Knowledge set and knowledge base

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.

1. If then .
2. If then for some .
3. If then or .

Then we have the following

Lemma 5. (1), (2) and (3) are equivalent.

We now study the semantical characterization of knowledge sets. Let be any KT5-model. For any and , we define by:

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.