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Discover LudwigThe phrase "cross purposed" is correct and usable in written English
It is used to refer to the state of having conflicting goals, ambitions, or motivations. For example, "The two sides seemed to be cross purposed, unable to agree on any of the points of negotiation".
Idiom
At cross purposes.
When people are at cross purposes, they misunderstand each other or have different or opposing objectives.
Similar(57)
"I'm at cross purposes," he lamented.
The two men talk at cross purposes.
Goodman then wrote: "Apols – cross purposes.
Moreover, some older policies work at cross purposes.
I think we may be at cross purposes here.
July 25 at 7 30 p.m. Cross Purpose Band, rock and blues.
However, critics say the government is working at cross purposes.
Sometimes these two roles work at cross purposes.
Often, the numbers work at cross purposes, as when increasing shield material decreases available payload.
In effect, the Treasury and the Fed are working at cross purposes.
In a tragicomic duet he and Ms. Gunstone talk at cross purposes.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com