There exists in predicate logic

Question

I've got this sentence:

$\exists x \forall y (U(y) \rightarrow ( y = x \vee y = root))$

where U(x) means the program is to be upgraded and root is a constant that is the root program.

I'm told the sentence means 'at most one program other than root can be upgraded', but I don't understand how. As I understand it $\exists x$ means there must be one or more x, so why doesn't the sentence mean 'at least one program other than root can be upgraded'?

Answer 1

The sentence says that there exist at least one such $x$, such that any upgradable program $y$ is either $x$ or $root$. So, let $x$ be that one, and let $x'$ be another program such that $U(x')$. By your formula (applied to $x$), $U(x')$ implies that either $x'$ is $x$, or it is $root$.

Answer 2

It reads "There exists a program $x$ such that every program that can be upgraded is either $x$ or $root$ (or both, i.e. $x=root$)."

Also, your "at least one program other than root can be upgraded" would be written:

$$\exists x (U(x) \wedge x \not = root)$$