Strength Factors: An Uncertainty System for a Quantified Modal Logic

TL;DR

This paper introduces a new uncertainty system for quantified modal logic grounded in probability and proof theory, enabling better handling of counterfactuals and human interaction.

Contribution

It develops a novel system S for uncertainty in first-order modal logic, grounded in epistemology, probability, and proof theory, extending reasoning capabilities.

Findings

Provides a solution to the lottery paradox.

Enables uncertainty quantification for counterfactuals.

Proposes a new reasoning algorithm extending first-order resolution.

Abstract

We present a new system S for handling uncertainty in a quantified modal logic (first-order modal logic). The system is based on both probability theory and proof theory. The system is derived from Chisholm's epistemology. We concretize Chisholm's system by grounding his undefined and primitive (i.e. foundational) concept of reasonablenes in probability and proof theory. S can be useful in systems that have to interact with humans and provide justifications for their uncertainty. As a demonstration of the system, we apply the system to provide a solution to the lottery paradox. Another advantage of the system is that it can be used to provide uncertainty values for counterfactual statements. Counterfactuals are statements that an agent knows for sure are false. Among other cases, counterfactuals are useful when systems have to explain their actions to users. Uncertainties for counterfactuals fall out naturally from our system. Efficient reasoning in just simple first-order logic is a hard problem. Resolution-based first-order reasoning systems have made significant progress over the last several decades in building systems that have solved non-trivial tasks (even unsolved conjectures in mathematics). We present a sketch of a novel algorithm for reasoning that extends first-order resolution. Finally, while there have been many systems of uncertainty for propositional logics, first-order logics and propositional modal logics, there has been very little work in building systems of uncertainty for first-order modal logics. The work described below is in progress; and once finished will address this lack.