The idea to use choice functions in the semantic analysis of indefinites has recently gained increasing attention among linguists and logicians. A central linguistic motivation for the revived interest in this logical perspective, which can be traced back to the epsilon calculus of Hilbert and Bernays (1939), is the observation by Reinhart (1992,1997) that choice functions can account for the problematic scopal behaviour of indefinites and interrogatives. On-going research continues to explore this general thesis, which I henceforth adopt. In this paper I would like to address the matter from two angles. First, given that the semantics of indefinites involves functions, it still does not follow that these have to be choice functions. The common practise is to stipulate this restriction in order to get existential semantics right. However, a so-far open question is whether there is any way to derive choice function interpretation from more general principles of natural language semantics. Another question that has not been formally accounted for yet concerns the relationships between choice functions and the ``specificity'' ``referentiality'' intuition of Fodor and Sag (1981) about indefinites. Is there a sense in which choice functions capture this popular pre-theoretical notion? In order to answer these questions, this paper proposes a revision in the treatment of choice functions in Winter (1997), leaving its linguistic predictions unaffected but changing slightly the compositional mechanism. This modification opens the way for proving the following theorem: function variables in the analysis of the noun phrase must denote only choice functions and can derive only the standard existential analysis by virtue of the conservativity, logicality and non-triviality universals of Generalized Quantifier Theory as proposed in Barwise and Cooper (1981), Van Benthem (1984), Thijsse (1983) and others. The same implementation also captures the ``specificity'' notion: indefinites with a non-empty restriction set denote principal ultrafilters in the revised formalization. These are the quantificational correlates to ``referential'' individuals..
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