Abstract
In many natural languages, there are clear syntactic and/or intonational differences between declarative sentences, which are primarily used to provide information, and interrogative sentences, which are primarily used to request information. Most logical frameworks restrict their attention to the former. Those that are concerned with both usually assume a logical language that makes a clear syntactic distinction between declaratives and interrogatives, and usually assign different types of semantic values to these two types of sentences. A different approach has been taken in recent work on inquisitive semantics. This approach does not take the basic syntactic distinction between declaratives and interrogatives as its starting point, but rather a new notion of meaning that captures both informative and inquisitive content in an integrated way. The standard way to treat the logical connectives in this approach is to associate them with the basic algebraic operations on these new types of meanings. For instance, conjunction and disjunction are treated as meet and join operators, just as in classical logic. This gives rise to a hybrid system, where sentences can be both informative and inquisitive at the same time, and there is no clearcut division between declaratives and interrogatives. It may seem that these two general approaches in the existing literature are quite incompatible. The main aim of this paper is to show that this is not the case. We develop an inquisitive semantics for a logical language that has a clearcut division between declaratives and interrogatives. We show that this language coincides in expressive power with the hybrid language that is standardly assumed in inquisitive semantics, we establish a sound and complete axiomatization for the associated logic, and we consider a natural enrichment of the system with presuppositional interrogatives.
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Notes
Like IEL, we take the symbols ‘\(\{\)’ and ‘\(\}\)’ to be part of the object language. This means that, e.g., \(?\{p,\lnot p\}, ?\{\lnot p,p\}\), and \(?\{p,p,\lnot p\}\) are three distinct interrogative formulas. Unlike IEL, we impose no requirement that \(n\ge 2\) or that the \(\alpha \)’s should be syntactically distinct. However, different choices in this respect would be just as compatible with the framework that we are going to propose, and would not impinge on the results that will be established here.
Instead of imposing this restriction, we could also think of a basic interrogative \(?\{\alpha _1,\ldots ,\alpha _n\}\) as presupposing that the actual world is one where the interrogative can be truthfully resolved, i.e., a world where at least one of \(\alpha _1,\ldots ,\alpha _n\) is true (see Hintikka 1981, 1983, 1999, 2007; Wiśniewski 1996, 2001). This alternative will be explored in Sect. 6. Yet another strategy would be to assume that a basic interrogative \(?\{\alpha _1,\ldots ,\alpha _n\}\) cannot only be resolved by establishing that one of \(\alpha _1,\ldots ,\alpha _n\) is true, but also by establishing that all of \(\alpha _1,\ldots ,\alpha _n\) are false. This also guarantees that basic interrogatives can be truthfully resolved in every world (see Groenendijk 2011).
The constraint that the antecedent of a conditional interrogative must be a declarative, is a bit arbitrary from a purely semantic perspective. In InqB, an implication is bound to be a question (i.e., non-informative) as soon as its consequent is. So, unlike the constraint on conjunction that both conjuncts must be interrogative, which is needed to guarantee that the conjunction as a whole expresses a question, the constraint on implication is not semantically motivated (see Groenendijk (2011) for more detailed discussion of this point).
This restriction will be lifted in Sect. 6, where we will consider interrogatives that can only be truthfully resolved in some worlds.
Throughout this section we will assume that both \(\mathcal {L}_{\mathsf{InqD}}\) and \(\mathcal {L}_{\mathsf{InqB}}\) are based on the same set of atomic sentences \(\mathcal {P}\).
If \(\varPhi \) is a finite set of formulas, we write \(\bigvee \varPhi \) to denote the disjunction \(\varphi _1\vee \dots \vee \varphi _n\), where \(\varphi _1,\dots ,\varphi _n\) is an arbitrary enumeration of the elements of \(\varPhi \). Similarly, later on we shall write \(?\varPhi \) for the interrogative \(?\{\varphi _1,\dots ,\varphi _n\}\), where \(\varphi _1,\dots ,\varphi _n\) is an arbitrary enumeration of \(\varPhi \).
It can be shown that this system coincides with the propositional fragment of the system presented in Velissaratou (2000), which amounts to an enrichment of partition semantics with conditional questions. This explains the C in InqC.
Disjunctive questions like (5), with rising intonation on all disjuncts, are called open disjunctive questions (Roelofsen and van Gool 2010). Open disjunctive questions are to be distinguished from alternative questions, which come with falling intonation on the final disjunct (Bartels 1999; Pruitt and Roelofsen 2013), and have different semantic characteristics as well. We will return to alternative questions momentarily.
Although see Isaacs and Rawlins (2008) for an analysis of conditional questions in a dynamic partition semantics that allows for hypothetical updates of the context of evaluation.
In the extended version of Groenendijk (2009) (url provided in the references), it is shown that the relational inquisitive semantics developed there allows for an alternative formulation that also assigns one of 5 values to each sentence in the language. A brief comparison with the 5-valued system of Nelken and Francez (2002) is also made. The two systems are closely related, but there is one value in each system that splits into two values in the other. It is this difference that causes the problem for Nelken and Francez noted in the main text, a problem that does not occur in the system of Groenendijk (2009). See also (Groenendijk (2008), §6.5) for more extensive discussion of this point.
This may also explain the fact why for Nelken and Shan \(4\) values are sufficient, whereas Groenendijk (2009) needs \(5\). The latter, unlike the former, does allow for disjunctive and conditional interrogatives, although we have seen above that its treatment of disjunctive interrogatives is problematic.
To avoid confusion, it should be noted that the syntax of Nelken and Shan’s system does allow for formulas of the form \(p\rightarrow {?q}\), corresponding to \(p\rightarrow (\Box q\vee \Box \lnot q)\). Since Nelken and Shan explicitly talk about such formulas as “conditional questions” it may be puzzling that we present them as restricting the expressive power of the interrogative fragment of their formal language to that of partition semantics. However, as remarked above, in proving their extensionality claim, Nelken and Shan do explicitly restrict themselves to conjunctions of basic interrogatives. This means that formulas like \(p\rightarrow {?q}\), at least for the purpose of Nelken and Shan’s central result, do not count as interrogatives. Furthermore, in our view, formulas like \(p\rightarrow {?q}\), as interpreted by Nelken and Shan, do not suitably capture the knowledge conditions of conditional interrogatives in natural language. In effect, the prediction is that a complete answer to \(p\rightarrow {?q}\) is known just in case \(p\) happens to be false, or a complete answer to \(?q\) is known. The right analysis, in our view, is that a complete answer to \(p\rightarrow {?q}\) is known just in case a complete answer to \(?q\) is known under the assumption that \(p\) holds. But this is impossible to express by a modal formula in the interrogative fragment of Nelken and Shan’s formal language, even if we extend this fragment with conditionals of the form \(\varphi \rightarrow \mu \), where \(\varphi \) is an arbitrary formula and \(\mu \) a basic interrogative. Rather, a new abbreviation would be needed for \(\Box ( p \rightarrow q) \vee \Box (p\rightarrow \lnot q)\).
It should be noted that, although this description of the semantic difference between open and alternative questions is good enough for our purposes here, it needs to be refined in view of cases like (i) below.
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(i)
Is Peter going to Italy\(\uparrow \), or France\(\uparrow \), or both\(\downarrow \)?
If (i) were to imply that exactly one of its disjuncts holds, then it would be contradictory, which is clearly not the case. For discussion of this issue, we refer to Roelofsen and van Gool (2010).
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(i)
Notice that we are not concerned here with how this type of meaning is constructed compositionally in natural languages. This is of course a very important issue, but it is beyond the scope of the present paper. All we want to show by means of this example is that, in terms of expressive power, the system \({\mathsf{InqD}}_{\pi }\) is rich enough to deal with alternative questions.
It is a distinctive feature of presuppositions that in compound sentences a presupposition of one of the components can be cancelled by the informative content of another component. It has proven to be notoriously difficult to accurately account for such cancellation phenomena, which is known as the projection problem for presuppositions. Since we only deal here with presuppositions of interrogatives, and since the only compound sentences in which interrogatives can combine with informative sentences are conditional interrogatives, projection poses no problem here.
Notice that, since \(\bigcup [\varphi ]\subseteq \pi (\varphi )\), a formula \(\varphi \) can only be supported in states in which it makes sense in the first place. Thus, the condition \(s\subseteq \pi (\varphi )\) need not appear explicitly in the definition of strong entailment, as it is implied by the condition \(s\models \varphi \).
We take it that as far as the division of labor is concerned InqD and \({\mathsf{InqD}}_{\pi }\) do not significantly differ, and that what we say in this section about InqD, by and large also applies to its presuppositional extension \({\mathsf{InqD}}_{\pi }\).
References
Aloni, M. (2007). Free choice, modals and imperatives. Natural Language Semantics, 15, 65–94.
Aloni, M., Égré, P., & de Jager, T. (2009). Knowing whether A or B. Synthese. doi:10.1007/s11229-009-9646-1.
Aloni, M., & Roelofsen, F. (2012). Indefinites in comparatives. Natural Language Semantics, 15(1), 65–94.
Alonso-Ovalle, L. (2006). Disjunction in Alternative Semantics. Ph.D. thesis, University of Massachusetts, Amherst, MA.
AnderBois, S. (2011). Issues and Alternatives. Ph.D. thesis, University of California, Santa Cruz, CA.
Åqvist, L. (1965). A new approach to the logical theory of interrogatives. Uppsala: University of Uppsala.
Bartels, C. (1999). The intonation of English statements and questions: A compositional interpretation. London: Routledge.
Biezma, M. (2009). Alternative vs polar questions: The cornering effect. In S. Ito & E. Cormany (Eds.), Proceedings of Semantics and Linguistic Theory (SALT XIX). Washington, PA: CLC Publications.
Biezma, M., & Rawlins, K. (2012). Responding to alternative and polar questions. Linguistics and Philosophy, 35(5), 361–406. doi:10.1007/s10988-012-9123-z.
Bolinger, D. (1978). Yes–no questions are not alternative questions. In H. Hiz (Ed.), Questions (pp. 87–105). Dordrecht: Reidel Publishing.
Ciardelli, I. (2009). Inquisitive semantics and intermediate logics. Master Thesis, University of Amsterdam.
Ciardelli, I. (2010). A first-order inquisitive semantics. In M. Aloni, H. Bastiaanse, T. de Jager, & K. Schulz (Eds.), Logic, Language, and Meaning: Selected papers from the seventeenth Amsterdam Colloquium. Berlin: Springer.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2012). Inquisitive semantics. NASSLLI lecture notes. www.illc.uva.nl/inquisitivesemantics.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2013a). Inquisitive semantics: A new notion of meaning. Language and Linguistics Compass, 7(9), 459–476. doi:10.1111/lnc3.12037.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2013b). Towards a logic of information exchange: An inquisitive witness semantics. In G. Bezhanishvili, V. Marra, S. Löbner, & F. Richter (Eds.), Logic, Language, and Computation: Revised Selected Papers from the Ninth International Tbilisi Symposium on Logic, Language, and Computation. Berlin: Springer.
Ciardelli, I. & Roelofsen, F. (2009). Generalized inquisitive logic: Completeness via intuitionistic Kripke models. Proceedings of Theoretical Aspects of Rationality and Knowledge.
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40(1), 55–94.
Ciardelli, I. & Roelofsen, F. (2012). Inquisitive dynamic epistemic logic. Manuscript, University of Amsterdam. http://www.illc.uva.nl/inquisitivesemantics.
Groenendijk, J. (2008). Inquisitive semantics and dialogue management. ESSLLI course notes. www.illc.uva.nl/inquisitivesemantics.
Groenendijk, J. (2009). Inquisitive semantics: Two possibilities for disjunction. In P. Bosch, D. Gabelaia, & J. Lang (Eds.), Seventh International Tbilisi Symposium on Language, Logic, and Computation. Berlin: Springer. Extended version: http://www.illc.uva.nl/Research/Reports/PP-2008-26.text.pdf.
Groenendijk, J. (2011). Erotetic languages and the inquisitive hierarchy. In a Festschrift for Martin Stokhof. http://dare.uva.nl/document/487828.
Groenendijk., J., & Roelofsen, F. (2009). Inquisitive semantics and pragmatics. Presented at the Workshop on Language, Communication, and Rational Agency at Stanford, May 2009. www.illc.uva.nl/inquisitivesemantics.
Groenendijk., J., & Stokhof, M. (1984). Studies on the Semantics of Questions and the Pragmatics of Answers. Ph.D. thesis, University of Amsterdam, Amsterdam.
Groenendijk., J., & Stokhof, M. (1997). Questions. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 1055–1124). Amsterdam: Elsevier.
Haida, A. (2010). On the semantics and pragmatics of yes/no-questions, yes/no-question disjunctions, and alternative questions: Evidence from Chadic. Manuscript, Humboldt University of Berlin, draft version of October 8, 2010.
Hamblin, C. L. (1973). Questions in montague english. Foundations of Language, 10, 41–53.
Hintikka, J. (1976). The semantics of questions and the questions of semantics. Acta Philosophica Fennica.
Hintikka, J. (1981). On the logic of an interrogative model of scientific inquiry. Synthese, 47(1), 69–83.
Hintikka, J. (1983). New foundations for a theory of questions and answers. Questions and Answers (pp. 159–190). Dordrecht: Reidel.
Hintikka, J. (1999). Inquiry as inquiry: A logic of scientific discovery. Dordrecht: Kluwer Academic Publishers.
Hintikka, J. (2007). Socratic epistemology: Explorations of knowledge-seeking by questioning. Cambridge: Cambridge University Press.
Isaacs, J., & Rawlins, K. (2008). Conditional questions. Journal of Semantics, 25, 269–319.
Karttunen, L. (1977). Syntax and semantics of questions. Linguistics and Philosophy, 1, 3–44.
Kratzer, A., & Shimoyama, J. (2002). Indeterminate pronouns: The view from Japanese. In Y. Otsu (Ed.), Proceedings of the Third Tokyo Conference on Psycholinguistics (pp. 1–25).
Kreisel, G., & Putnam, H. (1957). Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Archiv für Mathematische Logik und Grundlagenforschung, 3, 74–78.
Mascarenhas, S. (2009). Inquisitive Semantics and Logic. Master Thesis. University of Amsterdam, Amsterdam.
Nelken, R., & Francez, N. (2002). Bilattices and the semantics of natural language questions. Linguistics and Philosophy, 25, 37–64.
Nelken, R., & Shan, C.-C. (2006). A modal interpretation of the logic of interrogation. Journal of Logic, Language, and Information, 15, 251–271.
Pruitt, K., & Roelofsen, F. (2013). The interpretation of prosody in disjunctive questions. Linguistic Inquiry (in press).
Roelofsen, F. (2011). Algebraic foundations for inquisitive semantics. In H. van Ditmarsch, J. Lang, & J. Shier (Eds.), Proceedings of the Third International Conference on Logic, Rationality, and Interaction (pp. 233–243). Berlin: Springer.
Roelofsen, F. (2013). Algebraic foundations for the semantic treatment of inquisitive content. Synthese. doi:10.1007/s11229-013-0282-4. http://link.springer.com/article/10.1007%2Fs11229-013-0282-4.
Roelofsen, F., & van Gool, S. (2010). Disjunctive questions, intonation, and highlighting. In M. Aloni, H. Bastiaanse, T. de Jager, & K. Schulz (Eds.), Logic, language, and meaning: Selected papers from the seventeenth Amsterdam colloquium (pp. 384–394). Amsterdam: Springer.
Simons, M. (2005). Dividing things up: The semantics of or and the modal/or interaction. Natural Language Semantics, 13(3), 271–316.
Velissaratou, S. (2000). Conditional Questions and Which-Interrogatives. Master Thesis: University of Amsterdam.
Wiśniewski, A. (1996). The logic of questions as a theory of erotetic arguments. Synthese, 109(1), 1–25.
Wiśniewski, A. (2001). Questions and inferences. Logique et analyse, 173(175), 5–43.
Acknowledgments
We are grateful to Craige Roberts for critical discussion of the inquisitive semantics framework, which initially sparked us to develop the ideas that form the backbone of this paper. These ideas have been presented at several occasions over the last couple of years, including a Philosophy Colloquium at Carnegie Mellon University, November 3, 2011, a Linguistics Colloquium at Ohio State University, November 7, 2011, a Workshop on Questions, Logic and Games at the University of Amsterdam, December 1, 2011, a Workshop on the Interrogative Model of Inquiry in Paris, January 30-31, 2012, the Trends in Logic Conference in Bochum, June 3-5, 2012, and a Workshop on Inquisitive Semantics and Inferential Erotetic Logic in Poznań, April 18-19, 2013. We thank the organizers of these events, in particular Kevin Kelly, Craige Roberts, Stefan Minică, Yacin Hamami, Andrzej Wiśniewski and Mariusz Urbański, for inviting us, and the participants for helpful questions and comments. Some of the ideas presented here were published in preliminary form in Groenendijk (2011). We thank Jaap van der Does for helpful comments on that paper. Finally, we are especially grateful to Matthijs Westera for discussion of the ideas presented here as well as many closely related topics, to two anonymous Synthese reviewers for many insightful comments, and to the Netherlands Organisation for Scientific Research (NWO) for financial support.
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Ciardelli, I., Groenendijk, J. & Roelofsen, F. On the semantics and logic of declaratives and interrogatives. Synthese 192, 1689–1728 (2015). https://doi.org/10.1007/s11229-013-0352-7
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DOI: https://doi.org/10.1007/s11229-013-0352-7