Isn't allowing cladists to preach their belief in scientific institutions letting the fox into the hen-house?

It is impossible to find a non-contradictory dichotomous classification, because every dichotomous classification is contradictory.

The approach called cladistics is a belief in the contrary, but it is both internally and externally contradictory, and also falsified by facts.

So, how can science allow cladists to preach their contradictory belief in their institutions? Isn't it letting the fox into the hen-house?