# Vacuous truth

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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 inherently false. For example, the statement “all cell phones in the room are turned off” may be true simply because there are no cell phones in the room. In this case, the statement “all cell phones in the room are turned on” would also be true, and vacuously so, as would the conjunction of the two: “all cell phones in the room are turned on and turned off”.

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is “if Ayers Rock 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).

This notion has relevance in pure mathematics, as well as in any other field which uses classical logic.

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 or her parent “I ate every vegetable on my plate”, when there were no vegetables on the child’s plate to begin with.

## Scope of the concept

A statement $S$ is “vacuously true” if it resembles the statement $P \Rightarrow Q$, where $P$ is known to be false.

Statements that can be reduced (with suitable transformations) to this basic form include the following:

• $\forall x: P(x) \Rightarrow Q(x)$, where it is the case that $\forall x: \neg P(x)$.
• $\forall x \in A: Q(x)$, where the set $A$ is empty.
• $\forall \xi: Q(\xi)$, where the symbol $\xi$ 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 two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.

## Arguments regarding 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 is not immediately obvious whether all such statements are true, all such statements are false, or some are true while others are false.

### 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.

#### Arguments against taking all vacuously true statements to be false

##### Causing the Implies Operator and the Logical AND Operator to be logically equivalent

Second, the most obvious alternative to taking all vacuously true statements to be true — that is, 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:

 $P$ $Q$ $P \Rightarrow Q$ $T$ $T$ $T$ $T$ $F$ $F$ $F$ $T$ ? $F$ $F$ ?

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:

 $P$ $Q$ $P \Rightarrow Q$ $P \land Q$ $T$ $T$ $T$ $T$ $T$ $F$ $F$ $F$ $F$ $T$ $F$ $F$ $F$ $F$ $F$ $F$

Intuitively, this is odd, because it certainly seems like “if” and “and” ought to have different meanings; if that were not the case, then confusion can result as to 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, one can conclude that $P$ is true (or that $Q$ is true) based solely on the logical connection of the two.

A similar argument can be made for other possible truth assignments. If we decide that $F \Rightarrow T$ is false and $F \Rightarrow F$ is true, then $P \Rightarrow Q$ is equivalent to $P \equiv Q$. If instead we decide that $F \Rightarrow T$ is true and $F \Rightarrow F$ is false, then $P \Rightarrow Q$ is equivalent to $Q$. Thus, if we consider the statement $S$ = “if I am in Massachusetts, then I am in North America”, and assume any of these three alternate assignments of the truth table, the statement $S$ would mean, respectively, “Everyone is in both Massachusetts and in North America”; “Everyone in Massachusetts is in North America and vice versa”; or “Everyone is in North America”. None of these statements sounds like an appropriate interpretation of the original statement $S$.

##### Intuition from mathematical arguments

Making vacuous implications “true” 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 is not true, as we said before, but rather false. This does not seem to be how people intuitively 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 vacuous 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 vacuous, and hence false. Since Peter has spoken a falsehood, then Ned has every right to accuse Peter of having lied to him. On the face of it, this line of reasoning appears to be suspect.

### 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 consequence as a whole, and not merely to the case of vacuous truth. For example, it is 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 herbivores.

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.)

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 $(P \Rightarrow Q)\lor(Q \Rightarrow P)$ 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.[citation needed]

## Context of statements

Much of the confusion arises because the use of “if” and “then” in mathematics and formal logic is quite different from ordinary use of those words in daily life or science.[1] For example, experimental physics demonstrates with great accuracy (and with practical utility in the Global Positioning System) that the sum of the angles of a real-world triangle depends on the positions of its vertices with respect to aggregations of gravitational mass. Nevertheless, it is correct to say that if three line segments join three points and the axioms that form Euclidean geometry hold, then the measures of the angles produced add up to 180 degrees.

Similarly, within the mathematical formalism of Boolean algebra, it is always correct to say that “not-A” implies “if A then B” regardless of anything else about A and B. Here A and B are simply variables, with no more intrinsic meaning than x or y in a mathematical equation.

## 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 precisely 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 precisely seven elements.

Since no infinite subset of any set has precisely 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.