Thanks guys! I started late and didn't think anyone was still visiting the Chapter 1 threads.
That really opened my eyes. The use of all upper bounds as a way to get as tight a limit as possible is great, and so much better than my crude suggestion.
I'd never heard of completions before, so that's a great concept to add to my brain. The Dedekind-MacNeille completion is nifty. So from a quick skim of the Wikipedia page, it seems that the intuition is to take the identity \\(\wedge \uparrow a\\ = a\\) and "widen" it to talk about sets instead of elements, which forces new elements to appear as needed in the completed lattice.
I wonder if there's a way to use the completed lattice to back-transfer useful formulas from the completed lattice into the original poset, like my identity above. It would be cool if one could identify which formulas were "safe" to use in the poset -- technically incorrect but still formally usable. I guess the answer is to just use the completed lattice instead.
I appreciate the links and references.