People disagree; that's life. Sometimes we might describe those people as being at cross-purposes with each. But don't make the mistake of using "across purposes" instead.
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The Macquarie Dictionary provides a handy definition for 'be at cross-purposes' (note the hyphen which helps minimise ambiguity):
to be involved in a misunderstanding, such that each person makes a wrong interpretation of the other's interest or intentions
It's easy to see how someone who hears this expression (rather than reading it) might assume that the phrase is "across purposes". It's an innocent mistake, but that's no reason to perpetuate the incorrect version.
Language is often illogical, but in this case the correct version is also the more logical one. When two people start with incorrect assumptions, it's no surprise that their purposes become misaligned. Being "across purposes" simply makes no sense.
Lifehacker's Mind Your Language column offers bossy advice on improving your writing.