Incomplete formal proof that the KK-principle is wrong
(KK)If one knows that p, then one knows that one knows that p.
Definitions A0is the proposition that 1+1=2. A1is the proposition that Emil knows that 1+1=2. A2is the proposition that Emil knows that Emil knows that 1+1=2. … Anis the proposition that Emil knows that Emil knows that … that 1+1=2. Where “...” is filled by “that Emil knows” repeated the number of times in the subscript of A. Argument 1. Assumption for RAA (∀P∀x)Kx(P)→Kx(Kx(P))) For any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P. 2. Premise Ke(A0) Emil knows that A0. 3. Premise (∃S1)(A0∈S1∧A1∈S1∧...∧An∈S1)∧|S1|=∞∧S1=SA There is a set, S1, such that A0belongs to S1, and A1belongs to S1, and … and Anbelongs to S1, and the cardinality of S1is infinite, and S1is identicla to SA. 4. Inference from (1), (2), and (3) (∀P)P∈SA→Ke(P) For any proposition, P, if P belongs to SA, then Emil knows that P. 5. Premise ¬(∀P)P∈SA→Ke(P) It is not the case that, for any proposition, P, if P belongs to SA, then Emil knows that P. 6. Inference from (1-5), RAA ¬(∀P∀x)Kx(P)→Kx(Kx(P))) It is not the case that, for any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P. Proving it Proving that it is valid formally is sort of difficult as it requires a system with set theory, predicate logic with quantification over propositions. The above sketch should be enough for whoever doubts the formal validity.