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 (*KS*2) (or (*KB*2)) 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.

- If then .
- If then for some .
- 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.

Fri Jun 20 13:39:43 PDT 1997