"a dog" versus "the dog"

I thought I understood the difference between "a" and "the" as a difference in assumptions about the set the dog belongs to. I now have a better explanation in terms of my current ideas of topics, thesaurus's, and ledgers: 

When we begin a discussion we put a topic instance into a ledger. If we say "a dog" we are putting it into the ledger without a context. When we say "the dog" an empty context structure is created or assumed to be pre-existing.