An Order of Magnitude Calculus

TL;DR

This paper introduces a simple calculus for order of magnitude reasoning, providing semantics, probability functions, and decision theory, connecting to existing frameworks like kappa functions and Spohn's functions.

Contribution

It develops a new calculus for order of magnitude reasoning with soundness, completeness, and links to probability and decision theories, extending prior work.

Findings

Order of magnitude probability functions are equivalent to kappa functions.

The calculus supports an order of magnitude decision theory.

It justifies an amended version of Pearl's decision theory.

Abstract

This paper develops a simple calculus for order of magnitude reasoning. A semantics is given with soundness and completeness results. Order of magnitude probability functions are easily defined and turn out to be equivalent to kappa functions, which are slight generalizations of Spohn's Natural Conditional Functions. The calculus also gives rise to an order of magnitude decision theory, which can be used to justify an amended version of Pearl's decision theory for kappa functions, although the latter is weaker and less expressive.