[Woops, this doesn't answer the question.]

Checking every single pairing would be wasteful. If we know that a < b and that b < c, then we know that a < c. No need to check that one.

There are algorithms that effectively check every pairing in some circumstances, including Bubble Sort and so-called Quicksort.

Perl used to use Quicksort because it can be fast outside of the degenerate cases. But the degenerate case happens to be an already-sorted list, which is a common thing to sort.

Perl now uses Merge Sort, a much better algorithm. It always performs n log₂ n comparisons for an input of size n, which is far less than the n² approach of checking every pairing. (And it's also stable, meaning that items that compare equally will maintain their relative order. This is a very useful property.)

I linked to the Wikipedia pages of algorithms I mentioned, and they are great. They even include useful animations of the sorting algorithm in action.