Probability as a Modal Operator

TL;DR

This paper advocates for viewing probability as a modal operator, presenting a strong probability logic with semantics that unify probabilistic and logical reasoning, challenging traditional interpretations in AI.

Contribution

It introduces a formal probability logic with semantics for quantification and higher-order probabilities, framing probability as a modal operator rather than a simple continuum.

Findings

Probability as a modal operator is both natural and useful.

Provides semantics for quantification within probability logic.

Shows the continuum from impossibility to necessity rather than falsity to truth.

Abstract

This paper argues for a modal view of probability. The syntax and semantics of one particularly strong probability logic are discussed and some examples of the use of the logic are provided. We show that it is both natural and useful to think of probability as a modal operator. Contrary to popular belief in AI, a probability ranging between 0 and 1 represents a continuum between impossibility and necessity, not between simple falsity and truth. The present work provides a clear semantics for quantification into the scope of the probability operator and for higher-order probabilities. Probability logic is a language for expressing both probabilistic and logical concepts.