Rough Logics with Possible Applications to Approximate Reasoning

Extended Abstract Representations of the lower and upper approximations of a set in the context of an approximation space as modal operators in the first order language of modal logics, are quite natural and widely familiar now to the rough-set community. According to the perception of an observer, objects of a universe (of discourse) are clustered. These are the basic information granules (or quanta). With respect to the information available, objects belonging to the same cluster are indistinguishable. It may not always be the case that the clusters are mutually disjoint. Now, an indiscernibility relation I may be defined such that for any objects a, b of the universe, aIb holds if (and only if) a and b belong to the same cluster. It is quite reasonable to assume that I is at least reflexive and symmetric. The relation I may be formally interpreted as the (general) accessibility relation of the Kripke-models of modal logic-systems. Although the accessibility relation in the models of modal logics need not be symmetric (and even reflexive), if I is taken to be indiscernibility there should not be any valid reason of its being non-symmetric. The corresponding modal system turns out to be the system B. In Pawlaks rough set systems since I is defined in terms of information tables with respect to attribute-value systems, the indiscernibility relation I turns out to be transitive too. Thus, I being an equivalence relation, the corresponding modal logic-system has to be S5. In the present lecture, however, we shall concentrate on two basic types of clustering of the universe, viz. the covering based clustering and partition based clustering of the objects of the universe – the first giving rise to relations which are reflexive and symmetric, i.e. tolerances, while the second giving rise to equivalences (reflexive, symmetric and transitive). Now, the main support of the inference machine (methodology) in a logic or broadly speaking in reasoning, rests on the rule Modus Ponens (M.P.) which says that " If from a premise set X, the sentences A and A → B (If A then B) are both derivable then to infer B from X ". In the context of approximate reasoning, this rule of inference is usually relaxed in various ways. For example in the fuzzy logic literature one gets fuzzy modus ponens rule like " If A and A → B are derivable from X where A is …