Then, if I'm not mistaken, the point being made here is that our idea of beauty is not a single, specific concept, but a vague feeling that could correspond to any of myriad different concepts. Therefore, we can't give necessary and sufficient conditions for a thing to be beautiful—unless, as you mention, we determine which concept of beauty we're referring to.

(It sounds to me like "concept" means one thing when lukeprog says it and something else when you say it.)