28

A Beginner’s View on Fuzzy Logic

Itziar García-Honrado

28.1 First Touch with Fuzzy Logic

The first time I heard the term fuzzy logic goes together with the term fuzzy set [14].It was in the last year of my degree of Mathematics in the subject of Statistics, andmy partners and me have to complete an opinion pull answering through fuzzy sets.I heard a brief explication of what a fuzzy set is and I wrongly related it with adistribution function.

Immediately after my degree I entered the European Centre for Soft Computingto work on my PhD Dissertation under the advice of Professor Enric Trillas. Duringthis period I met several times Professor Lofti Zadeh, and I was also acquainted withtextbooks in Fuzzy Logic like [2] and [3], as well as with Trillas’ work on which Ifinally did my Dissertation.

Nowadays, I know that a fuzzy set is not related at all with probability. In fact,if we consider that a fuzzy set (p1 ∈ [0,1]X ) is a probability distribution, we realizewe can not compare these fuzzy sets under the typical pointwise ordering (p1 ≤ p2 ifand only if p1(x)≤ p2(x), ∀x∈ X), since two fuzzy sets representing two probabilitydistributions are identical or not comparable [12]. It is enough to follow the chain,p1≤ p2→ p1(a)≤ p2(a), and p1(a′)≤ p2(a′) → 1− p1(a)≤ 1− p2(a) and p1(a)≤p2(a) → p1(a) = p2(a), that is, functions p1 and p2 coincide.

Even more, probability is used to represent uncertainty and fuzzy sets impreci-sion. Both terms model unknown aspects, but under my view they model two kindsof lack of knowledge. Uncertainty is the one that goes together with an event beforeits realization and it is based on some background knowledge on the event, usuallyinformation on the results from the experience of previous realizations, and after therealization of the experience the uncertainty disappears. In the case of uncertainty,probability is devoted to model those situations. On the other hand, imprecision,under my view, is not only a matter of modelling experiences, or physical phenom-ena, it is mainly related to the human linguistic way of describing real situations.For instance, when a word is stated in a concrete language all people knowing thatlanguage understands its meaning. Following that idea a model for the meaning ofpredicates can be shown by using fuzzy sets.

Fuzzy logic is different from probability [16]. In fact, for defining a probabilityit is necessary a boolean structure and this is not the case of fuzzy logic: predicatesact on non necessarily structured universes of discourse. So, fuzzy logic is not thesame, and under my view it is not a kind of extension of the model of probability

R. Seising et al. (Eds.): On Fuzziness: Volume 1, STUDFUZZ 298, pp. 185–191.DOI: 10.1007/978-3-642-35641-4_28 © Springer-Verlag Berlin Heidelberg 2013

186 28 A Beginner’s View on Fuzzy Logic

for representing unknown events. Currently, no definitive theory of probabilities for‘imprecise events’ does exists [12].

Notwithstanding, there could be a relation between probabilities and fuzzy logic,but, which one? Models combining Fuzzy Logic and Probability. Although there aresome models of probability of fuzzy sets [15], [17], [12], considering fuzzy eventsor fuzzy representation of the events, a lot of studies in that field are yet necessary.

To conclude this section, I would like to remark the necessity of distinguishinguncertainty from imprecision in order to built different models depending on thephenomena.

28.2 Fuzzy Sets for Representing Predicates in the Language

Currently, the evolution of fuzzy logic arrives into the field of Computing WithWords [18]. Imprecision is in the language, and a possible way of representinggradable predicates is through fuzzy sets capable of building a model collecting theirimprecision [7].

In fact, Wittgenstein considered that “The meaning of a predicate is its use inlanguage" [13], and in order to built a model for a meaning of a predicate, followingthat definition, its use is analyzed, and is collected by a fuzzy set as it is shown inthis section.

When stating a predicate (P) in a universe of discourse (X), it is established a rela-tion, that allows to compare the elements in the universe, supposing that the relationis transitive and reflexive (mathematically it is a preorder), ≤P. If the predicate isP = tall, we can say that “the element x in the universe of discourse is less tall thanthe element y", and represent that fact by x≤P y. So, the relation≤P is known as theprimary use or meaning of the predicate in the universe of discourse.

Once a structure (X ,≤P) is established, an important question lies in how to mea-sure up to which extent x is P, allocating a value in the unit interval through a fuzzyset defined on the universe of discourse into the unit interval, μP : X → [0,1], andverifying that μP(x) ≤ μP(y) if x ≤P y. In the particular case that μP(x) ≤ μP(y) ifand only if x ≤P y, it is built an accurate representation, and it is said that μP per-fectly collects the meaning of P. Not always it is possible to have that sufficient andnecessary condition.

It is usual to have non-comparable elements under a predicate in X , but the fuzzyset forces them to be compared because they allocate a degree between 0 and 1 tomeasure up to which extent x is P, and the unit interval has a total order. In orderto avoid this problem, it could be enlarged the definition of fuzzy sets into an L-set,μ : X → L, where (L,≤) is any preorder. It is proven in [1], that there always exista preorder (L,≤), perfectly collecting the meaning of each P, and it is known as thenatural preorder associated to the predicate.

It is important to remark the importance of a careful design of fuzzy sets [10],or the L-set collecting the meaning of the predicate, it should be taken into accountthe population, the context, the purpose, the use,... So, depending on those variables

28.3 Ideas Related with Fuzzy Logic 187

different models of predicates could be built, that is, different fuzzy sets or L-sets,maybe analyzing the similarities keeping the idea of language games of Wittgenstein,as it is done in [11].

Here we show two possible representations of the predicate tall in two differentcontexts, a basketball team (μA1 ) and a school (μA2).

Fig. 28.1. Tall in a basketball team. Fig. 28.2. Tall in a school.

It is easy to check in a simple example how a fuzzy set representing a predicatecould vary depending on the population (Spanish people, a set of pygmy), the context(schools, adult population), the purpose (exaggerating what is considered tall),...

As well as a fuzzy set could represent a predicate, it also represents the collectivegenerated by the predicate. For instance, in a big horse’s farm, the predicate ‘short’allows to talk of the collective of short horses in the farm.

Therefore, fuzzy sets could be seen as ways of representing imprecise concepts,which is, under my view, the main idea of that Logic. Additionally, some light isshed on the problem of what can characterize a function μ : X → [0,1] to actuallyrepresent a fuzzy set.

28.3 Ideas Related with Fuzzy Logic

Fuzzy Logic collects, as a degenerated case, classical logic. In the jump betweenclassical to Fuzzy Logic, many classical principles supposed “always" true, fail, be-cause the flexibility of Fuzzy Logic breaks the rigid structure of Boolean algebras.This fact could seem to question the validity of Fuzzy Logic.

For instance, the flexibility of Fuzzy Logic allows elements to verify both a pred-icate and its negation, as it happens in real situations (“He is neither tall, nor nottall"). That idea represents the falsation of Non-contradiction (NC) principle in itsclassical form. Notwithstanding, going back into the Aristotelian Principle of NC,it is obtained a new interpretation under which Fuzzy Logic is not in contradictionwith this classical principle [6]. Indeed, all principles that were considered “always"valid could depend on its interpretation.

In the field of Fuzzy Logic, the possibility of obtaining new interpretations growsbecause Fuzzy Logic is not defined in strong structures, and depending of the chosenstructure for each model many properties could change, but this is not the case in

188 28 A Beginner’s View on Fuzzy Logic

classical Logic. This is, the positive side of Fuzzy Logic, but also it is necessarymuch more careful design when building models.

28.4 Models Following Those Ideas

Under my view, Fuzzy Logic allows us to build models closer to human activities,as it is mentioned in section 28.2. It allows us to represent, at least, a small part oflanguage, non ambiguous precise and imprecise predicates. That is, because humanbeings deal with imprecision, it appears in the language, and in others activities, infact, very often the transmission of knowledge is done through language, so it shouldnecessarily contain imprecise terms.

Anyway, although imprecision is an important characteristic of language, thereare other variables that are not included in the field of Fuzzy Logic, at least in thefield as it is understood nowadays. But, in any case, Fuzzy Logic opens a door thatallows us thinking in possible mathematical models more flexible and related withthe real world.

For instance, the meaning of the particle and in the language can not be modeledby the common representation of intersection in Fuzzy Logic, since in that case it ismandatory to be commutative and when times appears, the particle and in languageis not commutative (“He arrived and he left, or He left and he arrived”).

Therefore, the idea of collecting imprecision through Fuzzy Logic could allow usto build models translating, in some particular cases (predicates or collectives) theway of speaking. And maybe, in the future, the ways of thinking through Common-sense Reasoning (or Conjectural [4]) Models using fuzzy sets [9], [8]. These modelscontain the deductive models of reasoning, but they also include new ways of obtain-ing conclusions from a piece of information, for instance, non-monotonic reasoning,when if there is new information, the number of conclusions decrease. It is importantto notice that those models contain deduction since again it appears the idea of fuzzylogic with respect to the classical one, in the sense that it contains classical one butbreaking the boundaries and its inflexibility.

With fuzzy sets a small part of the language can be represented, but this part isindeed larger than the one representable with classical sets.

28.5 The Importance of Fuzzy Logic

After only few years working in the field of Fuzzy Logic, I consider very importantits development towards an experimental science. Since sciences usually requireconstruction of models of the real world, and we are immersed in it and describe it bymeans of imprecise terms, indeed, only artificial models of reality are precise. Insidethe concept of the real world, there are people which are also rounded of imprecision,nobody could define himself, his feeling, ways of reasoning, speaking,... into crispterms. The main characteristics of human beings are variability and originality, andwithout imprecision they will disappear.

28.6 The Future 189

So, instead of hiding this logic, I think it would be crucial to show its potential.And, as a previous stage, to show that Maths, although with a strong logic structure,allows to model imprecision. As Zadeh uses to say ‘Fuzzy Logic is not fuzzy’.

28.6 The Future

Following with the ideas shown in these lines and related with fuzzy logic, I proposeto further develop models of Language and models of Commonsense Reasoning.

Regarding the topic of language, fuzzy logic allows to represent predicates, alsoconnectives, conditional sentences [5], but it seems that fuzzy logic is not enoughto represent any piece of language. Collecting the meaning of language into mathe-matical expressions is a very long path, but in the future fuzzy logic can allow newcontributions of some interest for this purpose, such as, analyzing other particularwords different to predicates or collectives, analyzing the whole meaning of a sen-tence not based on a union of words, etc.

Relating with Commonsense Reasoning models, there are not many results whenthese models deal with fuzzy information. Therefore, as in the way of modelling hu-man ways of reasoning imprecision has to appear, this is why there is the necessityof improving these models in structures where fuzzy sets could exist, the strongeststructure of fuzzy set is a De Morgan Algebra where the classical form of the prin-ciple of Non-Contradiction is not verified. So, it is necessary to model reasonings instructures with fewer number of laws.

To conclude, I want to express the duty of showing the importance of Fuzzy Logicto approach reality through mathematical models. This idea could be interesting to betransmitted in different stages of education, not only in the graduate and postgraduatestudies. Nowadays, I have the opportunity of dealing with students that want to beteachers of primary school, and I think they should know the existence of this logic,in order to enlarge their vision of Mathematics, and not just teach mathematics tryingto built reasonings with their students closer to reasonings coming from classicallogic and deduction, but also taking into account, for example, real systems describedin Natural Language. Approaching maths in ways closer to the current world.

Acknowledgement. This work has been supported by the Foundation for the Ad-vancement of Soft Computing (ECSC) (Asturias, Spain), and by the Spanish Depart-ment of Science and Innovation (MICINN) under project TIN2011-29827-C02-01.

190 References

Fig. 28.3. Ana Belén Ramos, Ángela Blanco, Lotfi A. Zadeh and Itziar García-Honrado atNovember 2008 in Avilés (Spain) at the ceremony of Cajastur International Prize on SoftComputing

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