On rough set logics based on similarity relations

In this paper, dedicated to Professor Solomon Marcus on the occasion of His 80th birthday, we discuss the idea of intensional many-valued logic reflecting the logical content of rough set approach to analysis and treatment of uncertainty. In constructing the variety of logics presented in the paper, we make use of a certain kind of tolerance (similarity) relations called rough mereological tolerances. A Study of tolerance relations that arise in rough set environments was initiated in 1994, with the paper [23], in which basic ideas pertaining to tolerance relations in the rough set framework were pointed to. The analysis of the role tolerance relations may play in machine learning based on rough set-theoretic ideas was carried out by Professor Solomon Marcus in His seminal paper, written during His stay in Warsaw in December of the year 1994. At the same time the first author had first ideas related to the applicability of ideas of mereology in the rough set analysis Of uncertainty. In a later analysis it has turned out that mereological approach has led to a development of a new paradigm in reasoning under uncertainty, called rough mereology, proposed by Lech Polkowski and Andrzej Skowron. Within this paradigm, one is able to construct a variety of tolerance relations. Those tolerance relations, induced by rough mereological constructs called rough inclusions, serve as a basis for constructing a variety of logics, called rough mereological logics, that are related to the inherent structure of my rough set universe. In this paper, we introduce gradually all essential and necessary notions from the area of rough set theory, mereology and rough mereology, and then we discuss tolerance relations induced by tough inclusions alone with some methods for inducing rough inclusions with desired properties. The paper culminates with a discussion of intensional logics based on rough mereological tolerance relations. In this way, we explore one of so many paths in scientific research, that have been either pointed to or threaded by Professor Solomon Marcus.