The familiarity theory of definites
In this section, I want to offer a particular account of the familiarity theory, which is generally assumed by functionalists and cognitive linguists (although not Langacker 1991, Hawkins 1978); note though that there are formal approaches which also assume a familiarity view, including Dynamic Syntax (Cann et al 2005) and File Change Semantics (Heim 1983).
This approach assumes that a definite NP is "Hearer Old". It is straightforward to show how this approach to definiteness works out in a formalized cognitive theory, because Hudson (1990, 2007) assumes this approach for Word Grammar. In his most recent formulation, Hudson (2007: 226-227) assumes "conceptual binding", which is a way of formalising familiarity in a conceptual network. Conceptual binding works both for definite NPs (which are bound to a concept opened up in the conceptual network by earlier discourse) and reflexive pronouns (which have particular constraints on their distribution). Essentially, the claim is that the semantics for the inherits a particular link which has to connect to an antecedent concept. If there isn't an antecedent concept for this link to attach to, the utterance is ill-formed.
Hudson's bound concepts are related to Prince's "Hearer Old information", which Abbott (2004) describes in this way:
The concept of familiarity which Christophersen has in mind here seems quite similar to Prince (1992)'s concept of HEARER-OLD INFORMATION, which she aligns with the idea of information which is 'in the permanent registry' (Kuno 1972), or 'culturally copresent' (Clark and Marshall 1981)."
Hearer-old information subsumes discourse-old information. Hudson's analysis of the is given in Figure 2 below.
The best way to explain the diagram is to quote from Hudson's own account.
Definiteness comes from the determiners, which relate directly to the noun's referent. In 1990 I interpreted definiteness in terms of the addressee's knowledge (Hudson 1990: 293-302), with definite referents already known to the addressee and indefinites new to the addressee. This may be an accurate analysis, but a much neater way to explain the difference is now available thanks to the binding mechanism ... definite referents are bound. This is not only much easier to show in the analysis, but it also explains why the addressee is expected to know the referent
Figure 2. Hudson's treatment of definiteness
already. This binding is shown by the double arrow which means 'directed identity' linking one obligatory node to another, and is only 'potential' in this diagram because the object of the binding (the antecedent) is not available in this sentence. It must be available in the interlocutors' minds, so, at least in principle, we could complete the binding if we had a network analysis of the relevant parts of either the speaker's or the hearer's mind. (Hudson 2007: 226)
Hudson's account is explicitly cognitive he works with a notion of what is in the speaker's and hearer's minds, and how the relevant parts of their conceptual structures are activated in the course of a discourse. This description of a familiarity approach shows how it links to a cognitive theory, and also presents a formally precise account. We could look at other formal accounts Heim (1983) seeks to rehabilitate the familiarity theory within File Change Semantics as does Elbourne (2010), and Kamp (1981) presents an alternative Discourse Representation Theory account but as these theories are not cognitive, they are not directly relevant to my concerns here. I argue in the section below that the quantificational theory of definiteness also accounts for the notion that a definite NP is familiar.