Program for generating fuzzy logical operations and its use in mathematical proofs

Fuzzy logic is one of the tools for management of uncertainty; it works with more than two values, usually with a continuous scale, the real interval [0,1]. Implementation restrictions in applications force us to use in fact a finite scale (finite chain) of truth degrees. In this paper, we study logical operations on finite chains, in particular conjunctions. We describe a computer program generating all finitely-valued fuzzy conjunctions (t-norms). It allows also to select these t-norms according to various criteria. Using this program, we formulated several conjectures which we verified by theoretical proofs, thus obtaining new mathematical theorems. We found out several properties of t-norms that are quite surprising. As a consequence, we give arguments why there is no "satisfactory" finitelyvalued conjunction. Such an operation is desirable, e.g., for search in large databases. We present an example demonstrating both the motivation and the difficulties encountered in using many-valued conjunctions. As a by-product, we found some consequences showing that the characterization of diagonals of finitely-valued conjunctions differs substantially from that obtained for t-norms on [0,1].