Vacuous truth

Vacuous truth is a special topic of first-order logic. A conditional assertion is vacuously true if the assertion can already be shown to be true (often by the use of axioms) and the condition is logically unrelated to the assertion.

This notion has relevance in pure mathematics. One example is the empty product—the fact that the result of multiplying no numbers at all is 1—which is useful in a variety of mathematical fields including probability theory, combinatorics, and power series. Another example is that the elementary symmetric polynomial in no variables at all is 1. Yet another is the discovery that the Euler characteristic is one of the finitely additive "measures" treated in Hadwiger’s theorem.

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.

Compare: tautology, counterfactual.

Scope of the concept
The term "vacuously true" is generally applied to a statement $$S$$ if $$S$$ has a form similar to:


 * 1) $$P \Rightarrow Q$$, where $$P$$ is false.
 * 2) $$\forall x, P(x) \Rightarrow Q(x)$$, where it is the case that $$\forall x, \neg P(x)$$.
 * 3) $$\forall x \in A, Q(x)$$, where the set $$A$$ is empty.
 * 4) $$\forall \xi, Q(\xi)$$, where the symbol $$\xi$$ 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 conditional, but there are other logics which do not.

Arguments of the semantic "truth" of vacuously true logical statements
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 $$S$$ has the form $$P \Rightarrow Q$$, and $$P$$ 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.

Arguments that at least some vacuously true statements are true
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.

Making implies and logical AND logically equivalent
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 $$P \Rightarrow Q$$ should be true when both $$P$$ and $$Q$$ are true, and false when $$P$$ is true but $$Q$$ is false. That is, suppose we accept this as a partial truth table for implies:

Suppose we decide that the unknown values should be $$F$$. In this case, then implies turns out to be logically equivalent to logical AND ($$\land$$), 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:


 * 1) $$P \Rightarrow Q$$
 * 2) $$P \land Q$$
 * 3) $$P$$

and


 * 1) $$P \Rightarrow Q$$
 * 2) $$P \land Q$$
 * 3) $$Q$$

That is, we can conclude that $$P$$ is true (or that $$Q$$ is true) based solely on the logical connection of the two.

Intuition from mathematical arguments
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 to be how people use language, however.

A linguistic argument
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 $$S$$:
 * 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 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.

Arguments for taking all vacuously true statements to be true
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 $$P$$ is false, then $$P \Rightarrow Q$$ 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.

Arguments that only some vacuously true statements are true
One objection to saying that all vacuously true statements are true is that this makes the following deduction valid:


 * 1) $$\neg P$$
 * 2) $$P \Rightarrow Q$$

Many people have trouble with or are bothered by this because, unless we know about some a priori connection between $$P$$ and $$Q$$, what should the truth of $$P$$ have to do with the implication of $$P$$ and $$Q$$? Shouldn’t the truth value of $$P$$ 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 $$P$$ is known a priori to be relevant to the truth of $$Q$$.

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.

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 rhinoceros are carnivores.
 * All pink rhinoceros are vegetarians.

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.

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.

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 $$(P \Rightarrow Q)\lor(Q \Rightarrow P)$$ 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 in mathematics
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 $$R$$ on a set $$S$$ is antisymmetric if, for any $$a$$ and $$b$$ in $$S$$ with $$a\mathrel{R}b$$ and $$b\mathrel{R}a$$, it is true that $$a=b$$. The less-than-or-equal-to relation $$\leq$$ on the real numbers is an example of an antisymmetric relation, because whenever $$a\leq b$$ and $$b\leq a$$, it is true that $$a=b$$. The less-than relation $$<$$ is also antisymmetric, and vacuously so, because there are no numbers $$a$$ and $$b$$ for which both $$a< b$$ and $$b< a$$, and so the conclusion, that $$a=b$$ whenever this occurs, is vacuously true.

An even simpler example concerns the theorem that says that for any set $$X$$, the empty set $$\varnothing$$ is a subset of $$X$$. This is equivalent to asserting that every element of $$\varnothing$$ is an element of $$X$$, which is vacuously true since there are no elements of $$\varnothing$$.

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.

More disturbing are generalizations of obviously "nonsensical" statements which are likewise true, but not vacuously so:
 * There exists a set S such that every infinite subset of S has seven elements.

Since no infinite subset of any set has seven elements, we may be tempted to conclude that this statement is obviously false. But this is wrong, because we’ve failed to consider the possibility of sets that have no infinite subsets at all (as in the previous example—in fact, any finite set will do). It is this sort of "hidden" vacuous truth that can easily invalidate a proof when not treated with care.