Here’s another way to look at #2, and one that I think you’ll find helpful. Let’s go back to your example; I’ll cut and paste the way you phrased it:

“O will be performed if L is not.” (/L –> O) Not singing L triggers singing O; how can it then be guaranteed that not singing O will guarantee L?

So, our initial rule is “If not L, then O.” To analyze the contrapositive, take “not O” as your starting point:

YES :

NO: O

There are two possibilities for L; it will be sung, or it won’t. Let’s look at them separately. We’ll call the two possibilities “Variation 1” and “Variation 2,” respectively:

YES: L

NO: O

YES: (neither)

NO: LO

Notice that variation 2 violates our original rule, “If not L, then O”! So variation 2, just given the initial rule, is not a possibility. Which means that once O is in the “No” column, L must be in the “Yes” column.

In other words… /O –> L – the contrapositive.

Basically, putting both L and O out violates the original rule, so as soon as EITHER one is out, the other must be in. Hope this helps you get your head around it; these definitely take some getting used to.

-Dan