Characterization of interval fuzzy logic systems of connectives by group transformations
In commonsense and expert reasoning, we often use informal statements like ``zebra is a horse-like creature''. In contrast to traditional logic, these statements are true only to some degree; so, to formalize reasoning with such statements, it is desirable to describe these ``degrees of truth'' in precise terms. This precise description is one of the main reasons why fuzzy logic was invented in the first place. In fuzzy logic, the corresponding degrees of truth are usually represented either by values from the interval \([0,1]\) or by subintervals of this interval. There exist many different ways of transforming the expert's uncertainty into a precise number (or an interval); most of these methods are rather ad hoc. The authors have developed a technique called checklist paradigm that, crudely speaking, works as follows: we select a list of properties that describe a horse, and then for each object \(x\) we define its degree of horse-ness as \(d(x)=m(x)/n\), where \(n\) is the total number of properties defining a horse, and \(m(x)\) is the number of properties that are satisfied for the given object \(x\). In this formalism, to describe logical connectives, we must consider statements like ``both \(x\) and \(y\) are horse-like''. A natural checklist paradigm interpretation of the degree of truth of this statement is \(d(x,y)=m(x,y)/n\), where \(m(x,y)\) is the total number of properties that both \(x\) and \(y\) satisfy. The actual value of \(d(x,y)\) depends not only on \(d(x)\) and \(d(y)\), it also depends on the ``fine structure'': if both \(x\) and \(y\) satisfy exactly the same properties, then the value \(d(x,y)\) is larger, otherwise it is smaller. It is possible, for any given \(d(x)\) and \(d(y)\), to describe the exact bounds for \(d(x,y)\); these bounds correspond to what the authors call ``bottom'' and ``top'' operations. An arbitrary value of \(d(x,y)\) belongs to the interval formed by the results of these bottom and top operations. We can repeat this procedure for other logical operations, and get several bottom and top binary operations. The analysis of the resulting large set of possible binary operations can be simplified if we take into consideration that some of these operations can be transformed into each other by transformations from the following 8-element group: the identity operator \(f(x,y)\to f(x,y)\), duality \(\neg f(\neg x,\neg y)\) (that maps ``and'' into ``or''), \(\neg f(x,y)\), \(f(\neg x,\neg y)\), \(f(x,\neg y)\), \(f(\neg x,y)\), \(\neg f(\neg x,y)\), and \(\neg f(x,\neg y)\).
scientific article; zbMATH DE number 2081525