DE: Chomsky's theory of Universal Grammar has varied a great deal over the years. It began as a rather largish set of rules that were supposedly innate to all humans. Later it became a (n open-ended) set of "principles and parameters." Finally, after failing to develop proposals that were tight, made interesting predictions, or managed to keep the interest of most cognitive scientists, outside of a fairly narrowly focused cadre, Chomsky began to say that Universal Grammar is a "field of study," like biology. UG, as it is more commonly referred to, was claimed to be whatever it was about human biology that underwrote human grammar/language (Chomsky systematically used 'language' (a large and complex concept) when he in fact means 'grammar'. Then he settled on the distinction between E(xternal)-Language (sets of sentences actually spoken, let's say, or 'English', 'Spanish', and other natural languages--I put them in quotes because terms such as 'English' are abstractions--how could you say what English is in the world? It is impossible to define in such a way as to include the UK, India, Australia, USA, etc. and exclude, say, second-language speakers of the language vs. I(nternal)- Language, which is the grammar in an individual's brain/mind). I think most serious linguists began to be put off by all this terminology and switching, seeing it not as progress but as movement towards obsolescence. Pinker's idea of a Language Instinct fairs no better in offering precision or interesting predictions. Ultimately, it all becomes vague and definitional. 

So in 2002, Marc Hauser (Harvard), Tecumseh Fitch (now at Vienna) and Chomsky proposed to capture the uniqueness or basis of the human language faculty by distinguishing between FLB (the 'Broad Faculty of Language') and the FLN (the 'Narrow Faculty of Language'). They claimed that the FLB includes the things that are useful for language and exploited for human language, but aren't definitional or foundational for it, e.g. teeth and the tongue, which are necessary for speech but not sign language and which are shared with lots of other creatures. For the FLN they proposed that there was one human cognitive ability that was the crucial leap making human language possible, recursion--the ability of a process to apply to its own output, extended to grammar. Some examples of recursion in language would be (i) to place a sentence inside a sentence, as in 'The man [that you saw yesterday] came again today' ('that you saw yesterday' is part of 'The man came again today'; (ii) a phrase inside a phrase (The big man's little boy, where the noun phrase 'big man' is part of the larger noun phrase, 'The little boy'); (iii) a word inside a word (as in 'drunk-driving', where 'drunk' and 'drive' are both part of 'drunk-driving'). Most people who read this interpreted them as saying that all languages should have recursion. In fact, Chomsky in recent work, equates recursion to an operation specific to his most recent theory, Merge. I have claimed that Pirahã lacks evidence for recursion in its grammar. This has caused all sorts of nasty reactions. I think that the reactions come because recursion was in a way that last stand of Chomskyan innatism and if it isn't right the entire edifice crumbles. Here are some quotes from Hauser, Chomsky and critics of mine (and students and colleagues of Chomsky), Andrew Nevins, Cilene Rodrigues, and David Pesetsky:

From Hauser:

"It seems to me that your evidence suggests that the linguistic input provided to newborn Pirahã kids is insufficient to trigger the faculty of language." and further "if Pirahã lacks recursive operations altogether, then it lacks the quality of language that gives its characteristic quality of unbounded expressive power. Moreover, I assume that since a Pirahã child born in Paris would speak French with all its generative power from recursive operations, and would speak English if born in London, and again with all of its generative power, that something must be different about the input to Pirahã children, as well as perhaps the constraints operating on aspects of pragmatics. I don't see any of this evidence from the Pirahã, interesting as it is, as forcing a rejection of the hypothesis that FLN consists of the mechanisms that subserve narrow syntax (e.g., recursion), and the mappings to semantics and phonology." (personal emails from Hauser to me)

From Chomsky:

"Everett hopes that the readers do not understand the difference between UG in the technical sense (the theory of the genetic component of human language) and the informal sense, which concerns properties common to all languages. The speakers of Pirahã have all the same genetic components as us, so Pirahã children can create a normal language. Suppose that Pirahã doesn’t permit this. It would be the same as discovering a community that crawls but doesn’t walk, so that children that grow there only crawl and never walk. The implications of this for human genetics would be null." (interview Chomsky gave to the Folha de Sao Paulo, Latin America's largest newspaper, February 01, 2009)

And from Nevins, Pesetsky, and Rodrigues (I usually refer to them as NP & R; in the September 2009 issue of the journal Language): 

"If Pirahã really were a language whose fundamental rule is a nonrecursive variant of Merge, no sentence in Pirahã could contain more than two words." Nevins, et. al. (2009: 679)"

Here is what I said in the journal Language in reply to Chomsky, and the same reasoning applies to Hauser's quotes:

"Chomsky’s remarks deserve close scrutiny here because of their relevance to the demand by NP&R that I demonstrate how my claims falsify UG and because they show the difference between HC&F’s UG-1 and UG-2. Again, we see that UG-2 not only makes no predictions, but also has little if any connection to linguistic data. Chomsky allows in this latter quote that Pirahã could be as I describe it. Nothing in UG precludes this, he says. But then, of course, nothing in UG prevents a third, a half, or even all languages being like Pirahã, lacking recursion, and so forth. This means that there is no language nor any collection of languages that could possibly disconfirm UG in the ‘technical sense’. (Interestingly, if languages cannot disconfirm Chomsky’s view, then they also cannot support it.)"

For NP&R linguists who read this should be shocked that recursion, which is a process applying to its own output without bounds, is now reduced to saying that if a language has at least three words in a sentence it has recursion. All of these quotes show two things: (i) Pirahã is indeed a counterexample to the FLN and hence to all that remained of Chomsky's innatist program and that (ii) there is little empirical or mathematical content left in Chomskyan theory.