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Vacuous truth

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Informally, a logical statement is vacuously true if it is true but doesn't really say anything. For instance, the statement
All elephants inside a loaf of bread are pink.
is vacuously true; it is true, but there is really no consequence of it being true, since there are no elephants inside any loaves of bread. Another vacuously true statement is
If a prime number is even and bigger than two, then it must be divisible by three.
This is true, but, since there are no such numbers, it somehow doesn't "matter" that it is true.

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

Anyone who doubts the utility of the concept will find at least some evidence for it in the article titled empty product. [A stronger case can be made; perhaps I will add to this paragraph at a later date.]

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:

  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.

Note that these conditions on form show family resemblances to one another, roughly in the sense described by Ludwig Wittgenstein in his Philosophical Investigations; no obvious properties underly all these forms, but there are obvious connections between them. The first and second, for example, are both about implication. Both the third and the fourth can be transformed into the second, the third by setting P(x) to xA, and the fourth by setting P to a predicate introduced to indicate the type in question.

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?

At this point, we are left with one big question:

  • Why are statements of the above forms true?
It is hard to answer this question in general, but we can gain some insight by considering one of the particular statement forms above.

Let's consider 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. But should we necessarily call it 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, such as relevant logic, but there are a number of advantages to the classical approach, such as represententing 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. Using the specific value x = 3, we get a statement S of the above type, PQ with P (namely that 3 is even) being false. Consequently, this statement should be called true.

Another argument 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.

The claim that "it doesn't really matter" what we pick for S looks rather silly in this light. Such unfelicitous entailments do not arise if we choose that S is true. Thus that is what is generally picked.

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.

Statements of the other forms could be shown to be "vacuously true" for similar reasons; while there are good reasons for saying they are true, there is still some degree of arbitrariness or psychological discord involved.

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