The algebra of fuzzy logic

The algebra of fuzzy logic

Fuzzy Sets and Systems 1 (1978) 203-230. © North-Holland Publishing Company THE ALGEBRA O F F U Z Z Y LOGIC Peter ALBERT Department of Experimental P...

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Fuzzy Sets and Systems 1 (1978) 203-230. © North-Holland Publishing Company

THE ALGEBRA O F F U Z Z Y LOGIC Peter ALBERT Department of Experimental Physics, Facultyfor Natural Science, University of Szeged, Hungary

Received January 1977 Revised June 1977

In the following, human thinking based on premises with no complete truth value is reviewed for controlling the algebra of fuzzy sets operations. Assuming a system may be developed in this sphere, it should be considered as the algebra of fuzzy sets, as the same algebra is satisfied by classical logic and sets. As will be proved, this algebra is not a lattice and consequently the Zadeh definitions do not constitute an adequate representation. The binary operations of my algebra are "interactive" types. An axiom system is given that, in my opinion, is the foundation of the conception, adequately and without redundancy. The agreement of the theorems deduced from the axiom system with the intuitive expectations is shown. A special arithmetical structure satisfying this algebra is given, and the relation between this structure and the theory of probability is analyzed. Adapting a process of classical logics, fuzzy quantifiers are defined on the basis of the operations of propositional algebra. A "qualifier" is also defined. The qualifier is functional; applying it to Ax we get the statement "usually Ax'" as a middle course between the statements"at least once Ax'" and "always Ax". The concept of entailment of fuzzy logics is introduced. This concept is an innovative generalization of the classical deduction theory, opposite to the concept of entailment of classical multi-valued logics. An important error of the abbreviated system of notation of the fuzzy theory [e.g. re(x, A v B)] appears: the functional type operations (e.g. quantifiers) cannot be interpreted in propositional calculus. Therefore a new system of symbols is ptoposed in this paper.

1. Introduction A major part of human thinking lies outside classical formal logic. This is shown by the fact that a considerable part of everyday-life reasoning and scientific thinking is formed without knowledge of the rules offormal logic. Only a limited area of thinking is allowed by classical logic and on this basis the majority ofpractical decisions seem to be unfounded and of false logic. The human brain has only a few occasions in which to respect rigorously the ideal deduction rules of the science of logics. Real situations are in general complicated and with our slight knowledge we are unable to formalize them in their whole depth. In order to escape the fate of the man who, recognizing that he is unable to understand the complicated mechanisms of directing the going, could not make a single step, we should act on impulse with a light heart. Intuitive logic cannot be asserted in the algebra of classical logic, but, arising from the effect of the selection mechanism of evolution, it is never chaotic, it has its own rules and its algebra can be constructed. An attempt at the localization of a similar level of thinking is made by the author. Several attempts have been made to construct logics other than the classical logic. These logics could not gain the importance expected from the scale of the thought D



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processes outside classic logic. This may be explained by the fact that they still cover only a fragment of this immense area. What can be the aim of formalizing an intuitive logic? I would hope that this might support the many branches of science not using classic logic, as mathematics itself is supported by mathematical logic. However, facing the unexplored levels of our thinking is of interest in itself.

2. Symbols The fuzzy sets and old,rations given in papers about fuzzy theory can be generally represented in the following way"


("degree of membership" of element x to set A)

m(x, A u B ) , m(x, A n B ) , p

m(x,A). The symbols "'w " , " n " and "=" are of course not those of the classical theory of sets, as e.g., according to the general Zadeh definitions of the operation of fuzzy sets" -

m(x, A n ft )=/:m(x, O)-4-O, [0 is the empty set, 0 is numeric zero]. This is not valid if the operation signs come from the Boole lattice with complement of the classic algebra of sets, where the equation A r~ g = 0 is valid. Using a logical terminology, instead of that of the theory of sets, a disadvantage of these symbols appears. According to this terminology: the object "x" is an individual variable; the expression "x, A " is a propositional function; the object "A'" is a predicate and "re(x, A )" is a relative truth value function. Replacing x by a concrete xi individual, a statement xi, A and a related m(x~, A) truth value is obtained. It is clear that the operation signs of the fuzzy theory may only be considered as the abbreviation of the following:

m(x,A v x,B), m(x,A&x,B), m(x,A ). The symbols" v ", "&", "-" are operations of fuzzy logics. The symbol "A v B " in itself is meaningless to the same extent as the symbols seriesf + g in mathematical functions. This is merely a reference to the detailed expression f ( x ) + g(x). This is due to the fact that the operations made with functions are defined according to the concrete replacing values of the functions. Propositions are not dealt with in mathematical logics, only their truth values, i.e. operations are only defined among truth values. The above formulae cannot be directly

The algebra of fuzzy logic


interpreted, only as abbreviations of the following: m(x, A) v re(x, B ), m(x,A)&m(x,B), --m(x,A). The algebra of operations may be axiomatized with the" = " relation defined among the truth values, e.g.:

commutativeness, m(x, A) v m(x, B) = m(x, B) v re(x, A). An important error of the abbreviated system of notation of the fuzzy theory appears: our intention of combining the truth values of m(x,A) function with two different replacements xi and x~, cannot be stated. E.g. the series of signs m(xi, A ) v m(x i, A)

caanot be abbreviated. This combination has vital importance in the interpretation of the functional type operations, e.g. of quantifiers. This type of reduction cannot be employed, but of course the propositional functions and related truth value functions can be expressed in the form Ax, used in classical logics. (As the former do not appear in our formulae.) Thus the symbols of operations are the following: Ax v Bx, Ax & Bx, -Ax.

These will be used with the signs of operations" Ax + Bx, A x • Bx, Ax.

distinguished from the classical signs of operations. The above statements are not related to the terminology oflogics; similar results may be attained with considerations using that of the theory of sets. A problem occurs for the sets are denoted in general by a single type, omitting the individual variable, and we are predisposed to forget that the operations can be defined only with reference to the elements. It is also seldom formally expressed that e.g. the expression A _ B is obtained from the sets A and B by operations of a functional type. Thus the importance of the functionals is disregarded. The introduction of the proposed symbols is also justified by the necessary


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elimination of the obstacles between the fuzzy theory and the classical chapters of mathematics. In order to distinguish the fuzzy symbols from the symbols of classical logic and the algebra of sets, the latter are marked with "-" and " + " related to the signs of the operations: ~? +

universal quantifier, existential quantifier, implication,

~+ equivalence, 4-

_~ relation subset, +- equality of sets,

t: relation of entailment. According to the correspondence theorem this distinction is unnecessary, it was used for the sake of a better arrangement. The signs of the!~lassical logical operations are given by" v disjunction, & conjunction, -- ~egation, implication, equivalence, ¥ universal quantifier, 3 existential quantifier, ~- relation of entailment.

3. Analysis of intuitive thinking relying upon unstable premises

There are several examples in science of using everyday thinking and less mathematics for instinctive human behaviour in situations when the truth value of the premises serving as a base for the chain of thought is not complete, or if these are complicated to such an extent that there is no chance of producing a detailed formalization. The "law" of these patterns of thinking is expressed in a general way: "being logical does not always mean truth". We may also say with good reason that "being illogical does not always mean an error". This general statement is declared in some special cases; it is analysed in the following sections. Goguen[4] considers just this line of argument in a category-theoretic framework.

3.1. &. Conjunction of several premises Being an argumentation based on several premises, the conclusion is not app; :iated in everyday life--"he is arguing again"win biology, psychology, social sciences, sciences of literature, etc. The intricacy of the lengthy logical argumentation is always

The algebra of fuzzy logic


mentioned pejoratively, even supposing it is correct according to the rules of formal logics. However this cannot be the reason behind some people's eventual aversion to logics. Attention should be paid to the instinctive indisposition raised by chains of thoughts in those who are aware that the individual premises are uncertain and the determination of the units of thoughts (atomic propositions) are very difficult. See I-8, Ch. 8].

3.2. v . Disjunction of several premises Strangely, theorems proved by different primary hypothesis are assumed to be valid in these sciences. The related truth of everyday life is: "two heads are better than one". The mathematician is delighted when a theorem can be approved by several alternative ways, but in non-mathematical sciences the alternative approach is of vital importance. The mathematician is searching for the minimum group of axioms leading to the theorem in question, the alternative method of proving of other sciences is to confirm the existence of the theorem, thus usually all methods are mentioned. In everyday life this means that we take for granted the facts told by many of us, notwithstanding that much information may have a common source. Similarly, in non-mathematical sciences the logical independence of the single argumentations can rarely be detected.

3.3. I&. Invertibility of the conjunction If a conclusion requires the conjunctive fulfilment of several premises then the truth value of the conclusion is limited not only by the most uncertain premise. This may be illustrated with the example of the farmer who expects a rich crop as long as the following conditions hold: there should not be too much rain, no hail, adequate sunshine and finally that his land will not be damaged by birds. Out of these he considers the rain to be the most harmful however he erects scarecrows with great care. (I wish to mention that the situations are illustrated with naive and archaic examples in order to refer to the "common sense" of man instead of his "logic intelligence" eliminating the evaluation of the situation according to the sphere of thought of classical logics.)

3.4. I v . Invertibility of the disjunction If the conclusion is a function of the disjunction of the premises, then the truth value is not only based on the most certain premise, but the others also contribute to the trustworthiness of the reasoning. This is most apparent in the course of a debate when we use our weaker arguments as well as our best one. In Sections 3.1 and 3.3 the truth value reducing characteristic and invertibility of the conjunction were shown, while in Sections 3.2 and 3.4 the truth value increasing characteristic and invertibility of the disjunction were shown. The nature of disjunction may also be explained by the following example: According to an old observation the "illusion" of understanding a notion or phenomenon is produced by representing a complete!y unknown object as the union of several parts. We feel that by dividing something into several types or categories we may perceive, to some extent, a new unknown object. A popular play consists of


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narrating little stories about unknown things, describing them with made-up words, so we can shape an unknown thing with ideas, simply by mentioning in the course of the story its variants and characteristics. Evidently the impression of understanding is evoked by the association with the intelligible words used for the classification and characterisation. Supposing we do not want (and usually cannot) study these mechanisms in detail, we may describe, phenomenologically, the increase in our knowledge by the fact of classifying. This is the working method used by editors of explanatory dictionaries and it is often the essence of scientific cognition: the original undivided unit is built up from boxes of types and classes. 3.5. - . Complement When it is said "'not everything can be judged in black and white" this is of double meaning in the language of logics. On the one hand this means that the algebra of ihe logical operation has no complement: --VA 3/t[A + A =e'I, - V A 371[A'71=o].


(Where e and o denote universal bounds.) I.e. the logical operation which is denied is not an algebraic complement. 3.6. m. The number of truth values The above saying can, on the other hand, be interpreted as the negation of the theorem of the excluded third way. As can be seen, both statements are equal in bivalent logic. This problem is widely known; it is the starting point of many non-classical logics. 3.7. D e . Distributivity We shall say that there are two schemes of proving for a single conclusion. Firstly: Either A or B being realized and by combining with C the conclusion is reached. Secondly: Either A and C or B and C realized the theorem is satisfied. If premises A, B and C are not completely true then the latter scheme is preferred, as it provides a conclusion with greater truth value. As the distributive rule of classical logical algebra is present in our mind to such extent that we may feel the above distinction~unfounded, it is illustrated with an example. Income may be expected by the peasant supposing he has a good crop of wheat or corn and somebody buys it. But he has an increased feeling of security if he is contracting for both before the alternative condition is settled: ~A 3B 3C[(A + B).C < (A .C)+ (B.C)-I.


3.8. D&. Distributivity If a conclusion is valid to the extent of the fulfilment of A or B,, and A or C, it is taken less seriously than the conclusion depending on the validity of A, or B and C. Again mentioning agriculture: it is not the same that during spring, rain or irrigation is needed

The algebra of fuzzy logic


by a plant and during summer, rain or hoeing is needed; or that it requires once a year rain or irrigation and hoeing. The later may naturally be realized more easily: 3A 3B 3C[(A + B).(A + C ) < A + (B- C)].


The formulae of Section 3.7 and 3.8 may also be explained by the following reasoning: The meaning of the letters A, B, C denoting atomic statements is not fully localized, e.g. in Section 3.7 the letter C firstly denotes the purchase, and secondly it denotes a previous contract related to A and B. In Section 3.8 the letter A is represented twice on the left-hand-side: spring rain and summer rain: on the right-hand-side it denotes rainfall once a year. These motivations are not considered in the formulae, e.g. the two sides of the formula (3.3) may be written:




where the meaning of the symbols is strictly limited. We consider these formulae as classical logical ones (it is also emphasized with the operations signs) according to the classical propositional calculus these may be expressed as follows: (A&SP &SU) v (B&SP& A&SU ) v (C&SP&B&SU )

(left-hand-side) (A&SP) (right-hand-side)


(A&SU)v (C

&B&SU)v(B&C&SP) (3.5)

It may be seen that the right-hand-side is really broader, i.e. it may be fulfilled if the lefthand-side is not valid. This detailed structure could be written here in a simpler manner but in those cases which are not so clearly arranged and formalization is necessary; this specification leads to immense complications. Let us consider which kind of rain could be distinguished by a meteorologist and how many requirements could be enumerated by an agronomist. In this case, when formalizing the schematic notion of the rain by a single letter A, the undefined meaning of the letters is considered only when constructing the rules of algebra, the simple formulae are reached which are reflecting a surface, not deeply analyzing, which is the heuristic way of thinking. Beyond the, to some extent naive, sit~ations mentioned a similar logic may be used in: an experimental biological laboratory [8, Ch. 8], when analyzing a poem, etc., and in particular this way is used in thought concerning the trifles of human life. By thinking of the problems of the algorithms acting under undefined circumstances, we make a comparison between the results of character recognition processes and the similar capacity of man, then it is evident that we can also learn from these unscientific behaviour patterns. The "sliding of notion" may often be observed in inexact scientific papers; this means that the verbal determination of a theorem is substituted when used again with a wording of various nuances, i.e. the same grammatical form is motivated by different intuitive content. This is a simpler way to reach the desired (sometimes incorrect) result, taking advantage of the tolerance of the meaning of the

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grammatical units. The works of philosophy are criticized mainly for this by the scientists of the natural sciences, and this was the reason why Leibnitz wanted to construct a logical calculus.

3.9. ,-. Reversing the implication It is interesting how easily men are liable to give credit to statements based on the

A . ( B + A) P B scheme of entailment. In this respect Wason [9] is of special interest. Two chapters of this study ("Verification and exclusion in reasoning" and "The 'grammarama design': Perception and utilization of rules") describe experiments which show that men take the antecedent for granted if the consequent is true. In these experiments a rule was made by the experimenter and the participants have to discover it. Examples could be created by the participants who then asked the experimenter whether these satisfy the rule. It was proved by the experiment that the participants wanted to verify their hypotheses so they created examples to support them. Since several examples could be suitable, i.e. the experimenter accepted them, the participants then thought that their hypotheses were verified. Discovering the proper rule is evidently an inductive work, but an error in deductive thinking was used, as they wanted to verify the hypotheses and not to refute them. They supposed that both from the rule the correct examples may be found and similarly from the correct examples the correct rule may be deduced. This "error" proved obstinate, from the age of Aristotle till our days it serves as a general means of the artistical illustration evoking I-3, Ch. XXIV]. If you want to believe something, you have to show its consequence! This is the basic message of some proverbs: "Fortune favours the brave", "There's no smoke without fire". Therefore, we may reason and decide in undefined situations of everyday life on the basis of false schemes as to classical logics. According to classical laws of thinking we are not entitled to make a decision.

3.10. 3. Existential quantifier If we are searching among a number of individuals for a single one with specific characteristics which is very important for us, we are investigating with great endurance. This illustrates our intuitive optimism related to the truth of statements of type 3x Ax, when the activity of the quantifier is reaching a "subjectively infinite" set:

~xAx--e. Laymen consider the existency axioms of abstract algebras, geometries, etc.--"there are at least two points"was hair-splitting.

3.11. V. Universal quantifier Despite our intuitive optimism we are very pessimistic of statements of type VxAx: ~xAx = o.

The algebra of fuzzy logic


We are used to everyday-life situations where there ure no valid ru|es in every case. This general presentiment is strongly disappointed b) a universal mathematical theorem, e.g. the transfinite induction theorem of number theory. This is the reason why mathematics leaning from completely valid premises to universal theorems exerts irresistible influence upon us.

3.12. C. The principle of correspondence In so far as the truth value of the premises tends to t'qe strictly"true" value of classical logics, the formulae of the intuitive thinking must tend to the classicat formulae. In the case where the increase of the truth value of premises is not followed by the localization of atomic statements (localization of notions), we have no right to apply the formalism of classical logics. This problem lies outside the framework of our formal theory and is related to the process of formalizing. In the following two sections a formal system is described, serving as a basis for the previously outlined logic, by determining the algebra of truth values. The content of the formulae is not motivated towards the declaration of axioms and proving the theorems, for this would disturb the lucidity of formal structure. With the aid of the list of content of Section 7 the algebraic system and its interpretation may be continuously compared by the reader. Mnemotechnical signals are used instead of serial numbers for the designation of axioms, definitions and theorems. The meaning of them is evident from the following, but for practical purposes the arrangement of signals is given here too.

First sign d H M F T

closed ordered algebraic

definition, axioms, axioms, axioms, theorems.

Second sign Refers to the proposition in the definitions, axioms, theorems, e.g.: R reflexivity, T transitivity, K commutativeness, A associativity, etc. Finally, if necessary this is completed by a further sign referring to the logical operation related to the proposition: v: disjunction, & conjunction, etc.

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4. Axiom system v , &, - , --,, ,--~, -3, V are symbols of classical logics; a symbol of the classical theory of sets. Basic notions, undefined: M set, a, b, c,..., elements of M " t r u t h values", < order-relation on M, ' + • binary operations on M, -unary operation on M. d M = .VaVb[a = b,-,(a <=b & b <-a)].


:la3b[a e M & b ~ M & a--/: b].





H - . Va[a ~ M-,Ft ~ M]. M<. VaVb[a

FK v . VaVb[a + b < b + a]. FK&. VaVb[a'b< b'a]. 1 FA v . V a V b V c [ ( a + b ) + c < a +

(b + c)].

F D v .VaVbVc[ (a + b )'c < (a "c ) + (b .c )]. F E v . 3eVa[a + e = e & FE&. a "e = a].



F - - . Val-a = a]. TEE. Element " e " is unique. Proof. e' and e" denoting two elements e F E & Va[a.e'=a],

F E & Va[a.e"=a], e" "e' = e", e" e" = e', FK&, dM =, MT e'= e". May be proved by re,on of FK v, FDM, F - , MT.

The algebraoffuzzy logic


dFO. ~ = o. FOE, Va[o<_a<_e]. F D M . Y a Y b [ a . b = ~ + ~]. Fiv.

FI&. VaVbVcVd[(a.b
5. Theorems TIv.

Proof. By reason of MR, FI v . TI&. V(a.b<-a.c&a:/:o)--,b<-c.

Proof. D u a l of TI v .2



Proof. It should be d e m o n s t r a t e d only for < because by changing the signs > is also proved.

if a:/:e,

M=<,TI v Vc<__b-,a+c<__a+b, MT, F K v ~/c<__b-,c+a<=b+a,

if a = e, T&=.

MT, F E v V

c +a<_b4-a.


Proof. D u a l of T v = .



Proof. F - ,

dM =.

"liT. V(a=b&b=c)--,a=c. Prool. MT, dM = . ~May be proved by reason of FI v, FDM, F - , MT, dFO, aM =. 2The raeaning of duality, see in theorem TDUAL. Before proving its general validity, it relates only to the axioms: FI v, FI&, FK v, FI&, FK v, FK&, FD v, TO&. The demonstration of Theorem TO& is described later, but as there may be seen from the used t~neorems, it could prece~le the demonstration of T& =. The succession was chosen in order to separate distinctly the theorems determining the disposal of sign =.


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Proof. MR, d M = .



Proof. d M = . TK v.Va+b=b+a. Proof. Exchanging the symbol of a and b in FK v"


TK&. Va'b=b'a. Proof. Dual ofTK v . TA


Proof. Exchanging the symbol of a and c in F A v •

b)+a<_c+ (b + a), Va+ (b+c)<=(a÷b)+c, Va-I-(b+c)=(a+b)+c. V(c +








TK v dM=




FDM TV = , TT,



Va'5=a+b, Va'5=a+b.



V(a+b)+c=a+(b+c), TV(a+b)+c=a+(b+c), TT, T D M V(a+b)'c=a'(b÷c), T&=, Tr, T D M V(a'b)'c =a'(b ,c), TAr

by changing thesymbol leads to the proposition. TD&.


The algebra of fuzzy logic



T v =,


FD v

V(a + b)'c<= (a'c)+ (b.c),


V(a+ b)'c > (a.c)+ (b.c),




¥(a + b) + c >=(a'c) • (b.c),




V(a'b )+c >=(a+c )'(b +c ),

by changing the symbols leads to the proposition. T O y . Va+o=a. Proof.

F ~ ~9

T& =,











FE v







Va'o=o, Va.o=o.

TO&. Va. o = o. Proof.




TDUAL. If there is an exchange of the symbol + and ", of the formulae on both sides of"

symbol <, and of the symbol e and o; then we have valid formulae from valid formulae. Proof. With the above exchanges we are becoming always axioms or demonstrable theorems from axioms. The pairs: FK v,


FA v,


FD v,


FE v,


TO v,


FI v,




TOE. The element o is unique. Proof. Dual of TEE. TAA&. a . ~ ¢ o

if a~o&~=~o.


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Proof. Supposing that" V a - a : o,

TT, TO&, d=~o, d M = , TI&,

V a - ~ : o. ~, a.~-o

which is contrary to a ~ o. TA•v.


if a ¢ e & a ¢ e .

Proof. Dual of TAA&. T~&. a'b~a

if a~O&b--/:e, thereforee.g.:

--Va "a--a.

Proof. Supposing that:


a'b=a, TT, FE&, a'b=a'e, dM ~, TI&, b=e,

which is contrary to b =~e.

T=/=v. a + b ~ a if a ~ e & b ~ o , thereforee.g. -Va+a=a. Proof. Dual of T ~&.

T <&. Va'b



Ya+b >a'b, Va+b<~'~, Va+b<6.~+o,

F-, TT, ifb =~e, FOE,

TOy, FI v,


F-, which leads to the proposition with changing the signs.



T & = , TT, dM = ,

a=a'b a>a'b.

T> v. Va+b>a. Proof. Dual of T-<&. TO = v. a + b-- o only' being a - o & b - o.

The algebra of fuzzy logic

Proof. a+b=o,

T>-v, "IT,

Va+b>a, a<_-o~



In a similar way b o . T E = & . a. b = e only being a = e & b = e . Proof. Dual o f T O = v . T O = & . a : b = o only being a = o v b = o . Proof. a.b=o,







otherwise: T&=, 1. ifa:p o, FE&, 2. ifb4:o,






a ' b = b,


a'b =e.b,


a =e,

according to the above b = o which is contrary to 2 thus either 1 or 2 is not true. TE-- v . a + b = e only being a = e v b =e. Proof. Dual of TO = &. TABS v . V a < a + (a'b). Proof.


Va = e" a,



TT, T & = , MT, F D v , TK&, T v = , MT,


Va=(e+b)'a, Va<(a'e)+(b'a), Va
TABS&. V a > a . ( a + b ) . Proof. Dual ofTABS v . TABSq: v . a = a + ( a ' b ) o n l y being a = e v a = o v b = o .



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Proof. a=a+c,

T~ v,

a=evc=o, a.b=o,



TABS ~:&. a = a . ( a + b ) o n l y being a = o v a = e v b=e. Proof. Du~1 of TABS 4= v. TD¢ v. -V(a+b).c=(a.c)+(b'c).

Proof. Being c = a, b = e then if V(a + e)'a = (a "a) + (e.a),




which is contrary to T 4= v. TD¢&.


Proof. Dual ofTD4= v . T C O R R E S P O N D . I f Via = o v a = e l then elements " a " are forming Boolean algebra. Proof. In this case the preterm of the implication of axiom.- FI v and FI&, these became tautology. Only axioms of invertation are contrary to the Boolean algebra in the axioma system, so in this case the demonstration of all theorems became unfounded, which are forming statements contrary to the Boolean algebra. Further it may be proved with a simple finite process that a complete Boolean algebra is formed by the remaining system.

6. Propositional functions In the preceding two sections the axioms and theorems related to the M set of logical values were exposed. Now we have to introduce new basic notions: P universum A, B, C,..., elements of P: "propositional functions", X universum, x, y, z,..., elements ofX: "individuals". For simplification no difference is made between the symbol of concrete functions and the function variable. Similarly, either a concrete individual or an individual variable may be denoted by x. This does not lead to any misunderstanding. After this, the axiom related to the functions: PM.

VAVx[Ax ~ M],

TI.e algebra of fuzzy logic


where the expression "value of A for x " is used relating to the symbol Ax. Thus the axioms valid for set M are also valid for the symbols Ax. Second order formulae may be formed from these formulae in various ways. Here the following substitution is axiomatized: VAx may be substituted by VAVx, 3Ax may be substituted by 3AVx.

Thus all theorems are also valid for A x - s, and the symbol Vx may be reduced to the symbol V. Axioms related to the symbol " - " , which is between individuals PR. ¥ x [ x - x], P =. VAVxVy[x - y ~ A x - Ay].

From which may be proved: PS. V x V y C x - y - , y - x ] where Ax-*-- (x - x) is used, PT. V x V y V z [ x - y - , ( z - x ) = ( z - y ) ]

where Ax~=(z "-x)is used.

7. Comparison of Section 3 and the axiom system In the order of the subsections of Section 3 the agreement of the theorems deduced from the axiomatic system with the intuitive expectations are shown. The theorems not mentioned here are also valid in classical logics.


T__<& T:/:&









TI v



TD =~&


FD v

TD =~ v

see Section 11 3

T3E (see Section 9)


TVO (see Section 9}



8. A representation The algebra of M is on the one hand more general than the bounded lattice, since in E

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general the idempotential and semidistributivity is not fulfilled, on the other hand it is more special, as the axioms of inversion are valid and semidistributivity laws are the inverse of those of the lattice. The Zadeh conception of the fuzzy sets is the formation of a bounded distributive lattice. Evidently the axioms of inversion are not fulfilled in this system, as e.g. on the basis of max{a, c) < max(b, c) it does not follow, that: a<_b.

Besides the consideration of Section 3 the theory of deduction of fuzzy logic also supports the construction of the algebra in M. As shown in Section 11, an important type of the rul2,: of entailment is based explicitly on the axioms of inversion. Such a representation may be constructed in the interval [0.1] of real numbers, which satisfies our axiomatic system in all respects. The signs of operations on the righthand-side of the following formulae (rules of calculation) are to be read in usual arithmetical sense. [ 13, Appendix]. VaVb[a + b =a + b - a b ] ,

VaVb[a" b = ab], Va[ti= 1 - a ] , e=l, 0-- 9 .

It is easy to see that my theorems are valid in this representation. The characteristics of this representatiea may be made clear by Fig. 1 where the operations o f " a " with itself are presented in order to simplify the representation. As it is to be seen from K6czy [5], this system may be well applied in cluster analysis, in which the algorithms used are based explicitly on the invertibility of the system. It is also convenient that, contrary to the Zadeh system, ifa _4.Ax and b ~ Bx are everywhere differentiable functions, then a + b, a-b and t~ will also be everywhere differentiable. The algebra of the present paper originated from the analyses made together with K6czy which lead to the surprising observation that these rules of calculation correctly reflect a certain level of thinking. The above model of the algebra in M is isomorphic with the algebra defined among the probabilities of the independent events. This is made clear by the following equal terms: a=P(A), a+b=P(AvB), a'b=P(A&B),




T h e a l g e b r a o f f u z z y logic






I o


a,~,"a I





Fig. 1.

Since A is not independent from A, the symbol series of the t) ?e P(A v A) i.e. P ( A & A ) cannot appear on the right-hand-side. Evidently on the left-hand-side the symbol series a'a and a + a is also interpreted, thus our model may be characterized in terms of probability theory as an algebra of probabilities of events being "independent in a priori manner". 9. Quantifiers

The quantifiers interpreted in finite universum are defined on the basis of binary operations " + " " . " as in classical logics" (Wittgenstein[l 1]) "~xAx=Ax ~xAx

I + Ax 2 +'"+

A x n,

-" A x I " A x 2 " . . . "A x n.

I.e. n being finite, all can be said about the quantifiers based on the finite algebra in M.

P. Albert


The rules of entailment with expressions of quantifiers are listed at the end of Section 11. Replacing the sign I=by < and the sign =iI=by = , the formulae may be interpreted on the basis of the previous system. (Let us neglect the sign i-.) Their proof is not given in detail here. In order to interpret quantifications in an infinite universum with the previous axioms new axioms are needed, Similar to the process of classical logics, it is convenient to determine these axioms in such a manner that exactly the same characteristics are valid to quantifiets delined in an infinite universum as those which are valid in the finite ease, i.e. the two quantifier types must not be discerned. The problem of determining the minimal axiomatic system which results in the required theorems is not dealt with here. The following two theorems are necessary: Theorem 9.~. ~xAx.~IxBx = ;¢x[.Ax.Bx], Theorem 9.2. ~xAx + ~ixBx = ~x[Ax + Bx]. Let Ax be a function interpreted in a numerable individual region. The quantifier restrained to the fmite interval [xl, x j should be defined on it" n

Ax l "Ax2 .....

AXn = ~IxAx = an. 1

According to the definition of quantifier interpreted in a finite universum: an+ 1 "-an "Axn+ 1"

Thus, according to Theorem T < & it is evident that the series a. is decreasing raenotonously. We know from F O E that it is bounded. Due to this it will be convergent. The complete quantification of Ax is given by its limit value: lim an = 9xAx, a. being convergent, the following is valid: iim an = lim an + 1 / i ' ' ~ :.zO

n -'l' OO

in detail: lim 9xAx = lim W'*OD

xAx'Axn+ a

n"'b ~

(as n is finite within the brackets). Use the following signs:


ifx:~x.+ 1, A'x = [,e otherwise,

A,,x={AeX i f x = x . + , , otherwise.

The algebra of fuzzy logic


Our formula will be the following (using FE&): lim ~ x A ' x = lira


~ x A ' x . ¢~ x A " x .

n - " oo



Applying Theorem 9.1 for the quantifiers defined for the finite universum given in brackets: n+l


lira ) x A ' x = lim ) x [ A ' x . A " x ] .

n--* oo




Again applying Theorem 9.1, but now for the quantifier defined to the numerable universum" n+l



lim ) x A ' x = lira ) x A ' x " lim ~ x A " x .





n - - ' oo


Again writing our original signs: ~¢xAx =


lim Axn+ l"


According to Theorem T :/: & of our algebra: a=a.b

only if a = o v b = e .

The second consequence means in our case chat" lim Axn+ 1 = e.

n--# oo

With other terms: there are many such only finite Axn for which the following is valid" Axn < s < e where s is an arbitrary s :/: e value. If this is not satisfied, then" TVO

~xAx = o

Dual of this theorem"


~ x A x = e If there are many not only finite Axn values for which: Axn > s > o where s is an arbitrary s 4=o value.

It is easy to see that Theorems T¥O and T:lE are also valid in the special representation given in Section 8. The content of our theorems conforms+ to the statements of Section 3.10 and 3.11. From + , . , -, ~, ~ operations, the _ , = operations may be constructed similar to the cla.ssical theory (see e.g. [10, Volume I, Part I, Section B, .0]): Ax &_Bx = CCx[Ax + Bx] = gx[Tix + Bx], +

Ax-Bx=(Ax&__.Bx).(Bx+,fAx)= Cx[(3x + Bx). (Bx + Ax)].


P. Albert

Consequently e.g.: +

A x "B x -


O x = 3 x [ _ A x "B x ]


where Vx(Ox = o). If universum is infinite then, apart from extreme cases, we have o value: +

Axc_ Bx =o, +

A x - B x = o.

Similarly the value ofconjunctivity will be e. All these are intuitively the consequence of these statements made upon an infinite number of uncertain premises (see Sections 3.1 and 3.2). In order to construct more useful functionals, the definitions of the quantifiers should not be modified, but those operations should be detected which are applied by human thinking in uncertain situations. Quantifiers on a continuous universe are interpreted by considering those numerable universe parts which are characterized by the fact that the quantifier calculated for them is not altered when enlarging them with a new individual. It is easy to admit that the quantifiers calculated on such a universe part being equivalent, the complete quantifiers may be defined with these. Let us denote two such universe parts by X 1 and X2. Enlarging them in numerable steps we may arrive at the universe part X1 u X2 (u is the classical union). As during the process of enlarging, the value of quantification should not be altered, the quantifiers calculated for X 1 and X2 should be equivalent. Detailed discussion is given in [ 1]. It is evident that the quantifiers may be defined with this method not only for a continuous universe but also for a universe of any type. It is easy to see that in a universe of any kind a numerable part fulfilling the ~/xAx


~l x A x


may be found, then the quantifiers calculated for this fulfill the condition of the former section thus giving the quantifiers of the whole universe. By this Theorems TVO and ]'3E will be valid for all kinds of universes.

I0. Consistency of the relation +___ Similar to the function

_B xi B of the classical theory of sets; in our theory the function

Axix~Bx=Bx i

7 he algebra o l ./uz-y logic


is valid, where theset containing only xi is denoted by Axi"

Axix = {e o

ifx = xi, otherwise, +

Axi~Bx -'Vx[Axix + Bx] = (f4xixi + Bxi)'e= Bxi. where the definition of a quantifier interpreted in a finite region was used. This trivial relation was specially mentioned because the Zadeh [12] definition of ,elation +_ does not satisfy it, and I cannot detect any logical reason for giving up such an evident priaciple. Intuitively it is also satisfying to know that the equivality, subsetrelation, etc. of the "incorrectly defined" sets should not be characterised with concepts taken from classical theory. The equiva!ity, subset-relation, etc. of two sets cannot be s~ated categorically without knowing the sets exactly (i.e. we are unable to determine the elements of the sets with the divalued logic), merely based on the fact that according to the order of the truth values

Vx(Ax<=Bx), Vx(Ax=Bx), 3x{Ax:/:o), being valid or not.

1 I. The concept of entailment in fuzzy logic According to the principles of classical logic the F formula is the consequence of formulae Ai if demonstrated Ly analyzing the structure of formulae, that in all cases /I 1 & A 2 &/13 • " "


and at the same time F = true. The expression "in all cases" in the above sentence comes from the following quantification" V A V B " .VXVY'"


• • ",

where A,B,... denotes the propositional functions present in the formula; X, Y,... denotes the universum of the individual's; x, y,... denotes the individual's qx ~ X I. The concept of entailment of fuzzy logic may not be based on the notion "true" for these formulae are characterized by the less extreme value of truth. We cannot admit the wide-spread statement: taking a fuzzy formula as being true its truth value depasses


P. A l b e r t

a certain limit [7, 14]. It does not conform to the nature of many-valued logic to change to the notions of classical logic by the definition of a universal threshold. As the classical truth values for the l- relation form an ordered set: true t- true false ~-true false F-false the following formula A I & A 2 cY~ A 3 cY~ " " ~ F

may be generalized to fuzzy formulae in a most evident manner by replacing the sign tby the =< ordered relation in set M [2]" AI"A2"A

3 ....


In case the above relation can be proved in classical logic, i.e. from Ai, F may be demonstrated, then the formula F may be considered true, in all cases the conjunction of formulae Ai being true. In other words: the propositional function of F is "minorated" by the propositional function of the conjunction of premises: / " : - - A 1 &: A2 ~Y~A3 &: • • •

the meaning of sign ": = " "be equal". This conception of the entailment process may be extended to the fuzzy formulae: F:-A

1 "A 2 "A 3 . . . .


The minoration may be made with the algebra defined in M. This algebra is weaker than the Boolean algebra for the asymmetry of distributivity and negation behaviour. Thus instead of certain equalities only an asymmetric relation results, the idempotential and absorption having lapsed. The rules of entailment of fuzzy logics are stricter in this respect than those of classical logic, illustrating the advisable caution of our reasoning being based on not completely true premises. This group of the rules of entailment is not given here in detail, as, according to the minorating principle it is evident that the sign < may be replaced/mywhere by the sign t:. The propositional function of the formula on the right-hand-side of the sign may be minorated with the left-hand-side, thus we may consider it true due to the demonstration, at least in the value of truth of the left-hand-side. The deduction schemas containing quantifiers are listed at the end of the section, though these are easily demonstrable with our theorems and with the definition ofquantifiers. The inversion axioms give some entailment schemas not demonstrable with the above axioms, and which are not even allowable in classical logic. First are those entailments which are allowed in classical logic.

The algebra of fi4zzy logic


b may be minorated with a and a + b,

b'=(a+ b)+-li~

if a ¢ o and


where the inverse of + denoted by the sign + - a being in this case unique according to TI v . In our special representation: (the 4-, - , / signs are used here in the usual arithmetic sense)

(a+ bjq-a-1 (I

This deduction has the following course in classical logic: a, a--* b

with the conjunction of premises"

a & (a ~ b),

by reason of the definition of the --,"

a & (6 v b),

by reason of the distributivity"

( a & 6 ) v (a&b), being a & 6 = o and o v c = c " a & b,

by reason of the principle of minorating,

b As in the fuzzy algebra a" 6 = o is not usually satisfied, this process is impossible. The entailment a" b F~ b may also be realized in the fuzzy logic relying upon T < &. But in case the "a" is also known, by reason of the invertibility of the conjunction, better minoration may be found. Thus instead of b" =a'b, b may be determined as follows:




where the inverse o f " . " is denoted w i t h " - t,, sign.

In our special representation:

b.=(a'b). a N o w we are exposing those entailments which are not allowed in classical logic. F r o m the consequent of the implication on its antecedent may be inferred:





In our special representation" a'--

1-(a+b) 1-b

If a-% b = e, and as assumed b 4=e, then it is easy to see: a = o. This scheme of entailment is valid only if the premises are not fully true. Otherwise either the inversion axioma


P. Albert

shall lose the validity, or the entailment may remain valid, but the conclusion may be minorated only with o. Also a special fuzzy schema is the entailment to the terms of the disjunctions:


if a¢e.

In our special representation: b:=

(a+b)-a 1-a

Now the most important rules of quantifier-entailment of the classical function calculus are listed indicating those remaining valid in fuzzy logic too"

qx~/yAxy :tr-qyqxAxy

~/xAx "~xBx :I~ '~x[Ax'Bx]

. 3x 3yAxy:I~- ~y~xaxy

~txAx + ~txBx :II: ~]x[Ax + Bx]

~xqvAxyF ?y ~txaxy

9xAx + qxBx F 9 x[Ax + Bx]

fixAx" 3xBx q 3tx[Ax'Bx] VxAx ~-Axi A ~ '~xBx :IF Vx[A ~ Bx]

xAx :I Axi 9xAx I: ~xAx


9xAx :II" ~x~x . ~lxAx :It:9x.~x '(/xAx :It: ~xAx

A ~ ~lxBx -I~- ~]x[A +.LBx] , ¢¢xAx~ B q~ ~x[Ax +-,B] fix.4x~ e :IF ~/x[Ax +.-%B]

7. '(/xAx~ ~txBx :IF ~x[Ax~ Bx]

~xAx :II: 9xAx .

~/xAx" 3 xBx ~ ~ x[Ax" Bx] VxAx + ~xBx r-.~x[Ax + Bx]

' ( / x A x "B :IF ~ / x [ A x " B ] +


3xAx + B -I1'- ~tx[Ax + B] VxAx + B :IF ~/x[Ax + B]

'~xAx " ~]xBx ":t¢¢x[Ax "Bx] .

~]xAx'B -It: ~x[Ax'B]

VxAx + ~lxBx :'1¢¢xEAx+ Bx]

12. Qualifier For the concentration of the information of a propositional function as yet the quantifiers were applied. As manifested, in case of an infinite individual region these do not properly characterize the propositional function. This is related to the statements of

The algebra of ji~zzy logic


Sections 3.10 and 3.11, i.e. to the observation that in the course of heuristic thinking universal or existential judgements are rarely pronounced by us. Then the average, mostly valid theorems are by far more useful. In order to characterize the quality of a great number of mass-produced machine elements, we shall scarcely deliberate on the truth of such statements as: "there are some perfect machine elements" and "all machine elements are perfect". In practical cases a far more characteristic statement is" "machine parts are usually perfect". In order to measure the same quantity with several instruments, or to determine the average position of the oscillating pointer of an instrument, the values measured are combined so as to get the "usually" characteristic value. When we hear various information about somebody or something in everyday life, our opinion is formed by averaging these, rather than accepting extreme points of view. The opinion of a single person is rarely admitted, but we do not only believe those things which are confirmed by everybody. In other words the mental process leading to averaging is of a universal, logical nature. By this the logical functions are "estimated" and "qualified" by us. The "qualifier" is a functional, applying it to A x we get the statement: "usually Ax'" as a middle course between the statements "at least once Ax'" and "always Ax". Notation: UxAx. The qualifier is related to the fuzzy algebra by the following axioms: UZ.

VA[Vx(Ax ~ M )--,U x A x ~ M],


VA[Vx{Ax = a ) - - , U x A x =a],


V A V B [ U x A x + B = U x ( A x + B)],


V A V B [ U x A x "B = Ux( Ax "B )],


VA[UxAx =UxAx],


VA[UxU yAxy = UyUxAxy],

where UZ, U = and Uxy are the evident property of the mean value calculation. U v and U& express that the disjunction and conjunction with the logical constant may be interchanged with the qualifier. The U - ensures the interchangeability with the operation of negation. These properties seem to be natural and are useful in the theory of deduction. It may be seen easily, that in case quantifiers would be substituted by qualifiers in the formulae of entailment listed at the end of the previous chapter, conclusions may be made which could not be used with fuzzy quantifiers (e.g. groups 4 and 6). In the representation presented earlier the qualifier may be interpreted e.g. as follows: n

~" A x i UxAx-

~-- t



xIAx dx x A x =. - ~ x •

depending on whether set X is numerable or continuous. It i~ easy to see that our axioms are satisfied by this definition.


P. Albert

References [1] [2] [3] [4] [5] [6] [7] [8] [9]


[11] [12]

[13] [14]

P. Albert, The interpretation of quantifiers in the infinite logic. P. Albert, The multi-valued generalization of entailment concept. Aristotle, (~ -330) Poetica (A. Gudeman, Berlin, 1934). J. Goguen, Concept representation in natural and artificial languages: axioms extensions and applications for fuzzy sets, Int. J. Man-Machine Studies 6 (1974) 513-561. T.L. K6czy, Fuzzy Algebras and Some Questions of their Technical Applications, Dissertation (in Hungarian). University of Engineering, Budapest (1975). T.L. K6czy and M. Hajnal, A new fuzzy calculus and its applications as a pattern reccgnition technique, Proc. 3rd Int. Congress of Cybernetics and Systems, Bucharest, Romania (Augu~,t, 1975). E.T. Lee and C.I. Chang, Some properties of fuzzy logic, Information arid Control !9 (1971) 417-431. H. Selye, From Dream to Discovery on being a Scientist (McGra,,v-Hill, New Y,:,rk, 1964). P.C. Wason, Thinking, in Ne~ Horizons in Psychology, B.M. Foss, Ed. (Per~guin Books Ltd., Harmondsworth, U.K., 1966). A.N. Whitehead and B. Russell, Principia Mathematica (Cambridge University Press, London, 191013). L. Wittgenstein,Tractatus Logico-Philosophicus {1921). L.A. Zadeh, Fuzzy sets,Information and Control 8 (1965) 338-353. L.A. Zadeh, A fuzzy-algorithmicapproach to the definitionof complex or imprecise concepts. Int.J. Man-Machine Studies 8 (1976) 249-291. A.A. Zinovev, Philosophical Probiems of Many-valued Logic (D. Reidel, Dordrecht, 1963).