- Email: [email protected]

Which logic is the real fuzzy logic?夡 Vilém Nováka, b,∗ a Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic b Institute of Information and Automation Theory, Academy of Sciences of the Czech Republic, Pod vodárenskou vˇeží 4,

186 02 Praha 8, Czech Republic Available online 2 November 2005

Abstract This paper is a contribution to the discussion of the problem, whether there is a fuzzy logic that can be considered as the real fuzzy logic. We give reasons for taking IMTL, BL, Ł and EvŁ (fuzzy logic with evaluated syntax) as those fuzzy logics that should be indeed taken as the real fuzzy logics. © 2005 Elsevier B.V. All rights reserved. Keywords: Mathematical fuzzy logic; Fuzzy type theory; Fuzzy logic with evaluated syntax; Algebra of truth values

1. What makes many-valued logic a fuzzy logic In this position paper, I will discuss a question that arose during several discussions that took place at conferences in Vienna 2004 and Linz 2005: What, in fact, is fuzzy logic? After the famous book of Hájek [9], it turned out that there are many well established formal systems of fuzzy logic (FL) that may well claim to be “the real” fuzzy logic. I want to show that among many possible fuzzy logic systems we can ﬁnd most outstanding ones that are capable at fulﬁlling the “agenda of fuzzy logic”. Of course, one may hardly expect just one fuzzy logic but still, a few out of many, may be indeed picked up. The history of FL has been nicely summarized by Hájek in [10], and so I will not repeat it. Instead, I will start with the following informal characterization: Fuzzy logic is a special many-valued logic addressing the vagueness phenomenon and developing tools for its modeling via truth degrees taken from an ordered scale. It is expected to preserve as many properties of classical logic as possible. No concrete formal system is mentioned in this characterization, but it emphasizes well the aim of FL and pre-determines the way how a suitable system of initially many-valued logic should be developed. The leading concepts are degrees (of truth) and vagueness. The category of vagueness has been discussed enough already in [22] (cf. also [25]), and so we will only very brieﬂy remember the following.

夡 The

ˇ paper has been supported by the project MSM 6198898701 of the MŠMTCR.

∗ Corresponding author at: Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1,

Czech Republic. Tel.: +420 6961 60234; fax: +420 6961 20478. E-mail address: [email protected] 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.10.010

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The vagueness phenomenon rises when trying to group together objects that have a certain property . The result is an actualized grouping of objects; we can write it as X = {o | o is an object having the property }.

(1)

In general, X in (1) cannot be taken as a set since the property may be vague, i.e. it may be impossible to characterize the grouping X precisely and unambiguously, besides elements that either surely have the property (and thus, they surely belong to X) or surely do not have it; there can exist also borderline elements o for which it is unclear whether they have the property , or not. Consequently, it is not clear whether the borderline elements belong to X, or not. The assumption that X must be actualized (i.e. all its elements are at our disposal, at least in principle) is crucial since this is a core of how vagueness can be distinguished from uncertainty. Indeed, this difference corresponds to the difference between actuality and potentiality. While vagueness rises from the actualized non-sharply delineated groupings, uncertainty is encountered when dealing with still non-actualized groupings of objects. In the latter case, we speculate about the whole X, but only part of it truly exists. It makes sense to speak about the truth of the fact that some element belongs to a grouping of objects only when it is actualized (i.e. already existing). Indeed, let an object o be created (at least in our mind). If we learn that it has a property in (1), we know that it falls into (the existing part of) X, which means that we know the truth of the fact ‘o ∈ X’. In general, however, it is uncertain whether o will be created (will exist) or not, and so in the latter case we cannot speak about the truth of ‘o ∈ X’, but only about possibility, or probability of creation of o. 1 It follows from this discussion that the truth degrees provide a reasonable means for dealing with vagueness. On the other hand, they have little use alone: Imagine, e.g., the sentence “I love you in the degree 0.954867283. 2 ” Of course, nobody will ever say such a sentence. On the other hand, it is quite natural to say “I love you very much.” We may argue that while talking people implicitly use degrees. 3 Thus, it is important to compare the degrees. The concrete values are of little relevance; more important is their tendency, the shapes of the corresponding fuzzy sets. History teaches us that fuzzy logic offers a working mathematical model of the vagueness phenomenon and of situations where vagueness plays an important role. The following is required from FL: (a) FL should be a well established sound formal system to have its applications well justiﬁed. (b) FL should be an open system. It must be possible to extend it by new connectives and by generalized quantiﬁers. Moreover, some speciﬁc phenomena of natural language semantics should also be expressible, such as noncommutativity of conjunction and disjunction. (c) FL has a speciﬁc agenda, special techniques and concepts. Among them we can rank evaluating linguistic expressions, linguistic variable, fuzzy IF–THEN rules, fuzzy quantiﬁcation, defuzziﬁcation, fuzzy equality, etc. (cf. [23,30,31]). (d) FL should enable to develop special inference schemes including sophisticated inference schemes of human reasoning (e.g., compositional rule of inference, reasoning based natural language expressions, non-monotonic reasoning, abduction, etc.). Note that many of these requirements are already fulﬁlled by the available formal logical systems. The fundamental classiﬁcation of FL is fuzzy logic in narrow sense (FLn) and that in broad sense. The latter has been coined by Zadeh to denote all kinds of applications that use fuzzy sets. 4 Since this is too extensive, I have proposed in [19] (and elsewhere) a middle concept of fuzzy logic in broader sense (FLb) as an extension of FLn whose aim is to develop a formal theory of human way of reasoning that would include a mathematical model of the meaning of some expressions of natural language (evaluating linguistic expressions), the theory of generalized quantiﬁers and their use in human reasoning. The foreseen goal is to develop a formal logic that could be applied in human-like behaving robots.

1 Note that possibility means that o can be created, i.e. its creation is not in contradiction with some laws or principles, while probability means that o will be created though we may not be sure about it. 2 This part is based on the discussion with Hájek. 3 This argument is due to Di Nola (personal communication). 4 Personal communication as well as his talks at many international conferences.

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2. Two fundamental approaches to fuzzy logic in narrow sense We will now focus on fuzzy logic in narrow sense as a formal mathematical theory. Let us stress that the ﬁrst, mathematically deep and advanced formal system was published by Pavelka in [26]. 5 Surprisingly, his work went rather far and still it is a cause for dispute. Namely, though not stated explicitly in his work, he, in fact, established a limit generalization of logic by allowing evaluation of formulas also in syntax. After the seminal monograph of Hájek [9] (see also [24]), we now distinguish two fundamental approaches to the theory of mathematical fuzzy logic. 2.1. Fuzzy logic with traditional (classical) syntax This approach is less radical and it is promoted by many mathematicians, starting by Hájek and followed by Esteva, Gottwald, Godo, Montagna, Mundici, and others. The crucial question in fuzzy logic concerns the structure of truth values. It has been generally agreed that it must be a residuated lattice L = L, ∨, ∧, ⊗, →, 0, 1,

(2)

where L, ∨, ∧, 0, 1 is a bounded lattice, L, ⊗, 1 is a commutative monoid and → is a binary residuation operation joined with the product ⊗ by adjunction (a ⊗ b c iff a b → c). Adding various conditions, we obtain stronger structures (for details see [5,8,9,14,16,17,24,25] and elsewhere). Then, depending on the structure of truth values, we can distinguish basic fuzzy logic (BL), MTL-logic (MTL), IMTL-logic (IMTL), Łukasiewicz logic (Ł), Gödel logic (G), product logic , and others. A more signiﬁcant departure is Ł-logic whose structure of truth values has two products and two (different) implications. The list is by no means complete and there are various intermediate cases. All these logics generalize syntax of classical logic only in adding a new connective of strong conjunction (&) (and, possible, some other ones) and modifying special axioms. Furthermore, they have a ﬁnite list of schemes of logical axioms and inference rules of modus ponens and generalization. The fundamental concept of provability is classical, i.e. a formula A is provable if there exists its formal proof. Let V ⊆ FJ be a set of formulas and R be a set of inference rules. We say that V is closed with respect to r ∈ R if A1 , . . . , An ∈ V

implies r(A1 , . . . , An ) ∈ V

for all A1 , . . . , An ∈ dom(r). Then we may deﬁne a syntactic consequence operation by C syn (X) = {V ⊆ FJ | X ⊆ V , V is closed w.r.t. all r ∈ R}.

(3)

(4)

Theorem 1. For every formula A ∈ FJ , A ∈ C syn (X) iff A is provable from X. The above concepts are nicely generalized in Pavelka’s work outlined below. 2.2. Fuzzy logic with evaluated syntax As mentioned, Pavelka allowed in his work evaluation of formulas also in syntax simply by assuming that axioms may not be fully convincing, i.e., not fully true. Consequently, axioms may form a fuzzy set only. But this means a departure of syntax from the traditional conception. There is a good reason to call the resulting logic a logic with evaluated syntax. The fundamental concept of a formula is in this logic generalized to that of evaluated formula a A where A ∈ FJ is a formula and a ∈ L is its syntactic evaluation. Further basic principles are the following: (i) The designated truth values are replaced by the maximality principle: if the same formula is assigned more truth values, then its ﬁnal truth assignment is equal to the maximum (supremum) of all of them. Besides others, this means that all truth values are equally important. (ii) It is possible to make syntactical derivations concerning any truth value. 5 This was his dissertation defended already in 1976.

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It can be demonstrated that these principles lead to transparent generalization of classical logic both in semantics as well as in syntax. Let us stress that considering fuzzy sets of axioms is not merely a cheap generalization but it reﬂects the character of our knowledge when dealing with vagueness. This becomes apparent when analyzing the sorites paradox (a typical manifestation of vagueness) since this paradox is solved when considering the evaluated formula 1 − ε (∀n)(FN(n)⇒FN(n + 1)), ε > 0, as a special axiom where FN is a predicate expressing “n does not form a heap” 6 (for details see [12]). First of all, we must extend J by a set of logical (truth) constants that are names of all truth values {a | a ∈ L}. Note that this is a generalization of classical logic where we consider just the logical constants ⊥, . Furthermore, we must extend inference rules to manipulate with evaluated formulas. They assign a new evaluated formula to the given formulas ai Ai , i = 1, . . . , n, as follows: a1 A1 , . . . , an An r : evl , r (a1 , . . . , an ) r syn (A1 , . . . , An )

(5)

where r syn : FJn −→ FJ is a partial operation on formulas and r evl : Ln −→ L is a join-preserving evaluation operation on truth values. Note that in traditional syntax, we may also speak about evaluated formulas; however, each formula is evaluated either by 1 or by 0, i.e., it is either written down or not. Hence, we naturally introduce the concept of evaluated formal proof of an evaluated formula a A that is a sequence of evaluated formulas wA := a0 A0 , a1 A1 , . . . , an An (:= a A). The last evaluation an = a is a value, ValT (wA ) = an , of the proof wA . Let V ⊂ FJ be a fuzzy set of formulas and R be a set of inference rules of form (5). We say that V is closed with ∼ respect to r if V (r syn (A1 , . . . , An ))r evl (V (A1 ), . . . , V (An )) for all formulas A1 , . . . , An ∈ dom(r syn ). The fuzzy set of syntactic consequences of X ⊂ FJ is given by ∼ C syn (X)(A) = {V (A) | V ⊂ FJ , X V and V is closed w.r.t. to all r ∈ R} ∼

(6)

(cf. (6) with (4)). Theorem 2. Let X ⊂ FJ be a fuzzy set of formulas. Then ∼ C syn (X)(A) = {Val(wA ) | wA is proof of A from X}. This theorem is apparently a generalization of Theorem 1 which nicely generalizes the concept of provability. Indeed, it is clear that there exist more proofs of the same formula A. However, since each such proof has a value, in concord with principle (i) we take supremum of all of them. In traditional syntax, this is the same. However, since the only (non-zero) evaluation is 1, ﬁnding one proof is sufﬁcient. It is natural to call C syn (X)(A) a provability degree of A from the fuzzy set X. A fuzzy theory T is determined by a fuzzy set (LAx ∪ SAx) ⊂ FJ of logical and special axioms. Then a = C syn (LAx ∪ SAx)(A) is a provability degree of ∼ A ∈ FJ in the fuzzy theory T. Alternatively, we say that A is a theorem in the degree a in T and write T a A 6 This evaluated formula expresses that the implication “if n does not form a heap then n + 1 also does not form it” is not be fully convincing.

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to stress that this is a generalization of the classical concept. Note that Hájek in [9] uses the symbol |A|T for the provability degree and takes Theorem 2 as the deﬁnition. 7 The semantics is a straightforward generalization of classical deﬁnition. A model of T is a truth evaluation M of formulas such that SAx(A) M(A) holds for all A ∈ FJ . Then A is true in a degree a in T if a= {M(A) | M T } and we write again T a A to stress that this is a generalization of the classical concept. The following theorem holds both for propositional and for predicate case (see [25,26,29]): Theorem 3 (Completeness). For every fuzzy theory T and every formula A ∈ FJ (T ) T a A iff T a A. Note that a both in T a A and in T a A is uniquely deﬁned and so both symbols are sound. We prefer them to see that we deal with the nice generalization of the classical concept. Hájek in [9] writes AT instead of T a A and then Theorem 3 is written as |A|T = AT and called Pavelka-style completeness. A signiﬁcant limitation of FL with evaluated syntax is expressed in the following theorem (cf. [25,26]). Theorem 4. Let Theorem 3 hold for a formal logical system with evaluated syntax. Then the support L of the residuated lattice of truth values is either ﬁnite or it is isomorphic with [0, 1] and the interpretation → of the implication connective is continuous. Since the only residuated lattice (up to isomorphism) on [0, 1] with continuous residuation is Łukasiewicz MValgebra, fuzzy logic with evaluated syntax is bound to it. Therefore, we denote this logic by EvŁ . Let me comment on this result. From one side, this is unpleasant since we are very limited in choosing the structure of truth values. It demonstrates that we cannot go too far with the generalization. There are possibilities how to overcome this limitation by adding a special inﬁnitary inference rule but this is not too convincing. However, do we really need more general structures of truth values? MV-algebra is a beautiful structure since it nontrivially generalizes boolean algebra and keeps most of its important properties. Why are we not satisﬁed with it and search other possibilities? The uniqueness, on the other hand, together with the fact that a great deal of properties of classical logic are nicely generalized seems to be strong argument for taking EvŁ as the real fuzzy logic. It is also notable that, as Hájek has shown in [9], EvŁ is representable in Łukasiewicz logic (it is even its conservative extension—cf. [13]). On the other hand, Łukasiewicz logic is clearly a special case of EvŁ (because the provability degree includes also the classical provability). Another objection is that both in Pavelka’s work as well as in [25], the logical constants form a continuum. This has been simpliﬁed indirectly in EvŁ to rational constants only in [25], directly by Hájek in [9] in the Łukasiewicz representation of EvŁ , and recently also directly in EvŁ in [21]. Because of the outstanding position of EvŁ , fuzzy set theory developed from this unique formal point of view is presented in [18]. 3. What is the real fuzzy logic? Let us now turn to the main question of this paper. As one may expect, though, our answer will not be deﬁnite. One point of view is that the real FL should have evaluated both semantics and syntax. Then the real fuzzy logic is only EvŁ and we are done. Another more important point of view is to take as the real fuzzy logic such logic that fulﬁls the aims discussed in Section 1 in the best way. Then more logics can be found suitable. We may agree with the general principles of Cintula and Bˇehounek in [2] trying to deﬁne the class of fuzzy logics mathematically. From this point of view, it seems that the fundamental fuzzy logic is MTL. Consequently, we claim that fuzzy logic is any logic that is a schematic extension of 7 In fact, he identiﬁes evaluated formulas with formulas a⇒A and does not introduce them explicitly. This is possible since it is useful to introduce

a A using which evaluated formulas can be represented by ordinary ones. a special rule rLC : a → a a⇒A

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Evaluated syntax

Traditional syntax

Classical logic (+higher order)

EvLΠ L

A∨¬A

(+higher order)

Product logic

Gödel logic

EvL a / A ≡ (a⇒A) a/A ≡ (a⇒A)

Lukasiewicz logic (+higher order)

add provability degree

A ⇒A&A

¬¬A⇒((A⇒A&B )⇒ ⇒(¬¬A&A))

A&(A⇒ B)⇔A B

(+higher order) ¬¬A⇒A A&(A⇒ B)⇔A B

BL

IMTL (+higher order)

MTL

Fig. 1. Scheme of the most relevant fuzzy logics.

MTL. But not all of them can really serve well to fulﬁl the agenda of FL and to be a good basis for the development of FLb. The MTL itself seems to be rather weak. Much better for this purpose is IMTL, that is, MTL extended by the law of double negation (¬¬A⇒A). The latter is necessary if we want to prove various non-trivial properties. Another beautiful logic is BL. In a sense, this logic is essential logic for the development of fuzzy set theory (for a successful attempt see [11]) because it covers all continuous t-norms. Unfortunately, the double negation cannot be introduced in BL. When adding it to BL, we obtain Łukasiewicz logic where we have almost everything we should claim (and still not to collapse into classical logic). When compared with EvŁ , however, we see that EvŁ is stronger than Ł and so more convenient to fulﬁl the agenda of FL. It is notable that only EvŁ is capable at modeling of the concept of intension on the ﬁrst-order level (this is impossible in traditional syntax)—see the ﬁrst version in [19] and elsewhere (later also in [25]). An important operation needed, e.g. for considerations in fuzzy approximation theory (a concept signiﬁcantly elaborated in the frame of fuzzy logic by Perﬁlieva, cf. [27]) and elsewhere, is ordinary product of reals. A logic that uses it as interpretation of strong conjunction is product logic . But this logic has some strange properties; it is more special than BL but lacks important properties present in Ł or EvŁ (e.g. contraposition). A special case of BL is also Gödel logic. Some properties of it are very nice (e.g. it has classical deduction theorem). Unpleasant is its implication that has inﬁnite number of discontinuities; the negation is the same as in logic, i.e. it does not preserve double negation. A very nice possibility is to join everything good from all logics into one—the result is Ł logic. This logic has been established by Esteva et al. in [7] and further elaborated by Cintula [3,4]. Its only disadvantage is great complexity because of a plethora of connectives. But still, I think that this logic is the best to fulﬁl the agenda of FL, to be applied in fuzzy approximation (see [28]), and possibly also, to be used as the general frame for the theory of FLb. Whenever possible, of course, we may bind to some of its reducts (e.g. Ł (possibly EvŁ )). I am convinced that a prominent place in further development of fuzzy logic should be taken by the formal system of FLb; this automatically leads to fulﬁlling the agenda of FL. The most powerful logical system for this goal is fuzzy type theory understood as higher-order fuzzy logic (see [20]). Among others, it enables us direct generalization of various consideration from classical linguistic (e.g. Montague grammar [15]). An important possibility is logical theory of fuzzy IF–THEN rules construed as conditional linguistic statements (see [23]). We may expect among its results sophisticated inference schemes of human reasoning. We should also mention a recent attempt of Bˇehounek and Cintula to develop a sound mathematical basis for fuzzy mathematics (see [1]). Because of space limit, we cannot discuss all the raised questions in great detail. At the end, we present a general scheme of fuzzy logics in Fig. 1 that is a modiﬁcation of that from [6]. In the scheme, the most distinguished fuzzy logics discussed above are emphasized. Classical logic is also included as a limit crisp

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case of FL. Some of the intermediate logics are omitted. In the scheme, axioms that must be added to get to another logical system are (in most cases) depicted. The scheme depicts also the shift from Ł to evaluated syntax, and in a sense also similar shift from Ł (this requires the mentioned inﬁnitary rule). The dotted arrows express representability in the corresponding logic without evaluated syntax. Existence of higher order fuzzy logic (fuzzy type theory) for each of the emphasized logics (including, of course, classical one) is also captured; such logic is ready for extension to FLb. This discussion picks up some of the fuzzy logics that, according to their properties and strength, seem to ﬁt best the aims of FL in general. Of course, the future will only prove whether this picture is indeed relevant (and complete?). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

L. Bˇehounek, P. Cintula, Fuzzy class theory, Fuzzy Sets and Systems 154 (2005) 34–55. L. Bˇehounek, P. Cintula, Fuzzy logics as the logics of chains, Fuzzy Sets and Systems (this issue). P. Cintula, The Ł and Ł 21 propositional and predicate logics, Fuzzy Sets and Systems 124 (2001) 289–302. P. Cintula, Advances in the Ł and Ł 21 logics, Arch. Math. Logic 42 (2003) 449–468. F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. F. Esteva, L. Godo, A. García-Cerdana, On the hierarchy of t-norm based residuated fuzzy logics, in: M. Fitting, E. Orlowska (Eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic Series, Springer, Berlin, 2003, pp. 235–270. F. Esteva, L. Godo, F. Montagna, The Ł. and Ł 21 logics: two complete fuzzy systems joining Łukasiewicz and product logics, Arch. Math. Logic 40 (2001) 39–67. S. Gottwald, A Treatise on Many-Valued Logics, Research Studies Press Ltd., Baldock, Herfordshire, 2001. P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998. P. Hájek, What is mathematical fuzzy logic, Fuzzy Sets and Systems (this issue). P. Hájek, Z. Haniková, A development of set theory in fuzzy logic, in: M.C. Fitting, E. Orlowska (Eds.), More Than Two: Theory and Applications of Many-Valued Logic, Physica-Verlag, Heidelberg, 2003, pp. 273–285. P. Hájek, V. Novák, The sorites paradox and fuzzy logic, Internat. J. Gen. Systems 32 (2003) 373–383. P. Hájek, J.B. Paris, J.C. Shepherdson, Rational Pavelka predicate logic is a conservative extension of Lukasiewicz predicate logic, J. Symbolic Logic 65 (2000) 669–682. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. R. Montague, English as a formal language, in: Linguaggi nella società nella tecnica, Edizioni di Comunità, Milano, 1970. D. Mundici, R. Cignoli, I.M.L. D’Ottaviano, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000. H.T. Nguyen, E.A. Walker, A First Course in Fuzzy Logic, Chapman & Hall, CRC, Boca Raton, 2000. V. Novák, Fuzzy Sets and Their Applications, Adam Hilger, Bristol, 1989. V. Novák, Towards formalized integrated theory of fuzzy logic, in: Z. Bien, K. Min (Eds.), Fuzzy Logic and Its Applications to Engineering, Information Sciences, and Intelligent Systems, Kluwer Academic Publishers, Dordrecht, 1995, pp. 353–363. V. Novák, On fuzzy type theory, Fuzzy Sets and Systems 149 (2005) 235–273. V. Novák, Fuzzy logic with countable evaluated syntax, in: Fuzzy Logic, Soft Computing and Computational Intelligence II. Eleventh IFSA World Congress, Tsinghua University Press, Springer, Beijing, 2005, pp. 1264–1269. V. Novák, Are fuzzy sets a reasonable tool for modeling vague phenomena?, Fuzzy Sets and Systems to appear. V. Novák, S. Lehmke, Logical structure of fuzzy IF–THEN rules, Fuzzy Sets and Systems, submitted for publication. V. Novák, I. Perﬁlieva (Eds.), Discovering the World With Fuzzy Logic, Springer, Heidelberg, 2000 (Studies in Fuzziness and Soft Computing, vol. 57). V. Novák, I. Perﬁlieva, J. Moˇckoˇr, Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Boston, Dordrecht, 1999. J. Pavelka, On fuzzy logic I, II, III, Zeit. Math. Logic. Grundl. Math. 25 (1979) 45–52, 119–134, 447–464. I. Perﬁlieva, Normal forms in BL-algebra of functions and their contribution to universal approximation, Fuzzy Sets and Systems 143 (2004) 111–127. I. Perﬁlieva, Normal forms in BL and Ł algebras of functions, Soft Computing 8 (2004) 291–298. E. Turunen, Mathematics Behind Fuzzy Logic, Springer, Heidelberg, 1999. L.A. Zadeh, Fuzzy logic and its application to approximate reasoning, Inform. Process. 74 (1974) 591–594. L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, II, III, Inform. Sci. 8 (1975) 199–257, 301–357; L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, II, III, Inform. Sci. 9 (1975) 43–80.

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