The following excellent example comes from Wikipedia: suppose you're in a room full of people, and nobody has a phone (I know, when does that ever happen? But dream with me.) You're in a room full of people, and nobody has a phone. You now stand up, and say "all the phones in the room are off."
This is technically true, right? Because there are no phones, so any phone in the room is off.
This is an example of a vacuous truth: a logical statement that is true, but not in a meaningful way. It is true on a technicality, because it derives from a false premise.
To show you why this is "meaningless," you could also have said, "all the phones in this room are on," which is also true. You could have said, "all the phones in this room are both off and on." You could have said "all the horses in this room are 20 feet tall and covered with purple feathers." This is a true statement. Vacuously.
Why do we care?
Vacuous truth is actually very important in mathematics, computer science, formal logic and electrical engineering (digital logic design.) This is because a vacuous truth is really a gussied-up if-then statement; the phrase, "all horses in this room are 20 feet tall" can be written:
if (a horse is in this room) then (the horse is 20 feet tall.)
...a vacuous truth is an if-then statement where the if clause is false. In formal logic, if X is false, then "if X then Y" is a true statement.
I'll say that again: in formal logic, if X is false, then "if X then Y" is a true statement.
Wait, that can't be right. Why is this true?
First, you must understand how logicians see the statement "if X then Y". This is not meant to imply any sort of causal relationship or other specific physical connection between X and Y. It just means that, whenever X is true, we know Y is true also. That's it.
For example, we can say, "if human beings have 23 chromosome pairs, then gravity follows an inverse-square law." Both statements are true, but the former doesn't cause the latter or have anything to do with it. Nevertheless, we can confidently say that whenever the former statement is true, so is the latter statement; and so the overall implication is true.
We can formalize this by writing out the truth table for the statement if X then Y. A truth table simply enumerates all possible truth values for a logical statement. Here is the truth table for "(X is true) AND (Y is true)":
X |
Y |
X and Y |
True |
True |
True |
True |
False |
False |
False |
True |
False |
False |
False |
False |
"X and Y" is true only when both X is true and Y is true. In any other case it is false to say "X and Y". Let's write out the truth table for "if X then Y". Our requirement is that, whenever X is true, so is Y. Of all four possible truth values of X and Y, only one contradicts this statement: X being true but Y being false. Thus we can write:
X |
Y |
if X then Y |
True |
True |
|
True |
False |
False |
False |
True |
|
False |
False |
|
With a little work, you can deduce that the remaining three slots of this truth table have to be
True. Why? Because we can imagine all the (eight) ways we can fill in the rest of the table, then look at the whole truth table and ask ourselves what expression that table represents:
X=T, Y=T |
X=T, Y=F |
X=F, Y=T |
X=F, Y=F |
What expression has this table? |
True |
False |
True |
True |
(X is false) OR (Y is true) |
True |
False |
True |
False |
Y |
True |
False |
False |
True |
X is equivalent to Y |
True |
False |
False |
False |
X and Y |
False |
False |
True |
True |
X is false (not X) |
False |
False |
True |
False |
(not X) and Y |
False |
False |
False |
True |
(not X) and (not Y) |
False |
False |
False |
False |
False |
Now let's take an if-then statement: all crows are black (X="you are a crow", Y="you are black") and consider every statement in this table:
What expression has this table? |
Fact about crows |
(X is false) OR (Y is true) |
Everything is either black, or not a crow, or both. |
Y |
Everything is black |
X is equivalent to Y |
all crows are black, all black things are crows |
X and Y |
Everything is both black and a crow. |
X is false (not X) |
Nothing is a crow |
(not X) and Y |
Nothing is a crow, all is black |
(not X) and (not Y) |
There are no crows and nothing is black |
False |
Boo to everything |
Those last seven statements mean that we should all listen to Dead Can Dance and then die of ennui. Wait no, i mean those last seven statements don't really make sense.
None of them ring true, and certainly none of them seem to be the same as "all crows are black." Only the first statement is true if "all crows are black," and only the first statement seems to imply that "all crows are black." Therefore, the truth table for if(X) then (Y) is:
X |
Y |
if X then Y |
True |
True |
True |
True |
False |
False |
False |
True |
True |
False |
False |
True |
This is the truth table for implication (sometimes written X→Y) in digital logic. As I mentioned above, it can also be written as
(not X) OR (Y), which you can divine either by looking at the table and guessing, or by using
De Morgan's Law: the statement is False in the specific case when X is true and Y is false, and true otherwise, so this statement can be written:
not( X AND (not Y) ) = not(X) OR not(not Y) = not(X) OR Y
In short, then, X→Y is a true statement if X is false.
Humourous Interlude
There is an old story of John von Neumann explaining this concept to a class. A student objects: "wait, you're saying that 'if (1+1=3) then (I am the Pope)' is a true statement. That doesn't make sense."
von Neumann replies, "Why not? If 1+1=3 then we can subtract 1 from each side, giving 2=1. The Pope and I are two people. Since 2=1, then the Pope and I are one. Hence I am the Pope."
This is a procedural aspect of vacuous truth: often we can start with a false statement and prove anything we like, if we know how to pull the right implication from the initial falsehood. Many joke proofs in mathematics (yes, we have joke proofs) take advantage of a cleverly hidden false assumption to prove outrageous conclusions. Take, for example, the following bogus proof that all numbers are equal:
Let X and Y be any numbers, and let t=(X+Y):
X + Y = t
X2 - Y2 = (X + Y)(X - Y) = t(X - Y)
X2 - tX = Y2 -tY
X2 - tX + t2/4= Y2 -tY + t2/4
(X-t/2)2 = (Y-t/2)2
X-t/2 = Y-t/2
X=Y
There is one false assumption (in which line?), and a false assumption can often be worked to imply any statement, in this case 1=2 (and thus that I am the Pope.)
Vacuous Truth can be misleading
In particular, a vacuous truth can suggest a bogus causal relationship. If you've never tested a drug on humans, you can truthfully say, "every person who has taken this drug has gained 20 IQ points." This suggests that the drug actually does something, but the statement is only true in a vacuous sense.
Also, we can always wrap up any observation in a vacuous truth, even if it has no significance. We can observe that every left-handed president has been followed by a president of the opposite party---this is true, but there is no logical reason why this pattern should continue.
Applications for mathematics
The most immediate application in mathematics is thinking about the empty set. Consider the definition of a subset:
A ⊆ B means: if (x is in A) then (x is in B)
This is an if-then statement! By vacuous truth, then, we can therefore say that every member of the empty set is an integer, or every member of the empty set is a frog. This gives us, vacuously:
The empty set is a subset of every set.
Here's a weirder idea: suppose you have a set of numbers, like the interval [0,5). Is there a lower bound for this set---a number x that is ≤ every element of this set? Sure: -1 is ≤ every element of [0,5). This interval also has an upper bound, for example the number 6. [0,5) also has a
greatest lower bound, in this case 0, and a
least upper bound, 5.
Cool so far. But this leaves the obvious question: does an empty set of numbers have a lower bound, or an upper bound?
Vacuously, every number is a lower bound, and an upper bound, of the empty set: if x is in the empty set (always false), then 5≤x. Therefore 5 is a lower bound of the empty set. So is every other number.
More confusingly, the empty set may have a greatest lower bound or least upper bound depending on the set of numbers. An empty set of positive numbers has a least upper bound, because there is a smallest number: 1. There is no greatest lower bound. In the extended real numbers, an empty set has greatest lower bound infinity, and least upper bound negative infinity.
For this reason, we have to remember to account for the empty set when we talk mathematics. The completeness axiom of the real line states that
Every nonempty set of real numbers has a least upper bound and greatest lower bound.
Likewise, the well-ordering principle states:
Every nonempty set of positive integers has a smallest element.
I have occasionally forgotten the "nonempty" part when writing this on the chalkboard, and regretted it 20 minutes later. Oops! 1+1=5. Sorry, I just broke the natural numbers.
If you're wondering, I use the well ordering principle in my cryptography class. From this one apparently obvious fact we can prove the uniqueness of remainders under division, the consistency of modular arithmetic, derive the algorithm for greatest common divisor and computing inverses of numbers modulo N. With just a little more mathematics, we end up with everything we need to explain the RSA encryption algorithm and the Diffie-Hellman key exchange. That gives us the stuff that lets you shop at Amazon and the Apple store without your neighbors seeing your credit card number, so math equals more crap you don't need. Wait no, I mean math equals more privacy online.
It also gives us the fact that all natural numbers are kinda cool, because if there was a number that was boring, then the set of boring numbers would be nonempty; thus there would be a least boring number, which is kinda cool and therefore not boring, a contradiction.