Rough Set Semantics for Non-classical Logics

In the paper a general framework of defining information semantics for nonclassical logics is discussed. Rough sets theory (Pawlak 1992) has been an inspiration for logical investigations aimed at development of logical systems and deduction methods for representing and handling incomplete information and for reasoning in the presence of incompleteness. During the last decade a representation paradigm of rough-set-based information logics and models has received a lot of attention (Orlowska and Pawlak 1984a,b, Orlowska 1985, Farinas del Cerro and Prade 1986, Rasiowa and Skowron 1986, Nakamura 1988, Vakarelov 1988, Dubois and Prade 1992, Rauszer 1992). Information logics are formalisms for representation of and reasoning about data that consist of entities of the following two types: objects and properties of objects. According to the Tarskian tradition semantic structures for formalized languages are abstract relational systems consisting of a nonempty set and a family of relations of a finite arity over this set. In the paper we define a “more concrete”, in a sense, semantics provided by a suitable class of information models. In information models we introduce explicitly objects, properties and various relationships between them. Each information model is determined by what is called an information frame, in a similar way as possible world frames (Kripke 1963) determine models. Information frames are generated by nondeterministic information systems (Orlowska and Pawlak 1984a). Completeness of logical systems with respect to classes of information frames is discussed. The notion of informational representability of frames is introduced and investigated.