1. in a way that shows lack of thought, intelligence, or substance; emptily
2. in a way that is logically true but lacks meaningful content
Usage: logic; mathematics

Examples

• She smiled vacuously at the camera, clearly not understanding the question.
• The politician spoke vacuously about change without offering any specific plans.
• He stared vacuously out the window during the entire meeting.
• The statement is vacuously true since there are no counterexamples to consider.
• The celebrity answered vacuously, repeating the same meaningless phrases.
• In logic, the proposition is vacuously satisfied when the premise is false.