Hi, Evan! It's great to see you here, and great to see ideas are starting to bubble in this conversation. I"m hoping some people here will start to work in groups, or at least get into serious discussions that lead to new research.
I don't know much about Formal Concept Analysis but from my brief glances at it, it looked startlingly similar to classical predicate logic, where a model associates to each predicate a set of entities obeying that predicate. I was disappointed, in the treatments I saw, by a certain lack of mathematical sophistication. I felt that a little dose of category theory could work wonders here. But this was just a rough first impression, and I probably didn't read the best stuff. (That is, the stuff aimed at people who already know lots of math.)
> Prolog is usually interpreted in a special, fixed model (the minimum Herbrand model). But we even if we drop that restriction and interpret a Prolog program more liberally, like a typical first-order theory, it's not clear how regular logic and Prolog compare.
I've never studied [Herbrand semantics](http://logic.stanford.edu/herbrand/herbrand.html)? Is that the realm where the "minimum Herbrand model" lives?