On the Boolean-like Law I(x, I(y, x)) = 1

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 22, No. 2 (2014) 205–215 c World Scientific Publishing Company DOI: 10.1142/S0218488514500093 On the Boolean-like Law I(x, I(y, x)) = 1 Anderson Cruz Metrópole Digital Institute Federal University of Rio Grande do Norte – UFRN, Natal, Rio Grande do Norte, Brazil 59072-970 anderson@imd.ufrn.br Benjamı́n Bedregal∗ and Regivan Santiago† Group for Logic, Language, Information, Theory and Applications – LoLITA, Department of Informatics and Applied Mathematics – DIMAp, Federal University of Rio Grande do Norte – UFRN, Natal, Rio Grande do Norte, Brazil 59072-970 ∗ bedregal@dimap.ufrn.br † regivan@dimap.ufrn.br Received 6 July 2012 Revised 10 October 2013 The well-known Boolean-law “α → (β → α) = 1” can be generalized to fuzzy context as I(x, I(y, x)) = 1, where I is a fuzzy implication. In this paper we show the necessary and sufficient conditions under which this generalization holds in fuzzy logics. We focus the investigation on the following classes of fuzzy implication: (S, N )-, R-, QL-, D- and (N, T )-implications. In addition, we demonstrate that a fuzzy implication I satisfies such Boolean-like law if, and only if, its Φ-conjugate also satisfies it. Keywords: Fuzzy implication; Boolean-like laws; automorphisms. 1. Introduction Boolean laws have been generalized and studied as functional equations or inequations in fuzzy logics. These are called Boolean-like laws and they are not usually satisfied in any standard structure ([0, 1], T, S, N, I), where T is a t-norm, S is a t-conorm, N is a fuzzy negation and I is a fuzzy implication. In this scenario, researchers have proposed investigations about the conditions under which a Booleanlike law holds (e.g. Refs. 1–5). This paper reports a further investigation exposed in.6 Here, we will produce necessary and sufficient conditions under which the Booleanlike law (1) holds for the following classes of fuzzy implications: (S, N )-, R-, QL-, D- and (N, T )-implications. I(x, I(y, x)) = 1, for all x, y ∈ [0, 1]. 205 (1) 206 A. Cruz, B. Bedregal & R. Santiago From the formal logic point of view, this law is the fuzzy rewriting of the classical, intuitionistic, Lukasiewicz axiom “x ⇒ (y ⇒ x)” (*).a In Boolean logics, (*) is called “Weakening”, since it is said to be the weakening of “x ⇒ x”. In a similar sense, assuming “⇒” as the material implication, we observe that (*) is also a weakening formalization of Aristotle’s Law of Excluded Middle — “¬x ∨ x” — since, “x ⇒ (y ⇒ x)” would be equivalent to “¬x ∨ (¬y ∨ x)”.b Another theoretical justification to investigate (1) lies in the fact that definitions of fuzzy implication try to comprehend the correct notion (common sense) of the actual essence of a logical implication. Therefore, the structure of fuzzy implication classes will provide some information about what about the meaning of an implication in fuzzy context. Besides, the definition of such classes is generally motivated by the implication notion in distinct logics, for example: (S, N )-, Rand QL-implication generalize the implication of classical, intuitionistic and quantum logics, respectively. Hence, the investigation of conditions under which logical laws hold in each of those classes also contributes to characterize the approximate reasoning in accordance with each implication notion. Moreover, since (*) is valid in fundamental logical systems, it is likely that a Fuzzy Rule-Based System (FRBS) must consider such property true in every case. Consequently, this system must use fuzzy operators that guarantee the validity of (*). Therefore, an investigation about the validity of such statement in fuzzy logics, regarding distinct classes of fuzzy implications, also contributes the selection of a correct and more appropriate semantics (T, S, N, I) for a FRBS. The rest of this paper is organized as follows. Section 2 recalls basic definitions and some properties of t-norms, t-conorms and fuzzy negations. Section 3 does the same with respect to (w.r.t., for short) fuzzy implications, their classes — namely: (S, N )-, R-, QL-, D- and (N, T )-implications — and automorphisms on fuzzy implications. Section 4 works out on the sufficient and necessary conditions under which those classes and Φ-conjugate implications satisfy (1). Finally, we present a discussion about the results of this paper in Sec. 5. 2. Preliminaries In order to make this paper self-contained, this section summarizes some basic definitions. Definition 1. A function T : [0, 1]2 → [0, 1] is a t-norm if it satisfies Commutativity (T1), Associativity (T2), Monoticity (T3) and 1-identity (T4) — T (1, y) = y, where y ∈ [0, 1]. Remark 1. The minimun t-norm TM is the only idempotent t-norm.7 I.e. TM (x, x) = x, for all x ∈ [0, 1]. a “x ⇒ (y ⇒ x)” is also a theorem in basic and product logics. note that “x ⇒ x” would be equivalent to “¬x ∨ x”. b Also On the Boolean-like Law I(x, I(y, x)) = 1 207 Definition 2. A function S : [0, 1]2 → [0, 1] is a t-conorm if it satisfies Commutativity (S1), Associativity (S2), Monoticity (S3) and 0-identity (S4) — S(0, y) = y, where y ∈ [0, 1]. Remark 2. c Considering the partial order on the family of all t-conorms induced from the order on [0,1] on the family of all t-conorms, SM (x, y) = max(x, y) is the least t-conorm. So for any t-conorm S, S(x, y) ≥ SM (x, y) ≥ y. Moreover, SM (x, 1) = SM (1, x) = 1, so S(x, 1) = S(1, x) = 1 for any t-conorm S. Definition 3. A function N : [0, 1]2 → [0, 1] is a fuzzy negation if N (0) = 1 and N (1) = 0 (N1), and N is non-increasing (N2). Besides, a fuzzy negation is called strong, if it is involutive, i.e., N (N (x)) = x, such that x ∈ [0, 1]. As examples of fuzzy negations, we cite the least fuzzy negation N⊥ and the greatest fuzzy negation N⊤ : ( 1, if x = 0 N⊥ (x) = (2) 0, if x ∈]0, 1]; ( 0, if x = 1 N⊤ (x) = (3) 1, if x ∈ [0, 1[ . A t-conorm S is distributive over a t-norm T if S(x, T (y, z)) = T (S(x, y), S(x, z)), for all x, y, z ∈ [0, 1] . (4) An important result about distributivity in Boolean-like laws is given next. Proposition 1.7 Let T be a t-norm and S a t-conorm, S is distributive over T iff T = TM . Let S be a t-conorm and N a fuzzy negation, the pair (S, N ) satisfies the Law of Excluded Middle (LEM, for short) if S(N (x), x) = 1, for all x ∈ [0, 1]. (LEM) Note that any t-conorm S with N⊤ satisfies (LEM). Also, no t-conorm with the N⊥ satisfies (LEM).8 3. Fuzzy Implications Currently, there are different acceptable definitions of fuzzy implications (e.g. Refs. 4, 9 and 10). In this paper, we opt for a well-accepted definition: Definition 4. A function I : [0, 1]2 → [0, 1] is called a fuzzy implication if it satisfies: c Some demonstrations will refer this remark. 208 A. Cruz, B. Bedregal & R. Santiago I1. Boundary conditions: I(0, 0) = I(0, 1) = I(1, 1) = 1 and I(1, 0) = 0. Other potential properties are acceptable for some fuzzy implications: I2. I3. I4. I5. I6. Left antitonicity: if x1 ≤ x2 then I(x1 , y) ≥ I(x2 , y), for all x1 , x2 , y ∈ [0, 1]; Right isotonicity: if y1 ≤ y2 then I(x, y1 ) ≤ I(x, y2 ), for all x, y1 , y2 ∈ [0, 1]; Left boundary condition: I(0, y) = 1, for all y ∈ [0, 1]; Right boundary condition: I(x, 1) = 1, for all x ∈ [0, 1]; Identity property: I(x, x) = 1, for all x ∈ [0, 1]. There is also a property that relates fuzzy implications and negations: I7. Contrapositive: I(x, y) = I(N (y), N (x)), for all x, y, ∈ [0, 1] and N being a fuzzy negation. There are three main classes of fuzzy implications, namely: (S, N )-, R- and QL-implications. Other classes can be generated from those ones, namely: D- and (N, T )-implications are generated from QL- and (S, N )-implications, respectively. In the following definition we are going to recall them. Definition 5. Let T be a t-norm, S a t-conorm and N a fuzzy negation, then: • A function I : [0, 1]2 → [0, 1] is called an (S, N )-implication (denoted by IS,N ) if I(x, y) = S(N (x), y) . (5) • A function I : [0, 1]2 → [0, 1] is called an R-implication (denoted by IT ) if I(x, y) = sup{t ∈ [0, 1] | T (x, t) ≤ y} . (6) • A function I : [0, 1]2 → [0, 1] is called a QL-implication (denoted by IS,N,T ) if I(x, y) = S(N (x), T (x, y)) . (7) 2 • A function I : [0, 1] → [0, 1] is called a D-implication (denoted by IS,T,N ) if I(x, y) = S(T (N (x), N (y)), y) . (8) • A function I : [0, 1]2 → [0, 1] is called a (N, T )-implication (denoted by IN,T ) if I(x, y) = N (T (x, N (y))) . (9) Remark 3. (i) Every IT satisfies I1-I6.8,11 (ii) The concept of QL-implications and QL-operators can be found in a review of the literature. However, in this paper we do not have this distinction since our implication definition (Def. 4) considers only the boundary conditions (I1). (iii) Although D-implications are generally defined from strong negations,13–16 according to our definition (Def. 4) the D-operator defined from a non-strong negation is also a fuzzy implication, thus our D-implication definition does not require the strong negations. On the Boolean-like Law I(x, I(y, x)) = 1 209 (iv) Let N be a strong negation. IS,N = IN,T iff S is N -dual of T [pp. 139].17 D-implication is the contrapositive (I7) of QL-implication.12 Another way to obtain a D-implication from a QL-implication — or a QL- from a D-implication — is by (LEM) and (4) (see the following demonstrations). Lemma 1. Given a D-implication IS,T,N and a QL-implication IS,N,T . If (S, N ) satisfies (LEM ) and T = TM , then IS,T,N = IS,N,T . Proof. Since S is distributive over T (Eq. (8)) iff T = TM (Proposition 1), so: IS,T,N (x, y) = S(T (N (x), N (y)), y) by (8) = T (S(y, N (x)), S(y, N (y))) by (S1) and (4) = T (S(N (x), y), S(N (x), x)) by (S1) and (LEM ) = S(N (x), T (y, x)) by (4) = IS,N,T (x, y) by (T 1) and (7). Proposition 2. Given a D-implication IS,T,N and a QL-implication IS,N,T . If IS,N,T = IS,T,N then (S, N ) satisfies (LEM ). Proof. If IS,N,T = IS,T,N , then S(N (0), T (0, y)) = S(T (N (0), N (y)), y). On the left side, S(N (0), T (0, y)) = 1 and on the other side S(T (N (0), N (y)), y) = S(N (y), y). Hence S(N (y), y) = 1, i.e., (S, N ) satisfies (LEM). It is possible to generate fuzzy implications using automorphisms: Let Φ be the set of automormophisms, fuzzy implications I and J are Φ-conjugate,8 if there exists a ϕ ∈ Φ such that J = Iϕ , where Iϕ (x, y) = ϕ−1 (I(ϕ(x), ϕ(y))), for all x, y ∈ [0, 1]. (10) 4. Solutions of (1) in (S, N )-, R-, QL-, D- and (N, T )-Implications 4.1. (S, N )-implications Theorem 1. An (S, N )-implication IS,N satisfies (1) iff (S, N ) satisfies (LEM). Proof. ⇐: IS,N (x, IS,N (y, x)) = S(N (x), S(N (y), x)) by Eq. (5) = S(N (x), S(x, N (y))) by S1 = S(S(N (x), x), N (y)) by S2 = S(1, N (y)) =1 by (LEM) by Remark 2. ⇒: For all x ∈ [0, 1], I(x, I(1, x)) = 1 and I(1, x) = S(N (1), x) = S(0, x) = x, so S(N (x), x) = I(x, I(1, x)) = 1. Hence (S, N ) satisfies (LEM ). 210 A. Cruz, B. Bedregal & R. Santiago 4.2. R-implications Recognizably, the theory of residuated lattices demonstrates that when I is an Rimplication defined from a left-continuous t-norm T , in which, case (T, I) is an adjoint pair,8 then I satisfies (1). In the following theorem, we prove that the leftcontinuity of underlying t-norm is not required to an R-implication satisfy (1). Theorem 2. Every R-implication satisfies (1). Proof. Let T be any t-norm and IT an R-implication generated from it. Fixing arbitrarily x, y ∈ [0, 1], it is well known that T (y, x) ≤ min(y, x) ≤ x, so IT (y, x) = sup{t ∈ [0, 1] | T (y, t) ≤ x} ≥ x. We have already seen (Remark 3(i)) that Rimplications satisfy I3 and I6.8 Therefore IT (x, IT (y, x)) ≥ IT (x, x) = 1. Hence IT (x, IT (y, x)) = 1. 4.3. QL-implications Lemma 2. A QL-implication IS,N,T satisfies (1), whenever (S, N ) satisfies (LEM) and T = TM . Proof. Since T = TM iff S is distributive over T (Proposition 1), then I(x, I(y, x)) = S(N (x), T (x, S(N (y), T (y, x)))) by Eq. (7) = S(N (x), T (x, T (S(N (y), y), S(N (y), x)))) by (4) = S(N (x), T (x, T (1, S(N (y), x)))) = S(N (x), T (x, S(N (y), x))) by (LEM) by T 4 = T (S(N (x), x), S(N (x), S(N (y), x))) = T (1, S(N (x), S(x, N (y)))) by (4) by (LEM) and S1 = S(S(N (x), x), N (y)) = S(1, N (y)) by T 4 and S2 by (LEM) = 1 by Remark 2. Lemma 3. If a QL-implication IS,N,T generated by a strictly increasing t-conorm S in [0, 1[, a t-norm T and a fuzzy negation N satisfies (1), then (S, N ) satisfies (LEM ) and T = TM . Proof. Assume that IS,N,T is a QL-implication which satisfies (1), so IS,N,T (1, IS,N,T (y, 1)) = 1. Therefore, for any y ∈ [0, 1], S(N (1), T (1, S(N (y), T (y, 1)))) = 1 and: S(N (1), T (1, S(N (y), T (y, 1)))) = S(0, T (1, S(N (y), T (1, y)))) by N 1 and T 1 = S(N (y), y) by S4 and T 4. On the Boolean-like Law I(x, I(y, x)) = 1 211 Thus S(N (y), y) = 1. Hence (S, N ) satisfies (LEM). Moreover since IS,N,T is a QL-implication which satisfies (1) then IS,N,T (x, IS,N,T (1, x)) = 1. Therefore, for any x ∈ [0, 1], S(N (x), T (x, S(N (1), T (1, x)))) = 1 and: S(N (x), T (x, S(N (1), T (1, x)))) = S(N (x), T (x, S(0, T (1, x)))) by N 1 = S(N (x), T (x, T (1, x))) = S(N (x), T (x, x)) by S4 by T 4. Thus S(N (x), T (x, x)) = 1. Since (S, N ) satisfies (LEM), for any x ∈ [0, 1], S(N (x), T (x, x)) = 1 = S(N (x), x). Case x = 1, so, trivially T (x, x) = x. Case x < 1, since S is strictly increasing in [0,1[, S(N (x), T (x, x)) = S(N (x), x) implies T (x, x) = x. Therefore T (x, x) = x for any x ∈ [0, 1], i.e., T is an idempotent t-norm. By Remark 1, TM is the only idempotent t-norm. Hence, if I is a QLimplication — generated by a strictly increasing t-conorm S in [0,1[, a t-norm T and a fuzzy negation N — that satisfies (1) then (S, N ) satisfies (LEM) and so T = TM . Theorem 3. Let IS,N,T be a QL-implication generated by a strictly increasing tconorm S in [0, 1[, a fuzzy negation N and a t-norm T . I satisfies (1) iff (S, N ) satisfies (LEM ) and T = TM . Proof. Straightforward from Lemmas 2 and 3. 4.4. D-implications Corollary 1. Let IS,T,N be a D-implication generated by a strictly increasing tconorm S in [0, 1[, a t-norm T and a fuzzy negation N such that (I, N ) satisfies the contrapositive (I7). Then I satisfies (1) iff (S, N ) satisfies (LEM ) and T = TM . Proof. Since D-implication is the contrapositive of QL-implication and by Theorem 3. The result of the previous theorem is fairly trivial. But are there other conditions to guarantee that (1) will be satisfied for D-implications? The following lemmas and theorem answer this question positively. Lemma 4. Given a D-implication IS,T,N generated by a strictly increasing tconorm S in [0, 1[, a t-norm T and a fuzzy negation N . I satisfies (1), if (S, N ) satisfies (LEM ) and T = TM . Proof. Straightforward from Lemma 1 and Theorem 3. Lemma 5. Given a D-implication IS,T,N generated by a strictly increasing tconorm S in [0, 1[, a t-norm T and a continuous fuzzy negation N . If IS,T,N satisfies (1), then (S, N ) satisfies (LEM ) and T = TM . 212 A. Cruz, B. Bedregal & R. Santiago Proof. Let IS,T,N be a D-implication which satisfies (1), so IS,T,N (0, IS,T,N (y, 0)) = 1. Therefore, S(T (N (0), N (S(T (N (y), N (0)), 0))), S(T (N (y), N (0)), 0)) = 1 for any y ∈ [0, 1]. By N1, T4 and S4, S(T (N (y), N (0)), 0) = N (y) (∗), so: S(T (N (0), N (S(T (N (y), N (0)), 0))), S(T (N (y), N (0)), 0)) = S(T (N (0), N (N (y))), N (y)) by (∗) = S(N (N (y)), N (y)) by N 1 and T 4. Since N is continuous, then for all y ′ ∈ [0, 1] there exists y ∈ [0, 1] such that N (y) = y ′ . Therefore S(N (y ′ ), y ′ ) = S(N (N (y)), N (y)) = 1. Hence (S, N ) satisfies (LEM). Again by (1), IS,T,N (x, IS,T,N (1, x)) = 1, and IS,T,N (1, x) = S(T (0, N (x)), x) = x. Then IS,T,N (x, x) = 1 = S(T (N (x), N (x)), x). It is known that T (N (x), N (x)) ≤ N (x). If T (N (x), N (x)) < N (x) and since (S, N ) satisfies (LEM) and S is strictly increasing in [0,1[, then S(T (N (x), N (x)), x) < S(N (x), x) = 1. However, S(T (N (x), N (x)), x) = 1. So T (N (x), N (x)) must not be less than N (x). Thus T (N (x), N (x)) = N (x). Since N is continuous, for all x′ ∈ [0, 1] there exists x ∈ [0, 1] such that N (x) = x′ . Therefore T (x′ , x′ ) = x′ for all x′ ∈ [0, 1], i.e. T is an idempotent t-norm. Hence T = TM — by Remark 1. Theorem 4. Let IS,T,N be a D-implication generated by a strictly increasing tconorm S in [0, 1[, a t-norm T and a continuous fuzzy negation N . IS,T,N satisfies (1) iff (S, N ) satisfies (LEM) and T = TM . Proof. Straightforward from Lemmas 4 and 5. 4.5. (N, T )-implications Theorem 5. Let N be a strong negation and T a t-norm. An (N, T )-implication IN,T satisfies (1) iff for the N -dual t-conorm S of T , (S, N ) satisfies (LEM). Proof. Straightforward from Remark 3(iv) and Theorem 1. 4.6. Automorphisms Theorem 6. Let ϕ ∈ Φ. A fuzzy implication I satisfies (1) iff Iϕ satisfies (1). Proof. ⇒: Assume that ϕ ∈ Φ and a fuzzy implication I satisfies (1). Now by (10) we have that Iϕ (x, Iϕ (y, x)) = ϕ−1 (I(ϕ(x), ϕ ◦ ϕ−1 (I(ϕ(y), ϕ(x))))) and trivially ϕ−1 (I(ϕ(x), ϕ ◦ ϕ−1 (I(ϕ(y), ϕ(x))))) = ϕ−1 (I(ϕ(x), I(ϕ(y), ϕ(x)))). By (1), I(ϕ(x), I(ϕ(y), ϕ(x))) = 1, so ϕ−1 (I(ϕ(x), I(ϕ(y), ϕ(x)))) = ϕ−1 (1) = 1. Therefore, Iϕ (x, Iϕ (y, x)) = 1. Hence, Iϕ satisfies (1). The converse follows straightforward from ϕ−1 ((Iϕ )) = I. On the Boolean-like Law I(x, I(y, x)) = 1 213 5. Final Remarks This paper provides necessary and sufficient conditions under which the Booleanlike law I(x, I(y, x)) = 1, referred by (1), holds for (S, N )-, R-, QL-, D- and (N, T )implications. The main results of this paper are demonstrated in Theorems 1, 2, 3, 4, 5 and 6, and Corollary 1. These results show that a generalization of a classical implication to a fuzzy context — an (S, N )-implication — satisfies (1) iff (S, N ) satisfies (LEM). The dual implication of this generalization must obey similar necessary and sufficient conditions: A (N, T )-implication satisfies (1) iff N is strong and for a N -dual of T t-conorm S, (S, N ) satisfies (LEM). Every intuitionistic implication generalized to a fuzzy context — an R-implication — satisfies (1). However, a quantum implication in fuzzy context — QL-implication — satisfies (1) iff (S, N ) satisfies (LEM) and T = TM , regarding that its underlying t-conorm must be strictly increasing in [0,1[. The contrapositive of the QL-implication — D-implication — has an additional condition: its underlying negation must be continuous. Table 1 summarizes these results. Those observations lead us to conclude that (1) cannot be indiscriminately adopted in any computational system based on fuzzy concepts. In addition to the property (1) being adopted in Fuzzy-Based Systems, other properties must be considered. On the one hand, a relevant contribution of this paper is to show which properties and fuzzy operators, software engineers have to choose in order to consider (1) as true in their computational system. On the other hand, in a more general scenario, this paper contributes to understand what an implication actually means in the truth domain [0,1] and it also contributes to determine the meaning behind each fuzzy implication class, i.e., which properties they accept. In the introduction section, we describe the Boolean relation among LEM, “x ⇒ (y ⇒ x)” and “x ⇒ x”. In a fuzzy context, this is a relation among (1), (LEM) and (I6) which can be analyzed in each fuzzy implication class: • IS,N satisfies (I6) iff it satisfies (LEM) iff it satisfies (1). • Every R-implication satisfies (I6) and (1). • IS,N,T satisfies (I6), if N = NT or T = TM or S = SD .8 For any S, (S, N⊤ ) satisfies (LEM); and (SD , N) satisfies (LEM), such that N(x) = 0 iff x = 1.8 Moreover, IS,N⊤ ,T satisfies (1) and if (S, N ) satisfies (LEM) then IS,N ,TM also satisfies (1).d • IT,S,N satisfies (I6), if T = TM and (S, N ) satisfies (LEM). Regarding this same sufficient condition, IT,S,N also satisfies (1). • If N is strong and there is an S such that S is N -dual of T , then: (S, N ) satisfies (LEM) iff IN,T satisfies (1); and (S, N ) satisfies (LEM) iff IN,T satisfies (I6). dI SD ,N,T does not satisfy (1) for any T . 214 A. Cruz, B. Bedregal & R. Santiago Finally, we have also demonstrated that if a new implication is defined through an automorphism, this new implication will satisfy (1) iff the original fuzzy implication also satisfies it — Theorem 6. Table 1. Summary of paper results. Implication Class Sufficient and Necessary Conditions to Satisfy (1) (S, N )-implication (1) iff (S, N ) satisfies (LEM). R-implication Every R-implication satisfies (1). QL-implication S is strictly increasing in ]0,1]: (1) iff (S, N ) satisfies (LEM) and T = TM . D-implication S is strictly increasing in ]0,1] and (I, N ) satisfies (I7): (1) iff (S, N ) satisfies (LEM) and T = TM ; or S is strictly increasing in ]0,1] and N is continuous: (1) iff (S, N ) satisfies (LEM) and T = TM . (N, T )-implication N is strong: (1) iff (S, N ) satisfies LEM, for a N -dual of T t-conorm S. Φ-conjugate implications I satisfies (1) iff Iϕ satisfies (1). Acknowledgements This work was supported by the Brazilian Funding Agency CNPq (under the process numbers 480832/2011-0 and 307681/2012-2). We are very grateful to the anonymous referees for their valuable comments and suggestions that helped us to improve the paper. References 1. C. Alsina and E. Trillas, On iterative Boolean-like laws of fuzzy sets, in EUSFLAT Conf., 2005, pp. 389–394. 2. E. Trillas and C. Alsina, On the law [(p ∨ q) → r] = [(p → r) ∧ (q → r)] in fuzzy logic, IEEE Trans. Fuzzy Systems 10 (2002) 84–88. 3. J. Balasubramaniam and C. J. M. Rao, On the distributivity of implication operators over T and S norms, IEEE Trans. Fuzzy Systems 12 (2004) 194–198. 4. Y. Shi, D. Ruan and E. E. Kerre, On the characterizations of fuzzy implications satisfying I(x, y) = I(x, I(x, y)), Information Sciences 177 (2007) 2954–2970. 5. B. Jayaram, On the law of importation (x ∧ y) → z ≡ (x → (y → z)) in fuzzy logic, IEEE Trans. Fuzzy Systems 16 (2008) 130–144. 6. B. C. Bedregal and A. P. Cruz, A characterization of classic-like fuzzy semantics, Logic Journal of the IGPL 16 (2008) 357–370. 7. E. Klement, R. Mesiar and E. Pap, Triangular Norms (Kluwer Academic Publishers, Netherlands, 2000). On the Boolean-like Law I(x, I(y, x)) = 1 215 8. M. Baczyński and J. Balasubramaniam, Fuzzy Implications (Springer-Verlag, Poland, 2008). 9. J. Drewniak, Invariant fuzzy implications, Soft Comput. 10 (2006) 506–513. 10. E. Sainio, E. Turunen and R. Mesiar, A characterization of fuzzy implications generated by generalized quantifiers, Fuzzy Set and Systems 159 (2008) 491–499. 11. H. Bustince, P. Burillo and F. Soria, Automorphism, negations and implication operators, Fuzzy Sets and Systems 134 (2003) 209–229. 12. M. Mas, M. Monserrat, J. Torrens and E. Trillas, A survey on fuzzy implication functions, IEEE Trans. on Fuzzy Systems 15 (2007) 1107–1121. 13. E. Trillas, C. Alsina, E. Renedo and A. Pradera, On contra-symmetry and MPT conditionality in fuzzy logic, Int. J. Intelligent Systems 20 (2005) 313–326. 14. M. Mas, M. Monserrat and J. Torrens, QL-implications versus D-implications, Kybernetika 42 (2006) 315–366. 15. R. H. S. Reiser, B. C. Bedregal and G. P. Dimuro, Interval-valued D-implications, Tend. Mat. Apl. Comput. (T EM A) 10 (2009) 63–74. 16. R. H. S. Reiser, B. C. Bedregal, R. H. N. Santiago and G. P. Dimuro, Analyzing the relationship between interval-valued D-implications and interval-valued QL-implications, Tend. Mat. Apl. Comput. (T EM A) 11 (2010) 89–100. 17. B. Bedregal, A normal form which preserves tautologies and contradictions in a class of fuzzy logics, J. Algorithms 62 (2007) 135–147.