How is it possible to draw any meaningful statements in mathematics from vacuous truths?

Question

The material implication and its proof that the empty subset is a subset of every set is the classic example. Now I understand that $P$ does not cause $Q$ to be true and I understand why false antecedents still allow the conditional to be true.

However, what I don't understand is why we are able to conclude that the empty set is a subset of every set from the material implication since it is vacuously true? Is this just a consequence from the system of formal logic or is it because the empty set is a very special case and that we are allowed to draw a meaningful statement? Or is it both?

Answer 1

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Answer 2

At the most fundamental level, this is a question about what we want the word "subset" to mean.

We could, if we wanted, perfectly well decide to give the word "subset" a different meaning such that we don't call the empty set a subset of anything. (There's be chaos and confusion when we tried to communicate with mathematicians who have not heard of our decision or disagree with it, but that is merely a practical problem).

It turns out, from upwards of 100 years of experience with using sets to structure mathematical arguments, that it is more useful to have a "subset" concepts that considers the empty set to be a subset of whatever, than to have a concept that doesn't relate the empty set to anything. There's not a single slick reason why it is so -- and sometimes it does mean that we have to speak explicitly about "suppose $A$ is a non-empty subset ..." and so forth -- but the general opinion of mathematicians seem to be that there would be more pesky special cases to deal with in our theorems and arguments if the empty set was never a subset.

After we have a good intuitive idea about what we want "subset" to mean, we can proceed to phrase this definition in our formal logic. Here it is slightly fortunate that our favored logic allows vacuous truths and therefore makes it simpler to express the meaning we want than the alternative meaning that we don't want -- but that's just a bonus: If we wanted to we could have defined the other concept instead.

(There's a bit of circularity here, namely that one of the reasons that it is more useful to consider the empty set a subset of everything is that our logic and usual proof patterns makes it easy to work with that concept -- usually we don't have to consider the vacuous truth as a special case in proofs because our logic handles that automatically! For some cases other than "subset" we do want hard enough to get something else than what the logic most easily provides us with, though -- for example when we need to explicitly require that $0\ne 1$ in a "field").