Computing, in its usual sense, is centered on manipulation of numbers and symbols. In contrast, computing with words, or CW for short, is a methodology in which the objects of computation are words and propositions drawn from a natural language, e.g., small, large, far, heavy, not very likely, the price of gas is low and declining, Berkeley is near San Francisco, it is very unlikely that there will be a significant increase in the price of oil in the near future, etc. Computing with words is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Familiar examples of such tasks are parking a car, driving in heavy traffic, playing golf, riding a bicycle, understanding speech and summarizing a story. Underlying this remarkable capability is the brain’s crucial ability to manipulate perceptions – perceptions of distance, size, weight, color, speed, time, direction, force, number, truth, likelihood and other characteristics of physical and mental objects. Manipulation of perceptions plays a key role in human recognition, decision and execution processes. As a methodology, computing with words provides a foundation for a computational theory of perceptions – a theory which may have an important bearing on how humans make – and machines might make – perception-based rational decisions in an environment of imprecision, uncertainty and partial truth. A basic difference between perceptions and measurements is that, in general, measurements are crisp whereas perceptions are fuzzy. One of the fundamental aims of science has been and continues to be that of progressing from perceptions to measurements. Pursuit of this aim has led to brilliant successes. We have sent men to the moon; we can build computers

The paper considers the fundamental notions of many- valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to many-valued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into

this paper we would like to outline briefly another approach to these problems offered by the rough set theory Pawlak (1991). Although the proposed approach is somehow related to that offered by the fuzzy set theory Pawlak (1994) and the evidence theory, Skowron (1994) it can be viewed in its own rights. The rough set theory bears on the assumption that we have initially some information (knowledge) about elements of the universe we are interested in. Evidently to some elements of the universe the same information can be associated and consequently the elements can be similar or indiscernible in view of the available information. Similarity is assumed to be a reflexive and symmetric relation, whereas the indiscernibility relation is also transitive. Thus similarity is a tolerance relation and indiscernibility is an equivalence relation. It is worthwhile to mention in this context that the concepts of similarity and indiscernibility attracted attention of philosophers and logicians for many years e.g. Williamson (1990), nevertheless these concepts are still not understood fully. Interesting study of these problems can be also found in two recent papers of Marcus (Marcus 1994a; Marcus 1994b). We will refrain in this paper from philosophical discussions and simply give the definitions and properties necessary to explain the ideas of vagueness and uncertainty from the rough sets perspective. 1. VAGUENES AND THE BOUNDARY REGION