THE PRINCIPLES OF FUZZY LOGIC - Semantic Scholar

THE PRINCIPLES OF FUZZY LOGIC: ITS MATHEMATICAL AND COMPUTATIONAL ASPECTS VILÉM NOVÁK, IRINA PERFILIEVA University of Ostrava Institute for Research and Applications of Fuzzy Modelling Bráfova 7, 701 00 Ostrava 1, Czech Republic

Contents

1 Introduction 2 Vagueness, Uncertainty and Fuzzy Logic 2.1 2.2 2.3 2.4

Vagueness and Uncertainty Vagueness and Fuzzy Sets . What Is Fuzzy Logic . . . . Agenda of Fuzzy Logic . . .

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3 Algebraic structures for logical calculi 3.1 3.2 3.3 3.4

Boolean algebras . . . Residuated lattices . . MV-algebras . . . . . The theory of t-norms

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4 Fuzzy Logic in Narrow Sense 4.1 4.2 4.3 4.4 4.5

Graded formal logical systems . . . . . . . . . Syntax of predicate rst-order fuzzy logic . . Semantics of predicate rst-order fuzzy logic . Provability in fuzzy theories . . . . . . . . . . Some important theorems of fuzzy logic . . .

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5 Functional Systems in Fuzzy Logic Theories 5.1 5.2 5.3 5.4

Fuzzy logic functions and normal forms . . . . . . . . . . . . Fuzzy logic formulas as a tool for an approximate description Compositional rule of inference and the best approximation . Representation of continuous functions and its complexity . .

6 Fuzzy Logic in Broader Sense 6.1 6.2 6.3 6.4

Partial formalization of natural language . . Intension and extension in fuzzy logic . . . The meaning of simple evaluating syntagms Generalized inference . . . . . . . . . . . . .

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1 Introduction This is a study material for the introductory course in fuzzy logic. Our aim is to give a brief overview of the main aspects of fuzzy logic. We introduce the concept of fuzzy logic and discuss its philosophical background. We argue that people encounter a phenomenon of indeterminacy which has two complementary facets, namely uncertainty and vagueness. Fuzzy logic is then considered as a mathematical model useful for modelling of the latter. Furthermore, we outline the theory of special structures, which are suitable for representation of the structure of truth values. We distinguish fuzzy logic in narrow sense (FLn) and that in broader sense (FLb). Our further explanation continuse by an overview of some main concepts of FLn. In Section 5, we discuss important questions of the correspondence between formulas of FLn and special functions de ned in [0; 1], as well as the problem of approximation of continuous functions using functions derived from special formulas of FLn. The material is nished by a brief introduction to FLb, which is an extension of FLn enabling to capture the meaning of special natural language expressions, namely the so called evaluating syntagms, which characterize position on an ordered scale and are used especailly in the characterization of the behaviour of dynamic systems.

2 Vagueness, Uncertainty and Fuzzy Logic

2.1 Vagueness and Uncertainty

The phenomena of uncertainty and vagueness characterize situations in which we regard phenomena surrounding us; they are concerned with the amount of knowledge we have at disposal (or can have at disposal). We argue that these phenomena are two, rather complementary facets of a more general phenomenon called the indeterminacy. The phenomenon of uncertainty emerges due to the lack of knowledge about the occurrence of some event. It is encountered when an experiment (process, test, etc.) is to proceed, the result of which is not known to us. Let us stress that there is no uncertainty after the experiment was realized and the result is known to us. A speci c form of uncertainty is randomness, which has for many years been considered to raise mainly due to extensive number of outer in uences causing ambiguity in the result, thus preventing us to describe it as deterministic. As late as in the 20th century, mainly in connection with important discoveries in atomic physics and quantum mechanics, randomness had been recognized to be unavoidable and a deep feature of the nature. The mathematical model (i.e. quanti ed characterization) of the uncertainty phenomenon is provided especially by the probability theory. In everyday terminology, probability can be thought of as a numerical measure of the likelihood that a particular event will occur. Recall that the probability values are taken from the scale [0; 1] where values near 1 indicate that an event is likely to occur while those near 0 indicate the opposite. The probability of 0.5 means that an event is equally likely to occur as well as not to occur. There are also other mathematical theories addressing the uncertainty phenomenon, for example the possibility theory, belief measures and others. The vagueness phenomenon raises during the process of grouping together objects having some property ' (of objects). The result of this process will be called a grouping of objects. In a slightly formal way, the grouping can be written as

X = fx j x has the property 'g 2

(1)

where x varies over objects. It is important to stress that, in general, the grouping X cannot be taken as a set since the property ' may not make us possible to characterize the grouping X precisely and unambiguously; there can exist borderline elements x for which it is unclear whether they have the property ' (and thus belong to X ), or not. The objects, which can be unambiguously decided to have ', will be called prototypes. In general, we say that X , which is delineated using a property enabling borderline cases, has unsharp boundaries.

Example 1 A typical vague property is to be a small natural number. Can we imagine all

the small natural numbers? Clearly, 0 is small, 1 is small as well, etc. But where does this sequence nish? The only sure fact is that there exists a number, say 1; 000; 000; 000, which is not small. An attempt at sharp (exact) explanation would mean to be able to nd a small natural number n, 0 < n < 1; 000; 000; 000 such that n + 1 is not small. However, such a conclusion can hardly be defended. If n is small then n + 1 must be small, as well. Hence, there is no last small number before 1; 000; 000; 000 and no rst number n < 1; 000; 000; 000 which is not small. We can distinguish small numbers from big ones but we are not able to say unambiguously about each number whether it is small or not. The property of \small" is vague and small numbers disappear inside the sequence of numbers ranging from 0 to 1,000,000,000. Consequently, they form a vague grouping of numbers. 2

Vagueness is an opposite to exactness and we argue that it cannot be avoided in the human way of regarding the world. Any attempt to explain an extensive detailed description necessarily leads to using vague concepts since precise description contains abundant number of details. To understand it, we must group them together | and this can hardly be done precisely. It is likely that the explanation would signi cantly rely on the use of natural language since vagueness is often connected with its use. However, the problem lays deeper, in the way how people regard the phenomena around them. Vagueness is necessary to convey relevant information. Vagueness should be distinguished from generality and from ambiguity. In our terms, to be more general means to take into account more (various) groupings of objects, while ambiguity occurs in the language when more alternative meanings are assigned to the same word expression. A typical feature of vagueness is its continuity. This means that if an object has a vague property and another one diers very little from it, then it must have the same property. In other words, a small dierence between objects cannot lead to abrupt change in the decision of whether either of them has, or has not a vague property. The transition from having a (vague) property to not having it is smooth. A mathematical theory of the vagueness phenomenon is most successfully provided by fuzzy logic. Another deep mathematical theory addressing, among others, also vagueness is the Alternative Set Theory (AST) (see [12] and possibly also [9]).

2.2 Vagueness and Fuzzy Sets

Mathematics faces the problem, how vague groupings of objects can be characterized. Classical set theory does not carry any vagueness since there was no interest in it. Fuzzy set theory is an attempt to nd such an approximation of vague groupings which in situations where, for example, natural language plays a signi cant role, would be more convenient than classical set theory. 3

The principle of graded (fuzzy) approach is employed throughout fuzzy set theory and fuzzy logic. This means the use of a scale when characterizing a relation between object and its property. The graded approach will be mathematized by means of a special scale being an ordered set. In order to be suciently general and to be able to capture the continuity feature of vagueness, we will suppose that this set is uncountable. Furthermore, to be able to represent various manipulations with the properties partly possessed by objects, we will endow the scale by additional operations. The result is a speci c algebra. There are several classes of algebras used as scales in the graded approach (see Sction 3).

Fuzzy set theory The decision whether the concrete object x has the property ' (e.g.

\small", \tall", \light", \blue", etc.) is tantamount to the question whether it is true that x has it. However, such a question cannot be unambiguously answered. A reasonable solution consists in using a scale whose elements would express various degrees of truth of '(x). Let U be a suciently large set from which we take the objects x. This set is called the universe. Furthermore, let L be a scale of truth values having the smallest 0 and greatest 1 elements, respectively. We usually put L = [0; 1] but this is only an unnecessary commonsense caused by the fact that this interval is very natural and transparent.

De nition 1 The fuzzy set A is identi ed with a function A : U ?! L (2) assigning a value A(x) 2 L to each element x 2 U . The value A(x) 2 L is the membership degree of x in A.

The function (2) is also called the membership function of the fuzzy set A. We will therefore use the same symbol for both and write A U if A is a fuzzy set in the universe U . Explicitly, we will write down the fuzzy set as f A(x) x j x 2 U g (3) where the couple A(x) x means \the element x belongs to A with the membership degree A(x)", A(x) 2 L. The value A(x) expresses the degree of truth that the element x belongs to A.

1 0.5 10

80

150

N

-

Figure 1: Membership function of the fuzzy set of \small numbers".

4

Few basic concepts of fuzzy set theory Let the fuzzy set A U be given by the membership function (2) where L = [0; 1]. Let A; B U . Then A B if A(x) B (x) holds for all x 2 U . The set of all fuzzy sets on U is F (U ) = fA j A U g = LU :

Note that F (U ) contains also all ordinary subsets of U . Given the fuzzy set A U , it can be characterized in several ways. The following are important ordinary sets de ned on the basis of A. (i) Support Supp(A) = fx j A(x) > 0g;

(4)

(ii) a-cut

Aa = fx j A(x) ag;

a 2 (0; 1];

(5)

(iii) Kernel Ker(A) = fx j A(x) = 1g:

(6)

The fuzzy set A is normal if Ker(A) 6= ;. The empty fuzzy set is de ned by ; = f 0 x j x 2 U g: Obviously, this is the ordinary empty set. The basic operations with fuzzy sets are de ned using their membership functions. Given fuzzy sets A; B U. (i) Union of A and B is a fuzzy set A [ B U with the membership function

C = A [ B; i C (x) = A(x) _ B (x);

x2U

(7)

where _ denotes `supremum'. (ii) Intersection of A and B is a fuzzy set A \ B U with the membership function

C = A \ B; i C (x) = A(x) ^ B (x);

x2U

(8)

where ^ denotes ìn mum'. Both union as well as intersection of two fuzzy sets may be extended to arbitrary classes of fuzzy sets provided that suprema and in ma of arbitrary sets exist in L. (iii) Complement A is the fuzzy set A(x) = :A(x) = 1 ? A(x); x 2 U: (9)

De nition 2 Let U1 ; : : : ; Un be sets. An n-ary fuzzy relation R is a fuzzy set R U1 Un: 5

2.3 What Is Fuzzy Logic

The term \fuzzy logic" had rst the meaning of any logic possessing more than two truth values. Later on, after the famous paper of L. A. Zadeh [14] it received two other meanings, namely the theory of approximate reasoning and the theory of linguistic logic. Unfortunately, nowadays the meaning of the term \fuzzy logic" is often blurred by interpreting it as any kind of formalism based on the fuzzy set theory, which is used in the applications. In general, fuzzy logic can be characterized as the many-valued logic with special properties aiming at modeling of the vagueness phenomenon and some parts of the meaning of natural language via graded approach. We will distinguish fuzzy logic in narrow and broader sense. Fuzzy logic in narrow sense, FLn, is a special many-valued logic, which aims at providing formal background for the graded approach to vagueness. Fuzzy logic in broader sense, FLb, is an extension of FLn and it aims at developing mathematical model of natural human reasoning, in which principal role is played by the natural language. In general, fuzzy logic is the result of the graded approach to the formal logical systems. Due to the graded approach, fuzzy logic provides solution of some, classically non-solvable problems. For example, there are well known ancient paradoxes of sorites (heap) and falakros (bald men). One grain of wheat does not make a heap. Neither make it two grains, three, etc. Hence, there are no heaps. These paradoxes rise if we understand the properties \to be a heap" and \to be bald" sharply, i.e. if we neglect their vagueness. Let FN (x) denote the proposition \the number x is small" (x be natural). The problem lays in the truth veri cation of implication FN (x) ) FN (x + 1), i.e. classical induction cannot be applied to vague property. If the latter is taken classically (i.e. absolute true) then starting from FN (0) we necessarily arrive to the above paradoxes. However, for dierent subsequents of x, the veri cation of the proposition FN (x) ) FN (x + 1) is dierent. In general, veri cation that x is small does not imply that we will be able to verify that also x + 1 is small with the same eort. For example, if we verify that 1000 is small by counting one thousand lines then veri cation that 1001 is also small means that we must count one line more, i.e. our eort to verify that 1001 is small is a little (imperceptibly) greater than that for 1000. Consequently, the above implication is not fully convincing. A solution oered by fuzzy logic is to assume that the implication FN (x) ) FN (x + 1) is true only in some degree close to 1, say 1 ? " where " > 0. Then sorites (as well as falakros) paradox disappears. The formal apparatus of fuzzy logic provides us the following outcome. 1. Because the graded approach serves us for modeling of the vagueness phenomenon, we are able to deal with any possible truth value on both levels, namely syntactical (make graded syntactical derivations) as well as semantical (evaluate truth degrees of formulas in the interpretation). Note that unlike other multivalued logics, all truth values in FLn are equal in their importance, i.e. there are no designated truth values. 2. FLn and especially FLb as the extension of the latter seem to be working theories, which may be used as formal apparatus explicating the approximate reasoning schemes. 3. FLn is a generalization of classical logic. It opens dierent, more general way for explanation of, at least some of the classical problems. We may also expect new problems of classical mathematics in uenced directly by FLn. 6

The system of FLn is basically truth functional. There are reasons to assume truth values to form a complete, in nitely distributive residuated lattice, or an MV-algebra and several mathematically justi ed arguments we conclude that the structure of truth values should form the Lukasiewicz algebra LL . FLn aims at extension of the capability of classical logic to the directions where the latter cannot provide satisfactory solution. Hence, FLn neither mimics classical logic with slight generalizations, nor denies the latter.

2.4 Agenda of Fuzzy Logic

Linguistic variable This is one of the fundamental concepts introduced by L. A. Zadeh in [14] and is de ned as the quintuple

hX ; T (X ); U; G; M i; where X is the name of the variable, T (X ) is the set of its values (term set) which are linguistic expressions (syntagms ), U is the universe, G syntactical rule using which we can form syntagms

A; B; : : : 2 T (X ), and M is semantical rule, using which every syntagm A 2 T (X ) is assigned its meaning being a fuzzy set A in the universe U , A U. A typical example of the linguistic variable is X := Age. Its term set T (Age) consists of the

syntagms such as young, very young, medium age, quite old, more or less young, not old but not young, etc. Then U R is some set of real numbers, G is a context-free grammar and M assigns meaning to the terms from T (Age) being various modi cations of fuzzy sets depicted on Fig. 2. Examples of other linguistic variables are \height, size, temperature, press", etc. 1 \young"

\medium age" 50

\old" 90(years)

Figure 2: Possible forms of the fuzzy sets assigned as the meaning to the basic syntagms \small", \medium age", \old" from T (Age).

Approximate reasoning Linguistic variables have a quite wide scope of applications. The most important is their use in the approximate reasoning scheme, such as the behaviour of the car driver: Condition: IF the obstacle is near AND the car speed is big THEN break very much IF the obstacle is far AND the car speed is rather small THEN break a little Observation: the obstacle is quite near AND the car speed is big Conclusion: break quite much Fuzzy logic oers a model of this scheme. 7

Linguistic semantics and fuzzy logic in broader sense Since the agenda of fuzzy logic

is connected with the meaning of some expressions of natural language, it is desirable to be suciently precise, especially with respect to the results of the classical linguistic theory. One may see that the concept of linguistic variable presented above is not fully satisfactory. The meaning of syntagms depends also on the context, yet there are some general laws inside. One of the reasons for the criticism is neglecting of the concepts of intension and extension. Intension of the meaning, roughly speaking, is the property contained in it. It does not depend on the context and expresses the very content of the meaning. Extension is the re ection of the intension in the universe and it leads to groupings of objects. Thus, for example, the intension of \young" presents itself in various worlds equally but leading to various groupings of \young objects". Apparently, \young dog" has the meaning dierent from \young man" and even the latter may be dierent in Africa, Europe, or Japan. The concept of FLb outlined in Section 6 is an attempt to overcome this de ciency.

3 Algebraic structures for logical calculi 3.1 Boolean algebras

A Boolean algebra is a lattice with special properties (see e.g. [4]). The lattice is an ordered set containing with each pair of elements their least upper bound (supremum; sup) as well as the greatest lower bound (in mum; inf). They can be considered as two binary operations, namely _ (join ) and ^ (meet ) respectively and so, the lattice becomes an algebra endowed with these two operations sup(a; b) = a _ b; inf(a; b) = a ^ b: The following lemma exposes the basic lattice properties. Lemma 1 Let L be a lattice. Then the following identities (also called laws) are true: a _ (b _ c) = (a _ b) _ c; a ^ (b ^ c) = (a ^ b) ^ c; (associativity) (10) a _ b = b _ a; a ^ b = b ^ a; (commutativity) (11) a _ (a ^ b) = a; a ^ (a _ b) = a; (absorption) (12) a _ a = a; a ^ a = a: (idempotency) (13) If L = hL; _; î is an algebra with two binary operations ful lling (10){ (12) then an ordering cand be de ned on L using a b i a _ b = b; (equivalently a ^ b = a): Important instances of lattices are distributive lattices and lattices with complements. De nition 3 A lattice L is distributive if the following identities (distributivity laws) are true: a _ (b ^ c) = (a _ b) ^ (a _ c); (14) a ^ (b _ c) = (a ^ b) _ (a ^ c): (15) . It is a lattice with complements if it has the least element 0 and a unary operation of complement `0 ', so that the following identities are true: a00 = a; (law of double negation) (16) 0 0 0 (a _ b) = a ^ b ; (the rst de Morgan law) (17) 0 a ^ a = 0: (law of contradiction) (18) 8

Let us put 1 = 00 . It is easy to show that 1 is the greatest element of L. De nition 4 A Boolean algebra L is a distributive lattice with complements. In symbols L = hL; _; ^;0 ; 0; 1i: A Boolean algebra is also called a Boolean lattice. Example 2 (Boolean algebra for classical logic) This is based on the set consisting of two elements only, i.e. LB = hf0; 1g; _; ^; :i where the operations \_; ^; :" are de ned using the following tables:

_ 0 1 0 0 1 1 1 1

^ 0 1 0 0 0 1 0 1

: 0 1 1 0

Since these operations are heavily used in classical logic, they are known under the names corresponding to the respective logical connectives \disjunction", \conjunction" and \negation". The other two known logical connectives of \implication" and \equivalence" enrich the set of the basic operations above by ! and $, respectively and are de ned by:

! 0 1 0 1 1 1 0 1

$ 0 1 0 1 0 1 0 1

Note that the operations !; $ can be derived from the basic ones using the following expressions: a ! b = a0 _ b a $ b = (a ! b) ^ (b ! a):

2

Example 3 (Boolean algebra of subsets) Let A be an arbitrary nonempty set and P (A) = f0; 1gA be a set of all its subsets. The Boolean algebra based on this set is B(A) = hf0; 1gA ; [; \;; ;; Ai where \[",\\" and \" are the ordinary operations of union, intersection and complement, respectively. 2 Example 4 ((Algebra of Boolean functions)) Let P2n, n 1, be a set of all functions f (x1 ; : : : ; xn ) de ned on f0; 1g and taking values from this set. Such functions are called

Boolean. The respective Boolean algebra based on P2n is LnP = hP2n ; _; ^; :; 0; 1i where the operations \_", \^" and \:" are de ned pointwise, and 0; 1 denote the respective constant functions. 2 Example 5 (Boolean algebra of propositions) This algebra is based on the set of all propositions, that is, the sentences which can be evaluated either by \true" or by \false". Every two propositions P , Q are assigned the following three compound ones, namely \P or Q", \P and Q", \not P " being the result of the basic operations \_", \^" and \:", respectively. Two propositions are equal if their evaluations coincide. The least element 0 of this algebra is a proposition evaluated by \false". 2 9

Duality principle Note that the generating identities (10){(18) of a Boolean algebra are such that if we at the same time replace 0 by 1, 1 by 0, _ by ^ and ^ by _, then we again

obtain generating identities. This rule is known as the duality principle. Validity of this principle can easily be checked for (10){(15) since both, formula as well as its dual are there present. We show that (17) and (18) have also their duals and that they are identities. Indeed, from the identity (a _ b)0 = a0 ^ b0 (the rst de Morgan law) we can easily deduce the identity

a0 _ b0 = (a0 _ b0 )00 = (a00 ^ b00 )0 = (a ^ b)0 ; which is called the second de Morgan law. Important application of the duality principle gives us the representation of each element of a Boolean algebra in two normal forms.

Representation theorem and normal forms Among the examples of a Boolean algebra, the most important is Example 3 because it serves as a canonical representation of any Boolean algebra.

Theorem 1 Every complete and completely distributive Boolean algebra is isomorphic to B(A) for some set A.

The statement that every Boolean algebra is isomorphic to some algebra of sets is known as Stone representation theorem.

Corollary 1 Let L be a nite Boolean algebra and A be its set of atoms. Then each element x of L can uniquely be represented in two forms, namely _ ^ 0 x = (a ^ b ); (19) x=

a2A ax

b2Anfag _ (a0 _ b) a2A0 b2Anfag ax

^

(20)

which are called the perfect disjunctive normal form and the perfect conjunctive normal form, respectively.

3.2 Residuated lattices

In order to have more than two values for evaluation of formulas, many attempts have been done to generalize the classical algebra for logic. On this way, two special structures took a distinguished role, namely residuated lattices and MV-algebras. Suppose that the set L of truth values forms a lattice and contains the least 0 and the greatest 1 elements. Besides that we add two speci c operations of multiplication and addition . A residuated lattice can be placed between Boolean and MV- algebras. In the case that L = f0; 1g, these coincide with ^ and !, respectively.

De nition 5 A residuated lattice is an algebra L = hL; _; ^; ; !; 0; 1i:

(21)

with four binary operations and two constants such that (i) hL; _; ^; 0; 1i is a lattice with the ordering de ned using the operations _; ^ as usual, and 0; 1 are its least and the greatest elements, respectively;

10

(ii) hL; ; 1i is a commutative monoid, that is, is a commutative and associative operation with the identity a 1 = a; (iii) the operation is isotone in both arguments; (iv) the operation ! is a residuation operation with respect to , i.e.

a b c i a b ! c:

(22)

We see that in comparison with the ordinary lattice, this structure is enriched by a special couple h ; !i of operations called the adjoint couple. The de ning relation (22) is called the adjunction property. We will call the multiplication and ! the residuation.

Example 6 (Boolean algebra for classical logic) LB = hf0; 1g; _; ^; !; 0; 1i where ! is the classical implication, is the simplest residuated lattice (the multiplication = ^). In general, every Boolean algebra is a residuated lattice when putting a ! b = a0 _ b. 2 Example 7 (Gödel algebra) LG = h[0; 1]; _; ^; !G ; 0; 1i where the multiplication = ^ and ( (23) a ! b = 1 if a b; G

b

if b < a:

Gödel algebra is a special case of the Heyting algebra used in intuitionistic logic.

Example 8 (Goguen algebra) LP = h[0; 1]; _; ^; ; !P ; 0; 1i where the multiplication = is the ordinary product of reals and ( a ! b = 1 if a b; P

b a

if b < a:

Example 9 (Lukasiewicz algebra) L = h[0; 1]; _; ^; ; ! ; 0; 1i L

L

2

(24)

2 (25)

where

a b = 0 _ (a + b ? 1); a ! b = 1 ^ (1 ? a + b):

(Lukasiewicz conjunction) (Lukasiewicz implication)

L

(26) (27)

2 11

Example 10 (Residuated lattice of [0; 1]-valued functions) Let X be a nonempty set. For each two functions f; g 2 [0; 1]X we put (f _ g)(x) = f (x) _ g(x); (f g)(x) = f (x) g(x); (28) (f ^ g)(x) = f (x) ^ g(x); (f ! g)(x) = f (x) ! g(x) (29) for all x 2 X where _; ^; ; ! are the corresponding operations of the Lukasiewicz algebra (Example 9). Furthermore, let 1 and 0 be constant functions taking the values 1 and 0, respectively. Then

Lfunc(X ) = h[0; 1]X ; _; ^; ; !; 0; 1i

is a residuated lattice.

2

Lemma 2 Let L be a residuated lattice given by (21). Then for every a; b; c 2 L the following holds true. (a) a b a; a b b; a b a ^ b, (b) b a ! b, (c) a (a ! b) b; b a ! (a b), (d) if a b then c ! a c ! b (isotonicity in the second argument); a ! c b ! c (antitonicity in the rst argument); (e) (f) (g) (h) (i)

a (a ! 0) = 0, a ! (b ! c) = (a b) ! c, a b i a ! b = 1; a = 1 ! a, (a _ b) c = (a c) _ (b c), a _ b ((a ! b) ! b) ^ ((b ! a) ! a).

Lemma 3 Let L be a residuated lattice given by (21). If 1 = a _ (a ! 0) holds for every a 2 L then L is a Boolean lattice and = ^. Additional operations and their properties To be used as an algebra for many-valued

logic, residuated lattices should be completed by additional operations corresponding (among others) to the logical connectives. Analogously as in classical logic, they can be obtained as derived operations from the basic ones. :a = a ! 0 (negation), (30) a $ b = (a ! b) ^ (b ! a) (biresiduation), (31) a b = :(:a :b) (addition), (32) n (n-fold multiplication), (33) a = a| {z a} n?times

na = a| {z a}

(n-fold addition).

n?times

12

(34)

It follows from the antitonicity of ! in the rst variable (Lemma 2(d)) that the negation is order reversing, i.e. a b i :b :a: (35) Lemma 4 Let L be a residuated lattice given by (21). Then the following holds true for every a; b; c; d 2 L. (a) a $ 1 = a; a = b i a $ b = 1, (b) (a $ b) ^ (c $ d) (a ^ c) $ (b ^ d),

Linear ordering and completeness of residuated lattices De nition 6 Let L be a residuated lattice. (i) L is linearly ordered if a b or b a holds for every a; b 2 L. (ii) L is complete if the underlying lattice is complete. Lemma 5 Let L be a complete residuated lattice. Then the following identities hold for every a; b 2 L and sets fai j i 2 I g, fbi j i 2 I g of elements from L over arbitrary set of indices I . W (a) a ! b = fx j a x bg, V (b) a b = fx j a b ! xg. W W (c) ( i2I ai ) b = i2I (ai b), V V (d) i2I (a ! bi ) = a ! ( i2I bi ), W V (e) i2I (ai ! b) = ( i2I ai ) ! b, W W (f) i2I (a ! bi ) a ! ( i2I bi ), V W (g) i2I (ai ! b) ( i2I ai ) ! b.

3.3 MV-algebras

The notion of MV-algebra was introduced by C. C. Chang. MV-algebras have been developed as generalizations of Boolean algebras. However, it turned out that MV-algebras stand for the algebraic structures of truth values for various non-classical logical calculi including fuzzy logics. From the algebraic point of view, the MV-algebra diers from the Boolean one by absence of the idempotency law for their algebraic operations (addition and multiplication) and also, by the lack of the law of excluded middle for the lattice operations. De nition 7 An MV-algebra is an algebra L = hL; ; ; :; 1; 0i (36) in which the following identities are valid. a b = b a; a b = b a; (37) a (b c) = (a b) c; a (b c) = (a b) c; (38) a 0 = a; a 1 = a; (39) a 1 = 1; a 0 = 0; (40) a :a = 1; a :a = 0; (41) :(a b) = :a :b; :(a b) = :a :b; (42) a = ::a; :0 = 1; (43) :(:a b) b = :(:b a) a: (44) 13

The following are the examples of an MV-algebra. Example 11 (Lukasiewicz algebra) The Lukasiewicz algebra L from Example 9 is an MValgebra. It can be written as L

L = h[0; 1]; ; ; :; 0; 1i (45) where is the Lukasiewicz conjunction de ned in (26), is called the Lukasiewicz disjunction L

de ned by

a b = 1 ^ (a + b) and : is the negation operation de ned by :a = 1 ? a.

(46)

2

Example 12 (MV-algebra of [0; 1]-valued functions) The residuated lattice Lfunc(X ) of [0; 1]-valued functions on a nonempty X from Example 10 is an MV-algebra. 2 De nition 8 Let L be an MV-algebra given by (36). L is locally nite if to every a =6 1 there is n 2 N such that an = 0, or equivalently, to every a = 6 0 there is n 2 N such that na = 1. (In other terminology this property is called a nilpotentness.)

For example, Lukasiewicz MV-algebra LL is locally nite. Put

a b i :a b = 1:

(47)

It can be veri ed that (47) has the properties of partial ordering. The lattice operations can be introduced by

a _ b = :(:a b) b = (a :b) b; a ^ b = :(:a _ :b) = (a :b) b:

(48) (49)

Moreover, put a ! b = :a b and see that this is a residuation w.r.t. . Theorem 2 (a) Every MV-algebra is a residuated lattice. (b) A residuated lattice L is an MV-algebra i (a ! b) ! b = a _ b

(50)

holds for every a; b 2 L.

Corollary 2 An MV-algebra is a Boolean algebra if the operation (or, equivalently, ) is idempotent.

By Theorem 2 (b) both Gödel LG as well Goguen LP algebras are not MV-algebras.

Lemma 6 The following are identities in every MV-algebra. (a) (a ! b) _ (b ! a) = 1, (b) a ! b = :b ! :a. As usual, an MV-algebra is complete if the underlying lattice is complete. 14

Lemma 7 Let L be a complete MV-algebra. Then the following holds true for every a; b 2 L and sets fai j i 2 I g, fbi j i 2 I g of elements from L over arbitrary set of indices I . W V V W (a) i2I ai = : i2I :ai , i2I ai = : i2I :ai , W W V V (b) a ^ ( i2I bi ) = i2I (a ^ bi ), a _ ( i2I bi ) = i2I (a _ bi ), V V (c) a ( i2I bi ) = i2I (a bi ), W V (d) i2I (ai ! b) = ( i2I ai ) ! b, W W (e) i2I (a ! bi ) = a ! ( i2I bi ).

3.4 The theory of t-norms

The previous section we have presented three distinguished examples, namely, Lukasiewicz, Gödel and Goguen algebras, which are residuated lattices based on [0; 1] diering from each other by the chosen multiplication (and residuation). However, the class of such examples is signi cantly wider. In this section we will study the general class of multiplications known as triangular norms (brie y t-norms). These are binary operations t : [0; 1]2 ?! [0; 1], which have been introduced by K. Menger. They became interesting for fuzzy logic because they preserve the fundamental properties of the conjunction ànd' (to hold at the same time), and thus, they serve as a natural generalization of the classical conjunction to many valued reasoning systems. A concept associated with the t-norm is the triangular conorm (t-conorm) s : [0; 1]2 ?! [0; 1]. This corresponds to the behaviour of truth values when joined by the connective òr'. The other two concepts studied in this section are the residuation operation and the negation operation n : [0; 1] ?! [0; 1] generalizing behaviour of the logical negation. We will mostly con ne to the properties of the continuous t-norms because these are especially interesting due to the continuity property of the vagueness phenomenon.

Basic properties of triangular norms and conorms De nition 9 A t-norm is a binary operation t : [0; 1]2 ?! [0; 1] such that the following axioms are satis ed for all a; b; c 2 [0; 1]: (i) commutativity

a t b = b t a; (ii) associativity

a t (b t c) = (a t b) t c; (iii) monotonicity

a b implies a t c b t c; (iv) boundary condition

1 t a = a: Note that due to the commutativity, we also have a t 1 = a. Moreover, every t-norm ful ls the additional boundary condition 0 t a = a t 0 = 0 for all a 2 [0; 1]. Note also that all t-norms coincide on f0; 1g. 15

Example 13 The most important t-norms are minimum `^', product `' and Lukasiewicz conjunction ` ' (26). Other examples of t-norms are drastic product (

a tW b = min(a; b) if max(a; b) = 1; 0 otherwise; or nilpotent minimum (also called Fodor t-norm) (

a tF b = min(a; b) if a + b > 1; 0 otherwise:

2

The t-norms can be (partially) ordered as functions, i.e. for every t-norms t1 ; t2 we can put

t1 t2 i a t1 b a t2 b; a; b 2 [0; 1]: (51) If (51) holds then the t-norm t1 is weaker than t2 or, equivalently, t-norm t2 is stronger than t1. We write t1 < t2 if t1 t2 and t1 =6 t2. Lemma 8 For every t-norm t it holds that tW t ^: This means that the drastic product tW is the weakest and ^ the strongest t-norm. Furthermore, the following inequality holds for the t-norms introduced above:

tW < < < ^; Given a t-norm t, we put

ant = a| t {z t a} : n?times

A dual concept to t-norm is that of t-conorm. De nition 10 A t-conorm is a binary operation s : [0; 1]2 ?! [0; 1], which ful ls the axioms (i){(iii) from De nition 9, and for all a 2 [0; 1] it ful ls the following boundary condition: 0 s a = a:

(52)

A t-conorm is dual to the given t-norm t if

a s b = 1 ? (1 ? a) t (1 ? b) holds for all a; b 2 [0; 1].

Note that due to the commutativity, a s 0 = a holds for all a 2 [0; 1]. Moreover, every t-conorm ful ls the additional boundary condition 1 s a = a s 1 = 1.

16

Example 14 The most important t-conorms dual to the t-norms from Example 13 are the following. Dual to minimum is maximum `_', dual to product is probabilistic sum a sP b = a + b ? a b; dual to Lukasiewicz conjunction is Lukasiewicz disjunction

a b = min(1; a + b); dual to drastic product is drastic sum

(

a sW b = max(a; b) if min(a; b) = 0; 1 otherwise; and dual to nilpotent minimum is nilpotent maximum (

a tF b = max(a; b) if a + b < 1; 1 otherwise:

2

Lemma 9 For every t-conorm s it holds that _ s sW : This means that maximum is the weakest and the drastic sum the strongest t-conorm. For the t-conorms introduced above, we have

_ < sP < < s W : Given a t-conorm s, we put

nas = a| s {z s a} : n?times

Triangular norms with special properties The set of t-norms can be classi ed into var-

ious, partly overlapping groups according to their speci c properties. We will focus especially to three classes, namely continuous, Archimedean and non-Archimedean t-norms.

Example 15 The minimum, product and Lukasiewicz conjunction are continuous and, hence,

left- as well as right-continuous. An example of a non-continuous left-continuous t-norm is the nilpotent minimum tF . 2

De nition 11 (i) A t-norm t is called strictly monotone if atb < ctb holds for every a; b; c 2 (0; 1) and a < c. It is strict if it is strictly monotone and continuous. (ii) A t-norm t is Archimedean if for every a; b 2 (0; 1) there is n 2 N such that

ant < b: 17

(iii) A t-norm t is idempotent if a t a = a holds for all a 2 [0; 1]. (iv) A t-norm t is nilpotent if for every a 2 (0; 1) there is n such that ant = 0.

An element a 2 [0; 1] is called an idempotent if a t a = a. An element a 2 (0; 1) is nilpotent if there is n such that ant = 0. We may see that a t-norm is Archimedean if it has no other idempotents except for 0 and 1.

Example 16 The product, Lukasiewicz conjunction, and drastic product are Archimedean

while minimum is not. The Lukasiewicz conjunction is nilpotent and minimum is idempotent. The product is strict. 2

We obtain the following important classi cation of continuous t-norms: Every continuous Archimedean t-norm is either strict of nilpotent. Furthermore, every continuous t-norm is isomorphic either to product, or to Lukasiewicz t-norm, or it is equal to minimum `^', or a kind of `mixture' of all of them. Hence, when speaking about continuous t-norms, we may con ne ourselves to these three t-norms. Tehrefore, we will call them the basic t-norms.

Residuation and biresiduation We can introduce the residuation operation in the same

way as is done in Lemma 5.

De nition 12 Let t be a t-norm. The residuation operation !t: [0; 1]2 ?! [0; 1] is de ned by _ a !t b = fz j a t z bg: (53) It can be proved that if t is continuous, then a !t b = 1 i a b. Proposition 1 The following are the residuation operations corresponding to the basic tnorms: (a) Gödel implication !G given in (23) is a residuation operation corresponding to minimum `^'. (b) Goguen implication !P given in (24) is a residuation operation corresponding to multiplication `'. (c) Lukasiewicz implication ! given in (27) is a residuation operation corresponding to Lukasiewicz conjunction ` '. L

Theorem 3 Let t be a continuous t-norm and !t the corresponding residuation. Then a t (a !t b) = a ^ b. De nition 13 Let t be a t-norm. The biresiduation operation $t (cf. (31)) is the operation $t: [0; 1]2 ?! [0; 1] de ned by a $t b = (a !t b) ^ (b !t a): (54) Since [0; 1] is linearly ordered then for the continuous t, at least one of the arguments on the right hand side of (54) is equal to 1. Hence, the minimum operation can be replaced by an arbitrary t-norm.

Proposition 2 The following are biresiduation operations corresponding to the above introduced basic t-norms:

18

(a) For minimum ^:

(b) For product :

(

if a = b; a $G b = 1 a ^ b otherwise,

a; b 2 [0; 1]:

a $P b = aa _^ bb = exp (?j ln a ? ln bj) ;

where 0=0 := 1, resp. 1 ? 1 := 0. (c) For Lukasiewicz conjunction : a $ b = 1 ? ja ? bj;

a; b 2 [0; 1]

a; b 2 [0; 1]:

L

Negation In the de nition of t-conorm, we have used the operation :a = 1 ? a, which has

originally been used by L. A. Zadeh for the de nition of the complement of a fuzzy set. His choice was good since it turned out that this operation is a convenient interpretation of the negation in fuzzy logic. However, it is an interesting question, what properties must be ful lled by an operation to be considered as a many-valued negation. Since every generalization should preserve the original, it seems that the weakest requirement is merely to reverse 0 to 1 and vice-versa. De nition 14 The negation is a non-increasing operation n : [0; 1] ?! [0; 1] such that n(0) = 1 and n(1) = 0. The negation is involutive if n(n(a)) = a holds for every a 2 [0; 1]. The negation n is strict if it is continuous and a < b implies n(a) > n(b). It is strong if it is strict and involutive. Example 17 A typical example of the involutive negation is n(a) = 1 ? a. This is obtained from Lukasiewicz implication by putting n (a) = a ! 0. This negation is even strong. The negation nG (a) = a !G 0 is non involutive since nG (0) = 1 but nG (a) = 0 for every a 2 (0; 1]. 2 Let t be a t-norm and n be a negation. Put a s b = n(n(a) t n(b)); a; b 2 [0; 1]: (55) This de nes an operation s dual to the t-norm t. However, note that this needs not necessarily be a t-conorm. Lemma 10 Let n be an involutive negation. (a) If t is a t-norm then the operation s de ned in (55) is a t-conorm. (b) If s is a t-conorm then the operation t de ned by a t b = n(n(a) s n(b)); a; b 2 [0; 1] (56) is a t-norm. The operations t, s and n, where the latter is an involutive negation may form so called de Morgan triples. More precisely, if either (and consequently both) of the equations (55) or (56) holds then the triple of operations L

L

ht; s; ni

is called the de Morgan triple. It can be proved that there is no de Morgan triple with the operations t 6= ^ and s 6= _ ful lling both the distributive law as well as the law of the excluded middle with respect to the operations t; s. 19

4 Fuzzy Logic in Narrow Sense

4.1 Graded formal logical systems

We start with the theory of graded formal logical systems, which is based on the graded approach principle. The grades are supposed to be taken from the complete residuated lattice

L = hL; _; ^; ; !; 0; 1i:

(57)

At the same time, (57) is the set of truth values using which the semantics of formulas in fuzzy logic is de ned. This two-way view makes us nally possible to formulate generalization of the Gödel completeness theorem. Let us note that in comparison with the other kinds of many-valued logic, there are no designated truth values, i.e. all the truth values are equal in their importance. An important principle in fuzzy logic is the maximality principle according to which, if the same object is assigned more truth values then its nal truth assignment is equal to the maximum (supremum) of all of them.

Syntax First, let us consider some formal language J and a set FJ of well formed formulas,

which will be speci ed below. Let us only note that the language J is supposed to contain the logical constants a for all the truth values a 2 L. The speci c symbols ?; > denote logical constants for 0; 1 2 L, respectively. Furthermore, we will introduce the concept of evaluated formula. De nition 15 An evaluated formula is a couple

aA

(58)

where A 2 FJ is a formula and a 2 L is its syntactic evaluation.

De nition 16 An n-ary inference rule r in the graded logical system is a scheme a1 A1 ; : : : ; anAn ; r : evl r (a1 ; : : : ; an )rsyn(A1 ; : : : ; An )

(59)

using which the evaluated formulas ai Ai ; : : : ; an An are assigned the evaluated formula revl (a1 ; : : : ; an)rsyn (A1 ; : : : ; An ). The syntactic operation rsyn is a partial n-ary operation on FJ and the evaluation operation revl is an n-ary lower semicontinuous operation on L (i.e. it preserves arbitrary suprema in all variables).

De nition 17 An evaluated formal proof of a formula A from the fuzzy set X FJ is a nite sequence of evaluated formulas

w := a0 A0 ; a1 A1 ; : : : ; an An (60) such that An := A and for each i n, either there exists an n-ary inference rule r such that ai Ai := revl (ai1 ; : : : ; ain )rsyn (Ai1 ; : : : ; Ain ); i1 ; : : : ; in < i;

or

ai Ai := X (Ai )Ai: 20

The evaluation an of the last member in (60) is the value of the evaluated proof w. We will usually write the value of the proof w as Val(w). Axioms in the graded formal systems are sets of evaluated formulas. Since the evaluations can be interpreted as membership degrees in the fuzzy set, the axioms are at the same time fuzzy sets of formulas. Similarly as in the classical case, we distinguish logical and special axioms, namely LAx; SAx FJ . The we may introduce the following de nition. De nition 18 A formal fuzzy theory T in the language J of FLn is a triple T = hLAx; SAx; Ri where LAx FJ is a fuzzy set of logical axioms, SAx FJ is a fuzzy set of special axioms, and R is a set of sound inference rules. Given a fuzzy theory T . We denote its language by J (T ). If w is a proof in T then its value is denoted by ValT (w). De nition 19 Let T be a fuzzy theory and A 2 FJ a formula. We say that the formula A is a theorem in the degree a (or provable in the degree a) in the fuzzy theory T if _

a = fVal(w) j w is a proof of A from LAx [ SAxg: In this case, we will write T à A.

(61)

Note that unlike the classical syntax where nding one nite proof is sucient, the situation here is not so convenient. In general, a sequence (possibly in nite) of nite proofs must be considered, because maximum of their values does not need to exist, and so, we cannot con ne ourselves to one proof only.

Semantics A truth valuation of formulas from FJ is a function D : FJ ?! L being a homomorphism with respect to the connectives (see below). If a is a logical constant for some a 2 L then D(a ) = a. The truth valuation D is a model of the fuzzy theory T , D j= T , if SAx(A) D(A) holds for all formulas A 2 FJ . De nition 20 Let T be a fuzzy theory. we say that A is true in the degree a in the fuzzy theory T if ^ a = fD(A) j D j= T g (62) and write T j=a A.

Because all the truth values are equal in their importance, we may generalize the concept of tautology as follows. De nition 21 We say that a formula A is an a-tautology (tautology in the degree a) if ^

a = fD(A) j D is a truth valuationg and write j=a A. If a = 1 then we write simply j= A and call A a tautology.

(63)

Example 18 Let the set of truth values be the Lukasiewicz algebra L . Then the formula of the form A _ : A is a 0.5-tautology. Indeed, in any truth valuation D we have D(A _ :A) = D(A) _ :D(A) 0:5. 2 L

21

Soundness and completeness Recall that the classical syntactic and semantic consequence

operations are two dierent characterizations of true formulas. An important achievement of classical logic is the theorem stating that, even though each characterization is based on entirely dierent assumptions, both of them lead to the same result. We say that the graded formal logical system is sound if

T à A and T j=b A implies a b holds for every theory T and a formula A 2 FJ .

(64)

The graded formal logical system is complete if

T à A i T j=a A (65) holds for every theory T and a formula A 2 FJ . Note that due to the de nition, we always have T à A and T j=b A for some a; b 2 L, possibly equal to 0.

Bounds for completeness of the graded formal logical systems with implication Theorem 4 Given a graded formal logical system with the implication connective ) based on a complete residuated lattice (57). If the interpretation ! of ) does not ful l the equations _ _ (a ! bi ) = a ! ( bi ) (66) i2I _

i2I ^ i2I ^ i2I

^

i2I

(ai ! b) = ( ai ) ! b i2I

^

(a ! bi ) = a ! ( bi ) _

i2I

(ai ! b) = ( ai ) ! b i2I

(67) (68) (69)

for arbitrary subset of L then such a system cannot be complete. In the sequel, we will give detailed speci c de nition of the syntax and semantics of fuzzy logic.

4.2 Syntax of predicate rst-order fuzzy logic

The language J of predicate fuzzy logic consists of: (i) Countable set of object variables x; y; : : : . (ii) Finite or countable set of object constants u1 ; u2 ; : : : . (iii) Finite or countable set of n-ary functional symbols f; g; : : : . (iv) Nonempty nite or countable set of n-ary predicate symbols P; Q; : : : . (v) Symbols for logical constants fa j a 2 Lg. (vi) Symbol for the logical binary connective of implication ) . (vii) Symbol for the general quanti er 8. (viii) Various kinds of brackets as auxiliary symbols. Terms are symbols denoting objects. 22

De nition 22 (i) A variable x or constant u is a (atomic) term.

(ii) Let f be an n-ary functional symbol and t1 ; : : : ; tn terms. Then the expression f (t1 ; : : : ; tn ) is a term.

De nition 23 (i) A logical constant a is a (atomic) formula.

(ii) Let P be an n-ary predicate symbol and t1 ; : : : ; tn be terms. Then the expression P (t1 ; : : : ; tn ) is a (atomic) formula. (iii) If A; B are formulas then A ) B is a formula. (iv) If x is a variable and A a formula then (8x)A is a formula. (v) All formulas are formed by nite application of the previous four items.

A set of all terms of the language J is denoted by MJ and a set of all formulas by FJ . Furthermore, we will often use the symbol MV to denote the set of all closed terms of the language J . The concepts of the scope of the quanti er, sets FV (t); FV (A) of free variables, substitutible terms, open and closed formula are the same as in the classical logic. We will furthermore work with the following abbreviations of formulas.

:A :=A ) ? A _ B :=(B ) A) ) A A ^ B :=:((B ) A) ) :B ) A & B :=:(A ) :B ) ArB :=:(:A & :B ) A , B :=(A ) B ) ^ (B ) A) An := A {z & A} | & A& n?times

rA} nA := A | r Ar{z r

(negation) (disjunction) (conjunction) (Lukasiewicz conjunction) (Lukasiewicz disjunction) (equivalence) (n-fold conjunction) (n-fold disjunction)

n?times

(9x)A :=:(8x): A

(existential quanti er)

Inference rules (i) The inference rule of modus ponens is the scheme

aA; bA ) B rMP : a bB :

(70)

(ii) The inference rule of generalization is the scheme

aA rG : a : (8x)A

(71)

(iii) The inference rule of logical constant introduction is the scheme

aA rLC : a ! aa ) A 23

(72)

evl (note that the evaluation operation of rLC is rLC (x) = a ! x). Given an evaluated formula a A, the rule rLC enables to represent the evaluation a, which otherwise comes from the outside and does not belong to the language J , inside the latter. Consequently, we may identify the evaluated formula with the formula a ) A.

Logical axioms De nition 24 Given the following schemes of formulas. (R1) A ) (B ) A). (R2) (A ) B ) ) ((B ) C ) ) (A ) C )). (R3) (: B ) : A) ) (A ) B ). (R4) ((A ) B ) ) B ) ) ((B ) A) ) A). (B1) (a ) b ) , (a ! b ) where a ! b denotes the logical constant (atomic formula) for the truth value a ! b when a and b are given.

(T1) (8x)A ) Ax [t] for any substitutible term t.

(T2) (8x)(A ) B ) ) (A ) (8x)B ) provided that x is not free in A.

The fuzzy set LAx of logical axioms is speci ed as follows: 8 > 1 :

0

A := a ; if A has some of the forms (R1){(R4), (T1){(T3); otherwise:

4.3 Semantics of predicate rst-order fuzzy logic

De nition 25 A structure for the language J of FLn is D = hD; PD ; : : : ; fD ; : : : ; u1 ; : : : i where D is a set, PD D, : : : are n-ary fuzzy relations assigned to each n-ary predicate symbol

P; : : : , and fD are ordinary (crisp) n-ary functions on D assigned to each n-ary functional symbol f . Finally, u1 ; : : : 2 D are designated elements which are assigned to each constant u1 ; : : : of the language J , respectively.

A truth valuation of formulas in FLn is realized by means of their interpretation in some structure D in a way analogous to classical logic. For the given structure, it is a function FJ ?! L de ned below. Let D be a structure for the language J . We extend J to the language J (D) by new constants being names for all the elements from D, i.e.constants will be denoted by the corresponding bold-face letter, namely J (D) = J [ fd j d 2 Dg. 24

De nition 26 (i) Interpretation of closed terms: D(ui) = ui; ui 2 J; ui 2 D; D(d) = d; d 2 D; D(f (t1; : : : ; tn)) = fD (D(t1 ); : : : ; D(tn)): (ii) Interpretation of closed formulas: let t1 ; : : : ; tn be closed terms. Then

D(a ) = a; for all a 2 L; D(P (t1; : : : ; tn)) = PD (D(t1); : : : ; D(tn )); D(A ) B ) = D(A) ! D(B ); ^ D((8x)A) = fD(Ax [d]) j d 2 Dg; and in the case of the derived connectives,

D(A ^ B ) = D(A) ^ D(B ); D(A & B ) = D(A) D(B ); D(A _ B ) = D(A) _ D(B ); D(ArB ) = D(A) D(B ); D(A , B ) = D(A) $ D(B ); _ D((9x)A) = fD(Ax [d]) j d 2 Dg: Tautologies play central role in every logical system. The following lemma gives simple rules how to verify tautologies in fuzzy logic. Lemma 11 Let A; B be formulas in the language J . (a) j= A ) B i D(A) D(B ) holds in every structure D for the language J . (b) j= A , B i D(A) = D(B ) holds in every structure D for the language J .

4.4 Provability in fuzzy theories

Recall that due to our assumption about equal importance of all the truth degrees, every formula is a theorem (provable) in some degree, possibly equal to 0. Suppose we have learned that T à A. Then we may consider some proof wA of the formula A with the value Val(wA ) = a0 a. Furthermore, let a proof wB of some formula B be derived, which contains the formula A and which has the value Val(wB ) = f (a0 ; : : : ) (f is some operation). Using (61) we conclude that T `b B where _

b ff (a0 ; : : : ) j all proofs wAg: W

If f preserves suprema then we, moreover, know that b f ( a0 ; : : : ) = f (a; : : : ). The latter case is ful lled, for example, by the product .

Lemma 12 The following are schemes of eectively provable formal propositional tautologies in T , i.e. for each theorem there exists a proof with the degree equal to 1. (P1) T ` A ) A, 25

(P2) T ` A ) >,

(P3) T ` ::A ) A, (P4) (P5) (P6) (P7) (P8) (P9) (P10) (P11) (P12) (P13) (P14) (P15)

T ` A ) ::A, T ` (A ^ B ) ) A, T ` (A ^ B ) ) B , T ` (A & B ) ) A, T ` (A & B ) ) B , T ` A ) (A _ B ), T ` B ) (A _ B ), T ` (A & : A) ) B , T ` (A & B ) , (B & A), T ` ((A & B ) ) C ) , (A ) (B ) C )), T ` : (A ) B ) , (A & :B ), T ` (A ) B ) , (:ArB ), T ` (A ) B ) _ (B ) A), T ` (A ^ B ) , :(:A _ :B ), T ` (A ) (B ) C )) , (B ) (A ) C )), T ` (Am ) (B ) C )) ) ((An ) B ) ) (Am+n ) C )), m; n 2 N + ,

Lemma 13 Let T be a fuzzy predicate calculus. The following are schemes of the eectively provable formal predicate theorems in T , i.e. for each theorem there exists a proof with the degree equal to 1. (Q1) T ` (8x)(A ) B ) ) ((8x)A ) (8x)B ), (Q2) T ` (8x)(A & B ) , ((8x)A & B ) where x is not free in B ,

(Q3) T ` (8x)(A ) B ) , (A ) (8x)B ) where x is not free in A,

(Q4) T ` (8x)(A ) B ) , ((9x)A ) B ) where x is not free in B , (Q5) T ` (9x)(A ) B ) , (A ) (9x)B ) where x is not free in A,

(Q6) T ` (9x)(A ) B ) , ((8x)A ) B ) where x is not free in B , (Q7) T ` (8x)A , :(9x):A,

(Q8) T ` Ax [t] ) (9x)A for every term t.

Lemma 14 (a) To every proof wA with Val(A) = a there is a proof wb)A with the value Val(wb)A ) = b ! a. (b) In every fuzzy theory, T à A implies T ` a ) A for all formulas A and a 2 L.

26

4.5 Some important theorems of fuzzy logic

Due to the following theorem, we may derive formulas whose syntactic evaluation is at most as big as their semantic interpretation. Theorem 5 (validity theorem) Let T be a fuzzy theory. If T à A and T j=b A then a b holds for every formula A. As a special case, it follows from the de nition that T à a and T j=a a holds for every logical constant a , a 2 L, in every fuzzy theory T . The following theorem demonstrates that we may con ne ourselves only to closed formulas. Theorem 6 (closure theorem) Let T be a fuzzy theory. Let A 2 FJ (T ) and A0 be its closure. Then T à A i T à A0 . Theorem 7 Let T be a consistent fuzzy theory and T à A and T `b B . (a) T `c A ) B implies c a ! b. (b) T `c A & B implies c a b. V (c) If T à (8x)A then a fb j T `b Ax [t]; t 2 MJ (T )g. (d) If T `c A ^ B , T à A and T `b B then c = a ^ b.

Characterization of consistent theories De nition 27 A fuzzy theory T is contradictory if there is a formula A and proofs wA and w:A of A and : A, respectively, such that ValT (wA ) ValT (w:A ) > 0: (73)

It is consistent in the opposite case. Obviously, if T is contradictory then T à A, T `b :A and a b > 0. The following theorem states that a contradictory fuzzy theory collapses into a degenerated theory just as in classical logic. Theorem 8 (Contradiction) A fuzzy theory T is contradictory i T ` A holds for every formula A 2 FJ (T ) . One might think that De nition 27 is too strong and therefore, we may modify (73) by replacing by the in mum ^. Theorem 9 A fuzzy theory T is consistent i supfa j T à A ^ :A and A 2 FL(T )g 21 . In other words, if there is a formula A such that T à A ^ :A where a > 12 then T is again contradictory in the sense of De nition 27. The concept of the consistency threshold of a fuzzy theory is an interesting possibility speci c for the fuzzy logic, which, on the other hand, has no sense in classical logic. De nition 28 Let T be a consistent theory andA a formula. Then an element :a is a consistency threshold for A in T if T 0 = T [ f b :Ag is contradictory for all b > :a and consistent otherwise. Lemma 15 Let T be a consistent fuzzy theory and T à A where A is a closed formula. Let T 0 = T [ f b :Ag. Then T 0 is consistent i b :a. De nition 29 A language J 0 is an extension of the language J if J J 0 . A fuzzy theory T 0 is an extension of the fuzzy theory T if J (T ) J 0 (T 0 ) and T à A and T 0 `b A implies a b for every formula A 2 FJ (T ) .

27

Extension of fuzzy theories Occasionally, SAx may be extended by some fuzzy set of formulas ? FJ . Then by T [ ? we understand an extended fuzzy theory T [ ? = hLAx; SAx [ ?; Ri:

The extension T 0 is a conservative extension of T if T 0 `b A and T à A implies a = b for every formula A 2 FJ (T ) . The extension T 0 is a simple extension of T if J 0 (T 0 ) = J (T ).

The following theorem is a generalization of the classical important deduction theorem.

Theorem 10 (deduction theorem) Let T be a fuzzy theory, A be a closed formula and T 0 = T [ f 1 Ag. Then to every formula B 2 FJ (T ) there is an n such that T à An ) B i T 0 à B: Completeness theorem In this section, the analogy of the rst form of the completeness

theorem is presented. It is a nice generalization of the classical completeness theorem since it gives the provability and truth degrees of a formula into an exact balance, thus concluding the proclaim that these concepts nontrivially generalize the classical concepts of provability and validity of a formula. By the analogy with classical logic, there are two forms of the completeness theorem. It is surprizing that the formulation of the second form of the completeness theorem is exactly the same as in classical logic. However, its proof is not a trivial analogy.

Theorem 11 (Completeness theorem II) A fuzzy theory T is consistent i it has a model. Theorem 12 (Completeness theorem I) T à A i T j=a A holds for every formula A 2 FJ and every consistent fuzzy theory T . Sorites fuzzy theories The completeness theorem is a deep result with many consequences. In this subsection, we will demonstrate its application to the sorites paradox discussed above.

Theorem 13 Let T be a fuzzy theory in which all Peano axioms are accepted in the degree 1. Furthermore, let 1 " > 0 and H R 62 J (T ) be a new predicate. Then the fuzzy theory T + = T [ 1 H R (0); 1 ? " (8x)(H R (x) ) H R (x + 1)); 1 (9x):H R (x) is a conservative extension of T .

5 Functional Systems in Fuzzy Logic Theories This section relates to what is known as many-valued functional systems, i.e. systems of functions, whose domain and range are given by the same set of truth values. Functional systems relate to logical theories in such a way that formulas of the latter correspond to elements of the former. The following are special problems to be considered: (i) characterization of the functional systems related to propositional and predicate calculi, 28

(ii) construction of special formulas, which generalize Boolean disjunctive and conjunctive normal forms, (iii) investigation of what can be approximately described by fuzzy logic normal forms in both calculi, (iv) investigation of the quality of an approximation, which can be achieved by the special choice of a defuzzi cation.

5.1 Fuzzy logic functions and normal forms

The following de nition generalizes the notion of a 2-valued Boolean function. De nition 30 Let L be a support of an MV-algebra. A function f (x1; : : : ; xn), n 0, de ned on the set Ln and taking values from L is called an L-valued fuzzy logic function (or, shortly, an L-valued FL-function). The set of all the L-valued FL-functions will be denoted by PL . In the case that L = f0; 1g, Pf0;1g is a set of Boolean functions. In what follows we restrict ourselves to [0; 1]-valued FL- functions and thus drop the attribute \[0; 1]-valued". It is known that each Boolean function can be represented by a certain logical formula of the classical propositional calculus. It can be shown that this is not true for any L- valued FL-function when L = [0; 1] even if the set of constants is extended to L = [0; 1]. Precisely, due to McNaughton (1951), Mundici (1994), Per lieva, Tonis (1995) it has been proved that [0; 1]-valued FL-function can be represented by a formula of propositional calculus i it is a piecewise linear with integer coecients. Moreover, the special interest in fuzzy logic is paid to some distinguished formulas, which are linguistically expressed by \IF-THEN" rules (cf. Sections 2 and 6). Many applications are based on described in such a way models. Thus, it is imprecisely supposed that a certain class of dependences can be described by those rules appropriately. We will show that this presupposition is not an exaggeration. To do this, we translate \IF-THEN" rules into fuzzy logic formulas representable either in two forms, generalizing well known Boolean disjunctive and conjunctive normal forms. Then we investigate what kind of dependences (functions) can be represented by these generalized formulas. We will do it both in propopositional and predicate calculi as well.

Normal forms in fuzzy propositional calculus First, we recall the disjunctive and conjunctive normal forms for Boolean functions and explain our way of generalization. Let

f (x1 ; : : : ; xn ), n 1, be a Boolean function, i.e. its domain and range is the set f0; 1g. To represent the propositional variable x and its negation, we will write ( 0 x = x if = 0; x if = 1: It is easy to see that x = 1 i x = . Then f (x1 ; : : : ; xn ) can be represented in the disjunctive

normal form

f (x1 ; : : : ; xn) =

_

(x1 1 ^ ^ xnn ) =

f (1 ;:::;n )=1 _ = (x1 1 ^ ^ xnn ^ f (1 ; : : : (1 ;::: ;n )

29

; n ))

provided that f 6 0, and in the conjunctive normal form ^

f (x1 ; : : : ; xn) =

(x11 _ _ xnn ) =

f (1 ;:::;n )=0 ^ = (x11 _ _ xnn _ f (1 ; : : : ; n )) (1 ;::: ;n )

provided that f 6 1. A special interest will be payed to the right-hand sides of both expressions since they are prototypes of the suggested generalization. Analyzing them we see that they contain two parts, namely the characterization (positive or negative) of the set f(1 ; : : : n )g f0; 1gn or f(1 ; : : : n)g f0; 1gn respectively, and the value of the represented function de ned on it. Below, we suggest a generalization of the disjunctive and conjunctive normal forms for FL-functions de ned on [0; 1] over Lukasiewicz MV-algebra of operations LL . The induced MV-algebra of functions P[0;1] is then based on the set P[0;1] = ff (x1 ; : : : ; xn ) j f : [0; 1]n ?! [0; 1]; n 0g so that P[0;1] = hP[0;1]; ; ; :; 0; 1i and each operation is de ned pointwise. The constants 0; 1 are the respective constant FLfunctions. We will show how the generalized normal forms DNF and CNF look like and moreover, how the functions represented by them can be characterized. We will follow the explained construction of Boolean forms consisting in two parts and will preserve also two parts in each elementary conjunction or disjunction. The rst is the characterization (positive or negative respectively) of a certain set and the second is the average value of a certain function on it. Let m be a positive integer. De ne the interval functions expressed as follows: ( k x < (k+1) ; 1 ; if m m m Ik (x) = 0; otherwise where 0 k m ? 1, and ( Imm (x) = 1; if x = 1; 0; otherwise. As can be seen, each function Ikm , 0 k m ? 1, is a characteristic function of the subset fx j mk x < (km+1) g. Hence, we will use it and its negation as rst parts in the respective normal forms. De nition 31 Let f (x1; : : : ; xn) be an FL-function and m be a positive integer. The formula over P[0;1] DNF(x1 ; : : : ; xn ) =

m _

k1 =0

m _

(Ikm1 (x1 ) Ikmn (xn ) f ( km1 ; : : : ; kmn ))

kn =0

is called the MV-disjunctive normal form for f (x1 ; : : : ; xn ) and

CNF(x1 ; : : : ; xn ) =

m ^

k1 =0

m ^

(:Ikm1 (x1 ) :Ikmn (xn ) f ( km1 ; : : : ; kmn ))

kn =0

is called the MV-conjunctive normal form for f (x1 ; : : : ; xn ).

30

Let us apply the identity :a b = a ! b to the expression of CNF and thus derive the following equivalent form CNF(x1 ; : : : ; xn ) =

m ^ k1 =0

m ^

((Ikm1 (x1 ) Ikmn (xn )) ! f ( km1 ; : : : ; kmn )):

kn =0

If the normal forms include one variable then their representation is a simple reduction of certain formulas such that m _ m DNF(x) = Ik (x) f mk ; k=0 m ^ m CNF(x) = :Ik (x) f mk : k=0

Both these forms represent the same function (

k k k+1 fDNF (x) = fCNF(x) = f ( m ); if m x < m ; 0 k m ? 1; f (1); if x = 1:

Remark 1 The expressions DNF(x1 ; : : : ; xn ) and CNF(x1 ; : : : ; xn) are used to denote the corresponding formulas over P [0; 1]. When dealing with FL- functions represented by DNF and CNF, we will use the notation fDNF(x1 ; : : : ; xn ) and fCNF(x1 ; : : : ; xn ), respectively where f refers to the original FL-function.

Generally, the normal form (disjunctive or conjunctive) for the function f (x1 ; : : : ; xn ) does not represent the same function as f . We can only prove that there is an eect of approximation. In order to speak about it correctly we will use the notion which plays the role of a distance in MV-algebra. According to the de nition of C. C. Chang, the FL-function

d(x; y) = (:x y) (:y x) can be taken as a distance.

De nition 32 An FL-function f (x1; : : : ; xn) is uniformly continuous on L if for any u 2 L, [0] < [u] < [1], there exists v 2 L, [v] > [0], such that [d(x1 ; y1 )] < [v]; : : : ; [d(xn ; yn )] < [v] implies [d(f (x1 ; : : : ; xn ), f (y1; : : : ; yn ))] < [u].

Note, that in the case of L = [0; 1], the distance d(x; y) = jx ? yj and thus, the notion of uniform continuity coincides with the classical one for real valued real functions.

De nition 33 Let u 2 L and [u] < [1]. We say that an FL-function g(x1 ; : : : ; xn ) uapproximates the FL-function f (x1 ; : : : ; xn ), or in other words g(x1 ; : : : ; xn ) approximates f (x1 ; : : : ; xn ) with the accuracy u, if the inequality

[d(g(x1 ; : : : ; xn ); f (x1 ; : : : ; xn ))] [u] holds for all (x1 ; : : : ; xn ) 2 Ln .

Theorem 14 Let an FL-function f (x1; : : : ; xn) be uniformly continuous on L. Then for any u 2 L, [0] < [u] < [1], there exists v 2 L, [v] > [0], such that each function fDNF(x1 ; : : : ; xn) or fCNF(x1 ; : : : ; xn ) based on the parameter v u-approximates f (x1 ; : : : ; xn ).

31

Normal forms in fuzzy predicate calculus In this paragraph we introduce the disjunctive

and conjunctive normal forms and prove approximation theorems for uniformly continuous FLrelations. At rst, we recall what is understood by relations in connection with fuzzy logic. De nition 34 Let D be a nonempty set of objects and L be a support of a complete MVn algebra. An n-ary fuzzy logic relation R on D or an FL-relation, is a fuzzy set R n D , n 0, which is given by the membership function R(x1 ; : : : ; xn ) de ned on the set D and taking values from L. Note that according to this de nition, we identify an FL- relation with its membership function to be able to work with functions in the sequel. Furthermore, we will consider the case L = [0; 1] and the Lukasiewicz MV-algebra LL .

De nition 35 Let P1 ; : : : ; Pk be unary predicate symbols and Ei1 :::in , 1 ij k, 1 j n, n 1, be some atomic formulas. The following formulas of fuzzy predicate logic are called the MV-disjunctive normal form

DNF(x1 ; : : : ; xn ) =

k _ i1 =1

k _

& & Pin (xn )& & Ei1 :::in ) (Pi1 (x1 )&

in =1

and the MV-conjunctive normal form

CNF(x1 ; : : : ; xn ) =

k ^ i1 =1

k ^

r:Pin (xn )rEi1 :::in ): (: Pi1 (x1 )r r

in =1

As in the previous paragraph, the expression CNF can be rewritten into the following one based on implicaton, namely CNF(x1 ; : : : ; xn ) =

k ^ i1 =1

k ^

& & Pin (xn ) ) Ei1 :::in ): (Pi1 (x1 )&

in =1

In the sequel, we will use the shorts DNF and CNF. Contrary to the case of propositional fuzzy logic where we suggested the expressions for the normal forms as algebraic formulas, the latter expressions are formulas of fuzzy predicate logic. In what follows we suppose a support D of a structure D to be a compact uniform topological space. Let be a pseudometrics, which belongs to the gage of the uniformity on D, i.e. to the family of all pseudometrics which are uniformly continuous on D D.

De nition 36 An n-ary FL-relation R Dn is uniformly continuous on Dn if for any ",

0 < " < 1, there exists a positive number r, such that if (x11 ; x21 ) < r; : : : ; (x1n ; x2n ) < r then jR(x11 ; : : : ; x1n ) ? R(x21 ; : : : ; x2n )j < ".

Let us stress that this notion coincides with the analogous classical notion for real valued functions de ned on a compact uniform space.

De nition 37 An FL-relation S (x1 ; : : : ; xn) on D is said to "-approximate another FL-relation R(x1 ; : : : ; xn) on D if

jS (x1 ; : : : ; xn) ? R(x1; : : : ; xn)j " is true for all (x1 ; : : : ; xn ) 2 Dn .

32

Theorem 15 Let an FL-relation R Dn be uniformly continuous on Dn. Then for any ", 0 < " < 1, there exist a formula DNF(x1 ; : : : ; xn ) (CNF(x1 ; : : : ; xn )) and a structure D with the support D such that the FL-relation represented by DNF (CNF) w.r.t. D "-approximates R(x1 ; : : : ; xn).

5.2 Fuzzy logic formulas as a tool for an approximate description

In the previous subsection we have considered a class of FL-relations represented by normal forms (disjunctive or conjunctive) of predicate fuzzy logic and proved that any FL-relation with continuous membership function can be approximated by some element from this class. In this subsection, we will focus on analogous problem of approximation of any continuous function on a compact set in the same class. First, we will apply the above mentioned results. However, this is possible only if we take a function as a particular case of an FL-relation with continuous membership function. We will then obtain a non-direct solution of the problem. The second solution will be provided by a direct algorithmic construction of an FL-relation represented by a normal form, which approximates the given function. For simplicity, we will con ne ourselves to real valued real functions. Note, that the general case requires to consider many sorted relations, which might lead to unnecessary complexity. Moreover, throughout this subsection when speaking about a formula we will suppose some quanti er-free one of predicate fuzzy logic. Let us x some continuous real valued real function f (x1 ; : : : ; xn ), n 1, de ned on a compact set Dn 2 Rn . Obviously, f (x1 ; : : : ; xn ) is also uniformly continuous on Dn, i.e. for any " > 0 there exists > 0 such that jf (x11; : : : ; x1n ) ? f (x21 ; : : : ; x2n )j < " (74) whenever jx1j ? x2j j < , 1 j n, and (x11 ; : : : ; x1n ); (x21 ; : : : ; x2n ) 2 Dn . Moreover, f (Dn) is a compact set, as well. Let " > 0 be an arbitrary number, and let > 0 be as described above. Since D is a compact set there exists a nite covering of it by open - intervals I1 ; : : : ; Ik , k 1, respectively. It follows that the set Dn is also covered by all the direct products of those intervals. Since f is uniformly continuous, the set f (Dn) will be covered by intervals f (Ii1 Iin ), 1 ij k, 1 j n, whose lengths are not greater than ". Their centers will be denoted by yi1 :::in . For each -interval Ii , 1 i k, choose any closed subinterval ICi Ii and denote the closure of Ii by CIi. Furthermore, consider open "-intervals around the points yi1 :::in , 1 ij k, 1 j n, and denote them by Hi1 :::in . Observe that f (Ii1 Iin ) Hi1 :::in . Choose any closed subinterval HCi1 :::in Hi1 :::in containing the point yi1 :::in . By virtue of Urysohn lemma, for each i, 1 i k, there exists a continuous function gi (x) on D taking values from [0; 1], which is equal to one on ICi and equal to zero on D n Ii . At the same time, for each n-tuple (i1 : : : in ), 1 ij k, 1 j n, there exists a continuous function hi1 :::in (y) on f (Dn) such that it takes values from [0; 1], it is equal to one on HCi1 :::in and it is equal to zero on f (Dn) n Hi1 :::in . n n Now, for the given function f let us construct an (n +1)- ary FL-relation Rf D f (D )

Rf (x1 ; : : : ; xn ; y) =

k _ i1

k _

: : : (gi1 (x1 ) gin (xn) hi1 :::in (y)); in

(75)

which approximates f in a sense explained below. Observe, that `' denotes the ordinary product and thus, the expression (75) is not a formula of fuzzy predicate logic. 33

De nition 38 Let f (x1; : : : ; xn ), n 1, be a continuous real valued real function de ned on a n n compact set Dn 2 Rn and Q(x1 ; : : : ; xn ; y) D f (D ) be an FL-relation. We say that Q "approximates f if the condition Q(x1 ; : : : ; xn ; y) > 0 implies the condition jy ? f (x1 ; : : : ; xn )j ".

Lemma 16 The FL-relation Rf (x1 ; : : : ; xn; y) given by (75) is uniformly continuous on Dn f (Dn ).

Lemma 17 Let f (x1; : : : ; xn ), n 1, be a continuous real valued real function de ned on a compact set Dn 2 Rn . Furthermore, let " > 0 be an arbitrary number and > 0 be the number,

which provides the inequality (74). If Rf (x1 ; : : : ; xn ; y) is the FL-relation given by (75) with respect to the function f and values "; , then Rf 2"- approximates f . Theorem 16 Let f (x1; : : : ; xn ), n 1, be a continuous real valued real function de ned on a compact set Dn 2 Rn , " > 0 be an arbitrary number and > 0 be the number, which provides the inequality (74). Then there exists an FL-relation represented by a formula, which 2"-approximates f . The following constructive procedure gives the precise expression for the FL-relation, which "-approximates f and which can be represented by a formula in the disjunctive normal form. In comparison with the FL-relation Rf (x1 ; : : : ; xn ; y) given by (75), the one suggested below has not a continuous membership function. Let f (x1; : : : ; xn ), n 1, be a continuous real valued real function de ned on a compact set Dn 2 Rn and let " > 0 be an arbitrary number. Choose > 0 such that the inequality (74) holds and nd a nite covering of D by open -intervals I1 ; : : : ; Ik . As a result, the set Dn will be also covered by all the direct products of those intervals. Based on the uniform continuity of f on Dn nd the corresponding nite covering of f (Dn) by intervals f (Ii1 Iin ), 1 ij k, 1 j n, whose lengths are not greater than ". Denote their centers by yi1 :::in and consider open "- intervals Hi1 :::in around the points yi1 :::in , 1 ij k, 1 j n. De ne functions g1 (x); : : : ; gk (x) on D and h i1 :::in (y) on f (Dn), so that (

gi(x) = 1; if x 2 Ii ; 0; otherwise

and

(

h i1 :::in (y) = 1; if y 2 Hi1 :::in 0; otherwise; where 1 i k, 1 ij k, 1 j n.

Form the FL-relation

R f (x1 ; : : : ; xn; y) =

k _ i1

k _

: : : (gi1 (x1 ) gin (xn) h i1 :::in (y)): in

(76)

It can be shown that this FL-relation "- approximates f and can be represented by a formula in the disjunctive normal form w.r.t to some structure. Observe, that the described constructive procedure can be rearranged for the case of the conjunctive normal form. 34

5.3 Compositional rule of inference and the best approximation

A compositional rule of inference is a procedure, which generally speaking, corresponds a dependent value to a given relation and a certain restriction (fuzzy or crisp) to an independent value(s). Here we use the very special kind of a compositional rule of inference, which is nothing else as substitution. We will use it to extract a normal function from describing it FL-relation. This is eventually done by applying the defuzzi cation procedure. We can regard the defuzzi cation as a generalized fuzzy operation on a set of elements. De nition 39 A defuzzi cation of a fuzzy set A X given by its membership function A(x) : X ?! [0; 1] is a mapping : F (X ) ?! X such that A((A)) > 0: (77) The set of all the defuzzi cation operations on F (X ) will be denoted by DEFF (X ) . Let X be a compact set and A(x) 6 0 be a continuous membership function of a fuzzy set A X . Put x = (A) where 2 DEFF (X ) . The following speci cations of the defuzzi cation are used most frequently: x = inf fx j A(x) = sup A(u)g; (Least of Maxima) (78) u2X (Mean of Maxima) (79) x = xL +2 xG ; where xL = inf fx j A(x) = supu2X A(u)g, xG = supfx j A(x) = supu2X A(u)g, R xA(x)dx x = RX :

In particular, if X

X A(x)dx = fx1 ; : : : ; xn g then k X 1 x = k xij ; where j =1 Pn xi A(xi ) x = Pi=1 n A(x ) : i i=1

(Center of Gravity)

(80)

A(xij ) = max A(u); u2X

Remark 2 In the examples of the defuzzi cation given above the continuity of A(x) on the

compact set X guarantees validity of (77). If A(x) is not continuous then the defuzzi cation may be also realized in accordance with (78){(80), but the inequality (77) should be veri ed subsequently. In the sequel, we will not assume the membership functions of the defuzzi ed fuzzy sets to be necessarily continuous. Now, given a certain defuzzi cation we can uniquely correspond a function to an FLrelation. Denote the latter by R(x1 ; : : : ; xn ; y) and suppose that it is de ned on a set X1 : : : Xn Y . Choose a defuzzi cation 2 DEFF (Y ) and de ne the function fR; (x1 ; : : : ; xn ) : X1 : : : Xn ?! Y by putting fR; (x1 ; : : : ; xn ) = (R(x1 ; : : : ; xn ; y)): (81) The function fR; de ned by (81) is said to be adjoined to an FL-relation R by means of the chosen defuzzi cation . The following theorem shows the relation between functions f and fR; where R is an FL-relation, which approximates f .

35

Theorem 17 Let f (x1; : : : ; xn ), n 1, be a continuous real valued real function de ned on a compact set Dn 2 Rn and " > 0 be an arbitrary number. Furthermore, let an FL-relation R(x1 ; : : : ; xn; y) "-approximate f . Then for any defuzzi cation 2 DEFF (Y ) the function fR; de ned by (81) also "-approximates f as well, i.e. for any (x1 ; : : : ; xn ) 2 Dn jf (x1; : : : ; xn) ? fR;(x1 ; : : : ; xn)j ": We will consider the problem of the best approximation w.r.t. the choice of a defuzzi cation and in accordance with a certain criterion. To simplify the description, we will con ne ourselves to the case of functions with one variable. De nition 40 Let f (x) be a continuous real valued function de ned on the closed interval [a; b] and let G be a set of approximation, which consists of continuous real valued functions de ned also on [a; b]. Moreover, let d determine a distance on the set of all continuous real valued functions de ned on [a; b]. We say that g (x) 2 G is the best approximation from G to f if the condition d(f; g ) d(f; g) (82) holds for all g(x) 2 G. In the sequel, we will consider a set of approximation consisting of a set of real valued real functions, which are adjoined to some special FL- relations represented by formulas by means of some defuzzi cation operation. We will be interested in nding the best approximation to the given continuous real valued real function f (x) in a sense close to the above de ned one. On the account of results from the previous subsection we may regard those special FLrelations represented by normal forms. In particular, they are given by DNF(x; y) =

k _

& Qi (y)) (Pi (x)&

(83)

(Pi (x) ) Qi (y))

(84)

i=1

and CNF(x; y) =

k ^ i=1

where Pi and Qi , 1 i k, are unary predicate symbols. Let f (x) be a continuous real valued function de ned on the closed interval [a; b] so that f : [a; b] ?! [c; d]. Choose a structure D (follow the constructive procedure in the previous subsection including the notations) sucient for the interpretation of the normal forms (83) and (84) and such that the FL-relations RDNF (x; y) and RCNF (x; y) represented by these normal forms w.r.t. D are equal to R(x; y) where ( (85) R(x; y) = 1; if (9i)(1 i k and x 2 Ii ; y 2 f [Ii]); 0; otherwise: De ne the set of approximation Gf for f as the set of all functions adjoined to R(x; y) given by (85) by means of the defuzzi cations 2 DEFF (D) Gf = fg(x) j g(x) = fR;(x) = (R(x; y)); 2 DEFF (D) g: (86) In what follows we will x the structure D, which lead to R(x; y) and Gf and preserve all the intermediate notations. 36

Theorem 18 Let f (x) : [a; b] ?! [c; d] be a continuous function and Gf be the set of approximation for f given by (86). Furthermore, let gMOM (x) 2 Gf be the function speci ed by the defuzzi cation (79). Then for each closed interval CIi = [xi ; xi+1 ], 1 i k, the continuous restriction gMOM jCIi (x) is the best approximation from Gf;i to f jCIi(x) with respect to the distance d given by

d(u(x); v(x)) = max ju(x) ? v(x)j: CI i

5.4 Representation of continuous functions and its complexity

We are interested here in the precise representation of continuous functions by FL-relations in such a way that given a function nd the FL-relation, to which it can be adjoined by means of some defuzzi cation. In addition, the FL-relation is supposed to be represented by one of the normal forms. After a thorough analysis of the methodology of approximation described in the previous subsection we deduce that the problem of representation can be solved if a correspondence between function f (x) and the FL- relation R(x; y) adjoined to it, is more rigorous. We will suppose that the latter is built up in such a way that y = f (x) implies R(x; y) = 1: (87) Moreover, the defuzzi cation is supposed to be given by (78). As above, we will work with real valued real functions with one variable. Till now, both kinds of normal forms have been interchangable. In this section, we will see that the conjunctive normal form is more suitable for the solution of our problem. The justi cation of this follows from the subsequent lemma where the functions adjoined to FLrelations represented by DNF are shown to be very speci c. Lemma 18 Let f (x) be a real valued function de ned on [a; b], which is adjoined to the FLrelation R(x; y) [a; b] f ([a; b]) by means of the defuzzi cation (78) so that (87) holds. Suppose that R(x; y) is expressed in the disjunctive normal form

R(x; y) =

k _

i=1

(Pi (x) Qi (y))

where Pi (x); Qi (y), 1 i k, are membership functions of normal fuzzy sets de ned on intervals [a; b] and f ([a; b]), respectively. Moreover, suppose that those membership functions are continuous and their a-cuts are closed intervals. Then f (x) is a piecewise constant with nite number of steps (some steps may degenerate into points). Now, we will show that the same assumptions as in the previous lemma except the choice of the normal form lead to the wider class of real valued functions, which can be expressed. To show the precise expression, the lemma below will be given a proof. Lemma 19 Let f (x) be a continuous and strictly monotonous real valued function de ned on [a; b]. Then there exists an FL-relation R(x; y) de ned on [a; b] [f (a); f (b)] and expressed in the conjunctive normal form

R(x; y) =

2 ^

i=1

(Pi (x) ! Qi (y))

(88)

so that (87) holds, and f (x) is the function adjoined to R(x; y) by means of the defuzzi cation (78).

37

Let f (x) ful ls the assumptions. To obtain the proof, it is sucient to construct the membership functions Pi (x); Qi (y), i = 1; 2, de ned on [a; b] and [f (a); f (b)], respectively and then verify the conclusions. Since f (x) is continuous and monotonous on [a; b] it de nes a one-to-one correspondence between [a; b] and [f (a); f (b)]. Therefore, the inverse function f ?1 (y) exists. For the certainty, assume that f (x) monotonously increases. Put the above mentioned membership functions as follows:

Proof:

P2 (x) = xb ?? aa ; x 2 [a; b]; P1 (x) = 1 ? xb ?? aa ; ?1 ?1 Q1 (y) = 1 ? f b(?y)a? a ; Q2 (y) = f b(?y)a? a ; y 2 [f (a); f (b)]:

Let x0 2 [a; b] be an arbitrary element. We will show that

f (x0 ) = inf fy j R(x0 ; y) = 1g and R(x0 ; f (x0 )) = 1: Indeed,

P1 (x0 ) ! Q1 (y) = 1 i P1 (x0 ) Q1(y) 0 ?1 i 1 ? xb ??aa 1 ? f b(?y)a? a i y f (x0 ); ?1 0 P2 (x0 ) ! Q2 (y) = 1 i P2 (x0 ) Q2(y) i xb ??aa f b(?y)a? a i f (x0 ) y: Thus,

R(x0 ; y) = 1 i f (x0 ) = y what has been necessary. 2 The same technique can be applied for the representation of the more general class of real valued functions which are constituted of strictly monotonous restrictions. To be precise, we introduce the following de nition.

De nition 41 By a piecewise monotonous real valued function f (x) de ned on [a; b] we mean

a function, for which there exists a nite partition of [a; b] into subintervals such that the restriction of f to each subinterval from the partition is strictly monotonous.

Theorem 19 Let f (x) be a continuous and piecewise monotonous real valued function de ned on [a; b]. Then there exist a number N , and an FL- relation R(x; y) de ned on [a; b] f ([a; b]) and expressed in the conjunctive normal form

R(x; y) =

2 N ^ ^

(Pij (x) ! Qij (y))

i=1 j =1

(89)

so that (87) holds and f (x) is the function adjoined to R(x; y) by means of the defuzzi cation (78).

38

6 Fuzzy Logic in Broader Sense FLn should be a theoretical basis standing beyond methods and techniques of fuzzy logic. The mediator between the latter and FLn is natural language. Note that most of the applications of fuzzy logic are based on the generalized modus ponens mentioned in Section 2, which is a model of one of the fundamental principles of human reasoning. Since the success of applications of fuzzy logic is convincing, it is important to elaborate good formalization of natural language and to employ it in the model of the commonsense human reasoning. The fuzzy logic in broader sense is a contribution to this task. Of course, the formalization concerns only a small fragment of natural language, namely that which is used in evaluation of the behaviour of dynamic systems and various decision situations.

6.1 Partial formalization of natural language

Evaluating and simple conditional linguistic syntagms The considered fragment of natural language consists of a set S of linguistic syntagms speci ed below. They are constructed

from the following classes of words. (i) Nouns, which mostly will be taken as primitives without studying their structure and formally represented simply by variables denoting objects. (ii) Adjectives where we will con ne to few proper ones, mostly forming an evaluation linguistic trichotomy of the type \small", \medium" and \big". These words can be taken as prototypical for other similar linguistic trichotomies, such as \weak, medium strong, strong", \clever, average, dull", etc. (iii) Linguistic hedges, which form a special kind of the so called intensifying adverbs, such as \very, slightly, roughly", etc. We distinguish linguistic hedges with narrowing eect, such as \very, signi cantly", etc. and those with extending eect, such as \roughly, more or less", etc. (iv) Connectives \and", \or" and the conditional clause \if : : : then : : : ". Further class to be added in the future are linguistic quanti ers. For the time being, however, we will not consider them. A speci c is the class of evaluating syntagms, i.e., linguistic expressions which characterize a position on a bounded ordered scale (usually an interval of some numbers (real, rational, natural)). These are syntagms such as \small, medium, big, very small, roughly large, extremely high", etc. The symbol h : : : i used below denotes a metavariable for the kind of a word given inside the angle brackets. De nition 42 An evaluating syntagm is one of the following linguistic expressions: (i) Atomic evaluating syntagm, which is any of the following: An adjective \small", \medium", or \big", A fuzzy quantity \approximately z", which is a linguistic expression characterizing some quantity z from an ordered set. (ii) Negative evaluating syntagm, which is an expression not hatomic evaluating syntagmi:

39

(iii) Simple evaluating syntagm, which is an expression

hlinguistic hedgeihatomic evaluating syntagmi . (iv) Compound evaluating syntagms constructed as follows: if A; B are evaluating syntagms then À and B', À or B' are compound evaluating syntagms.

Example 19 Atomic evaluating syntagms are small, medium, big. Fuzzy quantities are, e.g.

twenty ve, the value z , etc. Simple evaluating syntagms are very small, more or less medium, roughly big, about twenty ve, approximately z , etc. Compound evaluating syntagms are roughly small or medium, quite roughly medium and not big, etc. 2 The \fuzzy quantity" is a linguistic characterization of some quantity | usually a number. This means that every linguistic characterization of a number is understood imprecisely. We will take the form \approximately z " as canonical.

De nition 43 Let A be an evaluating syntagm. Then the syntagm hnouni is A (90) is an evaluating predication. If A is a simple evaluating syntagm then (90) is a a simple

evaluating predication. Compound evaluating predications are formed from other ones using the connectives \and, or".

Example 20 Evaluating predications are, e.g. \ temperature is very high" (here \high" is

taken instead of \big"), \ pressure is not small", \ frequency is small or medium", \ cost is not small and not big", \ income is roughly three million", etc. Compound evaluating predication is, e.g. \ temperature is high and pressure is very high". 2

The following de nition characterizes the set of linguistic expressions which will be employed in FLb. De nition 44 The set S of syntagms consists of evaluating syntagms, evaluating predications and the conditional clause IF A THEN B

(91)

where A, B are evaluating linguistic predications.

6.2 Intension and extension in fuzzy logic

In the logical analysis of natural language, three concepts play a distinguished role, namely intension, extension and possible world. A possible world is a \way of things it could be in order for the expression to be true", or a \particular state of aairs" which determine truth of a sentence. Intension of a linguistic syntagm, sentence, or of a concept, can be identi ed with the property denoted it. Intension may lead to dierent truth values in various possible worlds but it is invariant with respect to them. Extension is a set of elements determined by an intension, which fall into the meaning of a syntagm in a given possible world. Thus, it depends on the particular context of use. 40

De nition 45 Let A(x1 ; : : : ; xn) 2 FJ be a formula with the free variables x1; : : : ; xn of the respective sorts 1 ; : : : ; n . Then the set

Ahx1 ;:::;xni = at1 ;:::;tn Ax1 ;:::;xn [t1; : : : ; tn] t1 2 M1 ; : : : ; tn 2 Mn

(92)

of evaluated formulas being closed instances of A(x1 ; : : : ; xn ) is called a multiformula.

Obviously, a multiformula A can be at the same time viewed as a fuzzy set A FJ of closed instances of the formula A(x1 ; : : : ; xn ). Recall that syntagms A 2 S are names of properties of objects. The latter can be represented in a formal language of logic using formulas A 2 FJ . At the same time, vagueness of syntagms must be taken into account and thus, a simple assignment of A to A is not sucient. The following de nition oers a solution. De nition 46 Let A 2 S be a syntagm and A(x1 ; : : : ; xn) 2 FJ be a formula assigned to it. Then the multiformula Ahx1 ;:::;xn i is the intension of A. A structure D for J is a possible world and the extension of A in a given possible world D is an FL-relation

n

RD;A = D(Ax1 ;:::;xn [t1; : : : ; tn ]) hD(t1 ); : : : ; D(tn )i ti 2 Mi ; i = 1; : : : ; n

where D(t1 ) 2 D1 ; : : : ; D(tn ) 2 Dn are interpretations of the terms t1 ; : : : ; tn , respectively in the structure D.

It is clear that one intension A may lead to (in nitely) many extensions D(A) with dierent truth evaluations.

Example 21 Let A :=Young. For simplicity, we will consider the background theory BT to be the theory of natural numbers and let MJ = ft0 ; : : : ; t100 ; : : : g be a set of terms representing years. We choose a formula Y (x) 2 FJ and assign it to Young. Then the following multiformula is the intension of Young:

Yhxi = 1 Yx[t0 ]; : : : ; 1 Yx[t20 ]; : : : ; 0:6 Yx[t30 ]; : : : ; 0:2 Yx[t45]; : : : ; 0 Yx[t60 ] : (93) Let D be a structure with D N . Then a possible extension of Young can be, for example,

the following.

D(Yhxi) = 1 1; : : : ; 1 20; : : : ; 0:6 30; : : : ; 0:2 45; : : : ; 0 60 (94) where D(t0 ) = 1; : : : , D(t20 ) = 20; : : : , D(t30 ) = 30; : : : , D(t45 ) = 45; : : : , D(t60 ) = 60 are

interpretations of the terms from MJ when representing age of people. It is clear that another extension can be obtained from this intension when representing ages of trees, dogs, or whatever else. Recall in these examples that if a A is an evaluated formula then only the inequality D(A) a should be ful lled. 2

6.3 The meaning of simple evaluating syntagms

There is a strong parallel between the concepts of nite numbers and the small ones. The nite numbers can be represented by a horizon running from 0 to large numbers, where it somehow \disappears". Also the meaning of the other two members of the evaluation linguistic 41

trichotomy, i.e. \medium" and \big", can be based on the same ideas and so, we can get a uni ed theory. There is a tendency of people to classify three positions on any ordered scale, namely \the leftmost", \the rightmost", and \in the middle". The horizon of small lays \somewhere to the right from the leftmost position". Small numbers are numbers which everybody easily understands and is able to verify that they are small. On the other hand, there is no last small number, i.e. small numbers form a class but not set and we may encounter only the horizon of small numbers running \somewhere to big ones". Thus, small numbers behave similarly as nite ones. In the same way, the horizon of big lays \somewhere to the left from the rightmost position" and there is no rst big number. In the case of medium, the horizon is spread both to the left as well as to the right from the middle | cf. Figure 3. It is possible to specify multiformulas RHRx and LHRx representing right

1

small

7

very small

roughly medium + .. s .. K ... 3 . medium .. .. .. .

roughly small

W

roughly big

U

big

I very big

.. .. .. .. .. .. .. .

Figure 3: Typical membership functions of simple evaluating syntagms. and left horizon, respectively on the basis of the concept of the Sorites fuzzy theory discussed in Section 4. Furthermore, we elaborate the concept of linguistic hedge (\very, roughly, signi cantly", etc.). Recall that L. A. Zadeh proposed to model their meaning using certain transformations of the fuzzy sets, which represent the meaning of the words standing with the linguistic hedges. From our point of view, the main principle of the model of hedges consists in shifting of the horizon. This is achieved by application of certain unary connective. In general, we speak about the linguistic hedges with narrowing eect (very, highly, etc.) and those with widening eect (more or less, roughly, etc.). The eect of both kinds of hedges when applied to the three atomic syntagms \small", \medium", and \big" is depicted on Figure 3. Let us suppose that the language contains a set of unary hedges = f ?k ; : : : ; ?1 ; 0 ; 1 ; : : : ; pg: being ordered as functions with the central hedge 0 . Then the hedges ?k ; : : : ; ?1 will be called the hedges with narrowing eect and 1 ; : : : ; p the hedges with widening eect.

De nition 47 Let #; be numerals representing the maximal number and some number between 0 and the maximal number, respectively.

42

(i) The multiformulas

Smx := 0 RH R x(0); Mex := 0 LH R x() [ 0 RH R x(); Bix := 0 LH R x(#):

(95) (96) (97) are intensions of the respective atomic evaluating syntagms \small", \medium" and \big". (ii) Let C := `hlinguistic hedgeiA' be a simple evaluating syntagm where the hlinguistic hedgei is assigned the hedge 2 nf 0 g and the intension A of its atomic evaluating syntagm A is one of (95){(97). Then the intension of C is C obtained from A by replacing the hedge 0 by .

Construction of intension of evaluating predications Since in the syntax of our logic, we have no means how to specify the objects concretely, hnouni is assigned a variable x (cf. also Example 21 where Yhxi can be understood as the intension of the evaluating predication \ noun is young" where noun may be \man" as well as \dog"). De nition 48 Let C := `hnouni is A' be an evaluating predication, x be a variable assigned to hnouni and the intension of A be Ahxi . Then the intension of C is Chxi := Ahxi : Using this de nition, we may construct also intension of of the compound evaluating predication when interpreting the connectives \and" and \or" using special logical connectives interpreted by t-norm or t-conorm, respectively. Moreover, the intension of the conditional clause C := ` IF A THEN B' is Chx;yi = Ahxi ) Bhyi = at ! bsAx[t] ) By [s] t 2 M1 ; s 2 M2 (98) where Ahxi and Bhyi are intensions of the corresponding linguistic predications A; B. Linguistic variable We have arrived to the slightly modi ed concept of the linguistic vari-

able in comparison with that presented in Section 2. The linguistic variable is given by a set T (X ) S of evaluating predications (90) containing the same hnouni, which is the name X of the linguistic variable. The syntactic rule is a rule according to which the evaluating syntagms are constructed (e.g. given by our De nition 43). The signi cant dierence lays in the semantical part since the de nition on page 7 is purely extensional considering some universe U . Hence, we may de ne the linguistic variable as follows. De nition 49 The linguistic variable is a tuple hX := hnouni; T (X ); G; M ; M; Pi where T (X ) S , G; M have been de ned above, M is a semantical rule assigning to each evaluating predication A 2 T (X ) its intension Ahxi = f at Ax[t] j t 2 Mg and

P = fD j Dis a structure for J g is a set of possible worlds for X . 43

6.4 Generalized inference

Fuzzy IF-THEN rules and the linguistic description An outstanding role in fuzzy logic

is played by the conditional clauses (91), which are implications characterized using natural language and used for description of some dynamic process or decision situation. A set of these statements is called the linguistic description. De nition 50 Let Aj ; Bj 2 S , j = 1; : : : ; m be evaluating predications with the respective intensions Aj , Bj . Then the linguistic description in FLb is either a nite set LDI or a nite set LDA of linguistic statements as follows. (i) LDI = fRI1 ; : : : ; RIm g where

RIj = IF Aj THEN Bj ;

(99)

j = 1; : : : ; m, are conditional clauses with the intension (98). (ii) LDA = fRA1 ; : : : ; RAm g where

RAj = Aj and Bj ;

(100)

j = 1; : : : ; m, are compound evaluating predications with the intension constructed using De nition 48.

The linguistic description de nes a theory of FLb.

Inference based on linguistic implications We will now consider a simple linguistic description LDI consisting of m IF-THEN rules and an evaluating predication A0k occurring in the antecedent of some rule from LDI . Theorem 20 Let T = fA0k ; LDI g; be a theory of FLb where LDI is a simple linguistic description and A0k the evaluating predication described above. Then we may derive a conclusion Bk0 with the intension o n W (101) B0k;hyi = b0k;s = t2M1 (a0k;t ck;ts)Bk;y [s] s 2 M2 such that all b0k;s for s 2 M2 in the multiformula B0k;hyi are maximal.

This theorem explicitly states the following: if we interpret IF-THEN rules as logical implications formed of the simple evaluating predications then the deduction based on them leads to the conclusion, which is the best possible one | in the sense of maximization of the provability degrees within the fuzzy theory determined by them. In fuzzy control, a certain precise value u0 is usually at disposal, which is obtained by measuring of some characteristics of the controlled process. From our point of view, this value is an interpretation of some closed term t0 in the language J . Given a linguistic description LDI , a question now arises whether it is possible to cope with it in such a way that Theorem 20 can be applied. To nd an answer, let us consider the following example. 44

Example 22 Given a linguistic description consisting of two rules R1 := IF x is small THEN y is big R2 := IF x is big THEN y is small : These rules are interpreted by the respective multiformulas Smhxi ) Bihyi and Bihxi ) Smhyi . Let us now consider a model D0 with the support U = V = [0; 1]. Onee may agree that small

values should lay around 0.3 (or smaller) and big ones around 0.7 (or bigger). Of course, given the measured value, say x = 0:3, we expect the result being \big" due to the rule R1 . Similarly, for x = 0:75 we expect the result being \small" due to the rule R2 . With respect to our theory, this means that the value 0:3 is an interpretation of some term t0 in D0 , i.e. D0 (t0 ) = 0:3 (and the like for 0.7). Then we have a speci c intension of the form Sm0hxi = f 1 Smx[t0 ]g and when applying Theorem 20, we expect the result to be a multiformula Bi0hyi . 2

The term t0 considered in this example can be understood as a typical example of the evaluating predication \hnouni is small". In general, given an evaluating predication A with the intension Ahxi , we derive a speci c syntagm Ex(A) :=\typical example of A" with the intension Ex(Ahxi ) = 1 Ax [t0 ] : (102) Determination of the term t0 , however, is out of logic. There are algorithms for solution of this task, implemented, e.g. in the software system LFLC [10]. The described situation is mathematically characterized in the following corollary of Theorem 20.

Corollary 3 Let

T = fEx(Ak ); LDI g

be a theory of FLb where LDI is a simple linguistic description and Ex(Ak ) is the syntagm \typical example of Ak " with the intension (102). Then we may derive a conclusion Bk0 with the intension o n 0 0 b = c (103) Bk;hyi = k;s k;t0 s Bk;y [s] s 2 M2 and all b0k;s for s 2 M2 in the multiformula B0k;hyi are maximal.

Since the intension of k-th rule of LDI has the form Ak;hxi ) Bk;hyi , the b0k;s are equal to b0k;s = ak;t0 ! bk;s.

Fuzzy interpolation of an imprecisely given function First, we will consider the following classical situation. Let f be a function to be interpolated. Our knowledge of f is limited only to m of its points (dj ; ej ), j = 1; : : : ; m. Furthermore, we know that if x is \near to dj " then y is \near to ej ", j = 1; : : : ; m. We are looking for some function g which interpolates f , i.e. it coincides with f in the points dj , g(dj ) = f (dj ), j = 1; : : : ; m and approximates it for x 6= dj . A question arises, how g can be speci ed. In the frame of fuzzy logic, the interpolation problem can be formulated when replacing the precise equality by the fuzzy one. We will speak about fuzzy interpolation. Then the function f is characterized imprecisely using fuzzy quantities in the points d1 ; : : : ; dm according to the table d1 approximately dm : (104) f : xy approximately approximately e1 approximately em . 45

Theorem 21 Let dj 2 M1 , ej 2 M2 be terms (corresponding to dj ; ej ) in the language J and let the fuzzy quantities \approximately dj ", \approximately ej " have the intensions Q[dj ]hxi and Q[ej ]hyi , respectively, j = 1; : : : ; m. Furthermore, we suppose that the fuzzy quantities \approximately dj "and \approximately dk " are disjoint for each j = 6 k. Then there is a simple linguistic description

LDA = f\approximately dj " and \approximately ej " j j = 1; : : : ; mg; (105) which determines the theory T = LDA of FLb such that the adjoint fuzzy theory T is consistent. Furthermore, we introduce a new binary predicate symbol F 62 J (T ) and extend the fuzzy theory T into

T0 = T

9 8 m = < _ [ : 1 (F (x; y) , ((x =: tj ) ^ (y =: sj )); : j =1

Then to every term t we can derive in T 0 the multiformula

Chyi;t =

n Wm j =1 (aj;t ^ bj;s) Fxy [t; s]

o

s 2 M2 ;

(106)

which is a fuzzy quantity Chyi;dj = Q[ej ]hyi for each j = 1; : : : ; m.

References [1] Black , M. (1937). \Vagueness: An Exercise in Logical Analysis," Philosophy of Science 4, 427{455. Reprinted in Int. J. of General Systems 17(1990), 107{128. [2] Butnariu, D. and Klement, E. P. (1993). Triangular Norm-based Measures and Games with Fuzzy Coalitions. Dordrecht: Kluwer. [3] Chang, C. C. and Keisler, H. J. (1973). Model Theory, Amsterdam: North{Holland. [4] Cohen, P. M. (1965), Universal algebra, New York: Harper & Row. [5] van Dalen, D. (1994). Logic and Structure, Berlin: Springer. [6] Gottwald, S. (1993).Fuzzy Sets and Fuzzy Logic. Wiesbaden: Vieweg. [7] Hájek, P. (1998). Metamathematics of fuzzy logic. Dordrecht: Kluwer. [8] Klir, G.J. and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. New York: Prentice-Hall. [9] Novák, V. (1992). The Alternative Mathematical Model of Linguistic Semantics and Pragmatics. New York: Plenum. [10] Novák, V. (1995). \Linguistically Oriented Fuzzy Logic Controller and Its Design," Int. J. of Approximate Reasoning 1995, 12, 263{277. [11] Novák, V., Per lieva, I. and J. Moèkoø (1999). Mathematical Principles of Fuzzy Logic. Boston: Kluwer. [12] Vopìnka, P. (1979). Mathematics In the Alternative Set Theory. Leipzig: Teubner. 46

[13] Zadeh, L. A. (1965). \Fuzzy Sets," Inf. Control. 8, 338{353. [14] Zadeh, L.A. (1975). \The concept of a linguistic variable and its application to approximate reasoning I, II, III," Inf. Sci., 8, 199{257, 301{357; 9, 43{80.

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