I'm planning a mountain hiking trip and one of the issues is that the trails that we're following will almost certainly be covered in ice. There are two of us and we'll both be wearing shoe traction equipment like this: YaxTrax Extreme Outdoor Traction. The trails follow ridge lines and moderate slopes on relatively low altitude mountains (5,945' or 1,812 m). However there is the possibility that a slip on the ice will lead to sliding right off the edge of a cliff or steep drop.
My question then is, should we rope ourselves together so if one of us started sliding the other could "self arrest" by grabbing on to a tree, or even throwing ourselves in the deep snow? We will not have ice axes or climbing harnesses. I was picturing a bowline tied around each waist. Does this seem smart or does it seem like a way to guarantee both of us dieing if one were to fall?