at cross-purposes

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There's never a good reason to argue at cross-purposes, unless your purpose is to make someone cross.
Often, when used at the same time, these two classes of drugs work at cross-purposes, says Ram Sasisekharan, a biological engineer at the Massachusetts Institute of Technology.
The statement of work (SOW) for the CESCO contract was written prior to the award of Task Order 2, so there was no knowledge of the coordination issues that must be solved for the two contractors to work in tandem rather than at cross-purposes (if the government shops were still operating, the same coordination would have to be accomplished).
The priorities of smooth implementation of the initiatives and the avoidance of duplication become tangled in the web of cross-purposes and counter-posing goals.
In France, the report points out, police and intelligence agencies have not only failed to coordinate but have sometimes worked at cross-purposes.
REACHING OUT TO THE GAY COMMUNITY HAS ALSO BEEN DIFFICULT, SINCE THE TWO GROUPS OFTEN FIND THEMSELVES AT CROSS-PURPOSES.
A hoard that includes directors who have no relationship with a CEO or other directors, who may lack critical competencies and who may perceive themselves as working at cross-purposes with management would be simply unworkable.
At illuminating cross-purposes with Virilio's desultory ruminations on an accident-induced "end of humanity," Artavazd A.
Despite advances in our understanding of heart disease, American culture still works at cross-purposes to health and well-being.
Well it may be that Coppe, Fox, and Winstanley were refashioning this idiom of rage; it is nonetheless true that they resorted to it even when they were at cross-purposes with its more orthodox applications.