vacuously

/ˈvækjuəsli/
adverb
  1. In a way that shows a lack of thought, intelligence, or expression; blankly or emptily.
    • He smiled vacuously, as if he had no idea what was happening around him.
    • She stared vacuously at the television, not registering anything on the screen.
    • The student nodded vacuously during the lecture, clearly not following the explanation.
  2. In a way that is meaningless, pointless, or lacking substance.
    • The movie's plot was vacuously simple, leaving audiences bored.
    • Many online comments are vacuously repetitive, adding nothing to the discussion.
    • The politician's speech was vacuously optimistic, offering no real solutions.
  3. In logic or mathematics, in a way that is true because the condition cannot be satisfied (e.g., a statement about all members of an empty set).
    • The theorem's premise was impossible, so the conclusion held vacuously.
    • The statement 'All unicorns are pink' is vacuously true because there are no unicorns.
    • In set theory, the claim that every element of the empty set has a certain property is vacuously true.