A Fundamental Probabilistic Fuzzy Logic Framework Suitable for Causal Reasoning

TL;DR

This paper develops a foundational probabilistic fuzzy logic framework that integrates probability theory with fuzzy logic, enabling causal reasoning through a measure-theoretic approach and application to causal inference.

Contribution

It introduces a novel probabilistic fuzzy logic framework with measure-theoretic foundations and formulas for conditional probabilities, specifically designed for causal reasoning.

Findings

Formulated a bridge between probability theory and fuzzy logic.

Provided measure-theoretic basis including Dirac delta functions.

Applied the framework to causal inference tasks.

Abstract

In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy attribute. To do so, we associate some specific crisp random variables to the random experiment. Then, several formulas are presented, which make it easier to compute different conditional probabilities and expected values of these random variables. Also, we provide measure theoretical basis for our probabilistic fuzzy logic framework. Note that in our theory, the probability density functions of continuous distributions which come from the aforementioned random variables include the Dirac delta function as a term. Further, we introduce an application of our theory in Causal Inference.