The Probability of a Possibility: Adding Uncertainty to Default Rules

TL;DR

This paper introduces a new semantics for default reasoning that integrates uncertainty using counterfactual probabilities, unifying probabilistic and possibility theories for belief revision.

Contribution

It extends conditional logic by incorporating zero-probability revision through counterfactual probabilities, unifying qualitative and quantitative belief update methods.

Findings

Accounts for belief revision with zero-probability events

Unifies probability and possibility theories

Connects to Lewis's imaging method

Abstract

We present a semantics for adding uncertainty to conditional logics for default reasoning and belief revision. We are able to treat conditional sentences as statements of conditional probability, and express rules for revision such as "If A were believed, then B would be believed to degree p." This method of revision extends conditionalization by allowing meaningful revision by sentences whose probability is zero. This is achieved through the use of counterfactual probabilities. Thus, our system accounts for the best properties of qualitative methods of update (in particular, the AGM theory of revision) and probabilistic methods. We also show how our system can be viewed as a unification of probability theory and possibility theory, highlighting their orthogonality and providing a means for expressing the probability of a possibility. We also demonstrate the connection to Lewis's method of imaging.