What is the significance of the phrase "for any two objects $x,y \in X$, the statement $x \leq_X y$ is either a true statement or a false statement."?
By the law of the excluded middle, I think that it's always the case that $x \leq_X y$ is either true or not true. Does that mean a "partially ordered set" is not more restrictive than simply a "set"?
On
You're right. That parenthetical statement by itself effectively says nothing; I think the authors put it there to help the reader conceptually think about the nature of the structure involved. But it is not what makes it a partial order.
It is the three properties listed after that that makes it a partial order ... and which makes it more than simply a 'set'. Indeed, the fact that we are dealing with a binary relation already makes it more than just any 'set'.
On
There is a word thus at the beginning of that sentence. That is, [$\leq_X$ being a relation on $X$] implies that [for all $x,y\in X$, $x \leq_X y$ is either true or false].
On
It just means that the $\leq_X$ binary operator must be a predicate function that is in fact defined for any pair of elements from $X^2$ (ie, it is surjective ). $~\leq_X : X{\times} X\twoheadrightarrow \{\bot,\top\}$
Which, yeah, should not really need to be stated, but better safe than sorry.
Not having more context I would have to guess. But it looks like instead of looking at relations as subsets of $X\times X$, the text is looking as relations as predicates: so $ x\leq y$ is established by means of a certain assertion.
The issue is that with that point of view it could be the case that the assertion is neither true or false, in the same sense that "this sentence is false" is neither true or false (i.e., Russell's Paradox).
So the text is saying that you have a relation when the $x\leq y$ is actually a statement (that is, true or false) for all $x,y$.