Warfield gets it right about the relationship between object languages and metalanguages. Something about the prefix meta- has contributed to a not uncommon misconception that metalanguages are formalized to a higher degree than the languages they objectify whereas in fact the opposite is true.
As it happens, the relation of informal contexts to formal contexts and what I’ve elsewhere called the formalization arrows between them are themes of major importance in my study of Inquiry Driven Systems. Being short on time at the moment, I’ll give just a pointer to one of many relevant discussions and hope to elaborate further at the next opportunity.
- Inquiry Driven Systems • Introduction