Table Of ContentJordiRecasens
IndistinguishabilityOperators
Studiesin Fuzziness andSoft Computing,Volume260
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FuzzyMathematics: Vol.260.JordiRecasens
ApproximationTheory,2010 IndistinguishabilityOperators,2010
ISBN978-3-642-11219-5 ISBN978-3-642-16221-3
Jordi Recasens
Indistinguishability
Operators
Modelling Fuzzy Equalities and Fuzzy
Equivalence Relations
ABC
Author
Dr.JordiRecasens
ETSArquitecturadelVallès
DepartmentofMathematicsandComputerSciences
C.PereSerra1-15
08190SantCugatdelVallès
Spain
E-mail:[email protected]
ISBN978-3-642-16221-3 e-ISBN978-3-642-16222-0
DOI10.1007/978-3-642-16222-0
StudiesinFuzzinessandSoftComputing ISSN1434-9922
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To my wife Mar´ıa
To my daughter Clara
Foreword
In1982,ata conference onLogicinBarcelona,I introducedthe conceptofa
T-Indistinguishability operator in fuzzy logic. As far as I know, this was the
firstpaper dealingwith a conceptI borrowedfromthe complementaryviews
of Henri Poincar´e’sthinking on the physical continuum’s intransitivity, Karl
Menger’sIndistinguishabilityRelationsinStatisticalMetricSpaces,andboth
LotfiA.ZadehandEnriqueH.Ruspini’spapersonFuzzySimilarityRelations.
Two years later, and with my then student Llorenc¸ Valverde, we published
ina bookanessayonthese operatorsinwhichweshowedseveralconceptual
examples, included their characterization by means of elemental fuzzy T-
preorders,andposedtheproblemwecalledofPoincar´e-Menger.Inthisbook,
SergeiOvchinnikovalsopublishedaveryinterestingcharacterizationofprod-
Indistinguishability operators by means of positive ’measurements’.
I think this was the beginning of a subjectwhich wasthen brilliantly con-
tinuedbyotherpeople,includingJoanJacas,Dion´ısBoixader,andJordiRe-
casensinBarcelona,starting withthe 1987Ph.DDissertationofJoanJacas,
whichis in anoutstanding contribution.I cordiallythank JordiRecasensfor
conferring me the honor of writing the Forewordto his book Indistinguisha-
bilityOperators.Modelling FuzzyEqualities andFuzzyEquivalenceRelations.
In what follows let me make a few remarks on what I consider to be most
important for conceptually capturing the essential treatment of the special
fuzzy relations this book considers.
I. The idea behind T-Indistinguishability operators is to give a definition
of a graded equivalence between the elements a,b,c,.. in a set X. To this
end, the problem is to establish a minimal number of properties a fuzzy re-
lation E : X ×X → [0,1] does satisfy in order to express that E(a,b) is
a ’degree up to which a is indistinguishable from, or equivalent, to b’, in
such a way that crisp equivalence relations could be obtained as a special
case. The typical reflexive and symmetric properties are easily translated
by E(a,a) = 1, and E(a,b) = E(b,a), for all a,b in X, but crisp transi-
tivity must be extended by means of a formula involving the three degrees
VIII Foreword
E(a,b),E(b,c),andE(a,c).Givenacontinuoust-normT ,thefuzzyrelation
E is said to be T-transitive if T(E(a,b),E(b,c)) ≤ E(a,c), for all a,b,c in
X,adefinitionclearlygeneralizingcrisptransitivitysinceT(1,1)=1.Notice
that the bounds r ≤ E(a,b), and r ≤ E(b,c),r > 0, give the weaker bound
T(r,r)≤E(a,c), with T(r,r)≤r. Of course, if T has not zero divisors then
T(r,r)isactuallyaboundfortheindistinguishabilitydegreebetweenaandc,
but ifT is in theL(cid:2)ukasiewicz family,itis requiredthatr satisfies0<T(r,r)
since, otherwise we obtain the non informative result E(a,c) ∈ [0,1]. That
is,if T =W ,it shouldbe r >ϕ−1(0.5)to haveE(a,c)∈[T(r,r),1](cid:2)[0,1].
ϕ
It is interesting to note that the crisp relations E(r), r > 0, defined by
’(a,b)∈E(r) iff r ≤ E(a,b)’, always reflexive and symmetric, are only crisp
equivalencesifT =min,inwhichcasethey facilitateanindexedtreeofcrisp
partitions of X.
II. Froma conceptualpointof view,it is relevantthat indistinguishability
operators E are directly related to the ’negation’ of distinguishability opera-
tors, or pseudo-distances. For instance,
• E is a min-Indistinguishability operator iff d=1−E is a ultradistance
• E is a W or prod -Indistinguishability operator iff d = N ◦E is such
ϕ ϕ ϕ
that ϕ◦d=1−ϕ◦E is a pseudo-distance,
with the strong negation Nϕ = ϕ−1 ◦(1−id)◦ϕ, where ϕ is an order-
automorphism of the unit interval. In the pseudo-metric space (X,ϕ◦d), it
holds that r ≤E(a,b) iff (ϕ◦d)(a,b)≤1−ϕ(r). That is, for each a in X it
holds that r ≤E(a,b) iff b is in the neighborhood of a with radius 1−ϕ(r).
Similarly, for each b in X, it is r ≤ E(a,b) iff a is in the neighborhood of b
with radius 1−ϕ(r).
III. Regardless if the connectives are functionally expressible, commuta-
tive, associative, dual, the negation is a strong one, etc., the only algebras
of fuzzy sets ([0,1]X,·,+,(cid:2)) that are lattices are those with · = min and
+ = max, none of them being a ortholattice and less again a boolean al-
gebra, but only De Morgan algebras if (cid:2) is a strong negation. This is an
important difference with the case of crisp sets, to which it should be added
the inexistence of an specification axiom simple as in the case of crisp pred-
icates, since the meaning of the imprecise ones is not definable by necessary
and sufficient conditions and is context and purpose dependent.
What is widely considered is the idea that the most salient feature dif-
ferentiating fuzzy from crisp sets is the failure of the principles of Non-
contradiction(NC),andExcluded-middle(EM),althoughthisisnotaintrin-
sic property of fuzzy sets but only a property of their algebras.For instance,
in the case of the standard algebras of fuzzy sets ([0,1]X,T,S,N):
• NC holds iff T =W , N ≤N , and S is any continuous t-conorm
ϕ ϕ
• EM holds iff S =W∗,N ≤N, and T is any continuous t-norm,
ψ ψ
Foreword IX
imply that both principles do hold if and only if T = W , S = W∗,
ϕ ψ
and N ≤ N ≤ N , and fail otherwise. Nevertheless, by interpreting the
ψ ϕ
principles `a la Aristotle:
• NC : μ·μ(cid:2) is always self-contradictory (impossible, in Aristotles terms),
that is, μ·μ(cid:2) ≤(μ·μ(cid:2))(cid:2),
• EM : (μ+μ(cid:2))(cid:2) is always self-contradictory (μ+μ(cid:2) is always the case, in
Aristotles terms), that is, (μ+μ(cid:2))(cid:2) ≤((μ+μ(cid:2))(cid:2))(cid:2),
for all μ in [0,1]X, both principles do hold in any algebra of fuzzy sets
([0,1]X,·,+,(cid:2)).
Hence,ifitisnotclearenoughthatthemostimportantdifferencebetween
fuzzy and crisp sets is the principles’ failure, it should be admitted that, a
clear difference lies, nevertheless, in the partial distinguishability between a
fuzzy set μ and its complement μ(cid:2). For instance, if X = [0,1], μ(x) = x,
μ(cid:2)(x)=1−x,andweconsidertheW-IndistinguishabilityoperatorE(x,y)=
1−|x−y|, it holds that E(μ(x),μ(cid:2)(x)) = 1−|2x−1|, which has value 1
iff x = 0.5, and 0 iff x = 0 or x = 1. That is, 0 < E(μ(x),μ(cid:2)(x)) < 1
iff x ∈/ {0,0.5,1}, although it holds NC: (μ·μ(cid:2))(x) = W(μ(x),μ(cid:2)(x)) = 0
and EM: (μ+μ(cid:2))(x) = W∗(μ(x),μ(cid:2)(x)) = 1, for all x in [0,1] . This idea of
intrinsicallyrelatingfuzzinesstothenon-completedistinguishabilitybetween
a fuzzy set and its complement, was proposed many years ago by Ronald R.
Yager, followed by myself, Claudi Alsina and Llorenc¸ Valverde, and it is yet
waitingtobedevelopedfurtherby,perhaps,consideringtheantonyminstead
ofthecomplement,sincetheseconddoesnotcorrespondtoalinguisticterm.
IV. For representing precise concepts, crisp equivalence relations are the
correctmathematicaltoolforthecomprehensionofthebipartitiongivenbya
predicateintheuniverseinwhichitisused.Inthe caseofthe representation
of imprecise words by means of fuzzy sets, T-Indistinguishability operators
are a mathematical tool for capturing the kind of imperfect fuzzy classifi-
cation generated by an imprecise predicate. A nice example of this, is that
of min-Indistinguishability operators that give, as described above, a tree of
indexed crisp partitions of the universe, and could help to pose finite fuzzy-
probability problems like classical partitions help to pose finite probabilistic
ones.
There is again another topic for which the operators Recasens takes into
account could be of a great interest. This `a la Wittgenstein topic is not
only important, but challenging for Computing with Words, and concerns
the degree of ’family resemblance’ existing between a predicate and their
migrates to different universes of discourse.
X Foreword
The author of this interesting book deserves a high recognition for his
effortinclearlypresenting the theory offuzzy equivalences andcollecting,in
a single volume, thatwhich is currentlydispersed in many articlespublished
either in edited books, conference proceedings, or journals. The book is well
organized,withveryclearmathematicalreasoning,andI amsureitwillhelp
peopleinterestedinfuzzylogictobetterunderstandtheworldofideasrelated
to imprecise partitionings.
May this book receive all the acknowledgments and success it deserves!
Enric Trillas
Emeritus Researcher
European Centre for Soft Computing
Mieres(Asturias), Spain
Preface
The ability to determine anequality is essentialto everytheory because it is
equivalent to the problemof distinguishing the objects that the theory deals
with. This need -and the importance of selecting between different possibili-
ties -arises from the fact that equality allows us to classify in the context of
the theory; and to classify is one of the most important processes of knowl-
edgesinceitallowsustorelate,organize,generalize,findgenerallaws,etc.In
fact,ascientificknowledgethatdoesnotneedanequalitycapableofclassify-
ingthe objectsitstudiesis unconceivable.Borgesillustratesthis veryclearly
in the following paragraphof Funes the Memorious (Ficciones).
E´ste, no lo olvidemos, era casi incapaz de ideas generales, plat´onicas. No
s´olo le costaba comprender que el s´ımbolo gen´erico perro abarcara tantos
individuos dispares de diversos taman˜os y diversa forma; le molestaba que
el perro de las tres catorce (visto de perfil) tuviera el mismo nombre que
el perro de las tres y cuarto (visto de frente)... Sospecho... que no era muy
capazdepensar.Pensaresolvidardiferencias, esgeneralizar, abstraer.En el
abarrotado mundode Funesnohab´ıa sino detalles, casi inmediatos. 1
According to the Identity of indiscernibles, or Leibniz’s law, two or more
objects or entities are identical (i.e. they are one and the same entity) if
they have all of their properties in common. In the abscence of uncertainty,
this principle establishes equivalence relations and, in particular, the first
Common Notion of Euclid’s Elements or transitivity:
Things which equalthe same thing also equalone another.
1 He was, let us not forget, almost incapable of general, platonic ideas. It was
not only difficult for him to understand that the generic term dog embraced so
many unlike specimens of differing sizes and different forms; he was disturbed
by the fact that a dog at three-fourteen (seen in profile) should have the same
name as the dog at three-fifteen (seen from the front)... I suspect... that he was
not very capable of thought. To think is to forget differences, to generalize, to
abstract. In the overly replete world of Funes there were nothing but details,
almost contiguous details.
