I agree that we do not need to know that we know in order to know, and we don't need to know what knowledge is in order to know, but we do need to know what knowledge is in order to know that we know,

Not withstanding the necessary condition that you know P is true, if you know that you know, then you also do know what knowledge is, for how can you know that you have knowledge unless you know what knowledge is?

"I know that I know that I know" approximately equals "I have a mental model of myself having a mental model of myself knowing something."

Easy. It is the same principle. Just as you can know P without knowing what knowledge is, so too can you know that you know P without knowing what knowledge is. Once again, all you're doing is knowing something else - that is the part you aren't grasping. There is nothing special about knowing that you know. You can go on as many times as you want with this knowing that you know that you know stuff. You have to look at the key beginning: K(*. It doesn't matter what comes after K(*, it's still knowing something. And you can always know something, anything, without first knowing that you know, or knowing what knowledge is. Period.

*First degree K(P)
Second degree K(K(P))
Third degree K(K(K(P)))
(-Taken from Emil-)

Although I agree that first degree doesn't imply second degree, I do believe second degree implies third degree. And, third degree implies fourth degree, and so on. But, that's a different issue, I suppose.

Unfortunately, I have failed to articulate some decent support of my case.

You're killin' me!

Because the subject matter has changed from knowledge of P to knowledge of knowledge of P. To say "I know P" is to say, "I have knowledge that P is true." This is what the subject matter has changed from. But, let's break it out even more. If I say, "I know P," then I'm saying, "I have knowledge, and what I have knowledge of is that P is true."

To say "I know that I know P" is to say something else much more elaborate. It's to say that I have knowledge, but it's not merely to say that I have knowledge that P is true; rather, it's to say that I have knowledge, and what I have knowledge of is: I have knowledge that P is true. So, I'm not merely saying that I have knowledge and that what I have knowledge of is that P is true but rather I'm saying, "I have knowledge and what I have knowledge of is that I have knowledge that P is true. Asking how do I know that P is true is different than asking how do I know that I know P is true. Not withstanding the necessary condition that you know P is true, if you know that you know, then you also do know what knowledge is, for how can you know that you have knowledge unless you know what knowledge is?

I don't think it's a case (this time) that I'm not grasping what is being said. What I'm doing is rejecting what you say as true. Unfortunately, I have failed to articulate some decent support of my case.

I'll tell you something else you may not be so quick to agree with me on--but then again, who knows. Although I agree that first degree doesn't imply second degree, I do believe second degree implies third degree. And, third degree implies fourth degree, and so on. But, that's a different issue, I suppose.

Everything after the first degree, implies the first degree. (As you cannot know that you know P, without first knowing P - and it would go on like this for each subsequent degree).

Consider a person who has never even heard of the word "know". Consider they have no taught understanding of what it means to know (they have never contemplated necessary or sufficient conditions, or anything critical). Despite this, they still have the capacity to know. Consider that this person knows P. I know you agree that this person can know P, without having any idea what knowledge is.

Consider now that this same person is taking a test, and P is one of the answers on this test. Consider that they come to the question and forget what the answer is. They still know what the answer is, and they know that they know, but they cannot remember. Said person need not understand what knowledge is to know that they have the knowledge in their mind, but just can't remember it at the moment.

I think it may come down to knowing not being a mental event, and fast mistaking that it is.

My reasoning is thus:

If he believes knowing is a mental event, he may believe knowing is a conscious activity, a thought or feeling. If he thinks this, then he may believe that to know you know is to say, "I know that I have this thought or feeling". He may think it is something akin to, "You can only know that you feel sadness until after you've experienced sadness". But knowing is not an experience, like feelings are, so this argument would not apply. And that is what I think he doesn't understand. He thinks you need to understand what knowledge is to know that you know, because knowing is an experience. But this is false.

I think that argument works for first degree knowledge, but I'm not so convinced about higher degrees. Taking the approach that knowing something is dispositional, which I think is a good one, an examiner may say to a student, "P?", and if the student answers positively, and he is justified in doing so, we should say K(p) is true (where K(x) means "The student knows that x). The examiner may follow up the original question with "K(P)?" And again, if the student answers positively, K(p) is true, and he is justified in doing so, we may say K(K(p)). But to even make sense of "K(p)?" in this formal language that we are using (let's call it L-k), and therefore to be justified in giving a positive answer, he would have to know something about the syntax of L-k, and have a reasonable comprehension of the semantics of L-k.

If the student knows that there is a rule saying that K(x) is a sentence of L-k if and only if x is a proposition, then we may be satisfied that he knows the syntactic rules, but he still may not know under what circumstances K(p) is true, in which case he cannot answer the question, "K(p)?", because he doesn't have a sufficient grasp of the semantics of the language. In English, if he doesn't know the syntax of our language, he might respond to "Do you know Paris is the capital of France?" by saying, "Do I what? What is this word know supposed to be? What is it doing there? Speak English, man!" If he doesn't have a proper grasp of the semantics he might reply, "Hmm, that is a sentence, and "know" is the verb, but what does it mean for me 'to know' that proposition? Under what circumstances can I answer yes to that question, and under what circumstances can I answer no? I understand the proposition 'Paris is the caoital of France', is that sufficient for me to answer positively?" It seems to me that one would need to at least understand to a reasonable degree the rules, conventions and habits governing the use of the word 'know' (or it's definientia) in order to have iterated knowledge.

Not too sure one can be J(K(P)) without having heard of the term knowledge. What kind of justification could that be? Some authority maybe?

My reasoning is thus:

If he believes knowing is a mental event, he may believe knowing is a conscious activity, a thought or feeling.

If you know that P is true; hence, if you have an adequately justified true belief that cats are mammals, then just how much of a leap am I taking to think that you know what a cat is or what a mammal is, given that you have adequate justification for thinking P is true?

But in the case of knowing that I know, do you really believe that my (second) knowing has to be justified? Why do you believe it must be justified?

I didn't explicitly say creatures, and I didn't mention definitions, and saying it would bolster my justification seems to avoid my question.

I think most children know what a cat is, and if a child is taught that a cat is a mammal, then a child will know that a cat is a mammal. They need not know the definition of either to know what either is.

Can one know that cats are mammals and not know what either is? I'm not saying it's logically impossible, for with logic, anything that's not a contradiction is logically possible. I'm just looking for a good reason to think it's a plausible real possibility

If it did not have to be justified, it would violate the principle that knowledge is justified true belief.

Consider this example. X believes, with justification, that cats are mammals. He therefore knows that cats are mammals. But he believes, erroneously, that knowledge is simply true belief, and that it does not have to be justified. He has therefore not considered the question of whether he is justified in believing that cats are mammals. (Although, as it happens, he is so justified.) He has therefore not considered all the relevant criteria for knowing that cats are mammals. Consequently he is not justified in believing he knows it; hence he does not know that he knows it.