Implication in intuitionistic fuzzy and interval-valued fuzzy set theory

Implication in intuitionistic fuzzy and interval-valued fuzzy set theory
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  Implication in intuitionistic fuzzy andinterval-valued fuzzy set theory:construction, classification, application Chris Cornelis  * , Glad Deschrijver, Etienne E. Kerre Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematicsand Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium Received 1 August 2002; accepted 1 July 2003 Abstract With the demand for knowledge-handling systems capable of dealing with and dis-tinguishing between various facets of imprecision ever increasing, a clear and formalcharacterization of the mathematical models implementing such services is quintessen-tial. In this paper, this task is undertaken simultaneously for the definition of impli-cation within two settings: first, within intuitionistic fuzzy set theory and secondly,within interval-valued fuzzy set theory. By tracing these models back to the underlyinglattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MValgebras), and on the other hand we are able to expose in a clear manner the two models  formal equivalence. This equivalence, all too often neglected in literature, we exploit toconstruct operators extending the notions of classical and fuzzy implication on thesestructures; to initiate a meaningful classification framework for the resulting operators,based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine theintuititive ideas giving rise to both approaches as models of imprecision and apply themin a practical context.   2003 Elsevier Inc. All rights reserved. * Corresponding author. Tel.: +32-9-264-47-72; fax: +32-9-264-49-95. E-mail addresses: (C. Cornelis), (G. Des-chrijver), (E.E. Kerre). URL:$ - see front matter    2003 Elsevier Inc. All rights reserved.doi:10.1016/S0888-613X(03) International Journal of Approximate Reasoning 35 (2004) 55–95  Keywords:  Intuitionistic fuzzy set theory; Interval-valued fuzzy set theory; Indetermi-nacy; Implicators; Smets–Magrez axioms; Residuated lattices; MV-algebras; Knowl-edge-based systems 1. Introduction Intuitionistic fuzzy sets [1] and interval-valued fuzzy sets ([54,67] and morerecently, [58]) are two intuitively straightforward extensions of Zadeh  s fuzzysets [66], that were conceived independently to alleviate some of the drawbacksof the latter. Henceforth, for notational ease, we abbreviate ‘‘intuitionisticfuzzy set’’ to IFS and ‘‘interval-valued fuzzy set’’ to IVFS. IFS theory basicallydefies the claim that from the fact that an element  x  ‘‘belongs’’ to a given degree(say  l ) to a fuzzy set  A , naturally follows that  x  should ‘‘not belong’’ to  A  to theextent 1  l , an assertion implicit in the concept of a fuzzy set. On the con-trary, IFSs assign to each element of the universe both a degree of membership l  and one of non-membership  m  such that  l þ m 6 1, thus relaxing the enforcedduality  m  ¼  1  l  from fuzzy set theory. Obviously, when  l þ m  ¼  1 for all ele-ments of the universe, the traditional fuzzy set concept is recovered. IFSs owetheir name [4] to the fact that this latter identity is weakened into an inequality,in other words: a denial of the law of the excluded middle occurs, one of themain ideas of intuitionism.  1 IVFS theory emerged from the observation that in a lot of cases, no ob- jective procedure is available to select the crisp membership degrees of elementsin a fuzzy set. It was suggested to alleviate that problem by allowing to specifyonly an interval [ l 1 ; l 2 ] to which the actual membership degree is assumed tobelong. A related approach, second-order fuzzy set theory, also introduced byZadeh [67], goes one step further by allowing the membership degrees them-selves to be fuzzy sets in the unit interval; this extension is not considered inthis paper.Both approaches, IFS and IVFS theory, have the virtue of complementingfuzzy sets, that are able to model  vagueness , with an ability to model  uncer-tainty  as well.  2 IVFSs reflect this uncertainty by the length of the intervalmembership degree [ l 1 ; l 2 ], while in IFS theory for every membership degree 1 The term ‘‘intuitionistic’’ is to be read in a ‘‘broad’’ sense here, alluding loosely to the denial of the law of the excluded middle on element level (since  l þ m  <  1 is possible). A ‘‘narrow’’, gradedextension of intuitionistic logic proper has also been proposed and is due to Takeuti and Titani[57]––it bears no relationship to Atanassov  s notion of IFS theory. 2 In these pages, we juxtapose ‘‘vagueness’’ and ‘‘uncertainty’’ as two important aspects of imprecision. Some authors [45,47,60] prefer to speak of ‘‘non-specificity’’ and reserve the term‘‘uncertainty’’ for the global notion of imprecision.56  C. Cornelis et al. / Internat. J. Approx. Reason. 35 (2004) 55–95  ð l ; m Þ , the value  p  ¼  1  l  m  denotes a measure of non-determinacy (or un-decidedness).Each approach has given rise to an extensive literature covering their re-spective applications, but surprisingly very few people seem to be aware of theirequivalence, stated first in [2] and later in [31,63]. Indeed, take any IVFS  A  in auniverse  X  , and assume that the membership degree of   x  in  A  is given as theinterval [ l 1 ; l 2 ]. Obviously,  l 1  þ 1  l 2 6 1, so by defining  l  ¼  l 1  and  m  ¼  1  l 2  we obtain a valid membership and non-membership degree for  x  in an IFS  A 0 . Conversely, starting from any IFS  A 0  we may associate to it an IVFS  A  byputting, for each element  x , the membership degree of   x  in  A  equal to the in-terval [ l ; 1  m ] with again  ð l ; m Þ  the pair of membership/non-membership de-grees of   x  in  A 0 . As a consequence, a considerable body of work has beenduplicated by adepts of either theory, or worse, is known to one group andignored by the other. Therefore, regardless of the meaning (semantics) that onelikes his or her preferred approach to convey, it is worthwhile to develop theunderlying theory in a framework as abstract and general as possible. Latticesseem to lend themselves extremely well to that purpose; indeed it is commonpractice to interpret them as evaluation sets from which truth values are drawnand to use them as a starting point for developing logical calculi. Let us applythis strategy to the formal treatment of IVFSs and IFSs: we will describe themas special instances of Goguen  s  L -fuzzy sets,  3 where the appropriate evalua-tion set will be the bounded lattice  ð  L  ;  6  L  Þ  defined as [14]: Definition 1  (Lattice  ð  L  ;  6  L  Þ )  L   ¼ fð  x 1 ;  x 2 Þ 2 ½ 0 ; 1  2 j  x 1  þ  x 2 6 1 gð  x 1 ;  x 2 Þ 6  L   ð  y  1 ;  y  2 Þ ()  x 1 6  y  1  and  x 2 P  y  2 The units of this lattice are denoted 0  L   ¼ ð 0 ; 1 Þ  and 1  L   ¼ ð 1 ; 0 Þ . A specialsubset of   L  , called the diagonal  D , is defined by  D  ¼ fð  x 1 ;  x 2 Þ 2½ 0 ; 1  2 j  x 1  þ  x 2  ¼  1 g . The shaded area in Fig. 1 is the set of elements  x  ¼ ð  x 1 ;  x 2 Þ belonging to  L  . Note . This definition favours IFSs as they are readily seen to be  L -fuzzy setsw.r.t. this lattice, while for IVFSs a transformation from  ð  x 1 ;  x 2 Þ 2  L   to theinterval [  x 1 ; 1   x 2 ] must be performed beforehand; this decision reflects thebackground of the authors. Nevertheless, it is important to realize that nothingstands in our way to define equivalently:  L  I  ¼ fð  x 1 ;  x 2 Þ 2 ½ 0 ; 1  2 j  x 1 6  x 2 g 3 Let  ð  L ;  6  L Þ  be a complete lattice. An  L -fuzzy set in  U   is an  U   !  L  mapping [36]. C. Cornelis et al. / Internat. J. Approx. Reason. 35 (2004) 55–95  57  ð  x 1 ;  x 2 Þ  6  L  I   ð  y  1 ;  y  2 Þ ()  x 1 6  y  1  and  x 2 6  y  2 and develop the theory in terms of   ð  L  I  ;  6  L  I  Þ . For compliance with the existingliterature, we denote the class of   L  -fuzzy sets in a universe  U   by F  L  ð U  Þ . Note . In this paper, if   x  2  L  , we refer to its and first and second componentsby  x 1  and  x 2  respectively. In case we want to refer to the individual componentsof an expression like  f  ð  x Þ , where in this case for instance  f   is an  L   !  L  mapping, we write pr 1  f  ð  x Þ  and pr 2  f  ð  x Þ , where the projections pr 1  and pr 2  mapan ordered pair (in this case an element of   L  ) to its first and second compo-nent, respectively.The lattice  ð  L  ;  6  L  Þ  is a complete lattice: for each  A    L  , sup  A  ¼ð sup f  x  2 ½ 0 ; 1 jð9  y   2 ½ 0 ; 1 Þðð  x ;  y  Þ 2  A Þg ; inf  f  y   2 ½ 0 ; 1 jð9  x  2 ½ 0 ; 1 Þðð  x ;  y  Þ 2  A ÞgÞ and inf   A  ¼ ð inf  f  x  2 ½ 0 ; 1 jð9  y   2 ½ 0 ; 1 Þðð  x ;  y  Þ 2  A Þg ; sup f  y   2 ½ 0 ; 1 jð9  x  2 ½ 0 ; 1 Þðð  x ;  y  Þ 2  A ÞgÞ .As is well known, every lattice  ð  L ;  6 Þ  has an equivalent definition as analgebraic structure  ð  L ; ^ ; _Þ  where the meet operator  ^  and the join operator  _ are linked to the ordering  6  by the following equivalence, for  a ; b  2  L : a 6 b  ()  a _ b  ¼  b  ()  a ^ b  ¼  a The operators  ^  and  _  on  ð  L  ;  6  L  Þ  are defined as follows, for ð  x 1 ;  y  1 Þ ; ð  x 2 ;  y  2 Þ 2  L  : ð  x 1 ;  y  1 Þ^ð  x 2 ;  y  2 Þ ¼ ð min ð  x 1 ;  x 2 Þ ; max ð  y  1 ;  y  2 ÞÞð  x 1 ;  y  1 Þ_ð  x 2 ;  y  2 Þ ¼ ð max ð  x 1 ;  x 2 Þ ; min ð  y  1 ;  y  2 ÞÞ This algebraic structure will be the basis for our subsequent investigations. Inthe next section, entitled ‘‘Preliminaries’’ the most important operations on ð  L  ;  6  L  Þ  are defined, notably: triangular norms and conorms, negators andimplicators. They model the basic logical operations of conjunction, disjunc-tion, negation and implication. Implicators on  L   will be the main point of in- Fig. 1. Graphical representation of the set  L  .58  C. Cornelis et al. / Internat. J. Approx. Reason. 35 (2004) 55–95
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