Probability Distributions Over Possible Worlds

TL;DR

This paper explores the semantics of probability distributions over possible worlds in first-order logic, highlighting expressive limitations and challenges in representing statistical assertions and default reasoning.

Contribution

It provides a detailed analysis of probabilistic semantics for first-order languages, revealing limitations in expressing statistical and default reasoning within this framework.

Findings

Probabilistic semantics face expressive limitations for statistical assertions.

Assigning probabilities to logical sentences complicates default reasoning.

The paper clarifies the features and constraints of probability logic over possible worlds.

Abstract

In Probabilistic Logic Nilsson uses the device of a probability distribution over a set of possible worlds to assign probabilities to the sentences of a logical language. In his paper Nilsson concentrated on inference and associated computational issues. This paper, on the other hand, examines the probabilistic semantics in more detail, particularly for the case of first-order languages, and attempts to explain some of the features and limitations of this form of probability logic. It is pointed out that the device of assigning probabilities to logical sentences has certain expressive limitations. In particular, statistical assertions are not easily expressed by such a device. This leads to certain difficulties with attempts to give probabilistic semantics to default reasoning using probabilities assigned to logical sentences.