Help understanding why truth table is the way it is (Very basic)

Question

I'm really embarrassed that I don't get such a simple concept but please bear with me. Also, sorry it is a bit long.

Ok, so I am studying elementary logic using the book Book of Proof. It gives the example statement If you pass the final exam, then you will pass the course.

Then gives the following table (sorry, it's an image because I don't know how else to do it):

I presume I know how to read it. You evaluate the truth of statement 1, then statement 2. Then you find the line that has the two first columns matching these respective truth values. Then you check the third column to see what the truth value of the final statement 3 is. Correct?

For the given example, the first line is easy to understand. If 1 and 2 are true, she was clearly making a true statement. A is sufficient for B. The second line is easy as well, she was clearly lying that A is a sufficient condition for B because even though A, B did not happen.

However, the third and fourth lines I have trouble understanding. For the third line, if you pass the course in spite of not having passed the exam (maybe there was another way to pass the course). However, we cannot possibly have evidence to verify that the professor was telling the truth and that, in fact, A is sufficient for B. Maybe you passed by other means and, meanwhile, someone who did pass the exam ended up not passing the course (Ok, it says in this example "you" but it could have been more general and said "if one passes..." and the problem would be the same). The author argues

"Now consider the third row. In this scenario you failed the exam but still passed the course. How could that happen? Maybe your professor felt sorry for you. But that doesn’t make her a liar. Her only promise was that if you passed the exam then you would pass the course."

I agree that we cannot say with certainty she lied, but we also cannot say she didn't. There simply was not sufficient evidence.

As for the 4th, he says

In that scenario you failed the exam and you failed the course. Your professor did not lie; she did exactly what she said she would do. Hence the T in the third column.

Can we really infer that if you didn't pass the exam, you necessarily wouldn't pass the course? Early on the author himself conjectures that there could be other ways of passing the course (giving the example of the teacher having pity). Did he just give a bad example to illustrate this or am I not getting it?

I'm very embarrassed by this, I'm sure I'm not understanding something very simple. For example, say someone made the following open sentence, which is not always true but could be, and we would like to test the truth of it using the very same table.

If a is a multiple of 2, then a is divisible by 8.

Let's make it a statement in order to test it,

If 16 is a multiple of 2, then 16 is divisible by 8.

In this particular case it is true, but can we really say that because 16 is a multiple of 2, it is divisible by 8? Isn't this what would follow from the table? I don't understand how that is a meaningful statement and not misleading, since really it is divisible by 8 because it is divisible by 2 three consecutive times, not only once.

When does the table hold and when does it not? Sorry for such a stupid question guys, and thanks for all your help!

IMPORTANT EDIT: littleO requested the introductory text and, as I was pasting it, I think I might have understood what the author meant, but I still don't really understand how to generalize this table and use it meaningfully with other examples. I think the table is supposed to represent strictly under which circumstances she would have technically "lied beyond any doubt". He says simply:

Your professor is making the promise (You pass the exam) ⇒ (You pass the course). Under what circumstances did she lie? There are four possible scenarios, depending on whether or not you passed the exam and whether or not you passed the course. These scenarios are tallied in the following table.

I'm not sure this starts making sense after this, but even if it does I don't see how this applies to the previous example he gave in the book.

P : The integer a is a multiple of 6. Q : The integer a is divisible by 2. R : If P, then Q.

How does ~P and ~Q being false makes P => Q true? For example, I know that "P:3 is divisible by 5" is false and that "Q:7 is divisible by 8" is also false. According to the table, I would have P => Q be true, which doesn't make sense to me.

Answer 1

The statements "if $a$ is a multiple of $2$, then $a$ is a multiple of $8$" and "if $16$ is a multiple of $2$, then $16$ is a multiple of $8$" are very different. You are correct that the first is false and the second true, but there is nothing paradoxical about this.

The first statement says that for every $a$, if $a$ is a multiple of $2$, then $a$ is a multiple of $8$. I know you don't see where it says, "for every $a$" but this is a common convention, that you are actually quite used to. What about $\sin^2 x + \cos^2 x = 1.$ This doesn't say, "for every $x$", but that's what it means.

As to the truth table, it takes some getting used to, and attempts to justify it by appeal to every day usage are beside the point, in my view. In everyday usage we don't reason from false premises, at least not intentionally. In formal argument though, the convention comes in very handy.

Let's take the true statement, "if $x$ is a multiple of $8$ then $x$ is a multiple of $2$. You'll agree that's true for every $x$, I think. Wait a sec! Is it true when $x=9?$ Is it true when $x=4?$ Yes and yes, if we adopt the convention in the table.