What is meant by “known” in “A knowledge-base exhibits complete knowledge if and only if, for every $P$, $P$ or $\neg P$ is known”?

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 Hector Levesque and Ronald J. Brachman, 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 $$\neg P$$ 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 $$\neq P$$ 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 $$\neg P$$, 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 helped clarify things.

3 Answers

It seems that they are stating that a knowledge base is consistent if and only if it never asserts the truth of both the truth and the negation of a particular P. In other words, a knowledge base is consistent if it never contradicts itself. Their definition allows incomplete knowledge bases to be considered consistent; by their definition, an empty knowledge base is still considered a consistent one.

I don't think they mean that "known" is equivalent to "entailed" --- in a reasonably complicated system, one cannot be expect to know every sentence which is entailed. Perhaps their example is just a bit lacking.

I guess in this context "known" means nothing but either $P$ or $\neq P$ is in the KB; further, exactly one of these two needs to be in the KB.

Just think about what it means if $P$ and $\neg P$ are in the KB, then the KB is obviously inconsistent.

And if neither of these two sentences is in the KB, then no information at all can be retrieved from the KB and it remains unclear whether $P$ or $\neg P$ is supposed to be a true statement; thus $P$ is "unknown".

However, if exactly one of these two is in the KB, then one has all information that one needs about $P$ (due to the excluded middle); $P$ is "known" and consistent.