Confused between if-then (→) and iff (↔)

Question

Consider the following argument from my assignment.

Tarzan will only be happy if his pet monkey comes home. The monkey will come home if it rains. It won't rain today. So Tarzan will be unhappy.

Let

• $H$ mean "Tarzan will be happy".

• $C$ mean "the monkey will come home".

• $R$ mean "it will rain today".

Question: Express the argument symbolically as a logical implication, using the symbol $\implies$.

Here’s the provided solution from the teacher.

$$(H \to C) \land (R \to C) \land \neg R \implies \neg H$$

I don’t understand why I can’t translate the first argument $(H \to C)$ as $(H \leftrightarrow C)$ since Tarzan will only be happy if his monkey comes home.

Answer 1

Just because the monkey is home, that doesn't automatically mean that Tarzan is happy. That's how "only if" is interpreted in conventional mathematical English. Tarzan can't be happy if the monkey is away, but when the monkey is there, Tarzan could be either happy or non-happy.

Answer 2

Think like this:

1) $\Rightarrow$ means implies, that is

It rains $\Rightarrow$ I take an umbrealla with me.

Observe that this is only in one way because the fact that I take an umbrella with me won't make it rain. So I cannot say

I take an umbrella $\Rightarrow$ It will rain.

2) An if and only if statement works in both ways

Raindrops are falling from the clouds $\iff$ It rains

Statements which have $\iff$ between them are called equivalent and are usually interchangeable.

Hope this helps clarify some things

Answer 3

Sentences containing 'only if' are always tricky.

To make some sense of these, let's use a different use of 'only if':

"You can be a bachelor only if you are a male"

I think this one is a little easier to analyze:

We clearly have $B \rightarrow M$: if you are a bachelor, you have to be male.

Or, analyzed differently: the statement says that if you are not a male, then you cannot be a bachelor, which is $\neg M \rightarrow \neg B$. But by contraposiiton, that's equivalent to $B \rightarrow M$

OK, but do we also have $M \rightarrow B$ (and thus effectively $B \leftrightarrow M$?)

No! Just because you are a male does not make you a bachelor.

OK, and so it is with Tarzan and his pet monkey:

Tarzan will only be happy if his pet monkey comes home

This clearly means that as long as his pet monkey is not coming home, Tarzan will not be happy, i.e. $\neg C \rightarrow \neg T$, which by Contraposition is equivalent to $T \rightarrow C$

OK, but does it also mean that Tarzan will be happy if his pet monkey does come home? Well, maybe Tarzan would only be happy if his pet monkey, while coming home, isn't bringing in any of his annoying friends. So no, just because his pet monkey comes home does not guarantee Tarzan to be happy so we don't have $C \rightarrow T$.

Indeed, if we want to say that Tarzan is happy exactly whenever his pet money is coming home, we should say something like "Tarzan will only be happy if and only if his pet monkey comes home" .. which they didn't say.