What are Implicit Definitions?

The paper surveys diferent notions of implicit defnition. In particular, we ofer an
examination of a kind of defnition commonly used in formal axiomatics, which in
general terms is understood as providing a defnition of the primitive terminology of
an axiomatic theory. We argue that such “structural defnitions” can be semantically
understood in two diferent ways, namely (1) as specifcations of the meaning of the
primitive terms of a theory and (2) as defnitions of higher-order mathematical concepts or structures. We analyze these two conceptions of structural defnition both in
the history of modern axiomatics and in contemporary philosophical debates. Based
on that, we give a systematic assessment of the underlying semantics of these two
ways of understanding the defniens of such defnitions, by considering alternative
model-theoretic and inferential accounts of meaning