To disagree, dispute, or quarrel. A logger was a heavy wooden block, and one meaning of “loggerhead” is “blockhead,” a stupid person or dolt. Possibly this meaning led to the phrase “at loggerheads,” with the idea that only dolts would engage in a quarrel. Shakespeare used the word as an adjective in The Taming of the Shrew (4.1): “You loggerheaded and unpolish’d grooms.” The full current expression appeared in the late seventeenth century.