,7,8,15,[17][18][19] I     (38) Lastly, we have By eorem 1, it follows that a D ′ -operation is generated by a unique negation. e reason for which we use the name D ′ -operations instead of D ′ -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 (T M , S P , N C ). e obtained D ′ -operation is
which does not satisfy (14) since (42) erefore, the first main problem is the characterization of those D ′ -operations, which satisfy (14). In this paper, we try to characterize these triples (T, S, N) that produce D ′ -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 D ′ -operation is a fuzzy implication, we use the term D ′ -implication. So, we have the following proposition, without proof. (14).

Proposition 2. A function
All the above are useful, but they do not ensure that the set of D ′ -implications is nonempty. is is ensured by the next example.
which is a D ′ -implication.
By Example 3, someone could automatically prove that the set of D ′ -implications is nonempty. Furthermore, it is very easy to observe and prove the following lemma. e proof is omitted due to its simplicity.
and N be a fuzzy negation.
Although Lemma 1 is simple, it is very useful for the following. It is T D ≤ T, for all t-norms T (see Remark 2.1.4.(ix), page 43, of [4]) and S M ≤ S, for all t-conorms S (see Remark 2.2.5.(viii), page 46, of [4]). us, it is easy to prove that the strongest D ′ -operation, which can be obtained by the fuzzy negation N D2 , is Journal of Mathematics which is a D ′ -implication, too. By this result, we can conclude that I 1 is not a D ′ -implication. at is, because its natural negation is N D2 and, furthermore, there are x, y ∈ [0, 1], such that I N D2 ,T D ,S M (x, y) � I 4 (x, y) < I 1 (x, y). I 1 is stronger than the strongest D ′ -implication that has N D2 as its natural negation, meaning that is not a D ′ -implication. e formula of D ′ -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 I N,T,S be a D ′ -implication and N be a nonfilling negation; then, it is
, that is, the pair (T, N) satisfies the law of contradiction (10).
Remark 5. By eorem 2, it is obvious that if N is a nonfilling negation and the pair (T, N) does not satisfy the law of contradiction (10), i.e., T(N(x), x) ≠ 0, for some x ∈ [0, 1], then the obtained I N,T,S D ′ -operation, for any t-conorm S, is not a fuzzy implication. Theorem 3. If ϕ ∈ Φ and I N,T,S is a D ′ -operation (respectively, implication), then (I N,T,S ) ϕ is a D ′ -operation (respectively, implication), and moreover, Proof. Let I N,T,S be a D ′ -operation (respectively, implication); then, (I N,T,S ) ϕ is a D ′ -operation (respectively, implication) according to Remark 4. Moreover, for all x, y ∈ [0, 1], we deduce that , ϕ(y)))), ϕ(ϕ − 1 (N(ϕ(y))))))) Since we have studied some general results for D ′ -operations (respectively, implications), we are going to study some more specific cases. is family has no interest if we use strong negations. is is explained by eorem 4 after the following Proposition 3.
On the contrary, (ii) If one of I T′,S,N and I T′,S,N is a fuzzy implication, then the other one is a fuzzy implication too, according to Proposition 3. If I N,T,S′ is a fuzzy implication, then because of (i), where I N,T,S′ � I T′,S,N , we conclude that I T′,S,N is a fuzzy implication and vice versa. So, the first question arising is whether D ′ -operations (respectively,, implications) are or not a new family of operations (respectively, implications) or they are simply D-operations (respectively, implications). Although we have mentioned so many theorems and we have presented the whole rationale of our study, this is a question that seems not to have been answered yet.
It is known that if I T,S,N is a D-operation (respectively, implication), then the pair (S, N) must satisfy (9) (see Lemma 3 of [20]). Additionally, there does not exist any tconorm S such that the pair (S, N D1 ) satisfies (9) (see Remark 2.3.10(iii) of [4]).
us, there is no D-implication generated by any triple (T, S, N D1 ). Furthermore, it is easy to prove that if I T,S,N is a D-implication, then N is its natural negation.
is proof is easy and similar to the proof of eorem 1. To sum up, there is no D-implication which has N D1 as its natural negation. On the contrary, we have the following proposition. [22]).

Proposition 4. By a triple of the form
Proof. By a triple of the form (T, S, N D1 ), where T is any tnorm and S is any t-conorm, we obtain Remark 7. e aforementioned Proposition 4 is very important, although it seems simple. Because of it, we conclude that the only D ′ -implication which has N D1 as its natural negation is I 3 . So, automatically, we conclude that I 0 , I GG , I G D , I RS , and I YG are not D ′ -implications since all of them have N D1 as their natural negation.
Proposition 4 is the answer of this first question. I 3 is a D ′ -implication, which is obviously not a D-implication since its natural negation is N D1 . So, it is proved that the family of D ′ -implications is not the same with D-implications' family.
Furthermore, due to tautology (5) and by our intuition, we expected duality properties of D ′ -operations and D-operations. So, the second question arising is whether or not this is true. For example, we mentioned that if I T,S,N is a D-operation (respectively, implication), then the pair (S, N) must satisfy (9). Does this imply that if I N,T,S is a D ′ -operation, then the pair (T, N) must satisfy (10)?
e answer seems to be negative, as we show in eorem 2, since (10) is satisfied if we have a nonfilling negation N and not for sure, for any negation N. Finally, the answer is negative, according to the following proposition.
Proposition 5. By a triple of the form (T, S, N D2 ), where T is any t-norm and S is any idempotent, positive, or strict tconorm, a D ′ -implication is obtained, which is I 4 (see [22]).
Proof. It is known and easy to prove that the only idempotent t-conorm is S M (see Remark 2.2.5(ii) of [4]), which is also positive (see Table 2.2, page 46, of [4]). erefore, anything is proved for positive t-conorms is valid and for the idempotent too. us, by the triple (T, S, N D2 ), where S is any positive or strict t-conorm, we obtain 1, if S(x, y) < 1 and y < 1, 0, if S(x, y) � 1 and y < 1, According to Proposition 5, I 4 is a D ′ -implication, which does not satisfy (10) since for any t-norm T, it is T(N D2 (x), x) � x ≠ 0, for any x ∈ (0, 1). is fact is exactly the reason why we have to study any new family generated by De Morgan's laws and, if necessary, the classical law of double negation from the beginning. e third question arising is what is the relation between these families (D and D ′ ). e answer is given partially in eorem 1, where we proved that if N is a strong negation, then I N,T,S satisfies (21). Also, it is easy to prove the next proposition. Proposition 6. Let I N,T,S be a D ′ -operation, which satisfies (21); then, N is a strong negation.

Journal of Mathematics
Proof. Since the D ′ -operation I N,T,S satisfies (21), we have (52) us, N is a strong negation. Clearly then, if N is a strong negation, then the corresponding D ′ -operation (respectively, implication) is a D-operation (respectively, implication) too, and vice versa.
erefore, in this case, every property of D-implications is valid to D ′ -implications. Since the properties are similar in both families, due to the duality of S, S ′ , T, T ′ , respectively, the existing theorems and propositions that have been studied so far (see [6]) are valid too, with some simple changes due to the mentioned duality.
For nonstrong negations, we have that D-operations (respectively, implications) are the operations (respectively, implications), which satisfy (21) (see Remark 1 of [6]), and on the contrary, D ′ -operations (respectively, implications) do not satisfy (21). All these results lead us to Figure 2.
In the following of this section, we present a useful theorem for the disqualification of some triples (T, S, N) that do not generate D ′ -implications. Finally, if T is any positive t-norm, then So, T(N(e), e) � T(e, e) ≠ 0, since e > 0, and again, according to Remark 5, any D ′ -implication cannot be obtained.
A very useful proposition is as follows. Proof. e only idempotent t-conorm is S M (see Remark 2.2.5(ii) of [4]), which is also positive (see Table 2.2, page 46, of [4]). erefore, by the triple (T D , S, N C ), where S is any positive t-conorm, we obtain Remark 8. It is easy to calculate D ′ -operations, which are obtained by their natural negation N D2 . ey are always two-valued of the form So, I WB , which has N D2 as its natural negation, is not a D ′ -implication since it is not two-valued.
At the end of this section, we present Table 5, which contains basic D ′ -implications.

QL9-Implications
Similarly, QL-operations (respectively, implications) are the generalization of the tautology and after the application of De Morgan's laws and the classical law of double negation, we get the following tautology: N is not a strong negation, and (21) is satisfied N is a strong negation N is not a strong negation, and (21) is not satisfied

Journal of Mathematics
which leads us to a new family of fuzzy operations (respectively, implications) obtained by the following formula.
By eorem 6, it follows that a QL ′ -operation is generated by a unique negation. Although it is not proved that I N,T,S satisfies (13), N I N,T,S is obviously a fuzzy negation since N I N,T,S � N. Moreover, we use the name QL ′ -operations instead of QL ′ -implications because they do not generally satisfy (13), so they are not always decreasing with respect to the first variable, as it can be seen in the next example.
(67) erefore, the first main problem is the characterization of those QL ′ -operations, which satisfy (13). Similar to the previous section, only partial results will be proved. Proof. Let I N,T,S be a QL ′ -operation generated by a nonfilling negation N and satisfy (20). erefore, it is since N is a nonfilling negation. us, the pair (T, N) satisfies (10). e set of QL ′ -implications is nonempty. is is ensured with the next theorem and some remarks.  N(N(x)), N T ′ (x, y) N(N(y) (ii) By (i) and Table 5, we deduce that us, the set of QL ′ -implications is nonempty. (N(x), x) � 0, x ∈ [0, 1], that is, the pair (T, N) satisfies the law of contradiction (10).

Theorem 8. Let I N,T,S be a QL ′ -implication and N be a nonfilling negation; then, it is T
Proof. If I N,T,S is a QL ′ -implication, then it satisfies (20), and if N is a nonfilling negation, then the proof is given by Proposition 10.  (N(x), x) ≠ 0, for some x ∈ [0, 1], then the corresponding QL ′ -operation, for any tconorm S, is not a fuzzy implication.

Theorem 9. If ϕ ∈ Φ and I N,T,S is a QL ′ -operation (respectively, implication), then (I N,T,S ) ϕ is a QL ′ -operation (respectively, implication), and moreover, (I N,T,S
Proof. Let I N,T,S be a QL ′ -operation (respectively, implication); then, (I N,T,S ) ϕ is a QL ′ -operation (respectively, implication), according to Remark 4. Moreover, for all x, y ∈ [0, 1], we deduce that S(N(ϕ(x)), N(ϕ(y)))))) (N(ϕ(x)))), ϕ(ϕ − 1 (N(ϕ(y)))))))))) (N ϕ (y)))))))) (73) Since we have studied some general results for QL ′ -operations (respectively, implications), we are going to study some more specific cases. Several questions arising are similar to those we answer for D ′ -operations (respectively, implications). Answers to many of them are given above, and they are similar to previous answers we gave for D ′ -operations (respectively, implications). Finally, we have to answer the next question. Are QL ′ -operations (respectively, implications) a new family of operations (respectively, implications)? It is known that there is no QL-implication generated by N D1 . Moreover, there is no QL-implication which has N D1 as its natural negation (see Remark 2.6.6(i) of [4]). On the contrary, we have the next proposition. (T, S, N D1 ), where T is any t-norm and S is any t-conorm, a QL ′ -implication is obtained, which is I 3 (see [22]).

Proposition 12. By a triple of the form
Proof. By a triple of the form (T, S, N D1 ), where T is any tnorm and S is any t-conorm, we obtain Remark 11. Because of Proposition 12, we conclude that the only QL ′ -implication, which has N D1 as its natural negation, is I 3 . So, automatically, we conclude that I 0 , I GG , I G D , I RS , and I YG are not QL ′ -implications since all of them have N D1 as their natural negation.
Although it seems simple, Proposition 12 is very important. I 3 is a QL ′ -implication, which is obviously not a QL-implication since its natural negation is N D1 . So, it is proved that the family of QL ′ -implications is not the same with the QL-implications' family.

Proposition 13. By a triple of the form (T, S, N D2 ), where T is any t-norm and S is any t-conorm, a QL ′ -implication is
obtained, which is I 4 (see [22]).
Proof. By a triple of the form (T, S, N D2 ), where T is any tnorm and S is any t-conorm, we obtain Remark 12. Because of Proposition 13, we conclude that the only QL ′ -implication, which has N D2 as its natural negation, is I 4 . So, automatically, we conclude that I 1 and I WB are not QL ′ -implications since all of them have N D2 as their natural negation. e question we need to answer is the relation between these families (QL and QL ′ ). An answer is given partially in Proposition 9 where it is proved that if I N,T,S satisfies (21), then N is a strong negation. Clearly by eorem 7, if N is a strong negation, then the corresponding QL ′ -operation (respectively, implication) is also a QL-operation (respectively, implication) and vice versa. Moreover, for nonstrong negations, the corresponding QL-operations (respectively, implications) satisfy (21) (see Proposition 2.6.2 of [4]), and the corresponding QL ′ -operations (respectively, implications) do not satisfy (21). All these results lead to Figure 3.
In the following of this section, we present a useful theorem for the disqualification of some triples (T, S, N) that do not generate QL ′ -implications. (T, S, N), where S is any tconorm, N is any continuous nonfilling fuzzy negation, and T is any idempotent, strict, or positive t-norm, any QL ′ -implication cannot be obtained.

Proof.
e proof is similar to the proof of eorem 5. It results by Remark 10.
A very useful proposition is as follows.

Proposition 14.
By a triple of the form (T D , S, N C ), where S is any idempotent or positive t-conorm, a QL ′ -implication is obtained, which is I DP .
Proof. As we mentioned before, the only idempotent tconorm is S M , which is also positive. By the triple (T D , S, N C ), where S is any positive t-conorm, we obtain Lastly, in this section, we present Table 6, which contains basic QL ′ -implications.

R9-Implications
e obtained generalization via De Morgan's laws of R-implications leads to the next definition, which refers to the family of R ′ -operations. is generalization is a counterexample for the fact that axioms (13)- (17) are not invariant via an application of De Morgan's laws. To be more precise, axiom (17) is not invariant since R ′ -operations violate it, as we will show in the following. is is the reason we use the term operations, rather than implications.  (N(x), N(t))) ≤ y , If I is an R ′ -operation generated by a t-conorm S and a fuzzy negation N, then we denote it by I N S . e induced family of R ′ -operations is a special case of residual implicator of the conjuctor C(x, y) � N(S (N(x), N(y)), where N is a fuzzy negation and S is a tconorm (see [23]). Also, this formula is a special case of N is not a strong negation, and (21) is satisfied N is a strong negation N is not a strong negation, and (21) is not satisfied formula (2) in Lemma 1 of [24]. However, we will study this special family because of the crucial results we will obtain.
which means that we have to show the inclusion So, for any t ∈ [0, 1] such that N(S (N(y), N(t)) ≤ z, it is obvious that ⟹ N(S(N(x), N(t))) ≤ N(S(N(y), N(t))).
Secondly, we have to prove (14); hence, for x, y, z ∈ [0, 1], we have to show that if y ≤ z, then I N S (x, y) ≤ I N S (x, z), which is equivalent with the inequality which is obvious that it holds. erefore, I N S satisfies (14).
I N S satisfies (16) since since N(N(1)) � N(0) � 1. e formula of R ′ -operations is very complicated and makes their study really difficult. Furthermore, R ′ -operations do not always satisfy (17) since So, it is obvious that there does not exist any R ′ -implication generated by N D2 . All these results above lead to the next proposition without the proof because it is obvious. (17). e question arising is when an R ′ -operation is a fuzzy implication. e next theorem is the answer.

Theorem 12. Let I N S be an R ′ -operation. I N S is an R ′ -implication if and only if the negation N satisfies the equivalence
Proof. If the negation N satisfies the equivalence N(N(x)) � 0 ⟺ x � 0, then I N S satisfies (17) since Hence, by Proposition 15, we deduce that I N S is a fuzzy implication. On the contrary, if I N S is an R ′ -implication, then (17) is satisfied. Hence, Remark 13. By eorem 12, we deduce one more time that there does not exist any R ′ -implication generated by N D2 since N D2 (N D2 (x)) � 0 ⟺ x < 1.

Proposition 16.
e R′-implication genrated from N D1 and any t-conorm S is I 0 . Proof.
e R′-implication genrated from N D1 and any tconorm S is Proof. Let I N S be an R ′ -operation, which satisfies (21); then, we have for any x ∈ [0, 1].

Theorem 14.
If N is a strictly decreasing nonstrong negation, then the obtained I N S is an R ′ -implication, which does not satisfy (21).
Proof. Let I N S be an R ′ -operation. Since N is a strictly decreasing nonstrong negation, we have that So, by eorem 12, we deduce that I N S is an R ′ -implication. If we assume that I N S satisfies (21), then by eorem 13, we have N (N(x)) ≤ x, for any x ∈ [0, 1]. Since N is not a strong negation, there is at least one x 0 ∈ (0, 1) such that N (N(x 0 )) ≠ x 0 , so N(N(x 0 )) < x 0 . N is a strictly decreasing function, so N(N(N(x 0 ))) > N(x 0 ), which means there is y 0 � N(x 0 ) ∈ (0, 1) such that N(N(y 0 )) > y 0 , which is a contradiction. us, I N S does not satisfy (21).

Theorem 15.
If N is a strong negation and I N S is an R ′ -implication, then I N S � I T , where the t-norm T is the Ndual of t-conorm S. Proof.
e proof is obvious.
eorem 15 leads us to have no interest about R ′ -implications generated by strong negations. Moreover, we have to mention that R-implications always satisfy (21) (see eorem 2.5.4 of [4]). So, some results are visualized in Figure 4.

Theorem 16. If ϕ ∈ Φ and I N
S is an R ′ -implication, then (I N S ) ϕ is an R ′ -implication. Moreover, Proof. Let I N S be an R ′ -implication; then, (I N S ) ϕ is an R ′ -implication according to Remark 4. Now, from the continuity of the bijection ϕ, we have for any x, y ∈ [0, 1].
Lastly, in this section, we present Table 7, which contains basic R ′ -implications. e formulas of R ′ -implications which are generated from N C are calculated by eorem 15 and Table 2

Some Results before Conclusions
According to Figures 1-3, we know that, for nonstrong negations, (T, N)-, D ′ -, and QL ′ -implications do not satisfy (21). e same result holds for R ′ -implications when we use a strictly decreasing negation N according to eorem 14. On the contrary, (S, N)-, D-, QL-, and R-implications always satisfy (21). Furthermore, there are other known families of fuzzy implications, such as (i) Yager's f-generated and g-generated implications as they are defined by Yager [12] (see also [4,9]), (ii) h-implications as they are defined by Jayaram [9,25,26], (iii) h-implications as they are defined by Massanet and Torrens [10], and (iv) Fuzzy implications through fuzzy negations as they are defined by Souliotis and Papadopoulos [11].
Lastly, we investigate the relation between the families of (T, N)-, D ′ -, QL ′ -, and R ′ -implications. As sets of operations, they are different since (T, N)-implications satisfy (13)- (17), D ′ -operations do not always satisfy (14), QL ′ -operations do not always satisfy (13), and R ′ -operations do not always satisfy (17). Now, let S λ SS be the Schweizer-Sklar t-conorm; then, for λ � 2, it is e obtained QL-implication by the triple (T P , S 2 SS , N C ) is Since the natural negation of I PC is the strong negation N C , obviously, I PC is not a (T, N)-implication, and it is a QL ′ -implication. More specifically, so the D ′ -implication I N C ,T 2 SS ,S P � I PC ′ is not an (S, N)-implication; hence, it is not a (T, N)-implication.
On the contrary, I LK belongs to all families that are generated and defined in this paper and those that are mentioned in Table 4 since (96) Moreover, it is obvious that D ′ -implications are the reciprocals of QL ′ -implications for strong negations since they are the same sets with their preliminary families, respectively. Surprisingly, sometimes, this property holds for nonstrong negations (for example, I 3 and I 4 in Tables 6 and  7). e exact relation of these two families needs more investigation. e same happens for the relation between Rand R ′ -implications.
Finally, some of these results are presented in Figures 5-7, where we can see that the intersection between (T, N)-, D ′ -, QL ′ -, and R ′ -implications is nonempty. Moreover, D ′ -, QL ′ -, and R ′ -implications contain at least one fuzzy implication that (T, N)-implications do not contain.

Conclusions
Many families of fuzzy implications can be produced by well-known generalizations of the notion of implication from classical to fuzzy logic. Moreover, there are fuzzy implications, which, in their formula, contain at least a tnorm (such as R-implications) or a t-conorm (such as (S, N)-implications). De Morgan's laws and, if necessary, the classical law of double negation are useful tools to transform these already known families of fuzzy implications. As a result, new families of fuzzy implications are arising. ese new families are the "closest" of their preliminary-"creator" families. At this point, we should remark that the preliminary and the induced family of fuzzy implications are different sets, at least in any family we mention in this paper. e induced family is an expansion of its preliminary family, in a field where all these fuzzy implications simulate the same (or a similar) way of classical thinking.
In this paper, we specifically mentioned a known family, the so-called (T, N)-implications, and we have studied three new families, the so-called D′-, QL ′ -, and R ′ -implications.  Figure 4: Intersection between families of R-and R′-operations (respectively, implications). e first three families give us the following results. For strong negations, the sets of (S, N)and (T, N)-implications are the same set. e same result holds for the sets of QL-and QL ′ -implications and for D-and D ′ -implications, respectively. However, if the negation we use is not strong, then the preliminary families (S, N)-, QL-, and D-implications satisfy (21), and the corresponding induced families (T, N)-, QL ′ -, and D ′ -implications do not satisfy (21). Although these new families are common with their preliminary families when N is strong, they have no common implications when N is not strong. ese families are very important since they are the "first" generalized generators from classical to fuzzy logic, where for nonstrong negations, (21) is not satisfied.
Furthermore, the last family, the so-called R ′ -implications, gives the following results. If we use a strong negation N in the formula of R ′ -implications, then R-and R ′ -implications are the same set. Moreover, for strictly decreasing and nonstrong negations N, the obtained R ′ -operations are fuzzy implications that do not satisfy (21), a property that R-implications always satisfy. is family is a counterexample that axioms (13)- (17) are not invariant, or, more specifically, axiom (17) is surely not invariant, via the application of De Morgan's laws.
However, the characterization of triples (T, S, N), such that a D ′ -operation or a QL ′ -operation is a fuzzy implication, is still unsolved. On the contrary, the condition under which an R ′ -operation is a fuzzy implication has been proved.
Another result is that the expected duality of the properties does not hold, in general, via this application of classical logic's laws, but under some conditions. Unfortunately, the induced families must be studied individually every time since there is no general theory that seems to hold for every induced family.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
D. S. G. contributed to writing the original draft. B. K. P. supervised the study and contributed to writing, reviewing, and editing. Both authors read and agreed to the published version of the manuscript.