Possibility as similarity; the semantics of fuzzy logic

This paper addresses fundamental issues on the nature of the concepts and structures of fuzzy logic, focusing, in particular, on the conceptual and functional differences that exist between probabilistic and possibilistic approaches. A semantic model provides the basic framework to define possibilistic structures and concepts by means of a function that quantifies proximity, closeness, or resem­ blance between pairs of possible worlds. The resulting model is a natural extension, based on multiple conceivability relations, of the modal logic concepts of necessity and possi­ bility. By contrast, chance-oriented proba­ bilistic concepts and structures rely on mea*This work was supported in part by the US Air Force Office of Scientific Research under Contract No. F49620-89-K-0001 and in part by the USArmy Research Office under Contract No. DAAL03-89-K0156. The views, opinions and/or conclusions con­ tained in this note are those of the author and should not be interpreted as representative of the official positions, decisions, or policies, either express or im­ plied, of the Air Force Office of Scientific Research ' the Department of the Army, or the United States Government. sures of set extension that quantify the pro­ portion of possible worlds where a proposi­ tion is true. Resemblance between possible worlds is quantified by a generalized similarity rela­ tion, i.e., a function that assigns a number between 0 and 1 to every pair of possible worlds. Using this similarity relation, which is a form of numerical complement of a clas­ sic metric or distance, it is possible to de­ fine and interpret the major constructs and methods of fuzzy logic: conditional and un­ conditional possibility and necessity distri­ butions and the generalized modus ponens of Zadeh.