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 $x>y$"?
Hasn't the possibility of $x=y$ already been negated when we said $x\not\leq y$? 
Thanks in advance. 

You are correct, if we were speaking of $\leq/\geq$ relations we know and love, as standard ordering relations on the reals: The negation of $x \leq y$ is exactly $x > y$, and that would be the correct assertion if we were talking about a "trichotomous" ordering, where we take that for any two real numbers, one and only one of the following hold. $x\lt y, \lor x = y, \lor x>y$.
But your text is not wrong that $x \nleq y \implies x\geq y$, (that is, the right hand side is implied by the left hand side, and this would be a valid implication in even in the standard real numbers). And it seems your text is using strictly $\leq$ and $\geq$ so that for any two numbers $x, y$, we have one and only one of the following relations to consider: $x \geq y$ or $x \leq y$, andthese relations do not necessarily hold the same properties we know and love with respect to their standard meanings on the reals.
