cross-purpose, n.
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A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory. Shaftesbury. [1913 Webster]
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A conversational game, in which questions and answers are made so as to involve ludicrous combinations of ideas. Pepys. [1913 Webster]
To be at cross-purposes, to misunderstand or to act counter to one another without intending it; -- said of persons.