Finitely generated sets of fuzzy values: if ``and'' is exact, then ``or'' is almost always approximate, and vice versa -- a theorem
This chapter deals with the problem of evaluating the compound sentences (in particular, built with conjunction, disjunction and negation) within fuzzy logic. The authors remind the reader that evaluations in traditional fuzzy logic represent degrees of confidence, which are indicated by numbers from the interval \([0, 1]\). Now, taking into account that there can only be countably many such degrees, the set \(S\) of the \textit{actual values}, which can be ascribed to sentences, should be a subset of the interval \([0, 1]\). Clearly, if conjunction and disjunction are defined as the functions of \(\min\) and \(\max\) respectively, and for any degree \(a_n \in S\) its negation is defined as \(1 - a_n \in S\), then the result of evaluating conjunctive and disjunctive sentences belongs to \(S\). However, this does not hold true in a general case. Namely, consider a set \(S\) of values of the type \(a^{k_1}_1 \ldots a^{k_{2n}}_{2n}\) from the interval \([0, 1]\), where \(a_1 , \ldots , a_n\) are given rational numbers, \(a_{n+i} = 1 - a_i\), and \(k_1 , \ldots , k_{2n}\) are arbitrary integers. Then, there are only finitely many element \(s \in S\) for which \(1 - s \in S\). As authors observe, it means that for all but finitely many values \(s \in S\), the negation \(1 - s\) is outside the finitely generated set \(S\). Hence, ``while for this set, `and'-operation is exact, the corresponding `or'-operation almost always leads us to a value outside \(S\)''. However, it can be shown that this observation cannot be generalized. Namely, it is proved in the chapter, that for each finitely generated set \(S\) -- with the only exception of the set generated by a single value \(1/2\) -- there exists a negation operation \(f_\neg(a)\) for which, for infinitely many \(s \in S\), we have \(f_\neg(s) \in S\). Moreover, it is also proved that the set \(S\) of degrees that can be obtained from \(0\), \(1/2\) and \(1\) using ``and''-operation \(f_\&(a, b) = a \cdot b\) and negation \(f_\neg(a) = 1 - a\) is the set of all binary-rational numbers, i.e., all numbers of the type \(p/2^k\) for natural numbers \(p\) and \(k\) for which \(p \leq 2^k\). For the entire collection see [Zbl 1467.62007].
scientific article; zbMATH DE number 7390603
scientific article