The partition semantics of questions, syntactically

Groenendijk and Stokhof (1984, 1996; Groenendijk 1999) provide a logically attractive theory of the semantics of natural language questions, commonly referred to as the partition theory. Two central notions in this theory are entailment between questions and answerhood. For example, the question Who is going to the party? entails the question Is John going to the party?, and John is going to the party counts as an answer to both. Groenendijk and Stokhof define these two notions in terms of partitions of a set of possible worlds. We provide a syntactic characterization of entailment between questions and answerhood. We show that answers are, in some sense, exactly those formulas that are built up from instances of the question. This result lets us compare the partition theory with other approaches to interrogation—both linguistic analyses, such as Hamblin’s and Karttunen’s semantics, and computational systems, such as Prolog. Our comparison separates a notion of answerhood into three aspects: equivalence (when two questions or answers are interchangeable), atomic answers (what instances of a question count as answers), and compound answers (how answers compose). We would like to thank Patrick Blackburn, Paul Dekker, Jeroen Groenendijk, Maarten Marx, Robert van Rooy, Stuart Shieber, and the anonymous referees for their useful comments and discussions. The 13th European Summer School in Logic, Language and Information, the 13th Amsterdam Colloquium, and Stanford University’s Center for the Study of Language and Information provided stimulating environments that led to this collaboration. Part of the work presented here was carried out by the second author during a visit to Nancy as part of the INRIA funded partnership between LIT (Language and Inference Technology, ILLC, University of Amsterdam) and LED (Langue et Dialogue, LORIA, Nancy). The first author is supported by the United States National Science Foundation under Grant IRI-9712068. Proceedings of the Seventh ESSLLI Student Session Malvina Nissim (editor) Chapter 1, Copyright c © 2002, Balder ten Cate 1 1 The partition theory of questions An elegant account of the semantics of natural language questions from a logical and mathematical perspective is the one provided by Groenendijk and Stokhof (1984). According to them, a question denotes a partition of a logical space of possibilities. In this section, we give a brief summary of this influential theory, using a notation slightly different from Groenendijk’s (1999) presentation. A question is essentially a first order formula, possibly with free variables. We will denote a question by ?φ, where φ is a first order formula. (We will also denote a set of questions by ?Φ, where Φ is a set of first order formulas.) An answer is also a first order formula, but one that stands in a certain answerhood relation with respect to the question, spelled out later in this section. For example, the statement Everyone is going to the party (∀xPx) will turn out to answer the question Who is going to the party? (?Px). We assume that equality is in the language, so one can ask questions such as Who is John? (?x≈j). We also assume that, for every function symbol— including constants—it is indicated whether it is interpreted rigidly or not. Intuitively, for a function symbol to be rigid means that its denotation is known. For example, under the notion of answerhood that we will introduce below, it is only appropriate to answer Who is going to the party? (?Px) with John is going to the party (Pj) if it is known who John is—in other words, if j is rigid. Also, for Who is John? (?x ≈ j) to be a non-trivial question, John must have a non-rigid interpretation. Questions are interpreted relative to first order modal structures with constant domain. That is, a model is of the form (W,D, I), where W is a set of worlds, D is a domain of entities, and I is an interpretation function assigning extensions to the predicates and function symbols, relative to each world. Furthermore, we only consider models that give rigid function symbols the same extension in every world. Relative to such a model M = (W,D, I), a question ?φ expresses a partition of W , in other words an equivalence relation over W : (1.1) [?φ]M = { (w, v) ∈ W 2 | ∀g : M,w, g |= φ ⇔ M,v, g |= φ } . Roughly speaking, two worlds are equivalent if one cannot tell them apart by means of the question ?φ, in other words if φ is never true in one world and false in the other. In general, any set of questions ?Φ also expresses a partition of W , namely the intersection of the partitions expressed by its elements: [?Φ]M = ⋂ φ∈Φ[?φ]M = { (w, v) ∈ W 2 | ∀φ ∈ Φ: ∀g : M,w, g |= φ ⇔ M,v, g |= φ } . (1.2) Entailment between questions is defined as a refinement relation among partitions (i.e., equivalence relations): An equivalence relation A is a subset