It Is Possible to Determine Exact Fuzzy Values Based on an Ordering of Interval-Valued Fuzzy Degrees

In the usual [0, 1]-based fuzzy logic, the actual numerical value of a fuzzy degree can be different depending on a scale, what is important – and scale-independent – is the order between different values. To make a description of fuzziness more adequate, it is reasonable to consider interval-valued degrees instead of numerical ones. Here also, what is most important is the order between the degrees. If we have only order between the intervals, can we, based on this order, reconstruct the original numerical values – i.e., the degenerate intervals? In this paper, we show that such a reconstruction is indeed possible, moreover, that it is possible under three different definitions of order between numerical values. I. FORMULATION OF THE PROBLEM Need for probabilities and need for fuzzy degrees. To describe how frequently different events occur, a natural idea is to use probabilities – i.e., in effect, frequencies with which this event has occurred. For example, if under certain weather conditions, in the past, rain happens in 30% of the cases, we say that under these conditions, the probability of rain is 30%. In general, if out of n cases, the event of interest happened in m of them, we say that the probability of the event is equal to m/n. Strictly speaking, this frequency is only an approximation to the actual probability; the larger the sample size n, the more accurate this approximation. So, if we want a more accurate estimate for the probability, we need to increase n, i.e., to consider a larger sample. In addition to this objective probability, there is also subjective uncertainty: experts are not 100% sure about their statements. For example, an expert may say that the probability of rain under certain conditions is small, without providing a precise numerical description of what this word means. To process this subjective uncertainty, we need to describe in computer-understandable terns, i.e., in terms of numbers. This description is one of the main objectives of fuzzy logic; see, e.g., [1], [4], [5]. To describe an imprecise (“fuzzy”) word like “small” in precise terms, we can, e.g., for each possible value x of the corresponding quantity (in the above example, for each possible probability value) to estimate, on a given scale, how certain this expert is that x is small. If the expert marked his or her degree of certainty as 7 on a scale from 0 to 10, we can then say that the expert’s degree of certainty is 7/10. In general, if an expert marked m on a scale from 0 to n, we can use the value m/n. An important difference between probabilities and fuzzy degrees. In both cases of probabilistic and fuzzy uncertainty, we have the same formula m/n for estimating the corresponding degree. However, there is a big difference between these values. Probabilities are objective. If two people use the same data, they will get the same probability value. In contrast, expert opinions are subjective. Based on the same evidence, and based on the same understanding of what is more probable and what is less probable, some experts will be more “optimistic” and mostly use values close to 10 on a scale from 0 to 10, while other experts may be more “pessimistic” and mostly use values close to 0 on the same scale. In this case, the values m/n corresponding to one expert can be transformed into values m′/n corresponding to another expert by an appropriate non-linear re-scaling. As a result, in fuzzy logic, the actual numerical value of a fuzzy degree can be different depending on a scale. What is important – and scale-independent – is the order between different values. Need for interval-valued fuzzy logic. Fuzzy logic deals with situations in which an expert uses imprecise words like “small” to describe his or her opinion. The expert uses imprecise words because he or she is unable to come up with an exact estimate. On the other hand, the traditional [0, 1]-based fuzzy logic requires the same expert to come up with an exact value of a scale from 0 to 10 that describes this expert’s degree of certainty. Of course, in practice, the expert is often unable to do it. To be more precise, the expect may be able to confidently say that his.her degree of certainty is 7 and not 6 and not 8, but if we try to get a more accurate description by taking a scale form, say, 1 to 100, it is doubtful that the expert will be able to mark his/her degree of confidence as 71 and not 70 or 72. At best, the expert would be able to mark a whole interval of possible values – e.g., from 65 to 75 – as describing his/her degree of certainty. This corresponds to the interval [0.65, 0.75] of possible degree. Such interval-valued fuzzy techniques have indeed been proposed. They are indeed more adequate in describing expert’s uncertainty, and they have led to many practical applications; see, e.g., [2], [3]. Interval values generalize the usual fuzzy logic: each degree a ∈ [0, 1] from the original fuzzy logic can also be viewed as a “degenerate” interval [a, a] in the interval-valued fuzzy scheme – but, of course, in interval-valued approach, we have additional degrees [a, b] with a < b. Formulation of the problem. In the interval-valued case, also re-scalings are possible. As a result, what is most important is the numerical values, but rather the order between the degrees. If we have only order between the intervals, can we, based on this order, reconstruct the original numerical values – i.e., the degenerate intervals? What we do in this paper. In this paper, we show that such a reconstruction is indeed possible, moreover, that it is possible under three different definitions of order between numerical values. II. FIRST ORDERING: LATTICE (COMPONENT-WISE) ORDER Component-wise order between intervals: a brief reminder. If for some statement, the expert’s degree of confidence is represented by an interval [a, b], and then we increase the lower bound, to make the interval [a′, b] with a′ > a, we thus increase our degree of confidence in this statement. Similarly, if we increase b to b′ > b, we thus increase our degree of confidence. From this viewpoint, it makes sense to say that the interval [a′, b′] represents a larger )or same) degree of confidence than the interval [a, b] if a′ ≥ a and b′ > b: [a, b] ≤ [a′, b′] ⇔ (a ≤ a′ & b ≤ b′). Formulation of the problem in precise terms. Suppose that on the set of all subintervals [a, b] of the interval [0, 1], we have the above ordering. Based on this ordering, can we uniquely determine degenerate intervals, i.e., intervals of the type [a, a]? In this section, we will answer that this is indeed possible. This determination will be done step by step. First step: it is possible to define the interval [0, 0] based only on the order. Indeed, [0, 0] is the only interval which is smaller (in the sense of the above relation ≤) than any other interval. In precise terms, the interval [0, 0] is the only interval I that satisfies the property