 This is “the paradox of analysis,” which can be seen as dormant in Frege’s own move from his (1884) focus on definitions to his more controversial (1892a) doctrine of “sense,” where two senses are distinct if and only if someone can think a thought containing the one but not other, as in the case of the senses of “the morning star” and “the evening star.” If analyses or definitions preserved sense, then, unlike the case of “morning star” and “evening star,” whenever one thought the definiendum, one should thereby be thinking the definiens. And perhaps one can’t think Bill is Bob’s brother without thinking Bill is Bob’s male sibling. But few of Frege’s definitions of arithmetic concepts are nearly so simple (see Gottlob Frege, §2.5). In their case, it seems perfectly possible to think the definiendum, say, number, without thinking the elaborate definiens Frege provided (cf. Bealer 1982, Michael Dummett 1991, and John Horty 1993, 2007, for extensive discussions of this problem, as well as of further conditions, e.g., fecundity, that Frege placed on serious definitions).