Returning to the mamzer and the hallal. Specifically, a widow who had been married to a man from a kohen family, but of questionable lineage, for all that. That questionable lineage is the potential problematic lineages that can get "mixed in" with those whose lineage is not problematic. Also, the mishnah at the end of the daf: the case of a young girl to fill up jugs with water - and is raped there, but the identity of the rapist is unknown. Depending on the status of the majority of the people in that locale, she will be able to carry on and marry anyone she might have married prior to the rape. Note the impact on the assumptions that are able to be made about the given populations.