1
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Phrase(s): at cross-purposes
with opposing viewpoints; with goals that interfere with each other. • We are arguing at crosspurposes. We aren’t even discussing the same thing. • Bill and Tom are working at cross-purposes. They’ll never get the job done right.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
With aims or goals that conflict or interfere with one another, as in I'm afraid the two departments are working at cross purposes. This idiom, first recorded in 1688, may have begun as a 17th-century parlor game called "cross-purposes," in which a series of subjects (or questions) were divided from their explanations (or answers) and distributed around the room. Players then created absurdities by combining a subject taken from one person with an explanation taken from another.
American Heritage Idioms