Deducation to do with knowledge and belief

Knowledge and belief are fairly intractable subjects. Nevertheless, we can give some properties that we believe knowledge and belief have, and deduce from that.

First, any statement about the world can be known or believed (or both, or neither). In addition, if A is something that can be known or believed, then so can the statements ‘I believe A’ and ‘I know A’. That is, it’s possible to have beliefs about your knowledge, or knowledge about your beliefs, etc.

Knowledge ought to be stronger than belief. In particular, if you know something you should also belief it. I cannot sensibly know that apples grow on trees but not believe it. If I don’t believe it, it is because I don’t know for sure.

Whilst ‘doublethink‘ is perfectly possible in reality, it is not a property of a good belief system. It is fair to assume that if you believe something, you do not believe the opposite to it.

If somebody asks you whether you know a statement is true or not, you’re aware if you don’t know, and can tell the person that. You know your ignorance. That is, if you don’t know something, you know you don’t know it.

And finally, the tightest coupling between believe and knowledge comes from the idea that if you believe something, you believe you know it. Let us give an example. I know that apples grow on trees. I also believe that it would be odd for something to both grow on trees and be an animal. Thus, I conclude in my head that I believe apples are not an animal. But I feel my believes are fairly secure, so although I don’t know apples are not animals, I believe I know it.

There are other rules one might want to add. For instance, it might be that knowledge is such a strong condition that it is impossible to know something that is false. Others are to do with you knowing what you believe, and others still to do with being able to deduce things from knowledge and believe of implications (as was done in the example to do with animals and apples). Nevertheless we do not need any more assumptions, so we stop here.

The four assumptions we have listed are enough to prove that if one believes something, then they know it. As one of the assumptions was that knowledge implies belief, together they imply that you believe something if and only if you know it. They are just two words for the same thing. It is possible to conclude different things from this, but I’d just say it implies our axioms do not model the system well. I feel it is the last statement – if you believe something, then you believe you know it. That is far too strong. There are plenty of things I believe – for instance, the truth of certain mathematical conjectures – but I do not believe I know such things. (In fact, I know I do not know them.)

Let us write ¬ for ‘not’, K(A) for ‘I know A’, and B(A) for ‘I believe A’, where A is an arbitrary thing which can be believed and known. The axioms become, in the order they were introduced above:

1. K(A) implies B(A)
2. B(A) implies ¬B(¬A)
3. ¬K(A) implies K(¬K(A))
4. B(A) implies B(K(A))

Let us also assume the usual laws of logic, e.g. that A implies B is the same as ¬B implies ¬A.

Suppose you believe a statement S. That is, B(S). Then you believe you know it, so B(K(S)) – that is rule 3. But B(K(S)) implies ¬B(¬K(S)), where it is an application of rule 2 with the parameter A = K(S). That is, follows that one does not believe that one does not know S.

The contrapositive of rule 2 is that if you don’t believe something, then you don’t know it. Therefore ¬K(¬K(S)) follows. You don’t even know that you don’t know S. But the rule of knowing your own ignorance – rule 4 – implies that this means you know S. Therefore K(S).

Hence, if you believe something, then you know it.

As mentioned above, I feel axiom 4 is the problem. Nevertheless, if one were to change ‘belief’ and ‘knowledge’ to ‘conjecture’ and ‘formal proof/strong evidence’, to model mathematical or scientific reasoning, one sees that axiom 3 is problematic too. Just because there is a lack of evidence for something, doesn’t mean there is evidence that there is a lack of evidence for something.