The core question behind this Frontiers research topic is whether explaining linguistic phenomena requires appeal to properties of human cognition that are specialized to language. We argue here that investigating this issue requires taking linguistic research results seriously, and evaluating these for domain-specificity. We present a particular empirical phenomenon, bound variable interpretations of pronouns dependent on a quantifier phrase, and argue for a particular theory of this empirical domain that is couched at a level of theoretical depth which allows its (...) principles to be evaluated for domain-specialization. We argue that the relevant principles are specialized when they apply in the domain of language, even if analogs of them are plausibly at work elsewhere in cognition or the natural world more generally. So certain principles may be specialized to language, though not, ultimately, unique to it. Such specialization is underpinned by ultimately biological factors, hence part of UG. (shrink)