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Chapter 15: Russell's Forest

Problem 1

a' a' = ~(a a') = ~(a a)

Contradiction.

Problem 2

Theta N = N (Theta N) = ~(Theta N)

Contradiction.

Problem 3

At first I thought this might not lead to a contradiction, but I peeked ahead and it did (but I didn't read how), and worked it out. I should have worked it out without the hint.

Theta (A A) = A A (Theta (A A)) = ~(A OR Theta (A A))

Which implies A. So, A A A is false.

Theta (A (A A A)) = A (A A A) (Theta (A (A A A)))
  = ~(FALSE OR Theta (A (A A A)) = ~(Theta (A (A A A)))

Contradiction.