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 just give a pointer
into one of many relevant discussions and hope to elaborate further at the next opportunity.