# 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.