A semantics for interrogatives is presented which is based on Karttunen’s theory,but in a flexible manner incorporates both weak and strong exhaustivity. The paperstarts out by considering degree questions, which often require an answer pickingout the maximal degree from a certain set. However, in some cases, depending onthe semantic properties of the question predicate, reference to the minimal degree isrequired, or neither specifying the maximum nor the minimum is sufficient. What isneeded is an operation which defines the maximally informative answer on the basisof the Karttunen question denotation. Shifting attention to non-degree questions, twonotions of answerhood are adopted from work by Heim. The first of these is weaklyexhaustive and the second strongly exhaustive. The second notion of answerhood isproven to be equivalent to Groenendijk and Stokhof’s interrogative semantics. Onthe basis of a wide range of empirical data showing that questions often are not inter-preted exhaustively, it is argued that a fairly rich system of semantic objects associatedwith questions is needed to account for the various ways in which questions con-tribute to the semantics and pragmatics of the utterances in which they appear.1. INTRODUCTIONIn this paper we propose a modification and extension of Karttunen’s (1977)semantics for interrogatives which incorporates a flexible approach to theproperty of questions called exhaustivity.1Karttunen’s original proposal wascriticized (in particular by Groenendijk and Stokhof 1982, 1984) for failingto account for what has been termed strong exhaustivity. This is a propertyof embedded questions that licenses inferences like (1), which Groenendijkand Stokhof claim are valid:(1) John knows who was at the party.Mary was not at the party. [ John knows that Mary was not at the party.SIGRID BECK AND HOTZE RULLMANN Natural Language Semantics 7: 249–298, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands.A FLEXIBLE APPROACH TO EXHAUSTIVITYIN QUESTIONS** While we wrote this paper, the first author was supported by a DFG grant to the SFB340 and the second author by a grant from the Dutch Organization for Scientific Research(NWO), as part of the PIONIER-project ‘Reflections of Logical Patterns in Language Structureand Language Use’ at the University of Groningen, and by an Izaak Walton Killam MemorialPostdoctoral Fellowship at the University of Alberta. We are grateful to Irene Heim, JackHoeksema, Paul Portner, Arnim von Stechow, Wolfgang Sternefeld, the audience at the TenthAmsterdam Colloquium, and two anonymous reviewers for comments and discussion.1Throughout this paper we use the terms ‘question’ and ‘interrogative’ interchangeably torefer to syntactic objects in natural language. When we want to refer to the correspondingsemantic objects we will use terms like ‘denotation’ or ‘intension’.They propose a different semantic analysis of interrogatives, which accountsfor (1) by virtue of the basic question interpretation. Their point is acceptedin Rullmann (1995), who proceeds to make a proposal that accounts forstrong exhaustivity within a Karttunen system. His means of implementingstrong exhaustivity is a maximality operator. Rullmann’s motivation forthe assumption of a maximality operator is not only exhaustivity, but alsoan effect in degree questions that we refer to as the ‘maximality effect’.Degree questions like (2) below seem to require an answer that is in somesense maximal:(2) How many books did John read?That is, someone who asks (2) will only be satisfied with an answer spec-ifying the largest number of books that John read. This effect is capturedin Rullmann (1995) by making maximality part of the basic question deno-tation by means of a maximality operator. While we agree that maximalityand exhaustivity should be viewed as one and the same phenomenon, wesuggest a different way of accounting for them.We will defend a Hamblin/Karttunen-style semantics for questions, inwhich the basic denotation of a question is a set of propositions whichintuitively constitute its possible answers. However, we will also incorpo-rate Groenendijk and Stokhof’s insights about what information questionsintroduce in embedded constructions by adopting a proposal by Heim(1994). She accounts for properties like strong and weak exhaustivity bydefining two semantic notions of answerhood (which she calls answer1and answer2), which create propositions from Hamblin/Karttunen questionintensions. We show that this also captures the maximality effect in degreequestions.Heim’s two notions of answerhood provide us with a fairly rich systemof semantic objects definable in terms of the basic question denotation.We argue that this rich system is needed in the analysis of interrogativeconstructions in natural language. Our proposal is superior to Groenendijkand Stokhof’s and Rullmann’s in that their interrogative semantics doesnot make available all semantic objects that the analysis of interrogativesrequires, as there appears to be considerable variation in what an inter-rogative contributes to a construction semantically.We will take as our starting point Rullmann’s (1995) proposal incorpo-rating strong exhaustivity with the help of a maximality operator (section2). Although part of the motivation for that operator (besides strong exhaus-tivity) comes from the maximality effect in degree questions, we will seethat once a broader range of degree questions is considered, the assump-tion of a maximality operator is actually problematic (section 3). On the250 SIGRID BECK AND HOTZE RULLMANNother hand, all types of degree questions can receive a satisfactory treat-ment in terms of Heim’s notions of answerhood. In section 4, we introduceHeim’s first notion of answerhood (answer1) and show in some detail howit can account for the data which are problematic for the maximality account.In section 5, we turn to exhaustivity. Heim’s approach reanalyzes exhaus-tivity not as a property of the question, but as a property of the notion ofa true, complete answer to the question. We prove that her second notionof answerhood (answer2) is equivalent to Groenendijk and Stokhof’s inter-rogative semantics, once certain complications dealing with so-called dedicto and de re readings of which-phrases are dealt with (section 6). Thismeans that we can account for exhaustivity and maximality effects evenwith a basic Hamblin/Karttunen