Fuzzy sets and residuated logic
Based on a logic which has residuated lattices as its intended truth value structures and using a modalized language, the author develops a formal set theory RZF, which is intended to act as a framework for fuzzy set theory and which has a natural embedding of the classical ``crisp'' sets. Its semantics is provided by a suitably modified ``residuated'' Kripke-Joyal semantics based on a forcing style definition of its basic relations. The presentation is in a clear style; a minor irritation is only caused on p. 212 by a proof starting within an actual proof and without a clear claim. Nevertheless it would have been helpful to see some actual semantic objects in an elementary presentation to give a better idea of the relation to usual sets. In comparison with the prevailing discussions in the field of fuzzy sets it is nice to have this attitude toward residuated lattices which continues work of \textit{J. Pavelka}, which is referred to, and \textit{V. Novak}, who has not been mentioned [cf., e.g., his book ``Fuzzy sets and their applications'' (1989; Zbl 0683.94018)]; in any case, it seems to be an approach which is superior to those ones using Heyting-valued structures or (usual) topos theory. Unfortunately, the author gives only very scarce hints to other approaches toward full systems of set theories/set theoretic universes for fuzzy sets. For approaches not based on residuated lattices or Łukasiewicz logic this may count as a minor deficiency, but in other cases, like the embedding of the crisp sets into the fuzzy sets of \textit{D. Klaua} [Monatsber. Deutsch. Akad. Wiss. Berlin 9, 258-272 (1967; Zbl 0221.02050)] or the reviewer's approach toward a cumulative universe of fuzzy sets based on Łukasiewicz logic and a generalized extensionality [Fuzzy Sets Syst. 2, 125-151 (1979; Zbl 0408.03042); ibid. 3, 49-74 (1980; Zbl 0421.03043)], this is a serious gap.
scientific article; zbMATH DE number 64932
scientific article