A note on the potentials of probabilistic and fuzzy logic

TL;DR

This paper explores advanced fuzzy set types, examines the relationship between fuzzy and probability logic, and discusses representing real-world uncertainties in data-driven problems.

Contribution

It provides a unified treatment of higher-type fuzzy sets, links fuzzy and probability logic, and addresses modeling uncertainties in practical data scenarios.

Findings

Generalized fuzzy sets of type n are mathematically characterized.

Potential connections between fuzzy and probability logic are identified.

Methods for representing real-life uncertainties in data are discussed.

Abstract

This paper mainly focuses on (1) a generalized treatment of fuzzy sets of type $n$, where $n$ is an integer larger than or equal to $1$, with an example, mathematical discussions, and real-life interpretation of the given mathematical concepts; (2) the potentials and links between fuzzy logic and probability logic that have not been discussed in one document in literature; (3) representation of real-life random and fuzzy uncertainties and ambiguities that arise in data-driven real-life problems, due to uncertain mathematical and vague verbal terms in datasets.