If $x\not\leq y$, then is $x>y$, or $x\geq y$? I'm currently reading about surreal numbers from here. 

At multiple points in this paper, the author has stated that if $x\not\leq y$, then $x\geq y$. 

Shoudn't the relation be "if $x\not\leq y$, then $x&gt;y$"?

Hasn't the possibility of $x=y$ already been negated when we said $x\not\leq y$? 

Thanks in advance.
The paper defines $\leq$ and it is not necessarily the "standard" $\leq$ that you and I are used to. It is often better to substitute some other symbol like $x\preceq y$, and realize we have to prove stuff about this symbol before we can use it.

In particular, in that paper, it is proved in Theorem 1.4 that $x\not\leq y$ implies $y\leq x$. It does not prove that $y&lt;x$. 

So trying to reason about $\leq$ as it if means "less than or equal" is a mistake. There is no definition of "less than" to use here.  There is only $\leq$.

There are examples where $x\leq y$ and $y\leq x$. This means that if we define $x&lt;y$ in terms of $x\leq y$ and $x\neq y$ then we get $x&lt;y$ and $y&lt;x$ and therefore we find that $&lt;$ defined this way is non-transitive.

I think the name "surreal" should give a strong hint at the non-standard nature of these "numbers."
Here's another way of looking at it.

$$(x\le{y})\leftrightarrow (x&lt;y \lor x=y)$$

$$(x\not\le{y})\leftrightarrow(x&gt;y\land{x\ne{y}})$$

$$x\not\le{y}\rightarrow{x\ne{y}}$$

Therefore:
$$x\not\le{y}\not\rightarrow{{(x\ge{y})}}$$