What are established and unestablished knights and knaves? I am reading Games for your mind by Jason Rosenhouse. In chapter 10, he defines established and unestablished knights and knaves: islanders bearing the label "established" can be thought of as having proved their identities.
Example (from puzzle 44). Anne says "I am not an established knight." If Anne were a knave, her statement would be true. Since knaves do not make true statements, we conclude that she is a knight, but not an established one.
What does it mean if a knight or a knave has or has not proved his/her identity?
 A: The term "proven" is somewhat confusing.  I suspect that the author intends this term to represent that statements made by this particular islander, or other islanders, have conclusively determined whether this particular islander is a knight or a knave.
For example, if you walk up to an islander whose identity has not yet been established, and the islander says, "2+2 = 4", then the islander has just established that he is a knight.
Peculiarly, there is something of a flaw in the analysis of the islander's statement of :
"I am not an established knight.".
As discussed, this statement could not have been made by a knave.  Therefore, at the time that the islander (who must be a knight) had this thought, the idea was truthful.
However, at the moment that the islander finished speaking the sentence, the sentence became a falsehood, since the act of speaking the sentence established that the islander was a knight and thus affected whether the spoken sentence was true or not.
This is similar to, but somewhat different from the notion of meta-statements, such as
"This statement is a lie".
The assumption that such a statement is true leads to a contradiction.  Similarly, the assumption that such a statement is false also leads to a contradiction.
