Logic: Formalization of statements and correct writing of answer

Question

I have following statements:

1. Everybody having a musical ear is able to sing properly
2. Nobody is a real musician if they can't electrify their audience
3. Nobody who does not have a musical ear can electrify their audience
4. Nobody, except a real musician, can compose a symphony

And need to answer this question:

Which properties does a person have who has composed a symphony?

I formalized these statements as follow

1. $M\implies S$
2. $R\implies E$
3. $E\implies M$
4. $C\implies R$

And then concluded that if a person composed a symphony, they have all properties ($R, E, M, S$).

My questions are: is my formalization correct? How can I mathematically correctly write my conclusion? (Since C implies R, and R implies E (...), a person who has composed a symphony satisfies all implied properties)

Answer 1

Yes, your formalisations are correct, if a little rough.

To be more precise, I would use predicates. For example, $$\forall x(Mx\to Sx)$$ for 1.

Answer 2

I agree with Shaun that you need to use quantifiers for these. Also, I would translate more directly, i.e include the negations that clearly appear in 2,3, and 4. So:

1. $\forall x (M(x) \rightarrow S(x))$

2. $\forall x (\neg E(x) \rightarrow \neg R(x))$

3. $\forall x (\neg M(x) \rightarrow \neg E(x))$

4. $\forall x (\neg R(x) \rightarrow \neg C(x))$

By contraposition, 2,3, and 4 are equivalent to:

2'. $\forall x (R(x) \rightarrow E(x))$

3'. $\forall x (E(x) \rightarrow M(x))$

4'. $\forall x (C(x) \rightarrow R(x))$

And now you see that you had all directions of the conditionals correct!