(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.