In this note we’ll introduce the notion of a “vacuous statement”—a statement that is true, but completely devoid of meaning. In particular, we’ll learn what it means for some statement to be “vacuously true”. Statements that are vacuously true come up from time to time in mathematics, and their existence (and truth) often are crucial for certain mathematical constructions to be logically sound. For example, we made use of a vacuous truth in our lesson on subsets, where we noticed that the empty set is a subset of every set. This is a great example of a vacuous statement, so let’s explore it in detail again.
Let us first remind ourselves briefly about subsets. Recall that if we have some set A, then a subset of A is some set whose every element is also in A. If the enemy hands me some set B and asks me if it’s a subset of A, all I need to do is consider every single element in B and ask if it’s also an element of A. If the answer is yes for every element in B, then the answer to the enemy’s question is yes. Suppose, however, that the enemy handed me the empty set and asked if it was a subset of A. What would be my answer? Well, what I would have to do is look at every element of the empty set and ask if it is also an element of A. But there are no elements in the empty set. Thus the statement “every element in the empty set is also in A” is true simply because there are no elements in the empty set to even consider!
Let’s take another example, which will hopefully make this notion clearer. This example won’t be as mathematically precise, but it will hopefully bring out the essential features of a vacuous statement, thus making the mathematically precise ones easier to handle. Consider the following statement: “Whenever there are cows on the moon, I can fly”. (I know this sounds totally absurd, but it’s actually mathematically relevant!) Now, I know that I can’t fly (despite my wishes). You know I can’t fly. Thus the statement “I can fly” is certainly false. However, it happens to be true that I can fly, provided that there are cows on the moon. Namely, every time in the history of the universe that there has been a cow on the moon, I have had the ability to fly. Of course (presumably), there has never been a cow on the moon. Also of course (at least I’m quite sure), I have never been able to fly. Thus it is indeed the case that the presence of cows on the moon has coincided perfectly with the instances of me being able to fly. Accordingly, the statement “Whenever there are cows on the moon, I can fly” is indeed true!
Now before you go crazy proving statements like “As long as pigs can speak French, Lebron is better than Kobe” (which would be true only in this circumstance), I should warn you that there is a reason why we call these statements “vacuous”. Namely, they are devoid of any content (like a vacuum). Thus, even though they’re true, they don’t tell us anything new. No one will be handed a Fields Medal for proving a vacuous statement, simply because a) they’re automatically true, and b) they don’t give us any new information. Just because I can prove that Lebron is better than Kobe whenever a pig can speak French, this does not mean that I can prove it in any scenario that might actually be relevant to the real world. The truth of the statement just kind of sits “out there” lacking any real substance. Nonetheless, the validity of some vacuous statements is important for certain mathematical results, and certain logical consistency. In particular, we used the fact that the empty set is a subset of every set (even itself) in order to establish the general pattern first described in lesson 3, and proved in lesson 15. These manipulations of the empty set will also come into play when we define topologies, as well as in category theory. Thus, despite the fact that the statement “the empty set is a subset of every set” is only vacuously true, its truth is necessary for much of the logical consistency of mathematics.
Now go have fun owning your friends in arguments by inserting various vacuously true statements in such a way that your argument is untouchable. Perfect this craft and you could seriously consider a career in politics!