1
general::
Phrase(s): at loggerheads (with someone) [and] at loggerheads over something
Fig. in conflict with someone; having reached an impasse (about something). • Tom is at loggerheads with Bill. • We are at loggerheads with each other. • The twins were at loggerheads over who should take the larger room.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
Engaged in a quarrel or dispute, as in The two families were always at loggerheads, making it difficult to celebrate holi days together. This term may have come from some earlier meaning of loggerhead, referring either to a blockhead or stupid person, or to a long-handled iron poker with a bulb-shaped end that was heated in the fire and used to melt pitch. If it was the latter, it may have been alluded to as a weapon. [Late 1600s] For a synonym, see AT ODDS.
American Heritage Idioms