The cases when "A implies B" is true

Question

I'm reading this book How to Read and Do Proofs. In "Preface to the Student" and "Preface to the Instructor", the author claims to keep the material simple and easy to be followed by students. I was so exciting until I have reached to this example

Suppose, for example, that your friend made the statement, If $\underbrace{\text{you study hard}}_{A}$, then $\underbrace{\text{you will get a good grade}}_{B}$. To determine when this statement “A implies B” is false, ask yourself in which of the four foregoing cases you would be willing to call your friend a liar. In the first case—that is, when you study hard (A is true) and you get a good grade (B is true)—your friend has told the truth. In the second case, you studied hard, and yet you did not get a good grade, as your friend said you would. Here your friend has not told the truth. In cases 3 and 4, you did not study hard. You would not want to call your friend a liar in these cases because your friend said that something would happen only when you did study hard.

Now cases 1 and 2 are clear but I didn't understand why cases 3 and 4 are true? To me at least cases 3 and 4 are unknown or there is no conclusion we can draw from the given info. Can anyone explain to me what does the author mean by the following sentence

You would not want to call your friend a liar in these cases because your friend said that something would happen only when you did study hard.

Answer 1

The idea is that $A\rightarrow B$ is automatically true when $A$ is false, regardless of the truth value of $B$.

For this specific example, the friend didn't say anything about what would happen should you not study hard, only when you did study hard, so you can't say that he lied.

Another example that might make this a little more clear is this: "If the traffic signal turns green, then the cars will go." We wouldn't say that this statement is false if we'd only encountered the situations in which the traffic signal hadn't turned green, since it only talks about when the signal is green, and not any other situation.

Answer 2

The friend made the implication $A \to B$. He did not make other claims, such as $\lnot A \to \lnot B$ (not studying implies bad grades).

i.e if you get good grades without studying you are just lucky and your friend did not lie.

Answer 3

Assume someone proves

Theorem. When $A$: "$\>\pi$ is normal", then $B$: "The Riemann Hypothesis is true".

Therewith the fact $A \Rightarrow B$ is established, and nobody would call this mathematician preposterous, even when in 2025 someone else proves that $\pi$ is not normal. Note that here $A$ as well as $B$ were simple propositions, both of them either true or false.

In the case of your friend's statement we actually have two predicates $A(x)$ and $B(x)$ about students $x$. For some students $x$ the predicate $A(x)$ is true, for other students it is false, and similarly for $B(x)$. The statement $\forall x: \ A(x)\Rightarrow B(x)$ has a simple meaning. It is falsified only if some student got a bad grade, even though he had worked hard.

Answer 4

The author means by // You would not want to call your friend a liar in these cases because your friend said that something would happen only when you did study hard. // that calling someone a liar requires a claim which evidence overturns.

The claim that studying hard leads to good marks strictly speaking doesn't make a claim about not studying hard.

Of course, in terms of other logics or ideas like Grice's "conversational implicature", when someone in everyday life says "if you study hard you'll get good marks" you are understood to be also saying that if you don't study hard you'll get bad marks.

But this is Grice's interpretation of how everyday language works, a kind of informal logic. In most formal logics, the statement says nothing about the consequences of not studying hard, so cases 3 and 4 fall in this area which formal logic judges to be outside the claim, therefore not something you'd accuse your friend of lying about.

Answer 5

You may like this example to understand the implication more clearer.

A father declared in public (or more formally he declared in a legal document),"I shall transfer my all money to my son if he performs outstanding in his job."

Now, considering all four possibilities regarding the Truth value of the antecedant and consequent part, we have:

• The son performs outstandingly and the father transfers all his money.
• The son performs outstandingly and the father doesn't transfer all his money.
• The son fails to perform outstandingly and the father still transfer all his money.
• The son fails to perform outstandingly and the father doesn't transfer all his money.

If one really do not know what happened but somehow the matter goes to the Court where you are the Judge. Now, think in which of the above four cases you are going to declare the father culprit? I think it is easy for you to believe that the father is innocent if cases 1, 3 and 4 occur as in case 1 he made his promise while in the last two cases, nothing has been committed (about the antecedant part) by the father.