Chomsky's competence-performance distinction led to his formal approach (study of competence). The formal approach focuses on the structure of the language, emphasising the deductive properties of the language system (generative rules, algorithms); looking a patterns within the linguistic elements.
On the other hand the Prague school (1926) (sec. ) emphasised a functional approach. The European tradition, influenced by de Saussure, branched into an English tradition led by Firth and Halliday.
London School: J. R. Firth (1890-1960) - Firthian linguistics; M. A. K. Halliday (1930- ) neo-Firthian, Systemic Functional Linguistics - more of an applied orientation. The empirical approach continues to develop in developmental, psycho- and sociolinguistics. For the last see William Labov.
The functional approach focuses on communicative properties, and the way that ideas are organized within the language, emphasising the inductive aspects of the language (rules-of-thumb, heuristics); linguistic elements are studied in terms of how they contribute to the functions of language (communication being the major one). This approach questions the autonomy of language
Some parallel dichotomies: competence vs. performance, model vs. data-oriented, mentalistic vs. sociological /functional/ situated (language in relation to the world), theoretical vs. applied linguistics.