> because it mistakenly treats a cause as having only one effect.

If A causes other things besides B, then knowing about those other caused things tells us nothing about whether C happened, because we already know it did. "no further information" is elided to things that are relevant to the statement being made. Please apply basic charity in interpreting ideas expressed in prose; LWers who want to express something precisely in logical or mathematical notation are certainly not afraid to do so.

> Less wrong is a bunch of people who think they understand Bayes better than they do.

The objection you point out is not relevant to demonstrating an understanding of Bayes' Law. It's just a semantic quibble.