vacuously

• Adverbe utilisé en logique, en mathématiques et dans le raisonnement formel, signifiant « sans contenu » ou « de manière vide ». Décrit une affirmation vraie par défaut, car la condition dont elle dépend ne peut pas être satisfaite (par exemple, « tous les licornes sont roses » est vrai au sens vide).

An adverb used in logic, mathematics, and formal reasoning meaning "in a vacuous manner" or "without content." It describes a statement that is true by default because the condition it depends on cannot be satisfied (e.g., "all unicorns are pink" is vacuously true).

1. In a mindless or empty wayC2formal

   In a manner that shows a lack of thought, intelligence, or purpose.

   de façon vide et sans esprit

   • He stared vacuously at the television screen.
   • She smiled vacuously, not understanding the joke.
   • The audience listened vacuously to the long lecture.

   Synonymesblanklyinanelymindlesslystupidlythoughtlesslyunintelligently

   Antonymesintelligentlymeaningfullypurposefullythoughtfully

2. In logic: true by defaultC2formal

   In logic, a statement is said to be vacuously true if its antecedent is false, making the implication true regardless of the consequent.

   vraie par vacuité / de façon vacante

   • The statement 'All unicorns are pink' is vacuously true because there are no unicorns.
   • In mathematics, a property holds vacuously for an empty set.
   • The theorem is vacuously satisfied when the condition cannot be met.

   Synonymestrivially