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A logic for rough sets. (English) Zbl 0896.03050
Summary: The collection of all subsets of a set forms a Boolean algebra under the usual set-theoretic operations, while the collection of rough sets of an approximation space is a regular double Stone algebra [J. Pomykala and J. A. Pomykala, Bull. Pol. Acad. Sci., Math. 36, 495-508 (1988; Zbl 0786.04008)]. The appropriate class of algebras for classical propositional logic are Boolean algebras, and it is reasonable to assume that regular double Stone algebras are a class of algebras appropriate for a logic of rough sets. Using the representation theorem for these algebras by T. Katriňák [Bull. Soc. R. Sci. Liège 43, 283-290 (1974; 314.06004)], we present such a logic for rough sets and its algebraic semantics in the spirit of H. Andréka and I. Németi [in: D. M. Gabbay (ed.), What is a logical system? Stud. Log. Comput. 4, 393-443 (1994; Zbl 0823.03040)].
MSC:
03G10Lattices and related structures (algebraic logic)
03B60Other nonclassical logic
06D15Pseudocomplemented lattices