Solution to Exercise 7.2Resolution 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