In the latter period of his writings, a chap by the name of Ludwig Wittgenstein looked a second time at his earlier work and rather drastically changed his mind. In his first book, the Tractatus Logico Philosophicus, he argued, among other things, that
A logical picture of facts is a thought (Proposition 3)
This, when one unpicks to a decent enough approximation what he means by the terms, seems to make some sense- after all, it seems we cannot think anything truly illogical*, and I certainly seem to have an internal model of the things I am thinking about.
Indeed, if one hunts for the intuitive appeal in the claim "knowledge => knowledge of knowledge", one sees something similar, for:
If I know that p, and I know what I am thinking (this much, by hypothesis or
*It is an immediate logical consequence* that I know that I know that p
Any logical picture shewing my knowledge of p occurring in my mind must logically shew its own presence.
As Wittgenstein demonstrated in his later work Philosophical Investigations, such accounts of thought are palpably false. To see why such accounts are false in full generality is rather a lot of work, so let's attack just the relevant bit of it. To adapt an example from section 60 (of PI): to bring a broomstick and brush attached as to make a broom, connotes a different thought than to bring a broom, although they are logically equivalent. That is to say exhaustive logical consequence is not a feature of thought.
Though one may derive second order knowledge of p from knowledge of p, one need not in knowing p.
To conclude, a thought experiment:
In a world where knowing state secrets is punishable by death; a man discovers, while walking in the park, an official document declaring the location of his children's school to be a nuclear testing site- at the bottom of the page; the words [top secret], the government seal, and a reminder of the law. Is his first thought for his own life, or his children's? Suppose somehow, both the reminder of the law, the secret itself and that it is a secret were conveyed simultaneously (for example, if there were a single monosyllabic word in the man's language containing all this information); what then?
*Although we may think plenty that is irrational or unreasonable. IMO the character Spock in the original Star Trek has a lot to answer for for confusing these three terms in the eyes of the public...