The Logic of Uncertainty and Geometry of the Worlds

Abstract

The classical logic is built in an axiomatic way without containing any uncertainty. Over the years, many attempts have been made to extend the classical logic framework to incorporate uncertainties. This work has produced different and apparently conflicting definitions of uncertainties. Previously we have argued that an extended modal logic framework can provide a unifying formal language for many formalizations of uncertainty such as probability, approximate probability, evidence theory, fuzzy sets, and rough sets.

In this chapter, we have shown how such an extended modal of logic framework provides a unifying formal language for formalizations of probability and fuzzy sets theories. We also show that this originally pure theoretical approach can be used to describe a mathematical model for linguistic uncertainty in a unified framework. The approach combines concepts of modal logic together the linguistic context space previously proposed. This combined approach is illustrated with an application to the question of economical preference between customers and goods.

A common language able to represent different types of uncertainties is useful because it allows us to define the uncertainties using simple entities and to compare different types of uncertainties as being different aspects of the same fundamental structure. The justification of different theories of uncertainty using a unified language is beyond the scope of this paper.

Our goal is to rebuild a foundation of the uncertainty concept by using a new interpretation of the modal logic structure. Using this new foundation we discover that disparate types of uncertainties and the idea of uncertainty itself can be understood. This new foundation opens opportunities to discover hidden connection between different types of uncertainties.