Before revision: If Axy is understood to mean "x admires y," and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is: That this formula has no first-order equivalent can be seen as follows .
After revision: If Axy is understood to mean "x admires y," and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is: That this formula has no first-order equivalent can be seen by turning it into a formula in the language of arithmetic .