### Theorem

Theorem.At least one of the following claims is false:^{1}

C1.$E$ favors $H_1$ over $H_2$ (where $H_1$ and $H_2$ are mutually exclusive) if and only if $\Pr(E|H_1)/\Pr(E|H_2)>1$, with $\Pr(E|H_1)/\Pr(E|H_2)$ measuring the degree of that favoring.C2.The degree to which one is justified in believing $H_1$ rather than $H_2$ is that degree to which one’s total evidence favors $H_1$ over $H_2$.C3.If the degree to which one is justified in believing $H_1$ is strictly positive, and $H_1$ and $H_2$ are mutually exclusive, then the degree to which one is justified in believing $H_1\cup H_2$ rather than $H_3$ is strictly greater than the degree to which one is justified in believing $H_2$ rather than $H_3$.C4.If $H_1$ is logically equivalent to $H_2$ and $H_3$ is logically equivalent to $H_4$, then the degree to which one is justified in believing $H_1$ over $H_3$ is the same as the degree to which one is justified in believing $H_2$ over $H_4$.C5. If one is justified in believing $H_1$ over $H_3$ and $H_4$ over $H_2$, then the degree to which one is justified in believing $H_1$ over $H_3$ is not the same as the degree to which one is justified in believing $H_2$ over $H_4$.

- C1-C5 are implicitly quantified over all $E$, $H_1$, etc. for which the relevant quantities are well defined. ↩