Vacuous Truth - Vacuous Truths in Mathematics

Vacuous Truths in Mathematics

Vacuous truths occur commonly in mathematics. For instance, when making a general statement about arbitrary sets, said statement ought to hold for all sets including the empty set. But for the empty set the statement may very well reduce to a vacuous truth. So by taking this vacuous truth to be true, our general statement stands and we are not forced to make an exception for the empty set.

For example, consider the property of being an antisymmetric relation. A relation on a set is antisymmetric if, for any and in with and, it is true that . The less-than-or-equal-to relation on the real numbers is an example of an antisymmetric relation, because whenever and, it is true that . The less-than relation is also antisymmetric, and vacuously so, because there are no numbers and for which both and, and so the conclusion, that whenever this occurs, is vacuously true.

An even simpler example concerns the theorem that says that for any set, the empty set is a subset of . This is equivalent to asserting that every element of is an element of, which is vacuously true since there are no elements of .

There are however vacuous truths that even most mathematicians will outright dismiss as "nonsense" and would never publish in a mathematical journal (even if grudgingly admitting that they are true). An example would be the true statement

Every infinite subset of the set has precisely seven elements.

More disturbing are generalizations of obviously “nonsensical” statements which are likewise true, but not vacuously so:

There exists a set such that every infinite subset of has precisely seven elements.

Since no infinite subset of any set has precisely seven elements, we may be tempted to conclude that this statement is obviously false. But this is wrong, because we’ve failed to consider the possibility of sets that have no infinite subsets at all (as in the previous example—in fact, any finite set will do). It is this sort of “hidden” vacuous truth that can easily invalidate a proof when not treated with care.