> One wonders why we have not developed something explicit like mathematical notations for legal stuff.

1. Laws are written or at least voted on by representatives, and they don't vote for things that they don't think they understand. Also, they're pretty regularly swapped out and often totally bonkers. Especially at the state level.

2. Things change. Look at how the right of search and seizure is applied to digital data and metadata.

3. Most importantly, the imprecision is intentional. "Beyond all reasonable doubt" has no definition because it is up to the person rendering judgement. The courts decide the bounds of the law, and within those bounds people decide how to apply them.