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# 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. For instance, 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.

 Table of contents 1 Scope of the concept 2 Why do we call vacuously true statements true? 3 Difficulties with the use of vacuous truth 4 Vacuous truths in mathematics 5 Further reading

## Scope of the concept

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

1. PQ, where P is false.
2. x, P(x) ⇒ Q(x), where it is the case that ∀ x, ¬ P(x).
3. xA, Q(x), where the set A is empty.
4. ∀ 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.

## Why do we call vacuously true statements true?

We will here consider only the case when S has the form PQ, and P is false. Should we say that S is true? That it's false? That it's something else? Should we not say anything?

For instance, consider this statement for S:

If Peter wins the lottery tomorrow, then he will buy a new house.
Now suppose that Peter doesn't win the lottery (i.e., P is false). No matter whether he buys a house or not, the original statement stands; it is certainly not false: the speaker cannot be accused of having lied. So a truth value of false for this statement S (or any other of the same form) is counterintuitive and is to be rejected.

Another argument against the falsehood of statements like S proceeds as follows. Suppose we were to make the general declaration that statements like S are always false. Then, using a truth table, we can show that PQ is precisely the same claim as P and Q, which is certainly unintuitive; we wouldn't even need the symbol ⇒ or the concept "implies" in this case.

But should we necessarily call statements like S true?

If we adopt the position that every statement S has to be either true or false, an assumption made by classical logic, then we are forced to call it true. Many people however feel uneasy with this and would rather call the statement "irrelevant" or "pointless", thus allowing a third truth value besides "true" and "false". Such logics have been studied, e.g. relevant logic, but there are a number of advantages to the classical approach, such as representing logical statements with a boolean algebra.

Another argument for picking "true" as the truth value for these implications is this: Most people will agree that the statement

If x is even, then x + 2 is even.
is true in the integers, which should mean that it's true for any integer x. In particularly, it should be true for x = 3; but the statement for x = 3 is a statement S of the above type, PQ with P (namely that 3 is even) being false. Consequently, this statement should be called true.

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?

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.