at cross purposes

cross-purpose

[kraws-pur-puhs, kros-]
noun
1.
an opposing or contrary purpose.
Idioms
2.
at cross-purposes, in a way that involves or produces mutual misunderstanding or frustration, usually unintentionally.

Origin:
1660–70

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Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
cross-purpose
 
n
1.  a contrary aim or purpose
2.  at cross-purposes conflicting; opposed; disagreeing

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
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American Heritage
Idioms & Phrases

at cross purposes

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.

The American Heritage® Dictionary of Idioms by Christine Ammer.
Copyright © 1997. Published by Houghton Mifflin.
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