---
abstract: 'A signed probability distribution may extend a given traditional probability from observable events to all events. We formalize and illustrate this approach. We also illustrate its limitation. We argue that the right question is not what negative probabilities are but what they are for.'
address:
- |
    Mathematics Department\
    University of Michigan\
    Ann Arbor, MI 48109–1043, U.S.A.
- |
    Computer Science and Engineering\
    University of Michigan\
    Ann Arbor, MI 48109-2121, U.S.A
author:
- Andreas Blass
- Yuri Gurevich
title: |
    Negative probabilities, II\
    What they are and what they are for 
---

> *What are numbers and what are they for? [^1]\
> — Richard Dedekind, 1888*

Introduction {#sec:intro}
============

The idea of negative probabilities arose in quantum mechanics [@Wigner; @Moyal; @Dirac; @Feynman]. This is not surprising. The weirdness of quantum mechanics required bold fresh ideas. The physicists have been using signed probability distributions primarily in connection with phase spaces for quantum systems [@Zachos] as suggested by Wigner in his 1932 paper [@Wigner].

The present article was provoked by the paper “An operational interpretation of negative probabilities and no-signaling models” by Samson Abramsky and Adam Brandenburger [@AB]. It occurred to us that the right question may not be 