Ewa Orlowska

We present an approach to nondeterministic time based on a concept of nondeterministic world. A nondeterministic world is a set of some equipos- sible states. Such a world is not treated as whole, it is neither atomic nor indivisible. Given a world representing the present, the past worlds are its subsets and the future worlds are sets in which it is contained. Thus we assume that worlds involve change and the way they change is expansion. In our approach indeterminism occurs in the two levels. First, we admit nondeterministic worlds, consisting of possibly many instances. Second, we treat the future as nondeterministically reachable. As a consequence, for any world there are many past worlds from which the world might be reached and many future worlds which might be reached from the world

One of the important issues in research on knowledge based computer systems is development of methods for reasoning about knowledge. In the present paper semantics for knowledge operators is introduced. The underlying logic is developed with epistemic operators relative to indiscernibility. Facts about knowledge expressible in the logic are discussed, in particular common knowledge and joint knowledge of n group of agents. Some paradoxes of epistemic logic are shown to be eliminated in the given system. A formal logical analysis of reasoning about knowledge is a subject of investigations both in logic and computer science , and several epistemic systems have been proposed to formalize the operator ‘an agent knows’. In the present paper we propose a formalization based on a semantic treatment of knowledge within the framework of rough set theory . The inspiration for the underlying epistemic logic came from the analysis of knowledge transfer in distributed systems developed in Orlowska and Sanders and from the author’s earlier work on indiscernibility and relative accessibility semantics

In Orlowska and Weingartner an approach to relevance was proposed based on the notion of sematic relevance criterion which is understood as a binary relation in the set of predicates of a language under consideration. Some general properties were investigated of sets of formulas relevant with respect to relevance criteria or sets of relevance criteria. Informally, a predicate P is relevant for a predicate Q whenever some information about instances of Q can be derived from information about instances of P. Relevance criteria given by means of binary relations reflect semantic relationship between predicates. We assume that relevance criteria are reflexive, that is every predicate is relevant for itself. In the present note we propose to consider propositional operations of implication which are relative to a semantic relevance criterion. The intuition for relevant implication is, that a necessary condition for the validity of an inference from F to G is that F be relevant to G. We define a logic with implications → , for a relevance criterion R, reflecting this intuition. We treat relevance criterion in the set of predicates as a primary notion, and next, we define its extension to arbitrary formulas. Our view is, that relevance relation between predicates reflects a background knowledge of a user of the language. This knowledge is relative to an investigator and to a theory within which he makes inferences, and in general it cannot be expressed explicitly as a formula of the language. Our semantics for relevant implication includes both validity of inference and its relativity with respect to some background information about relevant predicates