1. to be in an aggressive dispute or in disagreement with no end in sight
at loggerheads with authorities
If you are 'at loggerheads' with somebody it means you are in a debate, dispute or disagreement where no compromise can be reached. It is an expression often used to describe a quarrel between two groups or organisations. This is a formal phrase that you will often see in the newspaper or used to formally or politely describe a difference of opinion.