Possibility as Similarity: the Semantics of Fuzzy Logic

TL;DR

This paper explores the semantic foundations of fuzzy logic, contrasting possibilistic and probabilistic approaches, and introduces a model based on similarity relations to define fuzzy concepts and reasoning methods.

Contribution

It presents a semantic model using similarity relations to formalize fuzzy logic concepts, extending modal logic with a new perspective on resemblance between possible worlds.

Findings

Introduces a similarity-based semantic framework for fuzzy logic.

Defines possibilistic structures using a resemblance relation.

Provides a reinterpretation of fuzzy logic reasoning methods.

Abstract

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 resemblance 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 possibility. By contrast, chance-oriented probabilistic concepts and structures rely on measures of set extension that quantify the proportion of possible worlds where a proposition is true. Resemblance between possible worlds is quantified by a generalized similarity relation: a function that assigns a number between O and 1 to every pair of possible worlds. Using this similarity relation, which is a form of numerical complement of a classic metric or distance, it is possible to define and interpret the major constructs and methods of fuzzy logic: conditional and unconditioned possibility and necessity distributions and the generalized modus ponens of Zadeh.