The vacuous truth, a logic concept and not a nihilistic one, refers to a statement that is true because it is a statement about nothing. In my high school, there was a joke that our football team was the best in the state, the joke being that we had no football team. Another example is,

All the pigs in the sky are purple.

Such a statement is true because there are no pigs in the sky. In fact, any color or property could replace purple and the statement would still be true. Thus we can also have the true statement: All the pigs in the sky are not purple.

It seems strange that both statements could be true simultaneously, but there is some rationale to this; it is logic after all. Consider the general statement: All x are y. The statement’s negation is: There exists an x that is not y. One can see that if the former statement is true, then the latter must be false, and vica versa. Thus, the negation of our original statement is:

There exists a pig in the sky that is not purple.

Clearly, this is false, and thus the original statement about all pigs in the sky being purple must be true.

This explanation seems reasonable, but perhaps not entirely satisfying. Let us consider a more practical example of a vacuous truth. Suppose I have a website that demands, for some reason, a user’s password be in all caps. Furthermore, suppose that this website is not a great one, and so I was able to create an account despite accidentally leaving the password field blank. Then we must ask, does a blank password satisfy the property of being in all caps. Well yes, but vacuously so. It certainly does not contain any non-capital characters.

Now let us consider a more realistic website that requires a user’s password contain a capital letter. When I try to create an account with a blank password, the website gives me an error that there is not a capital letter in my password.

So now, I have a password that is in all caps, yet does not contain a capital letter. Such a statement is illogical, and yet logical.

A similar logical curiosity is the conditional operation, A→B, which can be read, “if A, then B.” The intuitive interpretation is that if A is true, then B must also be true.

If it is raining, then there are clouds.

We could also say:

Clouds are necessary for rain.

Rain is sufficient for clouds.

It is not raining or there are clouds.

One can check all scenarios and see that the four statements are equivalent. But consider the following conditional statements:

If pigs fly, then I am a wizard.

If pigs fly, then I am not a wizard.

Since “pigs fly” is false, the statement is true regardless of the clause following “then.” Thus both statements above are true, though it may seem contradictory. Such statements are vacuous: the “then” clause never obtains, because the “if” clause is always false.

Finally, I leave the reader to consider whether the following is true or false:

If pigs fly, then pigs do not fly.