(165) DOMAIN WIDENING

(165) DOMAIN WIDENING

Let g be an increasing function from sets into sets (i.e. for any set D, g(D) ⊇D). Then: ∃ D x [φ] entails ∃ g(D) x [φ]

where

D is a quantificational domain and φ is the plain meaning of the sentence without conversational implicatures.

According to Chierchia, this domain expansion function g(D) must be universally closed at some point in the derivation by an operator ∀g∈ , where is the possible excursion for g(D). At this point we have a very detailed characterization of the semantic properties of any. To summarize, any is an indefinite that must be existentially bound, but its special characteristic is that it forces the application of a domain widening function g(D) which applies on the domain of the existential quantifier, which requires universal closure. Given this analysis, widening could be always allowed in principle, but we know that this operation triggered by any is licit only in certain semantic contexts. We must ask now what is so special about these contexts.

The semantic status of the licensor. Many attempts have been made in order to capture the semantic properties of polarity licensing contexts, and much debate on their proper semantic characterization still exists. According to a trend started by Ladusaw (1979), Downward Entailingness is the appropriate notion to capture PIs distribution. Another influential proposal, due to Zwart

(1995) and further developed by Giannakidou (1997), instead assumes the concept of anti veridicality as the relevant one. It is not easy to empirically decide which notion is more appropriate, but given the assumptions on the semantic status of PIs previously made, DE offers a principled way to account for the dependency between the licensee and its licensor. Consider the following licensing condition from Ladusaw (1979):

(166) Licencing conditionfor PIs:

α is a trigger for negative polarity items in its scope if α is downward entailing where a DE function can be defined as follows: (167) DE function

A function f is downward entailing if for all arbitrary elements X,Y it holds that X ⊆ Y ⇒ f(Y) ⊆ f(X)

This function basically allows inferences from set to subset. In these contexts, expression denoting sets can be substituted for expression denoting subsets salva veritate. In order to illustrate this point, consider (168) and (169):

(168) Lucy does not like ice cream ||ice cream|| ⊆ || Italian ice cream|| ⇒

(169) Lucy doesn’t like Italian Ice cream In (168) we can substitute to the set ice cream the subset of Italian ice cream, in a way

that (168) entails (169). The inverse situation holds instead in UE functions. (170) UE functions

A function f is upward entailing if for all arbitrary elements X,Y it holds that X ⊆ Y ⇒ f(X) ⊆ f(Y)

Here, inferences go in the opposite direction, from subsets to sets, and (171) entails (172):

(171) Lucy likes Italian ice cream || ice cream|| ⊆ || Italian ice cream|| ⇒

(172) Lucy likes ice cream Now that the semantic properties of the trigger α are clearer, we have a way to explain

why domain widening functions are restricted only to DE contexts. We are ready to consider the last piece of the puzzle.

The relation which holds between the PI and its licensor According to Chierchia, the domain widening function has to be ultimately related to a

universal operator ∀g. The question, then, concerns the reasons why this operator is licit only in DE environments.

A way to answer this involves the "informativeness" or the "informational strength" of the sentence, where strength has to be related to more restrictive truth conditions: if α and β are two propositions, α is stronger than β if α asymmetrically entails β. In the case of DE contexts, where the entailment goes from set to subset, a larger set is more informative (stronger) than a smaller one. At this point, it is clear that a domain widening function g(D) which enlarges the set of reference of a quantifier introduces a stronger statement than the one conveyed by a plain existential. Thus, domain widening coincides with strength only in DE domain. The distribution of any can be ruled by the following condition proposed by K&L: