## Notre Dame Journal of Formal Logic

### Conditionals and Conditional Probabilities without Triviality

Alexander R. Pruss

#### Abstract

The Adams Thesis holds for a conditional $\rightarrow$ and a probability assignment $P$ if and only if $P(A\rightarrow B)=P(B\mid A)$ whenever $P(A)\gt 0$. The restriction ensures that $P(B\mid A)$ is well defined by the classical formula $P(B\mid A)=P(B\cap A)/P(A)$. 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
Accepted: 8 December 2017
First available in Project Euclid: 4 July 2019

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

Digital Object Identifier
doi:10.1215/00294527-2019-0019

Mathematical Reviews number (MathSciNet)
MR3985626

#### 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

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