[2][3] In integrational linguistics, a description of a language is understood as a theory (formulated by the linguist) of the language; similarly, for descriptions of varieties or individual idiolects.
"Grammar" is here used in a broad sense, covering not only morphological and syntactic but also phonological and semantic descriptions.
A description of the lexicon, i.e. a dictionary, is again construed as a theory of its object (Drude 2004).
For example, a traditional statement on a language variety, such as is construed as equivalent to which may be a sentence of a grammar that contains "Colloquial American English" as an axiomatic constant, denoting a certain subset of English.
Excepting the language needed for formulating the semantic part of a grammar, integrational grammars may be formulated using an appropriate version of set theory.