  The question of whether arithmetical truths are analytic then comes  down to the question of whether they can be derived purely logically. (Here we already have ‘transformation’, at the  theoretical level—involving a reinterpretation of the concept  of analyticity.) To demonstrate this, Frege realized that he needed  to develop logical theory in order to formalize mathematical  statements, which typically involve multiple generality (e.g.,  ‘Every natural number has a successor’, i.e. ‘For  every natural number x there is another natural number  y that is the successor of x’). This  development, by extending the use of function-argument analysis in  mathematics to logic and providing a notation for quantification, was essentially the achievement of his first book, the  Begriffsschrift (1879), where he not only created the first  system of predicate logic but also, using it, succeeded in giving a  logical analysis of mathematical induction (see Frege FR,  47-78).