GNU Collaborative International Dictionary of English

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Found one definition

1.       
   cross-purpose , n.
   A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory.
   Shaftesbury.
   
   1913 Webster
   pl. 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.
   1913 Webster