An Epistemic logic..

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Wed 26 May, 2010 08:48 am
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?

kennethamy

Wed 26 May, 2010 09:11 am
@Owen phil,
Owen;169044 wrote:
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?

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

HexHammer

Wed 26 May, 2010 09:13 am
@Owen phil,
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

kennethamy

Wed 26 May, 2010 09:20 am
@HexHammer,
HexHammer;169049 wrote:
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

Just marvelously relevant.

Owen phil

Wed 26 May, 2010 09:38 am
@kennethamy,
kennethamy;169048 wrote:
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.

kennethamy

Wed 26 May, 2010 09:45 am
@Owen phil,
Owen;169061 wrote:
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.

Why would you think I cannot know without knowing I know? Why would you think that all truths are believed?

For the first, on the contrary, I cannot know I know without first knowing? So, it would make no sense to think that I cannot know without knowing I know. For the second, it was true that the Earth was round, and it was not believed that the Earth was round.

Zetherin

Wed 26 May, 2010 09:58 am
@Owen phil,
Owen wrote:
6. p -> Bp.

If p is true, then it is believed that p is true.

But that's false. Why does something being true imply that it is believed to be true?

Quote:
Kp -> p
~p -> ~Kp

Both of these are true. If something is known to be true, it is true, if something is not true, it is not known to be true.

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

How do you make this leap?

Oh, this is how you did it (I think):

Quote:
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).

Why would you think that someone can believe p if and only if it is not known that p is not the case? People believe all sorts of things, even if they aren't true. (EDIT: For some reason I interpreted this as known by anyone, not simply the person doing the initial believing. Please refer to my latest posting.)

Emil

Wed 26 May, 2010 10:19 am
@Owen phil,
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.

Owen phil

Wed 26 May, 2010 10:43 am
@Emil,
Emil;169084 wrote:
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??

kennethamy

Wed 26 May, 2010 10:51 am
@Owen phil,
Owen;169094 wrote:
Why is it false?? Why is it funny??

Because (I think) each of your axioms is false.

Zetherin

Wed 26 May, 2010 10:56 am
@kennethamy,
Owen wrote:
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?

HexHammer

Wed 26 May, 2010 11:02 am
@Zetherin,
Zetherin;169102 wrote:
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.

Zetherin

Wed 26 May, 2010 11:02 am
@HexHammer,
HexHammer;169104 wrote:
Water can boil at a wide scale of temeperatures, it would be wise to use another anology.

No, that example is fine.

HexHammer

Wed 26 May, 2010 11:26 am
@Zetherin,
Fine I will let you have your exampe for youself.

Owen phil

Wed 26 May, 2010 11:43 am
@Zetherin,
Zetherin;169102 wrote:
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?

Yes, K(~p) -> ~Bp, is valid here.

Bp <-> ~K(~p).
~Bp <-> K(~p).
B(~p) <-> ~Kp.
~B(~p) <-> Kp.

"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?"

(K(~p) & Bp) <-> (~Bp & Bp) which is contradictory.
We cannot know that water does not boil at 37 degrees and believe that it does. (p & ~Bp) is also a contradiction.

B(p & ~p) <-> ~K(p v ~p) but all tautologies are knowable by 1. (p is a tautology) -> Kp.
That is K(p v ~p) is valid and therefore B(p & ~p) is contradictory.

There are no contradictory happenings that are believable or possible.

Zetherin

Wed 26 May, 2010 11:48 am
@Owen phil,
Owen;169117 wrote:

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?

Owen phil

Wed 26 May, 2010 12:09 pm
@Zetherin,
Zetherin;169120 wrote:
But what about cognitive dissonance? Doesn't this involve two contradictory beliefs held at once?

kennethamy

Wed 26 May, 2010 02:51 pm
@Owen phil,
Owen;169127 wrote:

Just what Zeth said. A person may believe both that a person is a prophet and knows when the world will end, but when the world does not end when the prophet prophesies it will, continue to believe the man is a prophet. See, When Prophets Fail by Leon Festinger.

Cognitive dissonance - Wikipedia, the free encyclopedia

Zetherin

Wed 26 May, 2010 03:32 pm
@kennethamy,
Owen wrote:
There are no contradictory happenings that are believable or possible.

I think your error is that you believe that for contradictory beliefs to be held, that there must be a contradictory happening (p and ~p). But this isn't true.

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, as we know, doesn't mean both propositions are true. We know that one is false.

Contradictory happenings are believable, but that doesn't imply that there are contradictory happenings. What we believe and what is, are two different things.

ughaibu

Wed 26 May, 2010 03:43 pm
@Zetherin,
Zetherin;169229 wrote:
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. . . . .
Are you serious, have you actually done this? If your example was believing a person to be both stupid and intelligent, I would accept it, but the example you gave. . . .

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