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

2. 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 ]