Why is this contrapostive assumed to be true?

Question

I have a problem with the following logical deduction:

$ incabal(Darren) \implies incabal(Martyna) $

This would read, "If Darren is in the cabal, then so is Martyna."

Later in the homework we learn that Martyna is not in the cabal. Our text makes the statement that

consider that (v) implies its contrapositive: if Martyna is not in the cabal, then neither is Darren. Therefore, since Martyna is not in the cabal, than neither is Darren.

From my brief understanding of this "mathematical game", I am willing to concede that this may be correct, however, it disagrees with my experience. All this statement covers is the case in which Darren is in the cabal. It does not cover the case in which Darren is not in the cabal. Let's say Martyna is always in the cabal. Then we still satisfy the original implication: Darren is not in the cabal but Martyn is. Why is the contrapositive assumed to always be valid, and does it really make sense from a rational standpoint?

Answer 1

"All this statement covers is the case in which Darren is in the cabal."

I think I would agree with this.

"It does not cover the case in which Darren is not in the cabal."

I think I would agree with this too, provided that you state it a bit more precisely: the statement does not permit any deductions if Darren is not in the cabal. However, in the follow-up question you have cited, there is no attempt to deduce anything from a supposition that Darren is not in the cabal. On the contrary, the deduction results in the conclusion that Darren is not in the cabal, and this is a very different thing.

It's important to realise that the mathematical use of "if... then..." is not entirely the same as its use in ordinary language. In ordinary language, "if... then..." frequently has a strong connotation of causality. A statement such as "if you study hard then you will pass the course" is taken to mean that studying hard will cause you to pass the course. From this point of view, the contrapositive, "if you do not pass the course then you didn't study hard" makes no sense at all, as it suggests that failing would cause you to not study hard, an effect preceding its cause.

The mathematical use of "if... then...", however, is purely defined by its truth table and has nothing to do with causality. I would suggest that the best way to think of it is in terms of knowledge. From this point of view, "if you study hard then you will pass the course" means the following: suppose you assure me that you studied hard: then I can be certain that you will pass. Consequently, if you admit that you didn't pass, then I can be sure that you didn't study hard. Which is to say, "if you didn't pass, then you didn't study hard", the contrapositive of the former statement.

Hope this helps!

Answer 2

Implication is a bit of a weird animal with regards to language and how we come to know implication outside of mathematics. A lot of students get tripped up on this early on. I'm of the opinion that it is best to think of the truth or falsity of an implication statement by considering when it is false. If you find the case when it is false, it must be true for all other cases by law of excluded middle.

Think about it this way: when can you say that $incabal(Darren)\Rightarrow incabal(Martyna)$ is not a true statement? Well clearly the only time this is false is if Darren is in fact in Cabal but Martyna is not. Thus it must be true otherwise.

Taking the contrapositive (converse inversed) says that if Martyna is not in Cabal, then Darren is not either. When is this statement false? Well by similar reasoning above, only when Martyna is not in Cabal but Darren is (which is exactly as above). Thus it is true otherwise. Hence they have the same truth tables and are equivalent statements.

Answer 3

Your understanding is incorrect. Let us take a look at what the statement $A \implies B$ means.

$$\text{$A \implies B$ means if $A$ occurs, then $B$ *has to occur*.} \tag{$\star$}$$

Now if we figure out that $B$ has not occurred, then we can conclude that $A$ has not occurred, since if $A$ had occurred then $B$ has to have occurred by $(\star)$.

Answer 4

In this case, if D is in the cabal, then M is def in the cabal. But if M is not in the cabal, then M is not in the cabal. Why can we make this assumption? Because suppose M was not in the cabal and D was in the cabal. Then that means M would have to be in the cabal since D is in the cabal, which is a contradiction because we were supposing that M was not in the cabal. Therefore, D cannot be in the cabal.

In general, if $A \implies B$, then $\neg B\implies \neg A$ because suppose we had $\neg B$ and $A$, then that means $B$ because $A \implies B$, which makes a contradiction.

Answer 5

You have to understand, semantically, what the implication means (this is the definition of implication):

$$ p \rightarrow q \equiv \neg p \vee q $$

This means that either $p$ is true and $q$ is true or $p$ is false (keep in mind that this means that $p$ and $q$ could both be false--this is what the english usually doesn't correctly represent). Therefore if $q$ is not true (i.e. $q = \text{false}$) then the only way for the implication to still be true is that $p$ must also be false.

Answer 6

You know that Martyna is not in the cabal. Can Darren be in the cabal?

The answer clearly can't be yes: if it were, then Martyna would be in the cabal.

Since it can't be yes, it must be no.