Branden Fitelson recently brought the following result to my attention.
Theorem: At least one of the following claims is false.
- Qualitative Law of Likelihood: $E$ favors $H_1$ over $H_2$ if and only if $\Pr(E|H_1)>\Pr(E|H_2)$.
- Likelihood-Ratio Measure of Confirmation: The degree to which $E$ confirms $H$ is $\Pr(E|H)/\Pr(E|\sim H)$.
- Favoring-to-Confirmation Bridge Principle: If $E$ favors $H_1$ over $H_2$, then it confirms $H_1$ more than $H_2$.