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Entry 2 senses Webster, 1913

cross-purpose

/(-p?r`p?s)/ · cross·-pur·pose · IPA
01 n. A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory.
  1. 1.
    A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory.
  2. 2.
    A conversational game, in which questions and answers are made so as to involve ludicrous combinations of ideas.
Phrases & compounds
To be at cross-purposes — to misunderstand or to act counter to one another without intending it; -- said of persons.