## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 60, Number 3 (2019), 551-558.

### Conditionals and Conditional Probabilities without Triviality

#### Abstract

The Adams Thesis holds for a conditional $\to $ and a probability assignment $P$ if and only if $P(A\to B)=P(B\mid A)$ whenever $P\left(A\right)>0$. The restriction ensures that $P(B\mid A)$ is well defined by the classical formula $P(B\mid A)=P(B\cap A)/P\left(A\right)$. Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results (Lewis, etc.), any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 551-558.

**Dates**

Received: 3 August 2016

Accepted: 8 December 2017

First available in Project Euclid: 4 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1562205627

**Digital Object Identifier**

doi:10.1215/00294527-2019-0019

**Mathematical Reviews number (MathSciNet)**

MR3985626

**Subjects**

Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

Secondary: 60A99: None of the above, but in this section

**Keywords**

conditionals probability measure theory measure algebra conditional probability

#### Citation

Pruss, Alexander R. Conditionals and Conditional Probabilities without Triviality. Notre Dame J. Formal Logic 60 (2019), no. 3, 551--558. doi:10.1215/00294527-2019-0019. https://projecteuclid.org/euclid.ndjfl/1562205627