Let $M(x, y)$ be “$x$ has sent $y$ an e-mail message” and $T(x, y)$ be “$x$ has telephoned $y$,” where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all e-mail messages that were sent are received, which is not the way things often work.)
a) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class.
b) Every student in the class has either received an e-mail message or received a telephone call from another student in the class.
The solution manual answers are as follows:
a) $$\exists x \forall y (y \neq x \rightarrow (M(x, y) \vee T(x, y)))$$
This makes sense to me, because we don't know if the person has sent an e-mail to him/herslef, or telephoned him/herself, and that's not emphasized in part a, and that's why even if $x = y$ the statement would still be true.
b) Here $y$ is "another student": $$\forall x \exists y (x \neq y \wedge ((M(y, x) \vee T(y, x)))$$
Here the word "another" is taken to mean that these students are different, and so $x = y$ would make the statement false, but we don't consider "else" in part a to be as strong as "another" in part b. So what rules are exactly to be followed when translating natural language to symbolic form? Is there a logic behind all this?
The words "everyone else" in part a corresponds to $y \neq x$ in the proposed solution. As for your second point, there are no rigid rules about translating natural language into formal logic. Natural language is riddled with ambiguities. The value of translating natural language statements into formal logic is to expose the ambiguities. For example, in your example a, it is just about plausible that "everyone else" means everyone apart from you (the speaker): the given solution would not agree with that reading.