The fact that the former symbol should be more specific than the latter can be represented using SPEC atoms like dog n 1 dog n. Note that even a deep grammar will not fully disambiguate to semantic predicate symbols, such as WordNet senses, and so dog n 1 can still be consistent with multiple symbols like dog n 1 and dog n 2 in the semantic representation.
unlike the output of a POS tagger, an RMRS symbol that's output by a deep grammar is consistent with symbols that all have the same arity, because a deep grammar fully determines lexical subcategorisation.