Let L be any language. We will make the notion of the totality of one's knowledge explicit by the following definitions:
Definition. is a knowledge set for St if K satisfies the following conditions:
(KS1) K is consistent.
(KS2) , where .
(KS3) If then for some .
Definition. is a knowledge base for St if B satisfies the following conditions:
(KB1) B is consistent.
(KB2) , where ,
(KB3) If then for some .
By (KS2) (or (KB2)) we see that any element in K (or B, esp.) has the form . It is easy to see that if B is a knowledge base for St then is a knowledge set for St.
Let be consistent. We compare the following three conditions.
Then we have the following
Lemma 5. (1), (2) and (3) are equivalent.
We now study the semantical characterization of knowledge sets. Let
KT5-model. For any and , we define
Since, as we will see below, is a knowledge set for St, we call it the knowledge set for St at w.
Lemma 6. is a knowledge set for St.
Let K be a knowledge set for St. We say characterizes K if .
Theorem 7. Any knowledge set is characterizable.