Refers to a logical or mathematical statement which is formally true but has no expressive power.
If Paris is in Italy, then Madrid is the capital of Sweden.
Regardless of in which country Madrid is located, the overall statement is true, paradoxically precisely because the antecedent (“Paris is in Italy”) is false.
By applying the rules of logic, one can also come to findings which are formally true but do not lead to further knowledge.
For example, in the case of a subjunction, a false (counterfactual) antecedent (in the above example: “Paris is in Italy”) always leads to a true overall statement - regardless of the truth value of the consequent.
This is not limited to subjunctions. Such “empty” statements are possible with (almost) all inferential forms. Similarly, tautologies are empty truths and one could call contradictions “empty falsehoods” (since they are always false).
With certain forms of inferences, it is also true that statements that refer to empty extensions are always true and thus also represent vacuous truths.
All unicorns are immortal.
This is true, because no unicorn exists that could ever die.
In colloquial language, a form of vacuous truth is sometimes used to point out the absurdity of a statement:
If Oslo is the capital of Sweden, then I am the emperor of China.
Although this is formally also only vacuously true, there is actually an information conveyed here, namely that the speaker considers the statement “Oslo is the capital of Sweden” to be counterfactual (read: false!).
- Vacuous truth on Wikipedia