at loggerheads
idiom

- to strongly disagree

Cambridge Advanced Learner's Dictionary

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ORIGIN

Etymologists are still at loggerheads about the origin of this word.

"Logger" is an archaic English dialect word meaning a heavy block of wood, especially one attached to the leg of a horse to prevent it from wandering away. Shakespeare used "loggerhead" the same way we would use "blockhead" today, meaning an idiot. Perhaps the phrase "to be at loggerheads" simply arose as a way of saying that people who get involved in long, stubborn arguments must be idiots.

Other suggests that the origin is nautical. Loggerheads were long-handled devices with a round cup at one end. These cups were filled with hot tar or pitch, which was thrown at enemy sailors, leading of course to a fight.

There are also three small towns in different regions of the UK with the name Loggerheads. Residents like to claim their town is the origin of the word, but it's likely the reverse. British stand-up comedians saw the name Loggerheads as an opportunity for a now common one-liner:

"I'm going on holiday - a fortnight at Loggerheads with the wife."

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SYNONYMS

going at it (tooth and nail); having a: drag-out fight, a to do, a row; being: aggressive antagonistic, at odds, at cross purposes, bellicose, combative, contentious, hostile, ornery, quarrelsome, cantankerous, hot-tempered; fighting, battling, clashing, locking horns

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Practice OWAD in a conversation today:

"Jim is AT LOGGERHEADS with Mary about  the budget allocation."