The following is written as if I were giving my own best answer to a student.
It's probably an accident of history that Aristotle defined classical logic in a specific way, and that classical logic has been almost exclusively used as the foundation of mathematics ever since. Common sense tells us that Aristotelian logic is an oversimplification of how we actually reason about the real world. For example, Aristotelian logic claims that every proposition is either true or false, but the following are statements that many reasonable people would say are neither true nor false:
Green is a nicer color than black.
Harry Potter had a great-great-great-great grandfather named Joe.
Based on the equal protection clause of the U.S. constitution, it's unconstitutional to deny gay people the right to get married.
My terrier is an intuitive Buddhist.
The vast majority of the literature of mathematics assumes classical logic, so for professional mathematicians in most fields, strong knowledge of classical logic is both necessary and sufficient.
But people didn't just invent fuzzy logic or multi-valued logic for no reason. They invented them because those logics were more suitable for what they wanted to do.