Pamplona Trillas

Fuzzy Logic:Looking at the future

Enric Trillas

European Centre for Soft-ComputingMieres (Asturias)

September 2009

EUROFUSE09, Navarra

Enric Trillas (ECSC) Fuzzy Logic: From past to future EUROFUSE09, Navarra 1 / 73

Introduction I

Karl Menger introduced statistical metrics, latter on called probabilisticmetrics, with the aim of helping to understand subatomic physics by meansof new mathematical ideas and models. At this respect and in practice, hisproposal was mostly ignored, and the Probabilistic Metric Spaces turned injust a new, although very interesting, part of mathematics that,nevertheless, deserved not too much attention by physicists.

Mengers basic idea is that the worlds comprehension comes parallel withthe introduction of new geometrical ideas that, in some form, contain asparticular cases the previous ones.

By the chance of a networking of people dealing with both statisticalmetrics and fuzzy logic, and some time latter, the redefinition of t-normsand t-conorms made by Schweizer and Sklar, resulted to be a sub-productof interest for modeling some uses of the linguistic connectives and, andor, respectively, in fuzzy logic.


Introduction II

Today there is a new subject that will require of new ideas and modelscoming from the field of mathematics, that is, from the mathematiciansbrains. This subject is the mathematical modeling of natural languagewith the goal of using it for the computational treatment of language.

Like in the case of Mengers statistical metrics, it is more than possiblethat these models will not be taken into account by linguists, at least fromthe beginning of the intents.

The most challenging aspect of language for these models is meaningunderstood a la Wittgenstein, that is as it is conveyed by the use of wordsand sentences, specially by imprecise ones.

In the last four centuries mathematical models did prove their usefulnessfor the comprehension of natural phenomena!


Introduction III

Currently, and from an experimental science point of view, language as awhole is largely unknown or, at least, scarcely described in typical scientificterms by not being considered like other natural phenomena. For instance,there are no enough mathematical models of language, perhaps by thelack of a hard-science experimental methodology in its treatment as, forexample, signals like it is done with the processing of perceptions andreasonings in the brain.Language is, like the brain, one of the most complex evolutive systemswith which science should deal with in the XXI Century.Today, only some models coming from fuzzy logic and, hence, usingmathematical analysis, can be taken into account as a basis from whichgrounding afterwards the analysis of the language dynamics.In a good part, language is the great challenge for reachingcomputers reasoning like people, a kind of reasoning that only in asmall part is deductive and that is mainly inductive, abductive andanalogical. It is, perhaps, deductive in, at most, only a 25%.


Introduction IVAs one of the consequences of brains activity, language that appearedafter brain is actually commanded through some brain functions and, inthis sense, perhaps language is no less complex than brain functioning is.

In addition, there is a relevant aspect that makes language less knownthan brain is. This lies in the amount of specific knowledge, expressed inscientific terms, neurobiologists do have on the functioning of the brain,something that is not the case with language since it is not currentlytreated as a natural experimental discipline concerning a special type ofliving beings.

Such a new discipline, provided it were created, could result of an upmostinterest for the advancement of Computing with Words and Perceptions(CWP).

At the end, brain functioning is closely related with the electro-chemicalprocessing of perceptions, and language with the representation andcommunication, or spreading, of such processing.


Introduction V

Concerning the mathematical modeling of language, and the possiblebenefits that can follow from some results in fuzzy logic, it could beinteresting to recall, for instance, the presence of symmetry in the way ofobtaining the membership function of an antonym, or opposite, of alinguistic term P from a membership function of P.

Antonymy is an important feature of language, and since symmetryis a pervasive concept in the world , and also in the brain, possiblyit should play some relevant role in language, and hence it coulddeserve to be studied for a deeper characterization of language.This is a challenging subject for mathematicians and computerscientists interested in CWP.


Introduction VI

There are two different lanes through which mathematicians canhelp the development of CWP.

The first corresponds to the typical one in which mathematics is useful asan archive of knowledge to be used when necessary, and consists on puremathematical results that, in principle, can be applied anywhere butwithout knowing when and where they will be applied. Its publication doesonly correspond to mathematical journals.

The second corresponds to the work of mathematicians on particularCWPs problems, and its publication should be done either inmathematical journals, when it does not refer directly to those problemsbut is only inspired on them, or in CWP journals when it directly shedslight on such problems.


Introduction VII

In any of those two lanes, some theoretical sub-products that can be usefulin other branches of science are expected to appear, as it happened notonly with statistical metrics but with many other parts of puremathematics. That is, new ideas showing their fertility in science.

In short, around the problems coming from CWP there is a new andexciting field of research for mathematicians to whom I encourage to dotheir work on the search for new models of some parts of language andordinary reasoning, and mainly for those parts where the currentarmamentarium of fuzzy logic is not usable, as it is the case with someuses of the linguistic particle and, as well as with those predicates thatare not directly gradable in the unit interval.


Introduction VIII

The current scientific challenge in Computer Science is natural languageand ordinary reasoning conveyed through it, that dynamically changes thecontents of the human mind. Among the most challenging topicsconcerning fuzzy logic let me quote the following ones:

1 The study of when and which classical schemes of deductivereasoning different of Modus Ponens, like the disjunctive mode, thereduction to absurd, etc., can be used within fuzzy logic.

2 To revisit the concept of degree of an imprecise predicate forcapturing the idea of fuzzy sets as representations of the use ofpredicates.

3 The analysis in fuzzy terms of larger linguistic expressions than thoseconsidered in fuzzy control.


Introduction IX

4 To revisit the current representation of and, or, not, if/then,byattending at their contextual uses.

5 To extend the current standard algebras (T ,S ,N) to algebras of thetype (Ti ,Sj ,Nk), with 1 i n, 1 j m, and 1 < k < p, forcapturing the representation of linguistic expressions with differentuses of the three basic connectives.

6 To formalize, in fuzzy terms, the Wittgensteins ideas of languagegames and family resemblances, with the goal of approaching thesimilarity and uses evolution of words, as well as to advance thestudy of analogical reasoning.


Introduction X

7 To introduce in fuzzy logic the concept of conjecture as the centralidea of reasoning, with the aim of considering not only deductiveschemes, but also inductive and abductive schemes.

8 To analyze in which problems, on which conditions, and with whichkind of benefit, the remarkable archive of mathematical knowledgeexisting on aggregation functions, could be applied to the CWWsproblems.

9 The mathematical study of the intriguing concept of fuzzyprobabilities posed by Lotfi A. Zadeh.


Introduction XI

Although in what follows I will only consider three of these nine points, letme proceed before that with a brief reflection on my first contact withZadehs fuzzy sets, to show how I always interpreted fuzzy sets.


My first steps in fuzzy logic I

In the summer of 1974, I was a recently appointed University Professor ofMathematics in Barcelona, was doing research on Generalized MetricSpaces (GMS) for around ten years, and was trying to take some distancefrom the, then and for me, too rigid and scholastic style of the prevalentbourbakism commanding both mathematical teaching and research in theSpanish universities. I arrived at GMS trough the curious following path.

In my last year before masters graduation, I was interested in theproperties of symmetric difference in boolean algebras, and aftergraduation I passed from the study of the symmetric difference to that ofordered semigroups with neutral, and in particular to those in the unitinterval that I called of type-1(t-norms), and of type-2(t-conorms).


My first steps in fuzzy logic II

After publishing some papers on ordered semigroups, I knew the paper ofMenger on statistical metrics, that by Abraham Wald using convolution forthe triangular inequality, as well as the first papers on probabilistic metricspaces by Bert Schweizer and Abe Sklar, in which they added theassociative property to Mengers t-conorms for reaching a polygonalinequality, and where I realized that the type-1 and type-2 semigroupswere re-christianized.

In this moment, around 1967, I started to correspond with Karl Mengerwho kindly posed me some difficult problems on finite geometry, and alsofacilitated me to contact Bert Schweizer. This contact allowed me toknow more on the work in probabilistic metric spaces, and also to knowthe works of Maurice Frechet on abstract metrics, and of Blumenthal onboolean metrics. All that conducted me to define the structure of GMS,on which I did my Ph.D. dissertation, and worked on up to 1989.


My first steps in fuzzy logic III

During my graduate student period, I read a lot on the work of KarlMenger, who kindly sent me his papers, and also on Bertrand Russell. Inparticular, I read the French written paper by Menger on Hazy Sets, andthe Russells ideas on perceptions, as well as his paper on vagueness, asubject that kept my interest since vagueness was, in that time, almostexcluded from scientific and philosophical consideration.

I dont remember how the work of Russell conducted me to the papers byMax Black on imprecise concepts and on reasoning with them. Anyway, atthat time I was unable to think on the Blacks profile functions as anythingelse than something related with probability distributions, surely under theinfluence of Mengers hazy sets and statistical distances.


My first steps in fuzzy logic IV

In the summer of 1974 I got my first notice on Zadehs fuzzy sets, throughthe first book on the subject by Arnold Kaufmann, and my first impressioncan be resumed with the word garbage.

Nevertheless, something called my curiosity since I immediately came toread the first paper on fuzzy sets by Zadeh. My first impression didimmediately change, in realizing that linguistic imprecision is not alwayssomething of a probabilistic character, and in linking the membershipfunctions with the profile ones.

The change was such that I decided to start some work on fuzzy sets, andthe first topic on which I tried to produce something was related with DeLuca - Termini fuzzy entropies, although not with the aim of controllingthe fuzziness of concepts, but with that of controlling its booleanity, andalso on obtaining different models of fuzzy entropies.


My first steps in fuzzy logic V

It was after 1978 that I was concerned with the representation ofconnectives in fuzzy logic, a subject for which the book on functionalequations by Janos Aczel gave me an important tool, since most of theconnectives properties come from the solutions of the functional equationsor inequations into which they can be translated. I was acquainted withthis book thanks to the study of t-norms and t-conorms.My first contribution on that topic was the characterization of strongnegations in 1979, and the second, that of non-distributive conjunctionsand disjunctions in 1980, a topic that was suggested by the paper ofBellman and Giertz on the characterization of the distributive pair(min,max).I think that after this moment my contributions to fuzzy logic aresufficiently well known for stopping at it, and taking into account for awhile the question of in which scientific and philosophical traditions fuzzylogic could be inscribed.


Thinking traditions and fuzzy logic I

Between the last eighties and nineties of XX Century, I heard or read toomany times from some people, mainly westerners, who tried to opposefuzzy logics ideas to the western traditions on science and philosophy.

Although, and for sure, I am not at all against the existence of influencescoming from other traditions like, for instance, some Eastern ones, I neverfully agreed with those views. At least in my case, but I think that also inthat of Lotfi Zadeh, the most relevant thinkers that influenced me docome from the western traditions.

To test that thought, I checked on my fuzzy logics papers between thethirty years from 1977 to 2007, which are the great thinkers, scientists orphilosophers, I quoted in them. I was surprised when in just a sample offifty of them I found the following thirty-eight names:


Thinking traditions and fuzzy logic II

Aristotle, John L. Austin, Alfred L. Ayer, Gaston Bachelard, Max Black,Georges Bodiou, George Boole, Mario Bunge, Rudolf Carnap, NoamChomsky, Albert Einstein, Federigo Enriques, Bruno de Finetti, JoseFerrater-Mora, Kurt Godel, Susan Haack, Paul Horwich, Jan Lukasiewicz,Ernst Mach, Karl Menger, George E. Moore, John von Neumann, WilliamOckham, Charles S. Peirce, Henri Poincare, Karl Popper, Hillary Putnam,Willard Van Orman Quine, Hans Reichenbach, Bertrand Russell, Luis A.Santalo, John Searle, Patrick Suppes, Alfred Tarski, Stanislav Ulam, JeanUllmo, Norbert Wiener, and Ludwig Wittgenstein.

From the work of all these authors in the scientific or the philosophicalside of thinking, I got insights for my own work. Hence, and withouttrying to judge the value of my work, I do consider it clearly influenced bythe thinking of western scientists and philosophers.


Thinking traditions and fuzzy logic IIIWhat I will recall, to end this section, is that people working in fuzzy logicwere aware from the very beginning, and perhaps by the first time, thatnot only predicates do have meaning, but that also connectives do havemeaning. This consideration places fuzzy logic inside the realm ofmodeling natural language, and makes of fuzzy logic a discipline differentof formal logic.

The difference is not only, as Zadeh says, that in fuzzy logic everything isa matter of degree, but also that everything is contextual and that fuzzyrepresentations are to be carefully designed by choosing therepresentations better reflecting the current use of predicates, connectives,relations, and so on.

Would they know it or not, people working in fuzzy logic arewittgensteinians in the sense that, for them, meaning is nothing else thanthe use of the terms to be represented. It is because of all that that, in myview, the grounds on which fuzzy logic is based are some parts of naturallanguage and ordinary reasoning: A Western thinking tradition!


Thinking traditions and fuzzy logic IV

To end this section, let me recall a pair of statements by two scientists inthe list I mentioned before.

The first is from John von Neumann, who in 1951 wrote what follows:Everybody who has worked in formal logic will confirm that it is one ofthe technically most refractory parts of mathematics. The reason for thisis that it deals with rigid, all-or-none concepts, and has very little contactwith the continuous concept of the real or the complex number, that is,with mathematical analysis (in The General and Logical Theory ofAutomata).

The second is from Karl Menger who in 1934 wrote:The only justification for scientific theories -for the way words are used inscience- lies in their fertility; that is, in the possibility of using them in agreat number of important propositions, preferably practicalpropositions(in Morality, Decision and Social Organization ).


Thinking traditions and fuzzy logic V

Who will doubt that Zadeh introduced logical concepts mixed withmathematical analysis?

Who will doubt that Zadehs ideas resulted to be fertile?

What follows is in these lines of thinking, corresponds to three of the ninepoints in section II, is presented from a mathematical point of view, andcomes from part of my own work.

In the way towards CWP, an to reach more relevant human-centricsystems, I think that the moment of revising and expanding fuzzy logic isarrived.


New schemes for fuzzy reasoning I

When representing fuzzy connectives, they should be designed notonly by capturing its use in the corresponding context, but also bytaking into account for which purpose the representation is needed.

For example, given a linguistic imprecise rule like If x is P, then y is Q,and provided its use is captured by the Larsen operatorJ(P(x),Q(y)) = P(x)Q(y), this operator is correct for the purpose ofdoing forwards reasoning since the corresponding Modus Ponensinequality T (a, J(a, b)) b, is verified for any t-norm T.

That is, in any case Q is a logical consequence of the set of premises{P Q,P}, with Q(y) belonging to an interval that, in general, isdifferent from the interval [0, 1].


New schemes for fuzzy reasoning II

Nevertheless, provided the reasoning should be a backwards one,then it should verify the corresponding Modus Tollens inequalityT (N(b), a.b) N(a), equivalent to T = Wf ,and N Nf . But thissolutions implies Wf (N(b), a.b) Wf (Nf (b), a.b)= f 1(max(0, f (Nf (b)) + f (a.b) 1)) = f 1(max(0, f (a.b) f (b)) = 0,since a.b b forces f (a.b) f (b) 0.

Hence, it is obtained the non-informative conclusion that N(a) is in theinterval [0, 1], that the degree up to which it is not(x is P), giventhe respective degrees of not(y is Q) and If x is P, then y is Q, isindeterminate.

In cases like this, either the rule can be captured by a different function Jallowing Modus Tollens, or the backwards reasoning cannot be done.


New schemes for fuzzy reasoning III

It is for that reason that the design of fuzzy systems should be doneby not only attending at the meaning, or use, of the terms in itslinguistic description, but also by considering the purpose to beattained with the representation.


The case of the disjunctive mode I

Suppose that a disjunction P or Q of two predicates P and Q (previouslydesigned by fuzzy sets, accordingly with their uses in a universe ofdiscourse X ) is needed for the purpose of doing the disjunctive reasoning

P or Q (= S (P Q))not Q (= N Q)

P

Which pairs (S ,N) can be taken?

That scheme can be compacted in the inequalityT (S(P(x),Q(x)),N(Q(x))) P(x), for all x in X , with the mainunknowns S ,N, and the auxiliary unknown T .That is, the functional inequality to solve is

T (S(a, b),N(b)) a (1)

for all a, b in the unit interval.


The case of the disjunctive mode II

Solution: Let f be any order-automorphism of the unit interval.

Theorem

Inequality (1) does hold if and only if: T = Wf ,N Nf , andS(a, b) W f (min(a,Nf (b)),Nf (N(b))), for all a, b in [0, 1].

In particular, (1) holds if, with T = Wf , is N Nf and S W f . Forinstance, with T = W , it can be taken S W and N 1 id .


The case of the resolution mode I

A disjunction S should be selected with the purpose of doing the reasoningby resolution

P or QNot Q or R

P or R.

Now, we should search for a triplet (T ,S ,N) such that the inequality

T (S(a, b),S(N(b), c) S(a, c) (2)

for all a, b, c in [0, 1] is verified.


The case of the resolution mode IIA solution: Theorem

A triplet (T ,S ,N), with the condition If T (a, b) = 0, then b N(a) ,verifies (2) if T = Wf , N = Nf , andS(a, b) = f 1(1 F (1 f (a), 1 f (b))), for all a, b in [0, 1], and somefunction F in the class of all binary operations in [0, 1] which arenon-decreasing in each variable, F W , and satisfiesF (a, b) + F (1 b, c) F (a, c) for all a, b, c in (0, 1).

For instance,

1 f = id , T = W , N = 1 id , with F = min, gives S = max: thescheme holds in the non-dual algebras with the triplet(W ,max, 1 id).

2 f = id , T = W , N = 1 id , with F = Prod , gives S = W : thescheme holds in the dual algebras with the triplet (W ,W , 1 id).

3 If W F Prod is a t-norm, is S = F : the scheme holds in thedual algebras with (F ,F , 1 id)


The case of reduction to absurd I

Not Q (N Q)If not P, then Q (J ((N P) (Q)))

P,

and the inequation to solve is

T (N(b), J(N(a), b) a (3)

for all a, b in [0, 1], and some continuous t-norm T, strong negation N,and conditional J.

A solution: Theorem

Provided J verifies J(1, b) b, for all b in [0, 1], (3) holds if and only ifT = Wf ,N Nf , and J(a, b) 1 (f (N(b)) + f (N(a))), for all a, b in[0, 1]

Condition J(1, b) b, is verified by S-implications, R-implications,Q-operators, D-operators, and ML-operators.


The case of reduction to absurd II

For instance,

1 If f = id , is T = W , N 1 id , and J(a, b) 1 (N(a) + N(b))2 If f = id , and N = 1 id , is T = W , and

J(a, b) min(1, 1 a + b), that is, J is any W-conditional.3 If f = id , and N = (1 id)/(1 + id), is T = W , and

J(a, b) 1 a(1 b) + 3b/(1 + a)(1 + b)


The case of the constructive dilemma I

The scheme,

P or QIf P then RIf Q then T

R or T ,

can be translated into

T (S(a, b), J(a, c), J(b, d)) S(c, d), (4)

for all a, b, c, d in [0, 1]. Some solutions,

1 If J is an S-implication S(N(a), b), and T verifies If T (a, b) = 0,thenb N(a) (4) holds if and only if T = Wf , N = Nf , and S = max.For instance, it can be chosen S = max and N = 1 id ,with theauxiliar T = W .


The case of the constructive dilemma II

2 If J is either an ML-operator, or an R-implication, it is J(a, b) b,for all a, b in [0, 1], and (4) holds for any triplet (T ,S ,N).

Remark. Notice that if J1(a, b) b , and J2(a, b) b, it also holdsthe inequality

T (S(a, b), J1(a, c), J2(b, d)) S(c , d)

and the constructive dilemma can be done with two differentconditionals,

P or QP 1 RQ 2 TR or T .


Conjectures, and the fuzzy case:Introduction I

Let us consider some modalities of ordinary reasoning, so in which, fromsome body of information, the following is done:

Going forwards, to obtain a conclusion of the given information.

Going backwards, to obtain an explanation, or hypothesis, for it.

To speculate, with or without a clear goal.

Concerning the body of information, it is usually on a particular subjectand supposedly consists on a finite number of statements (premises) oftenimprecise, partially true, perhaps not totally reliable, etc.

Concerning conclusions, they can be either necessary (safe), orcontingent (unsafe). The first are logical consequences, and the secondare guessings or conjectures.


Conjectures, and the fuzzy case:Introduction II

Suppose that the body of information is given by a set P = {p1, ..., pn} ofstatements. Call

p = p1 and p2 and ... and pn, the resumee of the information.

To avoid further complications, suppose that

it is not If p, then not p,

that is, p is not self-contradictory.


Conjectures, and the fuzzy case:Introduction IIIWith all that,

An statement q is a conjecture from P, if it is not If p, then not q.Any process allowing to pass from P to a conjecture of it, is aconjectural reasoning, or guessing.

A statement q is a consequence of P, if If p, then q.Any process allowing to pass from P to a consequence of it, is adeductive reasoning.

A statement q is a hypothesis for P, if If q, then p.Any process allowing obtain hypotheses for P, is an abductivereasoning.

A statement q is a speculation upon P, if q is a conjecture, but it isneither a consequence, nor a hypothesis.Any process allowing to obtain speculations for P , is an speculativereasoning.

With all that some mathematical models are possible.Enric Trillas (ECSC) Fuzzy Logic: From past to future EUROFUSE09, Navarra 36 / 73

A model in ortholattices I

Let (L; ,+, ; 0, 1) be an ortholattice in which

a b ab = a a + b = b.

Then, we can define:

The set of premises, P = {p1, ..., pn}, and its resumeep = p1p2...pn, with p p implying (p 6= 0).The set of conjectures,Conj(P) = {q L; p q} = {q L; p q}c = Ref (P)c , thecomplement of what refutes P.

The set of logical consequences, Cons(P) = {q L; p q}.The set of hypotheses, Hyp(P) = {q L; 0 < q < p}.The set of speculations, Sp(P) = {q L; p q, p q, q p}.


A model in ortholattices II

It is not difficult to prove:

P Cons(P) Conj(P), and Cons is a Tarskis operator ofconsequences.

Hyp(P) Cons(P) = , Hyp(P) Conj(P).Sp(P) Cons(P) = Sp(P) Hyp(P) = , Sp(P) Conj(P).If P1 P2, then Conj(P2) Conj(P1), Hyp(P2) Hyp(P1) andCons(P1) Cons(P2).That is, Cons is a monotonic operator, but both Conj and Hyp areanti-monotonic.

Sp is neither monotonic, nor anti-monotonic. It is non-monotonic.

Conj(P) = Cons(P) Hyp(P) Sp(P), is a partition of Conj(P)!Ref (P) = Conj(P)c , L = Conj(P) Ref (P).


A model in ortholattices III

If the ortholattice is an orthomodular one,

q Cons(P) p Sp(P), q = p + pq Hyp(P) p Sp(P), q = pp

this results seem to agree with the intuition that both to reach aconsequence, or an hypothesis, we speculate upon the availableinformation.

The speculative, non-monotonic, reasoning, could be identified withinduction when its goal is to find either a consequence, or a hypothesis.


A model in ortholattices IV

REMARKS.

1 If it were p = 0, then {q L; p q} = L Conj(P) = .No conjecture does exist if the resumee is zero (empty!)

2 If p 6= 0, neither Cons(P), nor Conj(P), can be empty. But eitherHyp(P), or Sp(P), can be empty.

3 If the ortholattice is a boolean algebra, p q pq 6= 0: Theconjectures are those elements matching the resumee.Analogously, p p p 6= 0.

4 Notice that nothing is stated on the possible truth of the elements inConj(P), or in its subsets!

5 Since Inf Cons(P) = p, it is clear that Cons(P) can be taken as aset of premises (for nothing, since Cons(Cons(P)) = Cons(P)).Inf Conj(P) = 0, Inf Hyp(P) = 0, or Inf Sp(P) = 0. Hence, thesesets cant be taken as a body of knowledge.


A model in ortholattices V

Graphically

For each P,


A model in ortholattices VI

Example

Let L be an orthocomplemented lattice with 3 elements, m, e, and smidday (m) eclipse (e) sunny (s)

Set of premises P = {m,ms , (es)}, p = mms (es) = ms 6= 0.

s is a consequence of P, since ms s , and s Conj(P) since ifms s , it would imply that ms = 0.ms e is a hypothesis of P, since ms e < ms .e is an speculation of P, since ms NC e.


A model for fuzzy sets I

In this case,

P = {1, ..., n} [0, 1]X , = T (1 n) with a continuous t-normT , = N with a strong negation N ( 6= 0).

Suppose P [0, 1]X = { [0, 1]X ; }.Normalized fuzzy sets are in [0, 1]X .

Consider Cons(P) = { [0, 1]X ; }. It is easy to prove that Consis a Tarskis operator of consequences:

P Cons(P)P1 P2 Cons(P1) Cons(P2). Cons is a monotonic operator.Cons(Cons(P)) = Cons(P).


A model for fuzzy sets II

Conj(P) = { [0, 1]X ; } :

Cons(P) Conj(P), since & , abs.Then, P Cons(P) Conj(P).P1 P2 Conj(P2) Conj(P1), since 2 & 2 1 &1 2 , abs. Conj is an anti-monotonic operator.


A model for fuzzy sets III

Hyp(P) = { [0, 1]X ;0 < < }

Hence, Hyp(P) Conj(P), since 0 < < & < ,abs.

P1 P2 Hyp(P2) Hyp(P1), since 2 1 &0 < <

2 0 < < 1. Hyp is an anti-monotonic operator.


A model for fuzzy sets IV

Since,Cons(P)Hyp(P) Conj(P) Conj(P) [Cons(P)Hyp(P)] = Sp(P),we get the partition

L = Ref (P) Conj(P) = Ref (P) Cons(P) Hyp(P) Sp(P).


A model for fuzzy sets V

Remarks:

Analogously to the case of ortholattices, neither Conj(P), norHyp(P), nor Sp(P), can be taken as bodies of knowledge, since theirresumees are easily empty.

The only needed for last definitions is1 The pointwise ordering ,2 The strong negation N, and the continuous t-norm T (for p)

In addition, it is

Conj(P) = {q L; q / Cons(P)},

showing that both hypotheses and speculations are defined fromconsequences.


A model for fuzzy sets VIEXAMPLE.

Lets consider a toy fuzzy system given by just a simple rule If , then .

Data: , [0, 1]X , [0, 1]X , the input [0, 1]X , the output [0, 1]X .

Remark: To work in the same universe of discourse of fuzzy sets, letschoose [0, 1]XX by defining

(x , y) = (x), x X , and each y X

(x , y) = (x), x X , and each y X


A model for fuzzy sets VII

Premises: P = {, }.

First Question: Is it Cons(P)?

If Modus Ponens is satisfied T : (x , y) = T ((x , y), ( )(x , y)) (x , y), x , y X

( (x , y) = T ((x), ( )(x , y)) (x), x , y X )

Cons(P).


A model for fuzzy sets VIII

Second Question: Is it the output by the CRI in Cons({})?

(x , y) = T ((x), ( )(x , y)) Sup

xXT ((x), ( )(x , y)) =

(y), the systems output Cons(P).

Take P1 = {}.Usually, it is not / Cons(P1).It is always = N Conj(P1), namely Sp(P1).That is,

The output is a logical consequence of the input and the rule

Once the rule is fixed, the output is only an speculation of the input.

Only in some rare cases the output is a hypothesis for the input.

Hence,Once the systems rules are fixed, fuzzy control is reached throughspeculations upon the inputs.Fuzzy control is a nice example of the guessing processes!


A model for fuzzy sets IX

As it is known, the reasonings made by people are deductive in, atmost, a 20%.

In conclusion, since fuzzy logic tries to modelize some treats of ordinaryreasoning and most of the reasonings people do are not deductive, it isimportant to extend the current fuzzy logic with the study of thespeculative and abductive modes of reasoning to complete fuzzy inferences.

In general, by considering conjectural reasoning or guessing. Somethingthat, in my view, is essential for dealing with CWW and Human-Centricsystems.


Language, fuzzy logic, and Wittgenstein. I

In my view, no static methodology is able to capture the most importantfeatures of language, and I think Ludwig Wittgenstein saw it clearly whenstating in Philosophical Investigations that language is nothing else thanplaying language games, where the upmost important criteria is to followfamily resemblances, that is, the property words do have of showing someair of family, and in identifying the meaning of words with its use inlanguage.

From its inception fuzzy logic dealt with natural language. For example,given a predicate (or a linguistic label) P on a universe of discourse X, amembership function P : X [0, 1] (provided it exists) is nothing elsethan an approximate description of the use of P in X .

In that direction, given P on X, and Q on Y, how is it possible to say thattheir uses, once described by P [0, 1]X and Q [0, 1]Y , show an air offamily? That is, how to define family resemblances and up to whichdegree P and Q do show an air of family?


Language, fuzzy logic, and Wittgenstein. II

I will show a possible definition in the case X and Y are bijective sets,that is, it exists a third set and two bijections,

1 : X , 2 : Y ,

allowing to pass P [0, 1]X and Q [0, 1]Y to P , Q [0, 1],through:

P(x) = P(11 (x)),

Q(y) = P(

12 (y)), x , y X

For instance, if X = [0, 10], Y = [0, 5], P [0, 1][0,10] can be translatedinto P [0, 1][0,5] by

P(x) = P(2x), x [0, 5],

with P(0) = P(0), P(2.5) = P(5),...,

P(5) = P(10), etc.


Language, fuzzy logic, and Wittgenstein. IIIWith this trick, lets define the relation of family resemblance FR forfuzzy sets in [0, 1]X

(, ) FR

if and only if

Z () Z () 6= , S() S() 6= , is non-decreasing in A X is non-decreasing in A, is decreasing in A X is decreasing in A,

where

Z () = {x X ;(x) = 0}, S() = {x X ;(x) = 0},

and decreasing is not understood in strict sense, that is, it allows forconstant piecer of .


Language, fuzzy logic, and Wittgenstein. IV

For instance,

(, ) FR


Language, fuzzy logic, and Wittgenstein. V

(, ) / FR


Language, fuzzy logic, and Wittgenstein. VI

(, ) FR


Language, fuzzy logic, and Wittgenstein. VII

It is easy to see that

(, ) FR, for all [0, 1]X . That is, FR is reflexive.(, ) FR (, ) FR, for all , [0, 1]X . That is, FR issymmetric.

(, ) FR & (, ) FR ; (, ) FR. That is, FR is nottransitive.

Since, FR is not an equivalence, the classes

[] = { [0, 1]X ; (, ) FR}, of the relatives of ,

do not give a partition of [0, 1]X .


Language, fuzzy logic, and Wittgenstein. VIII

It is also easy to prove that

(, ) = (,N ) / FR, N an strong negation in [0, 1].(, ) = (, ) / FR, a symmetry in X (opposite)If Q is a synonym of P, and provided Q = P u1:(P , Q) FR, with a convenient u : X Y .For any translation of keepingZ () Z () 6= , S() S() 6= :(, ) FR.


Language, fuzzy logic, and Wittgenstein. IXGraphically,

Figure: = 1 Figure: = (10 id)

Figure: (x) = (

x),u(x) = x2 Figure: translation


Language, fuzzy logic, and Wittgenstein. X

With X = [a, b] R, lets establish a degree up to which (, ) FR, doresemble, though a W -indistinguishability.

Ovchinnikov theorem (1984)

E : [0, 1]X [0, 1]X [0, 1]X is a prod-indistinguishability iff

there exist F = {f ; f : [0, 1]X R+0 } such that,

E (, ) = inff F

(f ()

f (),

f ()

f ()).

Since W prod , E is also a W -indistinguishability.

If F = {f1, f2}: E (, ) = min( f1()f1() ,f1()f1()

, f2()f2() ,f2()f2()

).


Language, fuzzy logic, and Wittgenstein. XI

Restrict to RF (X ) = { [0, 1]X ; is a Riemann-integrable}

Take, f1() = ba dx , f2() = | |, with the largest point in

[a, b] such that (x) > 0, and the smallest one, that means anavoidable restriction on the fuzzy sets. Then

Ovch(, ) = min(f1()

f1(),

f1()

f1(),

f2()

f2(),

f2()

f2())

Take I (, ) = 1 1ba ba |(x) (x)|dx , that is a

W -indistinguishability.

Usually, it is Ovch(, ) < I (, ). Take the degree of resemblance

= 1Ovch(,)+2I (,)1+2 , 1 + 2 = 1.


Language, fuzzy logic, and Wittgenstein. XII

All these degrees are reflexive and symmetric, but the only that isW -transitive is 1 = 2 = 1/2.

Take DR(, ) = Ovch(,)+I (,)2 , that is a W -indistinguishability, asthe Degree of Family Resemblance.


Examples. I

example 1)

(, ) frOvch(, ) = min( 54 ,

45 ,

89 ,

98 ) = 0.8

I (, ) = 1 110 |5 4| = 0.9

}DR(,)=0.85

Two uses of big in [0, 10]

example 2)

(, ) frOvch(, ) = min( 52 ,

25 ,

64 ,

46 ) = 0.4

I (, ) = 1 110 |5 2| = 0.7

}DR(,)=0.55 Two uses of aroundfive in [0, 10]


Examples. II

example 3)

(, ) frOvch(, ) = 0.88I (, ) = 1

}DR(,)=0.94

Two uses of big in [0, 10]

example 4)

(, ) frOvch(, ) = 0.8I (, ) = 1

}DR(,)=0.9

= is a use of around 5.5 = is a use of around five


Conclusion I

There are situations in which fr should be reconsidered. For example,and although I excluded the crisp sets, there are cases in which it ispossible to compute DR for {0, 1}X and F(X ).

For example,

= [4,6], the reds graphics

Ovch(, ) = 0.8I (, ) = 0.95

}DR(,)=0.875

There are cases in which the existence of prototypes is not clear. Forexample, if P = beautiful in a set of objects. In these cases, theexistence of family resemblance also deserves to be reconsidered.


Conclusion II

Hence,

1 DR does not seem too bad!

2 (, ) fr show HIGH FAMILY RESEMBLANCE whenDR(, ) 0.8 (perhaps a too big number)

3 There are Aggregation Functions different from x+y2 to aggregateOvch and I.

For example, DR(, ) = 3Ovch(,)+2I (,)5 gives, in some cases, moreaccurate values.

4 It lacks to aggregate DR and I, even without means !


Conclusion III

In natural language, the families [], of the relatives of , must besomething open, but not closed as they were defined.

Here these families are static (sets), but in NL they are dynamical.With time, an element that was not in [], but possibly with a not toolow degree DR(, ), could be included in [] and thus generate a newfamily, indeed, changing the relation FR.

Actually, in natural language FR is not permanent but changing, evolving.Only with families of relatives taken either as crisp sets, human thoughtdoes not seem possible.

What we presented today is only a tentative for continuing toreflect on the potentially interesting subject of family resemblancesfor CWW.


To end I

I referred here to some of the authors that influenced me, Menger,Wittgenstein, Zadeh, etc., but I said nothing on some others, for instance,Poincare, Tarski, Lukasiewicz, or Max Black who, like Zadeh was born inBaku, and whose writings were actually important for my acceptance ofZadehs ideas. Let me end with some return to Black.

What the so-called Sorites Paradox shows is that imprecisepredicates, those naming gradable properties, do not alwayscollectivize in the form of classical sets.

Black posed the question of the existence of a separation point, that is, forinstance, the number of grains for which a heap is no longer a heap ofgrains, and concluded that such a number cannot be determined.The problem is analogous to determine a number N of years for which isnot reasonable to say that a person whose age is greater than N could bequalified as young. It is a defuzzification.


To end II

It is a signal of the existing divorce between philosophers and scientiststhat in the big amount of papers the first wrote on Sorites, no onedescribes which is the meaning he/she attributes to the word heap, or, ifit were the case to the word young.

I think that such lack is due to the fact that the only way of describing themeaning of an imprecise word is by means of the fuzzy sets the use ofwords do generate, and that philosophers either dont know fuzzy sets, ordont believe in them.

In my view Sorites is not a paradox, but just a proof that there areconcepts not representable by crisp sets. What follows and inWittgensteins terms, dissolves it.


To end III

In the usual case, the Blacks separation point can be determined, once itis known a fuzzy set P [0, 1][a,b] for P, and a strong negation N forrepresenting not P by N P.

If P is non-decreasing, the number N = sup{x [a, b]; P(x) N(P(x))}is the separation point.

For instance, if [a, b] = [0, 100], P = old is represented by the fuzzy setP(x) = x100 , and not P is obtained by the strong negation N =

1id1+id , it is

P(x) = x100 N(P(x)) =1 x

1001+ x

100, or

(x 100 + 100

2)(x + 100 + 100

2) 0,

and N = 100(

2 1) = 41.42. Hence, under this conditions, after apersons age surpasses 41.42 years, this person can be properly startto be qualified as old.


To end IVAnother way of obtaining the separation point is through anantonym of P.

With P = old , and its antonym (aP = young),aP(x) = P(100 x) = 1 x100 , it is

x

100 1 x

100, or x 50,

and then when a persons age surpasses 50 years, this person canstarted to be properly qualified as old.

With all that it could be taken, for instance, as a good separation pointone of the following numbers

141.42+2501+2

(for example, 41.42+502 = 45.75)

min(41.42, 50) = 41.42

max(41.42, 50) = 50.


Albert Einstein: The more important thing is not to stopquestioning.

Nobel Laureate Isaac Rabis Mother: Did you ask today goodquestions, Isaac? (Each day, when returning School)

Thank you very much!


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