On Intuitionistic Fuzzy Negations and De Morgan Laws

Transcription

1 On Intuitionistic Fuzzy Negations De Morgan Laws Krassimir T. Atanassov CLBME-Bulgarian Academy of Sciences, P.O. Box 12, So a-1113, Bulgaria Abstract In a series of papers di erent intuitionistic fuzzy implications negations are introduced their properties are studied. It is well known that the classical negation : satis es De Morgan's Laws :(:X _:Y )=X ^ Y :(:X ^:Y )=X _ Y: Here it is shown that some of the new (intuitionistic fuzzy) negations do not satisfy De Morgan's Laws in the ordinary forms now, Modi ed De Morgan's Laws are constructed. Keywords: Intuitionistic fuzzy sets, implications, negations, De Morgan's Laws. 1 Introduction: On some previous results In a series of papers di erent variants of intuitionistic fuzzy implications (see [3, 7, 8, 9, 10]) negations (see [5]) are introduced their properties are studied. Let x be a variable. Then its truth-value is represented by the ordered couple V (x) =ha; bi; so that a; b; a + b 2 [0; 1], where a b are degrees of validity of non-validity of x. Any other formula is estimated by analogy. Everywhere below we shall assume that for the three variables x; y z equalities: V (x) = ha; bi;v(y) = hc; di;v(z) = he; fi (a; b; c; d; e; f; a + b; c + d; e + f 2 [0; 1]) hold. For the needs of the discussion below we shall de ne the notion of Intuitionistic Fuzzy Tautology(IFT,see[1,2])by: x is an IFT if only if a b, while x will be a tautology i a =1b =0. In some de nitions we shall use functions sg sg: 8 >< 1 if x>0 sg(x) = >: 0 if x<0 8 >< 0 if x>0 sg(x) = >: 1 if x<0 For two variables x y operations \conjunction" (&) \disjunction" (_) arede- ned by (see [1, 2]): or (see [3]): V (x&y) =hmin(a; c); max(b; d)i; V (x _ y) =hmax(a; c); min(b; d)i: V (x&y) =ha:c; b + d b:di; V (x_y) =ha + c a:c; b:di: Following [8], we shall mention that we can de ne at least 15 di erent implications - see Table 1.

2 Table 1: List of intuitionistic fuzzy implications Notation Name Form of implication! 1 Zadeh hmax(b; min(a; c)); min(a; d))! 2 Gaines-Rescher h1 sg(a c);d:sg(a c)i! 3 GÄodel h1 (1 c):sg(a c);d:sg(a c)i! 4 Kleene-Dienes hmax(b; c); min(a; d)i! 5 Lukasiewicz hmin(1;b+ c); max(0;a+ d 1)i! 6 Reichenbach hb + ac; adi! 7 Willmott hmin(max(b; c); max(a; b); max(c; d)); max(min(a; d); min(a;b); min(c; d))i! 8 Wu h1 (1 min(b; c)):sg(a c); max(a; d):sg(a c):sg(d b)i! 9 Klir Yuan 1 hb + a 2 c; ab + a 2 di! 10 Klir Yuan 2 hc:sg(1 a)+sg(1 a):(sg(1 c)+b:sg(1 c)); d:sg(1 a)+a:sg(1 a):sg(1 c)i! 11 Atanassov 1 h1 (1 c):sg(a c);d:sg(a c):sg(d b)i! 12 Atanassov 2 hmax(b; c); 1 max(b; c)i! 13 Atanassov Kolev hb + c b:c; a:di! 14 Atanassov Trifonov h1 (1 c):sg(a c) d:sg(a c):sg(d b), d:sg(d b)i! 15 Atanassov 3 h1 (1 min(b; c)):sg(sg(a c)+sg(d b)) (see below) min(b; c):sg(a c):sg(d b); 1 (1 max(a; d)):sg(sg(a c)+sg(d b)) max(a; d):sg(a c):sg(d b)i For them in [6] we constructed respective negations -see Table 2, using as a basis equality :x = x! 0 or :ha; bi = ha; bi!h0; 1i: In [8] we noted that: a) Zadeh, Kleene-Dienes, Lukasiewicz, Reichenbach, Willmott second Klir Yuan's negations coincide; b) Gaines-Rescher GÄodel's negations coincide; c) Wu rst negations of me coincide; d) Atanassov Trifonov, Third negation of Atanassov coincide. For the rst 10 of the above 15 implications, that are based on [12], the following three properties are checked in [4]: For the above negations the following three properties are checked in [5, 6]: Property P1: A!::A, Property P2: ::A! A, Property P3: :::A = :A. There are proved the following assertions: Assertion 1: Each of the negations from Table 2 satis es Property 1. Assertion 2: Negations of Zadeh, Kleene- Dienes, Lukasiewicz, Reichenbach, Willmott, KlirYuan2AtanassovKolev satisfy Property 2, while negations of Gaines- Rescher, GÄodel, Wu, Klir Yuan 1, the three Atanassov's, Atanassov Trifonov's implications do not satisfy it. Assertion 3: Each of the negations from Table 2 satis es Property 3. As it is noted in [6], we can number the different negations as it is shown on Table 3. In [6] the validity of the Law for Excluded Middle is studied in the forms: ha; bi_:ha; bi = h1; 0i (tautology-form) ha; bi_:ha; bi = hp; qi;

3 Table 2: List of intuitionistic fuzzy negations Name Form of negation Zadeh Gaines-Rescher h1 sg(a); sg(a)i GÄodel h1 sg(a); sg(a)i Kleene-Dienes Lukasiewicz Reichenbach Willmott Wu h1 sg(a); sg(a):sg(1 b)i Klir Yuan 1 hb; a:b + a 2 i Klir Yuan 2 Atanassov 1 h1 sg(a); sg(a):sg(1 b)i Atanassov 2 hb; 1 bi Atanassov Kolev Atanassov Trifonov h1 sg(a) sg(a):sg(1 b); sg(1 b)i Atanassov 3 h1 sg(sg(a)+sg(1 b)); 1 sg(a):sg(1 b)i Table 3: List of the di erent intuitionistic fuzzy negations Notation Form of negation : 1 : 2 h1 sg(a); sg(a)i : 3 hb; a:b + a 2 i : 4 hb; 1 bi : 5 h1 sg(sg(a)+sg(1 b)); sg(1 b)ii a Modi ed Law for Excluded Middle in the forms: ::ha; bi_:ha; bi = h1; 0i (tautology-form) ::ha; bi_:ha; bi = hp; qi; (IFT-form), where 1 p q 0 i =1; 2;:::;6 the following assertions are proved. Assertion 4: No one negation satis es the Law for Excluded Middle in the tautological form. Assertion 5: Negations : 1 ; : 3 : 4 satisfy the LEM in the IFT-form. Assertion 6: Only : 2 : 5 satisfy the Modi ed Law for Excluded Middle in the tautological form. Assertion 7: All negations satisfy the Modi ed Law for Excluded Middle in the IFTform. 2 Main results Usually, De Morgan's Laws have the forms: Theorem 1. forms x y: :x ^:y = :(x _ y); :x _:y = :(x ^ y): For every two propositional : i x ^: i y = : i (x _ y); : i x _: i y = : i (x ^ y) for i =1; 2; 4; 5, while negation : 3 does not satis es these equalities. We shall illustrate only the fact that the De Morgan's Laws are not valid for i =3. For example, if a = b =0:5;c=0:1;d=0,then V (: 3 x ^: 3 y)=0:5

4 V (: 3 (x _ y) =0:25: The above mentioned change of the Law for Excluded Middle inspire the idea to study the validity of De Morgan's Laws that the classical negation : (here it is negation : 1 )satis- es. Really, easy it can be proved that the expressions : 1 (: 1 x _: 1 y)=x ^ y : 1 (: 1 x ^: 1 y)=x _ y are IFTs, but the other negations do not satisfy these equalities. For them the following assertion is valid. Theorem 2. For every two proposiional forms x y: : i (: i x _: i y)=: i : i x ^: i : i y : i (: i x ^: i y)=: i : i x _: i : i y for i =2; 4; 5, while negation : 3 does not satis es these equalities. The proof of Theorem 2 is similar to the proof of Theorem 3 for this reason we will omit it. It is well know that De Morgan's Laws are manifested not only in the case of operations from conjunctive disjunctive type, but they are valid for the case of modal operators. Now, we shall discuss similar relations, starting with some de nitions. The evaluation function V can be extended for the modal operators \ "\}" asfol- lows V ( x) = V (x) =ha; 1 ai; V (}x) =}V (x) =h1 b; bi: It can be easily seen that for each proposition x such that V (x) =ha; 1 ai, i.e., the estimation is fuzzy, but not intuitionistic fuzzy, then V (}x) =V (x) = V (x): Theorem 3. For each propositional form x: (a) : i }: i x = : i : i x; i =1; 2; 3; 4; 5; (b) : i : i x = }: i : i x; i =1; 2; 4; 5; (c) : i }: i x = : i : i x; i =1; 2; (d) : i : i x = : i : i }x; i =1; 3; 4; 5: Proof: We shall check the validity of (d) for the most complex case: i =5. Weobtain: V (: 5 : 5 x)=: 5 : 5 ha; bi = : 5 h1 sg(sg(a)+sg(1 b)); sg(1 b)i = : 5 h1 sg(sg(a)+sg(1 b)); ; sg(sg(a)+sg(1 b))i = h1 sg(sg(1 sg(sg(a)+sg(1 b))) + + sg(1 sg(sg(a)+sg(1 b)))); ; sg(1 sg(sg(a)+sg(1 b)))i V (: 5 : 5 }x) =: 5 : 5 }ha; bi = : 5 : 5 }ha; bi = : 5 : 5 h1 b; bi = : 5 h1 sg(sg(1 b)+sg(1 b)); sg(1 b)i = h1 sg(sg(1 sg(sg(1 b)+ + sg(1 b))) + sg(1 sg(1 b))); ; sg(1 sg(1 b))i Let X 1 sg(sg(1 sg(sg(a)+ + sg(1 b))) + sg(1 sg(sg(a)+ + sg(1 b)))) (1 sg(sg(1 + sg(sg(1 b)+ + sg(1 b))) + sg(1 sg(1 b))) Y sg(1 sg(sg(a)+sg(1 b))) sg(1 sg(1 b)): If b =1,thena =0 X = sg(sg(1 sg(0)) + sg(1 sg(0)))+ + sg(sg(1 sg(0)) + sg(1)) = sg(sg(1) + sg(1)) + sg(sg(1) + 1) = = sg(2) + sg(2) = 1+1=0: If b<1, then X = sg(sg(1 sg(sg(a)+1))+ + sg(1 sg(sg(a)+1)))+ + sg(sg(1 sg(1+1))+sg(1 1)) = = sg(sg(1 1) + sg(1 1)) + + sg(sg(1)) = sg(0 + 0) + sg(sg(0)) = = 0+0 = 0:

5 If b =1,thena =0 Y =sg(1 sg(0)) sg(1) = 1 1=0: If b<1, then Y =sg(1 sg(sg(a)+1)) sg(1 1) = = sg(1 1) 0=0 0=0: Therefore, both terms coincide (d) for i = 5isvalid. Now, we shall discuss another form of the Law for Excluded Middle that in propositional calculus is equivalent with the stard Law for Excluded Middle, if De Morgan's Laws are valid. It is the following: :(x ^:x): Theorem 4: Negations : 2 : 5 satisfy it as tautologies, negations : 1, : 3 : 4 satisfy it as IFTs. 3 A new argument that the intuitionistic fuzzy sets have intuitionistic nature Now, following [6], let us return from the intuitionistic fuzzy negations to ordinary fuzzy negations. The result is shown in Table 4, where b =1 a: For example, having in mind the above mentioned equality b =1 a for the fuzzy case, we can see directly that 1 sg(sg(a)+sg(1 b)) = = 1 sg(sg(a)+sg(a)) = = = 1 sg(sg(a)) = 1 sg(a): Therefore, from the intuitionistic fuzzy negations we can generate fuzzy negations, so that two of them (: 3 : 4 )coincidewith the stard fuzzy negation (: 1 ). Therefore, there are intuitionistic fuzzy negations that lose their properties when they are restricted to ordinary fuzzy case. In other words, the construction of the intuitionistic fuzzy estimation hdegree of membership/validity; degree of non-membership/non-validityi that is speci c for the intuitionistic fuzzy sets, isthereasonfortheintuitionisticbehaviour of these sets. Over them we can de ne as intuitionistic, as well as classical negations. The other two intuitionistic fuzzy negations (: 2 : 5 ) also coincide in the fuzzy case the corresponding fuzzy negation satis es Properties 1 3 does not satisfy Property 2, i.e., it has intuitionistic character. In addition, for it : 2 (: 2 a _: 2 c)= = : 2 (1 sg(a) _ 1 sg(c)) = = : 2 max(1 sg(a); 1 sg(c)) = 1 sg(max(1 sg(a); 1 sg(c))) = = 1 max(sg(1 sg(a)); sg(1 sg(c))) = = min(1 sg(1 sga); 1 sg(1 sgc)) = = 1 sg(1 sga) ^ 1 sg(1 sgc) = = : 2 1 sga ^: 2 1 sgc = = : 2 : 2 a ^: 2 : 2 c: On the other h : 2 a _: 2 c = = 1 sg(a) _ 1 sg(c) = = max(1 sg(a); 1 sg(c)) = 1 min(sg(a); sg(c)) = = 1 sg(min(a; c)) = = : 2 (min(a; c)): 2 (a ^ c); analogously, we see that : 2 a ^: 2 c = : 2 (a _ c); i.e., negation : 2 satis es this form of De Morgan Laws. Thesameisthesituationswithformula:(x^ :x): For the fuzzy negation taht is analogous of : 1, the formula is not valid, while for the fuzzy analogous of negations : 2 : 5 the formula is a tautology. Finally, we must note that as the latter fuzzy negations, as well as the ve intuitionistic fuzzy negations are very simple. They can be extended essentially, if we use extended intuitionistic fuzzy modal operators this will be a subject of a next reseach.

6 Table 4: List of the fuzzy negations, generated by intuitionistic fuzzy negations Notation Form of the intuitionistic fuzzy negation Form of the fuzzy negation : 1 1 a : 2 h1 sg(a); sg(a)i 1 sg(a) : 3 hb; a:b + a 2 i 1 a : 4 hb; 1 bi 1 a : 5 h1 sg(sg(a)+sg(1 b)); sg(1 b)i 1 sg(a) 4 Conclusion Here it is shown that some of the new (intuitionistic fuzzy) negations do not satisfy De Morgan's Laws in the ordinary forms now, Modi ed De Morgan's Laws are constructed. In a next research these will be extended. The Modi ed De Morgan's Laws from above from the extended types will be checked not only for modal, but also for extended intuitionistic fuzzy modal operators for quanti ers. References [1] Atanassov K., Two variants of intuitonistc fuzzy propositional calculus. Preprint IM-MFAIS-5-88, So a, [2] Atanassov, K. Intuitionistic Fuzzy Sets. Springer Physica-Verlag, Heidelberg, [3] Atanassov, K. Remarks on the conjunctions, disjunctions implications of the intuitionistic fuzzy logic. Int. J. of Uncertainty, Fuzziness Knowledge- Based Systems, Vol. 9, 2001, No. 1, [7] Atanassov, K. A new intuitionistic fuzzy implication from a modal type. Advanced Studies in Contemporary Mathematics, Vol. 12, 2006, No. 1, [8] Atanassov, K. On some intuitionistic fuzzy implications. Comptes Rendus de l'academie bulgare des Sciences, Tome 59, 2006, No [9] Atanassov,K.,B.Kolev,Onanintuitionistic fuzzy implication from a possibilistic type. Advanced Studies in Contemporary Mathematics, Vol. 12, 2006, No. 1, [10] Atanassov, K., T. Trifonov, On a new intuitionistic fuzzy implication from GÄodel's type. Proceedings of the Jangjeon Mathematical Society, Vol. 8, 2005, No. 2, [11] Feys, F. Modal logics, Gauthier-Villars, Paris, [12] Klir, G. Bo Yuan, Fuzzy Sets Fuzzy Logic. Prentice Hall, New Jersey, [4] Atanassov, K. Intuitionistic fuzzy implications Modus Ponens. Notes on Intuitionistic Fuzzy Sets, Vol. 11, 2005, No. 1, 1-5. [5] Atanassov, K., On some types of intuitionistic fuzzy negations. Notes on Intuitionistic Fuzzy Sets, Vol. 11, 2005, No. 4, [6] Atanassov, K., On some intuitionistic fuzzy negations. Notes on Intuitionistic Fuzzy Sets, Vol. 11, 2005, No. 6,