  Multiple logic-based reconstructions of conceptual data modelling languages
such as EER, UML Class Diagrams, and ORM exist. They mainly cover various
fragments of the languages and none are formalised such that the logic applies
simultaneously for all three modelling language families as unifying mechanism.
This hampers interchangeability, interoperability, and tooling support. In
addition, due to the lack of a systematic design process of the logic used for
the formalisation, hidden choices permeate the formalisations that have
rendered them incompatible. We aim to address these problems, first, by
structuring the logic design process in a methodological way. We generalise and
extend the DSL design process to apply to logic language design more generally
and, in particular, by incorporating an ontological analysis of language
features in the process. Second, we specify minimal logic profiles availing of
this extended process, including the ontological commitments embedded in the
languages, of evidence gathered of language feature usage, and of computational
complexity insights from Description Logics (DL). The profiles characterise the
essential logic structure needed to handle the semantics of conceptual models,
therewith enabling the development of interoperability tools. There is no known
DL language that matches exactly the features of those profiles and the common
core is small (in the tractable DL $\mathcal{ALNI}$). Although hardly any
inconsistencies can be derived with the profiles, it is promising for scalable
runtime use of conceptual data models.
