Why is this predicate false?

Question

I am stumped at my professor's answer to this predicate logic. all x and y are natural numbers.

     ∃y∃x(x >= y)

I think it is true, since there is a pair $(Y, X)$ like $(1, 5)$ such that $X \geq Y$.

But my professor says it is false!

Why?

Thank you

Answer 1

I would ask what the domain of discourse is; however, the formula $$\exists y \exists x (x \geq y)$$ is true in every nonempty domain of discourse.

Let $\mathbb{U}$ be the (possibly empty) domain of discourse. Then, $a \geq a$ for all $a \in \mathbb{U}$. If $\mathbb{U}$ is nonempty, the existence of such an $a$ is guaranteed. However, if $\mathbb{U}$ is empty, then the existence of such an $a$ is not guaranteed.

In most settings, only nonempty domains of discourse are considered. Are you aware of an exception in this case?