The contrapositive is a logical truth. It means, in logic speak, that if A implies B, then NOT B implies NOT A.
(NOT A is true if A is false, and false if A is true. A implies B means that if A is true then B must be true, whereas if A is false, B could be true or false).
You can prove this with a truth table, but I won’t do it here. It is common sense, when you use an example.
Like if I say, “IF it is raining THEN I will use an umbrella.” You can write this in logic speak as “rain” implies “umbrella”. If you then see me without an umbrella, then you know that it must not have been raining. Otherwise my statement would have been false. So you can also conclude, from my original statement, that “IF I don’t use an umbrella THEN it is not raining.” That’s the contrapositive.
Why does this matter to Catholics? Well there are a lot of Protestants who say “You don’t need to do works to be saved.” So you say to them, so why do you bother doing good works? Why is Jesus telling us to do good works if He is gonna let us into heaven anyway, whether we do them or not? And the Protestant may reply, “People who are saved will do good works out of gratitude to God.” Or something like that.
So let’s convert their last statement into logic speak.
“Saved” implies “works”.
Therefore, by the contrapositive, “NOT works” implies “NOT saved”
In other words, if you don’t do good works then you’re not saved. They said it themselves.
Well lucky that’s true because that’s what Jesus was trying to say in Matt 25:31-46 ay? I guess Jesus knew logic.