Vacuous truth

Informally, a logical statement is vacuously true if it is true but doesn't say anything; examples are statements of the form "everything with property A also has property B", where there is nothing with property A.

Examples

The statement

All elephants inside a loaf of bread are pink.

is vacuously true since there are no elephants inside a loaf of bread; here property A is "being an elephant inside a loaf of bread", and property B is "being pink". Another example is

If a prime number is even and bigger than two, then it must be divisible by three.

There are no such prime numbers, so in a sense the truth of this statement "doesn't matter".

The statement "0 mathematicians can change a lightbulb" is not vacuously true (or, indeed, true at all); the lightbulb joke "in a group of 0 mathematicians, any one of them can change a lightbulb" however is vacuously true.

Vacuous truth should be compared to tautology, with which it is sometimes conflated.

The remainder of this article uses mathematical symbols.

Scope of the concept

The term "vacuously true" is generally applied to a statement S if S has a form similar to:

P ⇒ Q, where P is false.
∀ x, P(x) ⇒ Q(x), where it is the case that ∀ x, ¬ P(x).
∀ x ∈ A, Q(x), where the set A is empty.
∀ ξ, Q(ξ), where the symbol ξ is restricted to a type that has no representatives.

The first instance is the most basic one; the other three can be reduced to the first with suitable transformations.

Vacuous truth is usually applied in classical logic, which in particular is two-valued, and most of the arguments in the next section will be based on this assumption. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first 2 forms above will yield vacuous truth in any logic that uses material implication, but there are other logics which do not.

Does it make sense to call vacuously true statements true?

We will here consider only the case when S has the form P ⇒ Q, and P is false. This case strikes many people as odd, and it's not immediately obvious whether we should say that P ⇒ Q is true, say that it's false, or say something else altogether.

As for this third option, many people would rather say that P ⇒ Q, in this situation, isn't true or false, but rather "irrelevant", or "pointless". This idea has attracted some logicians, who have developed logics ( e.g. relevant logic) where "neither true nor false" is a possible truth value for a statement.

There are a number of advantages to two-valued logics (i.e. those where true and false are the only choices), however. Suppose we commit ourselves to two-valued logics, and to assigning the same truth value to all statements we have been calling "vacuously true". In this case, is it better to say these statements are true or false?

Most of the arguments one can give seem to suggest that true is the better choice. We might classify these arguments as arguments from linguistic usage and arguments from the nature of logic.

Arguments from linguistic usage

Consider these two arguments from how people seem to use the terms "true" and "false" in everyday speech:

First, calling vacuously true sentences false may extend the term "lying" to too many different situations. Note that lying could be defined as knowingly making a false statement. Now suppose two male friends, Peter and Ned, read this very article on some June 1, and both (perhaps unwisely) concluded that "vacuously true" sentences, despite their name, are actually false. Suppose the same day, Peter tells Ned the following statement S:

If I am female today, i.e., June 4, then I will buy you a new house tomorrow, i.e., June 5.

Suppose June 5 goes by without Ned getting his new house. Now according to Peter and Ned's common understanding that vacuously true sentences are false, S is a false statement. Moreover, since Peter knew that he was was not female when he uttered S, we can assume he knew, at that time, that S was vacuously true, and hence false. But if this is true, then Ned has every right to accuse Peter of having lied to him. This doesn't seem right, however.

Second, picking "true" as the truth value makes many mathematical propositions that people tend to think are true come out as true. For example, most people would say that the statement

For all integers x, if x is even, then x + 2 is even.

is true. Now suppose that we decide to say that all vacuously true statements are false. In that case, the vacuously true statement

If 3 is even, then 3 + 2 is even

is false. But in this case, there is an integer value for x (namely, x=3), for which it does not hold that

if x is even, then x + 2 is even

Therefore our first statement isn't true, as we said before, but false. This doesn't seem be how people use language, however.

Arguments from the "nature of logic"

These arguments also suggest that vacuously true statements should be true:

First, if we were to make the general declaration that statements like S are always false, then, using a truth table, we could show that P ⇒ Q is logically equivalent to P and Q. It certainly seems like "if" and "and" ought to have different meanings, however. If they didn't, then it's confusing why we should have a separate logical symbol for each one.

Second, consider the axioms of the propositional calculus, as outlined in that article. It seems quite plausible that the transformations indicated there are indeed truth-preserving. As explained in the article on logical conditionals, however, those axioms logically entail that, if P is false, then P ⇒ Q is true. That is, if we accept those axioms, we must accept that vacuously true statements are indeed true.

Summary

So there are a number of justifications for saying that vacuously true statements are indeed true. Nonetheless, there is still something odd about the choice. There seems to be no direct reason to pick true; it's just that things blow up in our face if we don't. Thus we say S is vacuously true; it is true, but in a way that doesn't seem entirely free from arbitrariness. Furthermore, the fact that S is true doesn't really provide us with any information, nor can we make useful deductions from it; it is only a choice we made about how our logical system works, and can't represent any fact of the real world.

Difficulties with the use of vacuous truth

All pink rhinoceri are carnivores. All pink rhinoceri are vegetarians.

Both of these contradictory statements are true using classical or two-valued logic - so long as the set of pink rhinoceri remains empty.

Certainly, one would think it should be easy to avoid falling into the trap of employing vacuously true statements in rigorous proofs, but the history of mathematics contains many 'proofs' based on the negation of some accepted truth and subsequently demonstrating how this leads to a contradiction.

One fundamental problem with such 'demonstrations' is the uncertainty of the truth-value of any of the statements which follow (or even whether they do follow) when our initial supposition is false. Stated another way, we should ask ourselves which rules of mathematics or inference should still be applicable after we first suppose that pi is an integer less than two?

The problem occurs when it is not immediately obvious that we are dealing with a vacuous truth. For example, if we have two propositions, neither of which imply the other, then we can reasonably conclude that they are different; counter-intuitively, we can also conclude that the two propositions are the same since this is a vacuous truth because (P⇒Q)∨(Q⇒P) is in fact a tautology in classical logic.

Avoidance of such paradox is the impetus behind the development of non-classical systems of logic relevant logic and paraconsistent logic which refuse to admit the validity of one or two of the axioms of classical logic. Unfortunately the resulting systems are often too weak to prove anything but the most trivial of truths.

Vacuous truths in mathematics

Vacuous truths occur commonly in mathematics. For instance, when making a general statement about arbitrary sets, we want the statement 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. Formally related is the approach to empty products: a product of no factors is defined to be 1 so as to make many general statements work without exceptions.

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 {1,2,3} has seven elements.