Hannah knows that the following two statements are true: If a student does not complete their homework, they will get detention. If the student gets detention, they will be late home. Jacob got detention and was late home. Can we conclude that he did not complete his homework?
What we have in these two statements in logic is called the law of syllogism. It goes like this: if p, then q; if q, then r. Therefore, if p, then r. We go from p to q and from q to r. And that means logically if event p happens, then event r will happen.
The key to working correctly with this logic though is remembering that it doesn’t work in the opposite direction. You can’t start with your conclusion and then go back to your conditional. These logic statements flow from left to right and not the other way around.
Jacob got detention and was late home. Can we conclude that he did not complete his homework?
Event r happened to Jacob. He was home late. Can we trace that backwards to event p? Can we say that he did not complete his homework? No, that’s faulty logic to say that. The only thing we can say is that he got detention and was late home. We cannot assume that he did not complete his homework.
The next statement says, Jennifer did not do her homework. Event p has occurred: a student did not complete their homework. Can we conclude that she was late home?
If event p happens — students don’t complete their homework — then event q will happen — they will get detention. If students get detention, they will be late home. And that means yes, absolutely we can conclude because Jennifer did not do her homework. And it’s true that students who do not complete their homework get detention. And students who get attention will be home late. Yes, that is a valid conclusion.