CROSS-PURPOSE

Cross″–pur′pose (–p?r′p?s), n. 1. 1. A counter or opposing purpose; hence, that which is inconsistent or contradictory. Shaftesbury.
2. 2. pl. A conversational game, in which questions and answers are made so as to involve ludicrous combinations of ideas. Pepys.
To be at cross-purposes, to misunderstand or to act counter to one another without intending it; — said of persons.