Pragmatic identification of the witness sets Livio Robaldo * and Jakub Szymanik + ∗ Department of Computer Science, University of Turin. + Institute of Artificial Intelligence, University of Groningen. robaldo@di.unito.it, jakub.szymanik@gmail.com Abstract Among the readings available for NL sentences, those where two or more sets of entities are independent of one another are particularly challenging from both a theoretical and an empirical point of view. Those readings are termed here as ‘Independent Set (IS) readings’. Standard examples of such readings are the well-known Collective and Cumulative Readings. (Robaldo, 2011) proposes a logical framework that can properly represent the meaning of IS readings in terms of a set-Skolemization of the witness sets. One of the main assumptions of Robaldo’s logical framework, drawn from (Schwarzschild, 1996), is that pragmatics plays a crucial role in the identification of such witness sets. Those are firstly identified on pragmatic grounds, then logical clauses are asserted on them in order to trigger the appropriate inferences. In this paper, we present the results of an experimental analysis that appears to confirm Robaldo’s hypotheses concerning the pragmatic identification of the witness sets. Keywords: quantifiers, pragmatics, witness sets 1. Introduction This paper is about the truth values of the Independent Set (IS) readings of NL sentences in the simple form ‘Subject- Verb-Object’. IS readings are interpretations where two or more sets of entities are independent of one another. Four kinds have been identified in the literature, since (Scha, 1981): (1) a. Branching Quantifier Readings, e.g. Exactly two students of mine have seen exactly three drug- dealers in front of the school. b. Collective Readings, e.g. Exactly three boys made exactly one chair. c. Cumulative Readings, e.g. Exactly three boys in- vited exactly four girls. d. Cover Readings, e.g. Exactly three children ate exactly five pizzas. The preferred reading of (1.a) is the one where there are exactly two students and exacly three drug-dealers and each of the students saw each of the drug-dealers. (1.b) may be true in case three boys cooperated in the construction of a single chair. In the preferred reading of (1.c), there are three boys and four girls such that each of the boys invited at least one girl, and each of the girls was invited by at least one boy. Finally, (1.d) allows for any sharing of five pizzas between three children. In Cumulative Readings, the single actions are carried out by atomic 1 individuals only, while in (1.d) it is likely that the pizzas are shared among sub- groups of children. For instance, the sentence is satisfied by the following extension of ate ′ (‘⊕’ is the standard sum operator, from (Link, 1983)): 1 In line with (Landman, 2000), pp.129, and (Beck and Sauer- land, 2000), def.(3), that explicitly define Cumulative Readings as statements among atomic individuals only. (2) ‖ate ′ ‖ M ≡{〈c 1 ⊕c 2 , p 1 ⊕p 2 〉, 〈c 2 ⊕c 3 , p 3 ⊕p 4 〉, 〈c 3 , p 5 〉} In (2), children c 1 and c 2 (cut into slices and) shared pizzas p 1 and p 2 , c 2 and c 3 (cut into slices and) shared p 3 and p 4 , and c 3 also ate pizza p 5 on his own. Branching Quantifier Readings have been the more contro- versial (cf. (Beghelli et al., 1997) and (Gierasimczuk and Szymanik, 2009)), as many authors claim that those read- ings are always sub-cases of Cumulative Readings. Collec- tive and Cumulative Readings have been largely studied; see (Link, 1983), (Beck and Sauerland, 2000), (Ben-Avi and Winter, 2003), and (Kontinen and Szymanik, 2008) to begin with. However, the focus here is on Cover readings. This paper assumes, following (van der Does and Verkuyl, 1996), (Schwarzschild, 1996), (Kratzer, 2007), that they are the IS readings, of which the three kinds exemplified in (1.a-c) are merely special cases. The name “Cover read- ings” comes from the fact that they are traditionally repre- sented in terms of Covers, a particular mathematical struc- ture. With respect to two sets S 1 and S 2 , a Cover is formally defined as: (3) A Cover Cov is a subset of Cov 1 × Cov 2 , where Cov 1 ⊆ ℘(S 1 ) and Cov 2 ⊆ ℘(S 2 ), s.t. a. ∀s 1 ∈ S 1 , ∃cov 1 ∈ Cov 1 s.t. s 1 ∈ cov 1 , and ∀s 2 ∈ S 2 , ∃cov 2 ∈ Cov 2 s.t. s 2 ∈ cov 2 . b. ∀cov 1 ∈ Cov 1 , ∃cov 2 ∈ Cov 2 s.t. 〈cov 1 , cov 2 〉∈ Cov. c. ∀cov 2 ∈ Cov 2 , ∃cov 1 ∈ Cov 1 s.t. 〈cov 1 , cov 2 〉∈ Cov. Covers may be denoted by 2-order variables called “Cover variables”. We may then define a meta-predicate Cover that, taken a Cover variable C and two unary predicates P 1 and P 2 , asserts that the extension of the former is a Cover of the extensions of the latter: 81