Solution to Exercise 7.2

Resolution of Russell's Paradox

Prove: ~EXIST(r):ALL(a):[ar <=> ~aa]

	
	Prove: ~ALL(a):[ar <=> ~aa]
	
	Suppose to the contrary, defining r as follows...

	1	ALL(a):[ar <=> ~aa]
		Premise

	Applying the definition of r to itself, obtaining the contradiction...

	2	rr <=> ~rr
		U Spec, 1

Applying the Conclusion Rule...

3	~ALL(a):[ar <=> ~aa]
	Conclusion, 1

Generalizing...

4	ALL(r):~ALL(a):[ar <=> ~aa]
	U Gen, 3

Switching quantifier...

5	~EXIST(r):~~ALL(a):[ar <=> ~aa]
	Quant, 4

As Required: 

Removing the double negation...

6	~EXIST(r):ALL(a):[ar <=> ~aa]
	Rem DNeg, 5