# Representing a sentence using propositional logic

I am confused regarding a propositional logic representation of a sentence.

Please note that this sentence is not realistic:

"A person who is male (M) is smart (S) if he is tall (T), but otherwise is not smart."

Which of the following propositional logic formulas correctly represents the above:

(a) $$(M \wedge S) \iff T$$

(b) $$M \Rightarrow (S \iff T)$$

Note that $$\Rightarrow$$ and $$\iff$$ represents "only if/implies" and "iff and only if," respectively.

The solution to the problem is (b)

The solution claims that (a) is NOT an option because it asserts, among other things, that all tall people are male, which is not what is asserted. This makes sense to me, I think (not confident, but this is not the main point of this post).

The solution claims that (b) IS a correct representation because it is saying that if a person is male, then they are smart IF AND ONLY IF they are tall. I don't understand that, and based on my current understanding, I don't agree with this because the "IFF" part is saying that if the person is tall, then they are smart, and that is not what the problem statement said. The problem statement only states that a person who is male is smart if he is tall.

So I feel like (b) is not correct, but if you swap out the $$\iff$$ for $$\Rightarrow$$ then it becomes correct.

Am I misunderstanding something here?

The sentence says that a male "is smart (S) if he is tall (S)", that gives $$T \Rightarrow S$$, but it also says "but otherwise is not smart", which gives $$\neg T \Rightarrow \neg S$$, and this is equivalent to $$S \Rightarrow T$$ by contraposition. Thus together this gives $$S \iff T$$ under the condition $$M$$. Therefore (b) is correct.