loggerhead: [16] Loggerhead originally meant much the same as blockhead – a stupid person with a block of wood for a head (in Shakespeare’s Love’s Labour’s Lost (1588), Berowne calls Costard a ‘whoreson loggerhead’). The first part of it probably represents logger ‘block for hobbling a horse’, which in turn was based on log, but how it came to be used in the phrase at loggerheads, meaning ‘in conflict’ and first recorded in 1831, is unclear.
Perhaps the underlying image is of two stupid people having an in-your-face argument, but loggerhead was also used over the centuries for various bulbous-ended objects, including a long-handled tool for melting pitch, and it could be that a fight is being invoked in which these fearsome articles are being used as weapons.
loggerhead (n.)
1580s, "stupid person, blockhead," perhaps from dialectal logger "heavy block of wood" + head (n.). Later it meant "a thick-headed iron tool" (1680s), a type of cannon shot, a type of turtle (1650s). Loggerheads "fighting, fisticuffs" is from 1670s, but the exact notion is uncertain, perhaps it suggests the heavy tools used as weapons. The phrase at loggerheads "in disagreement" is first recorded 1670s.
[W]e three loggerheads be: a sentence frequently written under two heads, and the reader by repeating it makes himself the third. [Grose, "Dictionary of the Vulgar Tongue," 1785]