Is it possible to characterize the sortal essence of Fs for a sortal concept F solely in terms of a criterion of identity C for F? That is, can the question ‘What sort of thing are Fs?’ be answered by saying that Fs are essentially those things whose identity can be assessed in terms of C? This paper presents a case study supporting a negative answer to these questions by critically examining the neo-Fregean suggestion that cardinal numbers can be fully characterized as those things whose identity can be assessed in terms of one-one correspondence between concepts.

According to philosophical orthodoxy, there are informative criteria of identity over time. Anti-criterialism rejects this orthodoxy and claims that there are no such criteria. This paper examines anti-criterialism in the light of recent attacks on the thesis by Matt Duncan, Sydney Shoemaker and Dean Zimmerman. It is argued that those attacks are not successful. Along the way, a novel strategy to defend anti-criterialism against the critics’ most challenging objection is developed. Under-appreciated difficulties for criterialism are also raised which, I claim, there is no obvious way to solve. It is concluded that anti-criterialism may be a much stronger rival to criterialism than is often supposed.