> The DBMS doesn't interpret relations at all. It performs relational operations on them; humans interpret them (often using other software as a tool to present an interpretation.)
A database schema is a multisorted axiomatic system (each base relation is its own sort), of which the database's state is a model (in the logician's sense: a concrete interpretation of a theory). Unlike the interpretation in your head, whose only justification is your personal wishes, the DBMS' interpretation is backed by a concrete formal system: the language in which your database schema is expressed.
> In the constructed language of a particular library or database, that the instances of a Customer class or members of a Customer relation are representations of customers is part of the semantics of that language, even if it isn't part of the semantics of the host language.
This is only true if you can actually prove that your database or program's “Customers” actually behave like customers in your problem domain.
A database schema is a multisorted axiomatic system (each base relation is its own sort), of which the database's state is a model (in the logician's sense: a concrete interpretation of a theory). Unlike the interpretation in your head, whose only justification is your personal wishes, the DBMS' interpretation is backed by a concrete formal system: the language in which your database schema is expressed.
> In the constructed language of a particular library or database, that the instances of a Customer class or members of a Customer relation are representations of customers is part of the semantics of that language, even if it isn't part of the semantics of the host language.
This is only true if you can actually prove that your database or program's “Customers” actually behave like customers in your problem domain.