A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is false. For example, the statement "all cell phones in the room are turned off" may be true simply because there is no cell phone in the room.
More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Uluru is in France, then the Eiffel tower is in Bolivia". Such statements are considered vacuous because the falsity of the antecedent prevents one from using the conditional to infer the consequent. They are true because a material conditional is defined to be true when the antecedent is false (or the conclusion is true).
Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his parents "I ate every vegetable on my plate," when there were no vegetables on the child’s plate.
The term "vacuously true" is generally applied to a statement if has a form similar to:
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 conditional, but there are other logics which do not.
This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i.e., the case when has the form , and is false. This case strikes many people as odd, and it’s not immediately obvious whether all such statements are true, all such statements are false, some are true while others are false, or what.
Consider the implication "if I am in Massachusetts, then I am in North America", which we might alternatively express as, "if I were in Massachusetts, then I would be in North America". There is something inherently reasonable about this claim, even if one is not currently in Massachusetts. It seems that someone in Europe, for example, would still have good reason to assert this proposition. Thus at least one vacuously true statement seems to actually be true.
Second, the most obvious alternative to taking all vacuously true statements to be true — i.e., taking all vacuously true statements to be false — has some unsavory consequences. Suppose we are willing to accept that should be true when both and are true, and false when is true but is false. That is, suppose we accept this as a partial truth table for implies:
Suppose we decide that the unknown values should be . In this case, then implies turns out to be logically equivalent to logical AND (), as we can see in the following table:
Intuitively this is odd, because it certainly seems like "if" and "and" ought to have different meanings; if they didn’t, then it’s confusing why we should have a separate logical symbol for each one.
Perhaps more disturbing, we must also accept that the following arguments are logically valid:
That is, we can conclude that is true (or that is true) based solely on the logical connection of the two.
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
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 4, 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 :
The main argument that all vacuously true statements are true is as follows: As explained in the article on logical conditionals, the axioms of propositional logic entail that if is false, then is true. That is, if we accept those axioms, we must accept that vacuously true statements are indeed true. For many people, the axioms of propositional logic are obviously truth-preserving. These people, then, really ought to accept that vacuously true statements are indeed true. On the other hand, if one is willing to question whether all vacuously true statements are indeed true, one may also be quite willing to question the validity of the propositional calculus, in which case this argument begs the question.
One objection to saying that all vacuously true statements are true is that this makes the following deduction valid:
Many people have trouble with or are bothered by this because, unless we know about some a priori connection between and , what should the truth of have to do with the implication of and ? Shouldn’t the truth value of in this situation be irrelevant? Logicians bothered by this have developed alternative logics (e.g. relevant logic) where this sort of deduction is valid only when is known a priori to be relevant to the truth of .
Note that this "relevance" objection really applies to logical implication as a whole, and not merely to the case of vacuous truth. For example, it’s commonly accepted that the sun is made of gas, on one hand, and that 3 is a prime number, on the other. By the standard definition of implication, we can conclude that: the sun’s being made of gas implies that 3 is a prime number. Note that since the premise is indeed true, this is not a case of vacuous truth. Nonetheless, there seems to be something fishy about this assertion.
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 is vacuously true; it is true, but in a way that doesn’t seem entirely free from arbitrariness. Furthermore, the fact that 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.
Both of these seemingly contradictory statements are true using classical or two-valued logic – so long as the set of pink rhinoceros remains empty. (See also Present King of France.)
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 (proof by 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 if we suppose that pi is an integer (which it is not).
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 implies the other, then we can reasonably conclude that they are different; counter-intuitively, we can also conclude that the two propositions are the same. The reason for this is that is a tautology in classical logic, so every assertion that is made about "two propositions, neither of which implies the other" is an assertion about nothing, hence vacuously true. Although such a fact that "two propositions, neither of which implies the other, are both different and the same" poses no theoretical problems, it can easily be disturbing to the human mind.
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 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