MichaelLewis:
I mean a world where your belief set, along with the degree of certainty at which you hold each of your beliefs, is just the same as it is in this world. Whatever you believe you know now, you would still believe that you know it in this possible world, but you may not really know it (because if the truth of a proposition necessarily entails that Catholicism is true, then that proposition is not true in the world in question).
I don’t just believe that I know that Catholicism is true; I know that I know that Catholicism is true. So while there are such worlds where Catholicism is false, these would not be worlds in which I am epistemically situated the exact same way that I am in this world.
In my experience of the common use of the word, (both outside and inside epistemology class) knowledge entails subjective certainty at least within a relevant context (for example when I say, while not in sight of my car, that “I know where my car is parked” I’m assuming that nothing my addressee or I would consider extraordinary has happened, for instance I assume that it has not been stolen), as well as, traditionally, the justification and truth of a belief (setting aside Gettier counterexamples). It seems contradictory for someone to say, “I know that my car is parked out front, though I’m not certain that my care is parked out front.”
The reason why it seems contradictory is
not because it is not possible to know that X while not being subjectively certain that X, but
rather because it is improper to assert that X while not knowing that X. Even if X happens to be true, one is not to assert it without knowing it to be true. Thus in your statement, taking the proposition that your car is parked out front to be P. you are not only asserting that you know that P but also implicitly asserting that you know that you know that P. Knowing that one knows that P is indeed incompatible with not being subjectively certain that P and that is the reason for the incongruity. IOW where K(x) is knowing that x and S(x) is being subjectively certain that x:
K(T) does not entail S(T)
However:
K(K(T)) does entail S(T)
To be subjectively certain that some proposition is true and
that you are justified in believing it
is to believe that you know that it is true.
No it’s not considering that justified true belief does not entail knowledge as has been proved and considering that one can believe that something is true without being subjectively certain that it is true.
To assert that you “know that you know” something is to assert that you are certain that you are justified in believing
that you are epistemicly infallible with respect to the proposition in question, and that you
are epistemicly infallible with respect to the proposition in question.
Again that’s not what it means since justified true belief does not entail knowledge.
But in principle you could never have justification for your own epistemic infallibility, so this claim must be false. (It would be fine, on the other hand, to claim simply that it is true
that you know something.)
As I explained above, to assert that you know that X is to implicitly assert that you know that you know that X. For any proposition X it is only proper to assert it if one knows that X. So if one were to want to assert:
K(X)
it would only be proper to do so when one:
K(K(X))
So your claim that you could never:
K(K(X))
while at same time being able to assert:
K(X)
is a self-contradictory claim.