Enric Trillas - Fuzzy Logic

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Springer International Publishing Switzerland 2015

Enric Trillas and Luka Eciolaza Fuzzy Logic Studies in Fuzziness and Soft Computing 10.1007/978-3-319-14203-6_1

1. On the Roots of Fuzzy Sets

Enric Trillas 1

(1)

European Centre for Soft Computing, Mieres, Asturias, Spain

(2)

European Centre for Soft Computing, Calle Gonzalo Gutirrez Quirs S/N, Mieres, Asturias, Spain

Enric Trillas (Corresponding author)

Email:

Luka Eciolaza

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As fuzzy sets were introduced by Zadeh in 1965, they were born closely linked with imprecise predicates, that is, with names of non-precisely defined classes of objects. Even more, most of the applications of Zadehs ideas are made with properties the objects do verify in some degree between the two classical extremes 0 and 1, respectively. Because of that, it is not at all odd to introduce fuzzy sets from some considerations on how predicates are used in language. We will follow Wittgensteins statement The meaning of a word is its use in language.

1.1 A Genesis of Fuzzy Sets

Usually, isolated words mean nothing. To mean something, words are to be used in a given context and in a known way. Words do serve to describe perceptions, to translate reasoning, and to show the reasons for judgements.

For example, what it is meant by the predicate unleaty ? It is impossible to answer this question, since nobody has never heard something like this is unleaty, such is leaty, etc. Neither unleaty, nor leaty, are English words, nobody has used them and they dont appear in English dictionaries. Meaning is inherited by predicates only after being used, in some ground, by means of elemental statements is . By now, unleaty has no meaning.

Words are introduced in a language by using them in concrete ways. For example, the Spanish word madre comes from the latin mater that, at its turn, came from one in older indo-european languages, but always used to name someones mother. Later on, the word could take, by analogy, new meanings as it is, for example, mother country, goodmother, mother in law, mother of all wars, mother Nature, mother-of-pearl, etc. The predicate leaty neither appears in English, nor in Spanish, French, German, ..., because it was never before used to name a property of the elements in some class, like it is with tall, young, middle-aged , with people, high with buildings or mountains, or heavy with metals, for example. It is only through its use that predicates do acquire meaning.

What is it meant by the predicate odd? In principle, it depends on where it is used. With natural numbers , is odd is used accordingly with the mathematical definition/rule is odd if and only if once divided by 2 the rest is 1. With people, things, or situations, odd does coincide with the meaning of strange, separated, and not often. In the first context, the predicate odd is precise or crisp, since natural numbers are only odd or not odd, but in other contexts the predicate is imprecise or fuzzy, since for example, this is an odd book, or that is an odd event, could admit degrees depending on to what extent the book or the event can be qualified as odd. When the predicate is imprecise there is not a perfect classification of the objects to which it refers to.

Only after a predicate acquires meaning, concepts like tallness, oddity, heaviness, etc., appear in the corresponding language. Predicates do appear in language after being used, and only after being used they can evolve in new contexts and give birth to concepts. Like it happened with high and highness, and royal highness. Concepts like uncertainty come from a mother-predicate as it is uncertain in this case.

Notice that there are not natural numbers that can be qualified as very odd, or more or less odd, but there are buildings that can be called to be so. If the predicate is crisp, that is, it names just an either yes, or not, property, the application to it of a linguistic modifier needs to be newly defined, but if the predicate is imprecise it does not, since people immediately understand what it is meant by, for instance, very odd.

In what follows we will only deal with predicates , the name of a property on a previously given set , and considering the use of through the elemental statements is , for all in , and accepting ( la Wittgenstein) that the meaning of is its use in the current language. Hence, the first problem to tackle is placed by the following question, How the use , or meaning, of on can be mathematically represented or modeled ?

First, and to distinguish between two statements is and is , for in , let us suppose it is possible to decide when is is less than , where shows the property named less than shows it.

Let us call

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