Discrete Math - Rule of Inference

I have to determine whether or not the statement is valid. If yes, state the rule of inference.However, I have one small problem in this problem. Let me state the problem first.

I.) If I study hard, then I ace the quiz. I ace the quiz. Therefore, I study hard.

My work:

Let:
p = I study hard
q = I ace the quiz

p -> q
q
-------
Therefore, p


It's very similar to Modus Ponus inference: Except that the last two lines are changed. p for q ; q for p

p -> q
p
-------
Therefore, q


There are no other inferences that are exactly similar to the format I've made. Does that mean that this is NOT valid? Or could this possibly be Modus Ponens despite the change.

• You are right : modus ponens does not license this inference. It is not valid as you can easily see with some counter-examples. – Mauro ALLEGRANZA Oct 6 '15 at 7:06

1 Answer

Yes, your inference is invalid. For instance, you could just be very lucky with the questions. Then it is possible to ace the test without studying (or at least without studying hard).

However, if you failed the test, we can conclude that you didn't study hard. This inference is called modus tollens, or contraposition.

• Thanks Arthur! One question, so a statement is always valid as long as it matches at least one of the inferences right? – Xirol Oct 6 '15 at 7:10
• @Xirol Yes. However, if you want to apply your rules of inference to the real world (like here, with studying for and taking a test), then you have to be absolutely certain that your assumtions are true, or otherwise get a result that is somewhat certain but not an absolute fact. For instance, it might be that your teacher feels sadistic that morning and give you a test that you have no chance of even answering a question. Then you could study for it all month, but you still wouldn't ace it. – Arthur Oct 6 '15 at 7:36