Can we extend propositional logic to inlude the epistemic operators K and B, such that propositions about knowledge and belief can be decided ?

Kp, is read, it is known that p is the case. (Known)p.
Bp, is read, it is believed that p is the case. (Believed)p.

If we assume (Known)p as primitive, we can define:
(Believed)p as ~(Known)(~p).

It is believed that p is the case iff It is not known that p is not the case.
Bp <-> ~K(~p).

Kp <-> ~B(~p), is also a theorem.
It is known that p is the case, iff, It is not believed that p is not the case.

The axioms are:

1. |-p -> Kp.
If p is a tautology then p is known.

2. Kp -> p.
If p is known to be true then it is true.

3. K(p -> q) -> (Kp -> Kq).
If it is known that (if p then q) then, if p is known then q is known.

4. Kp -> K(Kp).
If p is known, then It is known that p is known.

5. p -> K(Bp).
If p is true then it is known that p is believed.

Example theorems:

6. p -> Bp.
7. Kp -> Bp.
8. (Kp & K(p -> q)) -> Kq.
9. (Bp & K(p -> q)) -> Bq.
etc.

Is this epistemic logic a useful instance of C.I. Lewis' (S5) modal propositional logic ?

whatdoyouthink?

I think that all modern logic is poor, sure it has it's few uses, but most logic does not account for uncertain phallacies.

If you master greater abstract logic, solve my story.

http://www.philosophyforum.com/lounge/general-discussion/7744-greater-logic.html

4. Is quite clearly false. I can know without knowing that I know.
5. Is clearly false. P may be true without being believed, and therefore, p can be true and p not known to be believed.
6. is false

Why do you say that 4 and 5 are clearly false?

6. p -> Bp.

Proof:

Kp -> p
~p -> ~Kp
~~p -> ~K~p
p -> Bp.
QED.

6. p -> Bp.

Kp -> p
~p -> ~Kp

~~p -> ~K~p
p -> Bp.
QED.

If we assume (Known)p as primitive, we can define:
(Believed)p as ~(Known)(~p).

It is believed that p is the case iff It is not known that p is not the case.
Bp <-> ~K(~p).

It is believed that p is the case iff It is not known that p is not the case.

That is also false. So even though it is funny and interesting playing along with such systems, this one is clearly false.

Why is it false?? Why is it funny??

It is believed that p is the case iff It is not known that p is not the case.
Bp <-> ~K(~p).

If it is known that p is not the case, then p cannot be believed to be the case.

This is assuming, I think, that a person cannot have a contradictory belief. I know that water doesn't boil at 37 degrees F., and if I believed water boiled at 37 degrees F, I would be contradicting myself, yes? Is this plausible?

Water can boil at a wide scale of temeperatures, it would be wise to use another anology.

If it is known that p is not the case, then p cannot be believed to be the case.

This is assuming, I think, that a person cannot have a contradictory belief. I know that water doesn't boil at 37 degrees F., and if I believed water boiled at 37 degrees F, I would be contradicting myself, yes? Is this plausible?

There are no contradictory happenings that are believable or possible.

But what about cognitive dissonance? Doesn't this involve two contradictory beliefs held at once?

Could you expand on this? I am not clear about this point.

There are no contradictory happenings that are believable or possible.

For instance, I can believe that the chemical composition of water is H2O, and I can believe that the chemical composition of water is not H2O. But that I believe both. . . . .