Let us consider the case of the “heap” to understand the idea of vagueness in fuzzy logic. What is a heap? What do we mean when we are talking about a heap of dresses, for example? Assume that we have a heap of clothes and we begin to put them into a wardrobe by taking each item of clothing separately out of the heap. We take first the socks, then the shirts, and so on. In doing this, we observe that the heap disappears slowly. Finally, when we take the last of the items of clothing, there is nothing left. So, at what stage of this activity does the heap disappear? Does the heap disappear when we have taken away half of the clothes, or one third of them, or all of them? These questions are very important in terms of the logical characterization of this case. Classical logic, which depends on two truth values, namely true-and-false, does not answer these questions because there is no clear boundary between being a heap of clothes and not being a heap of clothes in this case. If we consider the constantly changing flow of events in the universe, we face similar cases all the time. Another outstanding example is the spectrum of light. There are no clear boundaries between colors. Where exactly is the boundary between yellow and light yellow? The indeterminacy of boundary can be observed in social area as well. Political ideologies range from the most radical right wing to the most radical left wing, and we can observe many different variations of ideologies in this range. For example, sometimes we may have difficulty in identifying the position of a particular political movement if it is between moderate right wing and moderate left one.
This is a special case of indefiniteness which we call “fuzziness” or “vagueness,” but there are different types of indefiniteness. For instance, when we use the term “bank” unqualifiedly, we can either mean a financial institution or a place to sit. We call such cases “ambiguous.” Whether it will rain next week or not is another indefinite case related to our ignorance of future.
The indefiniteness, which we deal with in fuzzy logic, is vagueness. The problem of vagueness is as old as humanity. It is discussed under the name of “sorite paradoxes”in ancient Greek and under the term “fuzzy logic” today.
Fuzzy logic proposes a systematic model to solve problems in vague cases. This model depends upon mathematical symbolization. It seems that fuzzy logic presents a better way to deal with vague cases than the way suggested by classical logic. Let us see why this is so.
Classical two-valued logic
Classical logic has two truth values, namely true and false. Both Aristotelian logic and symbolic logic have two truth values even though there are important differences between them. For that reason, both of them are treated under the name of “classical logic.” Aristotelian logic analyzes sentences by dividing them into subject and predicate. Consider the following sentence: Tom played tennis with John. “Tom” is the subject in this sentence and “played tennis with John” is the predicate. On the other hand, in symbolic logic, this sentence is analyzed in a different way. According to symbolic logic, there are two subjects in this sentence, namely Tom and John. The predicate of the sentence is a relation between these two subjects, namely “played football.” Furthermore, since Aristotelian logic is very close to metaphysics, it has a very strong existential import. For instance, if we assert the sentence “All dogs bark,” we also imply that there are dogs, according to Aristotelian logic. However, there is no such existential import in symbolic logic. In the view of the latter, we cannot say anything about the existence of dogs when we assert the above-mentioned sentence. What we mean by that sentence is that if something is a dog, then it barks. Nothing more, nothing less.
Although there are some differences between Aristotelian and symbolic logic, both of them are two-valued logics. However, we cannot call classical logic the logic of “white-and-black” because it has two values. This is a common mistake stemming from not appreciating exactly to what these two values refer. White-black duality ignores the grey tones between white and black. However, classical logic covers all tones between white and black. The duality here is between whiteness and non-whiteness or between blackness and non-blackness. So non-whiteness encompasses the grey tones.
The problem in classical logic is not exclusion of the grey tones but its insufficiency in representing them adequately. We have a problem in determining the boundary between whiteness and non-whiteness, and this is the problem of vagueness. Therefore, classical logic is inapplicable to vague cases, but clear cases lie in the area to which classical logic is applicable. Think of mathematics. A number is either even or odd, and this is clear for any number in question. There is a clear boundary between odd and even numbers. In comparison to classical logic, fuzzy logic claims that it has wider range of applicability. Let us examine fuzzy logic in detail.
There is a discussion as to whether a logical system for vague concepts can be established or not. Gottlob Frege, one of the founders of symbolic logic, thought that logic cannot be applicable to fuzzy concepts and restricted logic to the area including clear concepts. The fundamental problem in vague concepts is determining the boundary between a concept and its negation. Assume that this problem cannot be solved by a two-valued logical system. Does it mean that the problem in question cannot be solved by three- or many-valued logical systems either?
Now, let us examine a three-valued logical system and see whether it solves the problem or not. In 1920, a Polish logician, Lukasiewitcz, presented a three-valued logical system whose values are true, false and indeterminate.
Assume that “This object is white” is true, “This object is black” is false and “This object is grey” is indeterminate in this logical system. Does presenting the grey area as indeterminate solve the problem of vagueness or make it more complex? In fact, the problem becomes more complex. The complexity arises from the question as to whether there is a certain, clear boundary between white and grey or between black and grey. Again here, we cannot claim that there is such a clear boundary. Vagueness is observed not only between black and non-black but also between black and grey. The problem of vagueness does not disappear if you increase the number of truth values in a logical system. For instance, postulating twenty truth values, instead of three, does not bring a solution to this problem. However, if you establish a logical system that includes infinitely many truth values, then the problem of vagueness seems to disappear. In that case, all colors with their tones can be analyzed by a scale of real numbers between 1 and 0. The infinitely many real numbers in this scale correspond to a truth value and so we can postulate a truth value for every tone of a color. For instance, “1” refers to pure blackness, “0” refers to pure whiteness, “0.5” refers to whiteness and blackness, “0.2” refers to a color whose whiteness overweighs its blackness. Since there are infinitely many truth values, you can give a truth value for any place in the light spectrum. In this way, we can get rid of determining clear boundaries.
The first serious study on fuzzy logic was carried out by Lotfi Askerzade Zadeh in 1965. He published an article called “Fuzzy Sets” when he was a professor of electrical engineering in the USA. Fuzzy logic was then improved by Zadeh, Joseph Gougen and many other scholars.
Fuzzy logic has a wide area of applicability in technology today ranging from washing and dishwasher machines to computers. Since it brings the advantage of flexibility, it increases productivity. Fuzzy logic is also used in the research program of artificial intelligence and modeling the human mind in computers.
If we come back to the internal structure of fuzzy logic, we observe that some of the fundamental principles of classical logic, such as the principle of non-contradiction and the law of the excluded middle, are not valid in fuzzy logic. For instance, if the proposition that this man is bald is true to a degree of 0.3, we say that this man is neither completely bald nor completely non-bald. The position of this man is the intersection point of baldness and non-baldness to a degree of 0.3. This man belongs to the set of baldness to a degree of 0.3 and to the set of non-baldness to a degree of 0.7.
An Islamic evaluation of vague cases
Said Nursi was one of the outstanding Muslim scholars of the twentieth century. He considers vague cases which give rise to relative truths to be important in terms of understanding divine activity in nature and knowing God’s Names and Attributes:
. . . God, the Eternal All-Wise and as the requirement of His Eternal Grace and Wisdom, has created this world as a testing arena, a mirror to reflect His Beautiful Names, a vast page upon which to write with the Pen of His Destiny and Power. People are tested here to develop their potentialities and to manifest their abilities. This emergence of abilities causes relative truths to appear in the universe, which, in turn, causes the Majestic Maker’s Beautiful Names to manifest their inscriptions and make the universe a missive of the Eternally-Besought-of-All.1
So, relative truths play an important role in knowing God. They are relative because they do not correspond to clear and fixed cases. As we remember from the analysis of fuzzy logic, these truth values are between 1 and 0. All values between 1 and 0 are relative because the case represented with such values belongs to two different sets with different values and the value depends on the perspective from which you are looking at it. If the truth value of “This man is bald” is 0.3, then the truth value of “This man is not bald” is 0.7.
According to Nursi, God gives rise to vague cases by mixing opposites. He mixes goodness with badness, beauty with ugliness, and so on. By this mixture, there occur infinitely many levels of beauty, goodness, and other qualities.
On the other hand, there is a famous saying of the Prophet Muhammad, peace be upon him, which suggests a criterion for human actions in fuzzy cases: “What is permitted is clear, what is forbidden is clear as well. There are doubtful cases between these two cases. Whoever avoids the doubtful things, his religion remains clean. Whoever approaches the doubtful cases, he approaches the forbidden cases as well.” Prophet Muhammad points out the vague cases between forbidden and unforbidden cases. In Islamic jurisprudence, scholars suggested many categories to represent these vague cases. Some cases are close to the forbidden ones, and some are not. They named each case separately. Fuzzy logic can be used in this regard to represent these different levels better than classical logic’s representation. Think of the following case. Grape juice is permitted in Islam. However, wine, which is prepared from grapes, is forbidden. More interestingly, there is a stage of fruit juice which is permitted but very close to the forbidden case. This stage is called “must” (unfermented grape-juice). After this stage, it gradually becomes closer to the forbidden case and we cannot exactly determine where the boundary between permitted and forbidden cases is. The Hadith emphasizes this point and suggests that we avoid such cases.
Fuzzy logic represents a wider area of cases than classical logic does. By representing vague cases in a symbolic and systematic way, fuzzy logic functions better than classical logic. However, fuzzy logic is not the ultimate system of thought which has been established by human mind. We should not exaggerate its importance by claiming that it can represent any case no matter what it is. For instance, fuzzy logic is inadequate to represent the logical relations between concepts including the notion of infinity. To deal with cases of infinity in mathematics, a different type of logic is proposed, namely intuitionistic logic. All of these systems show that the human mind is powerless or unable to grasp or establish a system which represents all cases and any case no matter what it is.
1. Nursi, Bediuzzaman Said, The Words, Twenty-ninth Word, The Light, Inc. NJ: 2005, p. 546.
Williamson, Timothy. Vagueness. New York: Routledge. 1994.
Mukaidono, Musao. Fuzzy Logic for Beginners, New Jersey, World Scientific, 2001.