A new axiom system of fuzzy logic

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 77 (1996) 203-205

A new axiom system of fuzzy logic Hubert Frank Department of Mathematics, University of Dortmund D-44221 Dortmund, Germany

Abstract It is shown that there are different fuzzy logical systems on the unit interval [0, 1] which are not compatible like Euclidean and non-Euclidean (for instance hyperbolic) geometries in the plane. Any fuzzy logic belongs to one of two different classes which are characterized by the value of the implication 0 ~D 0. We name the class of (0 "~D0 = 1) the class of Lukasiewicz and gather in it well-known fuzzy logics, especially the continuous generalization of the classical mathematical logic given by Lukasiewicz logic. Then we sum up all other fuzzy logics with the property (0 ~~>0 = 0) in a different class which we call the class of Zadeh-Mamdani F-logics. Any Mamdani controller is a decision-making system and must be based on a logic which has to belong to the last class. Proofs in the Lukasiewicz logic do not automatically generate true results in other multivalued logics like ZM F-logics. Here we give axiom systems for the different classes of fuzzy logics and stress the point of separation.

Keywords: Fuzzy logic; Lukasiewicz and Non-Lukasiewicz logics; Approximate reasoning; Mamdani fuzzy controller

1. Introduction Till now it is an unsolved problem to integrate the logic of a Mamdani fuzzy controller into the Lukasiewicz multivalued logic. We found a system of axioms for all multivalued logics appropriate for Mamdani controllers and for expert systems. Any Mamdani controller must be based on a logical system. Therefore, we have to describe fuzzy logics, shortly called F-logics, belonging to Mamdani controllers, These logics we sum up to a class which we name the class of Z M F-logics after Zadeh and Mamdani or sometimes Non-Lukasiewicz logics. All other logics belong to a second class which we call the class of Lukasiewicz logics. The main example of the last class is the well-known Lukasiewicz logic (cf. [6]) defined as the continuous generalization of the classical mathematical logic. Other members of

the last class are the logics of Dienes, G6del, Heyting, Kleene, and the intuitionistic logic. These two classes are separated by the characterizing property that the value of the implication 0 ~ D 0 is 0 or 1. The separating conditions for continuous logical systems on the unit interval are the antitonicity and the isotonicity. This important fact is shown in Section 3. The following axiom system for fuzzy logic is given in such a way that the first part describes the common properties of all fuzzy logics. Then we point out the separation between the two classes by the conditions (7) and (8).

2 A new axiom system Since the time of the famous mathematician David Hilbert (cf. [1]) it is a well-known tradition

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H. Frank / Fuzzy Sets" and Systems 77 (1996) 203--205

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3. Different fuzzy logics

to describe logical systems by means of systems of axioms. Thus it was done by Lukasiewicz (1920) (cf. [3]). In the propositional logic it suffices to consider the algebraic systems of truth values (cf. [5]). In opposition to the classical logic system which contains only two truth values false = 0 and true = 1, for all multivalued logics (finite or continuous) we take the unit interval [0, 1]. Then we

It is easy to show that a Lukasiewicz logic, finite or continuous, and a logic of a Mamdani fuzzy controller fulfill the above axiom system. But now there is an important difference between these logics. The Lukasiewicz logic fulfills the condition (in [5, ch. 4], called the adjunction property)

define

(6)

Definition. The algebra of truth values of a fuzzy logic is a set V together with four connectives: AND by A (intersection), OR by V (union), context operator by ® and implication operator by ~D fulfilling the following conditions called axioms:

and the Mamdani fuzzy controller the condition of isotonicity

(1) the system (V, A, V ) is a lattice with zero

(7)

c®a~b

if a ~< b then a "~Dd ~ b ,--Dd.

Assuming the conditions (1)-(6) we find the condition of antitonicity (cf. [6]) (8) if a ~< b then b "-~Dd ~< a "-~Dd.

element 0 and greatest element 1 concerning the ordering relation ~< defined by

Now we will compare different fuzzy logics with the conditions (6) and (7). We receive in formula (7) for

a <~ b: if a V b = b

d = O a n d a = O <<, l = b

f o r a, b ~ V,

(1.1) a A b = b A a

and

(1.2) a A ( b A c ) = ( a A b ) A c

aVb=bVa,

and

(9.1)

and referring to the modus ponens axiom (5) (9.2)

aV(bVc)=(aVb)Vc,

0 ~ D 0 ~< 1 ,-,DO,

1 --~D>0= 1 ® (1 "-,DO) ~< 0,

we get the final result (1.3)

a A (a V b) = a

and

a V (a A b) = a,

(9.3) 0 ~ D 0 = 0 . (2.1) a ® b = b ® a , (2.2) a ® ( b ® c ) = ( a ® b ) ® c ,

This is different from the result following from formula (6) w i t h c = l a n d a = b = 0 :

(2.3) a ® l = a ,

(9.4) 0 ~ D 0 = I .

(3)

if a ~< b then a ® d ~ b ® d (isotone),

(4)

if a ~ b then d ~D>a ~< d "--Db,

Thus it is not possible to find a multivalued logic simultaneously fulfilling both conditions (6) and (7). Obviously, we have a situation like that of different geometries in the real plane: The Euclidean geometry with the parallel axiom, the hyperbolic geometry with infinitely many parallels (and the elliptic geometry without parallels). On the same basic system it is possible to consider different algebraic systems that are not compatible. Thus we have the two different classes of Lukasiewicz logics and ZM F-logics. For fuzzy control and fuzzy expert systems we usually need ZM F-logics. Evaluating c = 1 in (6) we get the condition

(5) modus ponens axiom: a ® ( a "-~Db)~< b for all a, b e V. The modus ponens is the inference rule: a ~Db a

From a follows b a is given

b

Replace a by b

Without such a rule no logical system is possible.

(10)

a ~Db = 1 if an only if a ~< b.

H. Frank / Fuzzy Sets and Systems 77 (1996) 203-205

So we find the result that the fuzzy logic characterized by the axioms (1)-(6) is the well-defined continuous Lukasiewicz logic (cf. [6, Definition 1.2, p. 11]). We note that conditions (6) and (7) separate logics of the Lukasiewicz class and ZM F-logics into two disjoint classes. The reason for not using Lukasiewicz logic in fuzzy control is based on the fact that the Lukasiewicz t-norm operator a' b = max(0, a + b - 1) and the corresponding s-norm operator a + b -- rain(l, a + b) are working like filters for small, respectively, for

large values. Most of the problems are not solvable by means of filters. The implication operator in the well-defined Lukasiewicz logic is given by a ,-~>b := min(1, 1 - a + b). 4. Summary There are different fuzzy logical systems on the unit interval [0, 1] which are not compatible like

205

Euclidean and hyperbolic geometries in the plane. Proofs in the Lukasiewicz logic do not automatically generate true statements in other fuzzy logics. In this paper we give an exact mathematical foundation of fuzzy logic which is not of Lukasiewicz type.

References [1] G. Boehm, Einstieg in die Mathematische Logik (Miinchen,

Wien, 1981). [2] A. Kaufmann, Theory of Fuzzy Subsets Vol. I: Fundamental Theoretical Elements (New York, 1975). [3] J. Lukasiewicz,Logike trojwartoscieweg, Ruch. Filosofiece 169 (1920). [4] E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Int. J. Man-Machine Stud. 7 (1975) 1-13. [5] v. Novak, The Alternative Mathematical Model of Linguistic Semantics and Praomatics (New York and London, 1992).

[6] w. Wechler, The Concept of Fuzziness in Automata and Lanouage Theory (Akad. Verlag, Berlin, 1978). [7] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. 3 (1973) 28-45.