Knowledge and belief implication

Question

Does the fact that I do not know that I do not know a fact imply that I believe the negation of that fact?

For instance, John does not know that John does not know that water is wet implies that John believs that water is not wet?

Answer 1

Interesting question. I've two reasons for saying "no".

First, if the implication you've presented holds, it implies ignorance amounts to inconsistent beliefs. Consider:

(1) John doesn't know that John doesn't know P implies John believes not P

(2) John doesn't know that John doesn't know not P implies John believes P

If John is ignorant about P, then (1) and (2) seem true of John, but that means ignorance implies inconsistent beliefs. That seems mistaken.

Second, suppose John doesn't know that he doesn't know that polonium is radioactive, because he's never cared about chemistry. Rather than saying John believes polonium is not radioactive, it seems more plausible to say one of the following:

(A) John might not have any belief about the radioactivity of polonium.

(B) John might have a belief about the radioactivity of polonium, but be indeterminate between polonium being radioactive or not. In this latter case, John might believe the logical truth that polonium either is or is not radioactive, but it wouldn't follow that he believed either disjunct.

In any event, these are the reasons for thinking not knowing you don't know doesn't imply believing not. Thanks for the interesting question.

Answer 2

In October 2019, people did know about Covid 19.

And obviously, because Covid 19 was not even a thing, they did know they didn't know.

Yet, I don't think anybody would say we believed Covid 19 did not exist. We simply had no belief about it for having never thought of it.

It's yet another state than "not believing", since the person never ever gave any consideration to the subject. It is different for a western kid to not believe in Santa after carefully considering the question and a mesoamerican kid who never had a concept of Santa to begin with.

One could object that if at that time I were to ask people "do you believe covid 19 exists ?", some would reply "never heard of it, I believe it does not exist". But in fact rigorously they should answer "what is this Covid 19 you're talking about ? I never heard of it", and those who formulated a belief only did because I asked them about it. I kind of forced them into taking a position by introducing the concept to them.

Edit: I just realized that by introducing the new concept of Covid 19 to someone, I would in fact make them realize there is a thing called Covid 19 they don't know about. Thus, they now know they don't know, which takes us out of the question's scope.

Answer 3

It's not clear from the q if you're making any distinction between knowledge and belief as modals. Assuming (e.g. based on your math SE questions) that this was some kind of homework/study/exam question, it was probably intended to use a [uni]modal logic in which these concepts are the same (K = B). In that you're asserting (for all p):

T = ¬B¬Bp → B¬p

(were B is the single belief/knowledge modal.)

It's easy to see that this is not valid in all frames even in (the modal logic) S5. (Simplified) proof by tableaux: show that the negation of T is satisfiable, thus T is not valid in all frames:

¬T = ¬(¬B¬Bp → B¬p) = ¬(B¬Bp ∨ B¬p) = ¬B¬Bp ∧ ¬B¬p

Introducing the usual notation M = ¬B¬, we have ¬T = MBp ∧ Mp, which is clearly satisfiable by p being true at all worlds. So T is not valid (at all frames) in S5 and thus also not valid (on all frames) in any other [uni]modal logics with fewer theorems than S5.

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If this was somehow meant as more research-y question, in which we're using a bimodal epistemic-doxastic logic, i.e one in which knowledge and belief are not interfinable (remember I just took K = B)... then I suggest you read some papers like Halpern et al. which have have some interesting results like:

It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief.

It is slightly complicated what they mean by that:

In first-order logic, the notions of implicit definability and explicit definability of predicates are standard, and are known to be equivalent by Beth’s (1953) theorem. These notions can be lifted to the definability of modalities in modal logics in a straightforward way. We explain the definitions in the context of knowledge and belief.

Consider a logic L for knowledge and belief. Knowledge is explicitly defined in L if there is a formula DK (for “definition of knowledge”) in L of the form Kp ↔ δ, where δ is a formula that does not mention the knowledge operator. Knowledge is implicitly defined in L if, roughly speaking, L “determines” knowledge uniquely. Syntactically, this determination means that any two modal operators for knowledge that satisfy L must be equivalent. Semantically, this means that two Kripke models of L with the same set of worlds that agree on the interpretation of belief (and on the interpretations of all primitive propositions) must agree also on the interpretation of knowledge.

Unlike the case of first-order definability, these two notions of definability do not coincide for modal logics; implicit definability is strictly weaker than explicit definability.

Also, in order to show their results they basically take belief to satisfy KD45 (this seems to be a standard assumption in this field). Also interestingly...

Finally, we address the question of defining KD45 belief in terms of knowledge. It is shown that it can be neither explicitly not implicitly defined for any logic of knowledge contained in S5, and hence for any weaker logic. Yet KD45 belief can be reduced to S4.4 knowledge by the axiom Bp ↔ ¬K¬Kp. The fact that adding this formula to S4.4 generates the logic of KD45 was already observed by Lenzen (1979).

I'll leave it as (trivial) exercise to check whether in S4.4 with that "Lenzen belief" the formula closer to (your) natural language formulation is valid or not:

¬K¬Kp → B¬p = ¬K¬Kp → ¬K¬K¬p

Looking through SEP page on this and the Handbook chapter I don't see any [discussed] system that could plausibly prove your proposition... generally the connection admitted [by most researchers] between K and B is weaker in the direction of Bp → KBp, but as strong as replacement in the other direction i.e Kp → Bp. (It can be a more tedious to show a countermodel with these assumptions though; perhaps best use some software.)