Whenever I glanced at it, [formal concept analysis](https://en.wikipedia.org/wiki/Formal_concept_analysis) seemed like unimpressively obvious application of classical logic and a bit of set theory. Quite possibly I've missed the interesting parts. But throwing a Galois connection into the stew would certainly liven it up!

One of my favorite Galois connections is the one due to Galois, and this again has an "op" in it. He was classifying subfields \$$K\$$ of a given field \$$F\$$, by associating to each subfield \$$K \subseteq F\$$ the subgroup \$$G\$$ of the group \$$\mathrm{Aut}(F)\$$ of symmetries of \$$F\$$ that act as the identity on every element of \$$K\$$. The bigger the subfield \$$K\$$ is, the smaller the subgroup \$$G\$$ is. So, we get a monotone function from the poset of subfields of \$$K\$$ to the _opposite_ of the poset of subgroups of \$$\mathrm{Aut}(F)\$$. We also get a map going back, and these form a Galois connection, and that's how Galois theory works.