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.
- If a prime number is even and bigger than two, then it must be divisible by three.
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.
Table of contents |
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:
- 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.
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?
For instance, consider this statement for S:
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 P ⇒ Q 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.
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.
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.
Further reading
- When is truth vacuous? Is infinity a bunch of nothing?: a transcript of a discussion in which some professional and amateur mathematicians try to find a definition for vacuous truth and debate its properties