It’s a question from a Mathematical Olympiad of public schools. I’m pretty sure first-order logic is assumed here.
However, you’re correct that there are other valid logical systems in which different interpretations are possible. These are not idle curiosities, either; they’re active areas of research in foundations; moreover, there’s no obvious “natural” choice of which of them is “correct” - they’re all just different sets of rules for playing a game. Mathematicians have used (without always realizing it) first-order logic for centuries, and it’s permitted the proofs of some amazing and beautiful things. But some are uncomfortable with the extremely abstract nature of those conclusions (for example, they allow you to assert the existence of things with certain properties even though there’s no way to give a specific example of something with those properties). Constructive mathematics is growing in popularity as an alternative to this.
