Given many partial orders, check them for consistency and report any that are not consistent
This sounds like weighted FAS (Feedback Arc Set).  Find a minimal (weight) feedback arc set in a directed graph and report the assertions that contributed to the wrongly directed arcs.

For each pair of symbols s and t, there is an arc saying how many assertions have s &lt; t and how many assertions have t &lt; s.

This is a known NP-complete problem.

On a different note, your problem is a classic social choice problem where each of the "assertions" are votes.