Previous/next posts links showing when I don't need them to

I can’t seem to track down what I’m after, which convinces me I’m going about it the wrong way.

I’m developing a theme with accessibility foremost in mind. It’s very simple so far, and I’m currently adding to it. But first I need to take something away. Namely the code WordPress outputs, in this particular circumstance. I’m specifically attempting to address pagination — I’m not entirely sure that is the proper term. But the previous/next post links that appear on the index, archive and search pages?

I’m attempting to have said links sit in an unordered list. Trick is how do I have the list not appear when there isn’t other pages? Plus not have the previous posts link list item output on the last page and same deal with the next posts link list item on the first page? Is this even possible?

I’ve managed to dig up some code that makes this behaviour happen sans the list, but that isn’t what I’m trying to accomplish. I’ve found a solution for my single template and, for consistencies sake a least, I’d really like it be the same. Well it would be working if I had more than one post published. But same goes for that instance. How would I make the unordered list not show when it’s not needed.

My site is http://abledaccess.com. Any help would greatly be appreciated. Thanks.