Approximate similarities and Poincaré paradox (Q929640)

The Poincaré paradox was discussed by many authors and it has inspired several new ideas. \textit{M. De Cock} and \textit{E. Kerre} [Fuzzy Sets Syst. 133, No. 2, 137--153 (2003; Zbl 1020.03049)] have observed that the intuitive notion of approximate similarity cannot be adequately represented by the fuzzy equivalence relations. In this paper, the author presents a fuzzy logic based deduction apparatus giving a tool for a formal presentation of the Poincaré argument preserving its intuitive content but avoiding its paradoxical character. Several examples of approximate similarities are given and their relation to the Poincaré paradox is discussed. A connection between a first-order theory (whose fuzzy models are plausible candidates for the notion of approximate similarity) and the point-free metric spaces is also given.