Toward a probability theory for product logic: states, integral representation and reasoning

TL;DR

This paper develops a probability theory framework for product logic, establishing a correspondence between states and measures, and introduces a modal logic for probabilistic reasoning within this setting.

Contribution

It axiomatizes a generalized probability notion called state, proves its integral representation, and links it to modal logic for probabilistic reasoning in product logic.

Findings

States are represented as Lebesgue integrals over Borel measures.

Extremal states correspond to truth-value assignments.

A two-tiered modal logic for probabilistic reasoning is sound and complete.

Abstract

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.