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A knowledge base (KB) is complete if there is no formula α such that KB ⊭ α and KB ⊭ ¬α.

Example of knowledge base with incomplete knowledge:

KB := { A ∨ B }

Then we have KB ⊭ A and KB ⊭ ¬A.

In some cases, you can make a consistent knowledge base complete with the closed world assumption - that is, adding all not-entailed literals as negations to the knowledge base. In the above example though, this would not work because it would make the knowledge base inconsistent:

KB' = { A ∨ B, ¬A, ¬B }

In the case you have KB := { P(a), Q(a), Q(b) }, you have KB ⊭ P(b) and KB ⊭ ¬P(b), so with the closed world assumption you would get KB' = { P(a), ¬P(b), Q(a), Q(b) } where you have KB' ⊨ ¬P(b).




Pretty obvious, when you think about it.