I have a question as to what it means for a knowledge-base to be consistent and complete. I've been looking into non-monotonic logic and different formalisms for it from the book "knowledge Representation and Reasoning" by Brachman and Levesque, but something is confusing me. They say:
We say a KB exhibits consistent knowledge iff there is no sentence P such that both P and ~P are known. This is the same as requiring the KB to be satisfiable. We also say that a KB exhibits complete knowledge iff for every P (within its vocabulary) P or ~P is known"
They then seem to suggest that by "known" they mean "entailed". They say
"In general, of course, knowledge can be incomplete. For example suppose KB consists of a single sentence (P or Q). Then KB does not entail either P or ~P, and so exhibits incomplete knowledge."
But when dealing with sets of sentences, I usually see these terms as being defined w.r.t. derivability and not entailment.
So my question is, what exactly do these authors mean by "known" in the above quotes?
edit: this post the math stack exchange helped clarify things.