Cn is a specific notation used in your particular textbook or course. I agree that a person familiar with logic should know what "logical consequence" is, but he/she does not have to know what Cn is.
Further, proof theory is one of the four main branches of mathematical logic (see "Handbook of Mathematical Logic" edited by Barwise). Logic programming (LP) is a quite small topic in proof theory. Concepts like "product of a program" and "closed under a program," which are particular to LP and possibly to a particular source about LP, are in no way basics of logic.
I've studied logic programming, but it was quite a few years ago, and I can't help with your question without being reminded of the definitions.