Abstract
Information exchange can be seen as a dynamic process of raising and resolving issues. The goal of this paper is to provide a logical framework to model and reason about this process. We develop an inquisitive dynamic epistemic logic (IDEL), which enriches the standard framework of dynamic epistemic logic (DEL), incorporating insights from recent work on inquisitive semantics. At a static level, IDEL does not only allow us to model the information available to a set of agents, like standard epistemic logic, but also the issues that the agents entertain. At a dynamic level, IDEL does not only allow us to model the effects of communicative actions that provide new information, like standard DEL, but also the effects of actions that raise new issues. Thus, IDEL provides the fundamental tools needed to analyze information exchange as a dynamic process of raising and resolving issues.
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Notes
Either of these conditions may be dropped or weakened to model scenarios of false information or not fully introspective agents (see, for instance, Fagin et al. 1995). The system considered here is usually taken to be the most basic variant of epistemic logic. For this reason we take it as a point of departure here, but we do not expect to encounter particular difficulties in adapting our proposal to weaker variants.
Public announcement logic was first proposed by Plaza (1989) and was further developed by Gerbrandy and Groeneveld (1997), Baltag et al. (1998), and van Ditmarsch (2000), among others. Recent overviews of the system and its role in the general dynamic epistemic logic landscape are provided by van Ditmarsch (2007) and van Benthem (2011).
Actually, restricting the epistemic maps to \(\varphi \)-worlds would be all we need to model the intended change, which is a merely epistemic one. The only reason why the \(\lnot \varphi \)-worlds also have to be eliminated from the model is that, if we did not eliminate them, the resulting model would no longer be an epistemic model in the sense of our definition, since the new epistemic maps would not satisfy the factivity requirement in the \(\lnot \varphi \)-worlds.
We thank an anonymous reviewer for emphasizing this.
A detailed exposition of inquisitive semantics, in particular the notion of issues that we will adopt here, can be found in Ciardelli et al. (2012, 2013a), Roelofsen (2013a). Earlier expositions of the framework can be found in Groenendijk and Roelofsen (2009), Ciardelli (2009), Ciardelli and Roelofsen (2011). The notion of issues that will play a crucial role here is already implicit in these earlier expositions, but is not explicitly defined and motivated there. Yet earlier expositions of the framework can be found in Groenendijk (2009), Mascarenhas (2009). However, the notion of issues that is implicit in this early work is really different from the one adopted here, and, as argued in Ciardelli (2009), Ciardelli and Roelofsen (2011), Ciardelli et al. (2013b), not general enough to suitably capture the issues that are expressed by certain types of questions in natural language (e.g., disjunctive questions and mention-some wh-questions).
Notice that this means that the empty information state, \(\emptyset \), is an element of every issue. Intuitively, \(\emptyset \) models the absurd, inconsistent information state, in which any candidate world is discarded. In this limit state, any piece of information is established, and any issue is resolved. This may be regarded as a generalization to issues of the usual ex falso quodlibet principle.
This is not the only way in which inquisitive sentences can be introduced in the picture. In inquisitive semantics, it is actually common practice to assume a language that does not make a categorical distinction between declaratives and interrogatives (see, e.g., Ciardelli 2009; Groenendijk and Roelofsen 2009; Ciardelli and Roelofsen 2011; Ciardelli et al. 2012). However, the inquisitive semantic framework can equally naturally be applied to a bi-categorical language (Groenendijk 2011; Ciardelli et al. 2013b). A detailed comparison of the two approaches, as well as meaning-preserving translations between the two resulting formal systems, are provided in Ciardelli et al. (2013b). Here, we choose to spell out our proposal for a bi-categorial language for two reasons. First, it seems that the intuitions are somewhat easier to get across this way. And second, assuming a distinction between declaratives and interrogatives makes it easier to compare our proposal to others, in particular that of van Benthem and Minică (2012), which will be done in Sect. 4.
Since our system assumes a strict partition of sentences into declaratives and interrogatives, hybrid conjunctions like \(p\wedge {?q}\) are not included in our logical language. Such conjunctions do in fact occur quite widely in natural language, both as standalone sentences and embedded under modal operators (e.g., Ann is coming, but is Bill coming as well?, I know that Ann is coming and whether Bill is coming as well), and can be handled straightforwardly in the standard hybrid system of inquisitive semantics (see the references in footnote 7).
We will see in a moment that, while we can make sense of the notion of truth for interrogatives, the truth conditions of an interrogative sentence—unlike those of a declarative sentence—do not completely determine its semantics.
In the case of the connectives, this uniformity is brought out in even fuller generality in the standard hybrid system of inquisitive semantics (see the references in footnote 7).
This complies with the suggestion that Belnap ended his 1966 paper Questions, answers, and presuppostions with: “I should like in conclusion to propose the following linguistic reform: that we all start calling a question ‘true’ just when some direct answer thereto is true.”
If we were to restrict the interrogative fragment of our system to this class of interrogatives, we would arrive at (a modal extension of) a system known as InqC, whose properties are discussed in some detail in Ciardelli et al. (2013b).
For a simple propositional language augmented with interrogatives, this cross-categorial notion of entailment has been investigated in detail and axiomatized in Ciardelli et al. (2013b).
Notice that two states supporting \(p\rightarrow q\) and \(p\rightarrow \lnot q\), respectively, may overlap (they may both contain worlds where \(p\) is false). For this reason, conditional questions have always been notoriously problematic for theories of questions that model issues as partitions of the logical space (e.g. Groenendijk and Stokhof 1984). This problem no longer arises in inquisitive semantics since its notion of issues is more general than the partition notion. Conditional questions have played an important motivational role in the development of inquisitive semantics (see, e.g., Mascarenhas 2009; Groenendijk 2011; Ciardelli et al. 2013a). We will return to this point when comparing our proposal with that of van Benthem and Minică (2012) in Sect. 4.
Of course, we would also not naturally say that \(a\) entertains \(\mu \) in that case: although we read \(E_a\varphi \) as “\(a\) entertains \(\varphi \)”, this should be understood as technical terminology.
We give truth conditions here, rather than support conditions, to bring out the analogy with standard modal logic more clearly. Since \(K_a\varphi \) and \(E_a\varphi \) are declaratives, Fact 5 ensures that they are supported by a state \(s\) just in case they are true at every world in \(s\).
This means that, after the announcement, it is common knowledge that the world is located in \(|\varphi |_M\). Technically, for any world \(w\) in the model \(M^\varphi \) resulting from the announcement, we will have \(\sigma _*^\varphi (w)\subseteq |\varphi |_M\). As discussed in Sect. 2, this does not mean that the formula \(K_*\varphi \) necessarily holds in the updated model. For, after the update, \(\varphi \) may come to express a different proposition than it previously did, that is, we do not necessarily have \(|\varphi |_{M^\varphi }=|\varphi |_{M}\cap W^\varphi \).
For the update of a single inquisitive state, understood as the public state, this dynamic picture is put forward and motivated in detail in Ciardelli et al. (2013a).
Notice that, like in standard DEL, an update does not change the truth value of atomic sentences at worlds, which reflects the fact that atoms are intended to model facts that are not themselves epistemic in nature.
In Sect. 3.2.4 we will develop a more realistic dynamic picture, in which the announcement of an interrogative does not provide the information that its presupposition is true, but rather requires such information to be publicly established prior to the announcement.
Notice that the principle as formulated here incorporates a non-redundancy requirement: an announcement is only appropriate at a world in case it enhances the public state. We could choose to separate out non-redundancy from division of labor proper, which would then amount simply to the following: the announcement of a declarative \(\alpha \) is appropriate in a world only if \(\alpha \) is non-inquisitive, while the announcement of an interrogative \(\mu \) is appropriate in a world only if \(\mu \) is non-informative.
Notice that on this approach, a public announcement never removes any world from the model. This has the puzzling consequence that in a \(\lnot \varphi \)-world, announcing \(\varphi \) has the effect of making \(\lnot \varphi \) common knowledge. This treatment of public announcements of declarative sentences is clearly different from the one we gave. However, since both systems are in principle compatible with either account of public announcements of declaratives, we do not take this difference to reflect an essential discrepancy between the two approaches.
This point has been argued forcefully by Stalnaker (1998).
Readers may wonder—with one of the reviewers of this paper—whether the notion of issues adopted in DELQ may be generalized suitably by weakening the condition that \(\approx \) be an equivalence relation. Early versions of inquisitive semantics did indeed seek to overcome the limitations of the partition theory of questions by modeling issues as reflexive and symmetric, but not necessarily transitive relations (Groenendijk 2009; Mascarenhas 2009; Sano 2009). However, in more recent work (Ciardelli 2009; Ciardelli and Roelofsen 2011; Ciardelli et al. 2013a) it has been argued in detail that such a notion is still not general enough to model certain natural types of issues, such as the ones expressed by open disjunctive questions and mention-some questions. More generally, the arguments in Ciardelli et al. (2013a) can be phrased in such a way as to show that the relevant issues do not correspond to a binary relation on worlds at all, that is, they are not of the form \(I_\approx \) for any binary relation \(\approx \) on \(\mathcal {W}\).
The same point is made by Aloni et al. (2013), who also propose a compositional interpretation of questions embedded under a knowledge operator.
In the liguistic literature, the point that a proper treatment of questions, especially embedded questions, requires inquisitiveness to enter the picture at the semantic level, and not just at the speech act level, has been made in much detail by Groenendijk and Stokhof (1997). At the time, it was directed mostly at the speech act treatment of questions proposed by Searle (1969) and Vanderveeken (1990), and at the imperative-epistemic treatment of questions proposed by Åqvist (1965) and Hintikka (1976, 1983). The argument we just gave is similar, but now directed specifically at the speech act treatment of questions in DELQ.
It is perhaps worth emphasizing that the declarative fragment of the logic cannot be separated out and studied in isolation. For instance, it can be seen that \(E_a\varphi \models E_a\psi \) holds if and only if \(\varphi \models \psi \), where \(\varphi \) and \(\psi \) may be sentences of either category. This shows that the logic of declaratives is inextricably entangled to that of interrogatives, and that a unified account of entailment is called for.
References
Ågotnes, T., van Benthem, J., van Ditmarsch, H., & Minică, Ş. (2011). Question–answer games. Journal of Applied Non-Classical Logics, 21(3–4), 265–288. doi:10.3166/jancl.21.265-288.
Aloni, M., Égré, P., & de Jager, T. (2013). Knowing whether A or B. Synthese, 190(14), 2595–2621.
Åqvist, L. (1965). A new approach to the logical theory of interrogatives. Uppsala: University of Uppsala.
Baltag, A. (2001). Logics for insecure communication. In Proceedings of theoretical aspects of rationality and knowledge (TARK) VIII (pp. 111–122).
Baltag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In Proceedings of the 7th conference on theoretical aspects of rationality and knowledge (pp. 43–56). Morgan Kaufmann Publishers.
Belnap, N. D. (1966). Questions, answers, and presuppositions. The Journal of Philosophy, 63(20), 609–611.
Ciardelli, I. (2009). Inquisitive semantics and intermediate logics. Master Thesis. University of Amsterdam. www.illc.uva.nl/inquisitivesemantics.
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40(1), 55–94.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2009). Attention! might in inquisitive semantics. In S. Ito & E. Cormany (Eds.), Proceedings of semantics and linguistic theory (SALT XIX). Cornell University, Ithaca, NY: CLC Publications.
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). On the semantics and logic of declaratives and interrogatives. Synthese. doi:10.1007/s11229-013-0352-7.
Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995). Reasoning about knowledge. Cambridge, MA: MIT Press.
Gerbrandy, J., & Groeneveld, W. (1997). Reasoning about information change. Journal of Logic, Language and Information, 6(2), 147–169.
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.
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. PhD thesis, University of 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.
Hintikka, J. (1976). The semantics of questions and the semantics of questions: Case studies in the interrelations of logic, semantics, and syntax. Acta Philosophica Fennica, 28(4), 200.
Hintikka, J. (1983). New foundations for a theory of questions and answers. In F. Kiefer (Ed.), Questions and answers (pp. 159–190). Dordrecht: Reidel.
Liu, F., & Wang, Y. (2013). Reasoning about agent types and the hardest logic puzzle ever. Minds and Machines, 23, 123–161.
Mascarenhas, S. (2009). Inquisitive semantics and logic. Master Thesis, University of Amsterdam.
Minică, Ş. (2011). Dynamic logic of questions. PhD thesis, ILLC, University of Amsterdam. www.illc.uva.nl/inquisitivesemantics.
Pelis̆, M., & Majer, O. (2010). Logic of questions from the viewpoint of dynamic epistemic logic. In M. Pelis̆ (Ed.), The logica yearbook (pp. 157–172). London: College Publications.
Pelis̆, M., & Majer, O. (2011). Logic of questions and public announcements. In N. Bezhanishvili, S. Löbner, K. Schwabe, & L. Spada (Eds.), Logic, language and computation: Selected revised papers from the eighth international Tbilisi symposium (pp. 145–157). Berlin: Springer.
Plaza, J. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, & Z. W. Ras (Eds.), Proceedings of the fourth international symposium on methodologies for intelligent systems (pp. 201–216). Oak Ridge National Laboratory. Reprinted as Plaza (2007).
Plaza, J. (2007). Logics of public communications. Synthese, 158(2), 165–179.
Roelofsen, F. (2011). Information and attention. Manuscript, ILLC University of Amsterdam. www.illc.uva.nl/inquisitivesemantics.
Roelofsen, F. (2013a). Algebraic foundations for the semantic treatment of inquisitive content. Synthese, 190, 79–102. doi:10.1007/s11229-013-0282-4.
Roelofsen, F. (2013b). A bare bone semantics for attentive might. In M. Aloni, M. Franke & F. Roelofsen (Eds.), The dynamic, inquisitive, and visionary life of \(\varphi ,\; ?\varphi \) , and \(\Diamond \varphi \) : a festschrift for Jeroen Groenendijk, Martin Stokhof, and Frank Veltman (pp. 190–215). Amsterdam: ILLC Publications.
Sano, K. (2009). Sound and complete tree-sequent calculus for inquisitive logic. In Proceedings of the sixteenth workshop on logic, language, information, and computation.
Searle, J. R. (1969). Speech acts: An essay in the philosophy of language. Cambridge: Cambridge University Press.
Stalnaker, R. (1998). On the representation of context. Journal of Logic, Language, and Information, 7(1), 3–19.
Unger, C., & Giorgolo, G. (2008). Interrogation in dynamic epistemic logic. In K. Balogh (Ed.), ESSLLI student session (pp. 195–202). Amsterdam: ILLC Publications.
van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logics, 17(2), 129–155.
van Benthem, J. (2011). Logical dynamics of information and interaction. Cambridge: Cambridge University Press.
van Benthem, J., & Minică, S. (2012). Toward a dynamic logic of questions. Journal of Philosophical Logic, 41(4), 633–669. doi:10.1007/s10992-012-9233-7.
van Benthem, J., van Eijck, J., & Kooi, B. (2006). Logics of communication and change. Information and Computation, 204(11), 1620–1662.
Vanderveeken, D. (1990). Meaning and speech acts: Principles of language use. Cambridge: Cambridge University Press.
van Ditmarsch, H. (2000). Knowledge games. PhD thesis, University of Groningen.
van Ditmarsch, H. (2005). Prolegomena to dynamic logic for belief revision. Synthese, 147(2), 229–275.
van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic. Berlin: Springer.
van Eijck, J., & Unger, C. (2010). Computational semantics with functional programming. Cambridge: Cambridge University Press.
Westera, M. (2013). Exhaustivity through the maxim of Relation. In Proceedings of logic and engineering of natural language semantics (LENLS 10).
Acknowledgments
We are grateful to two anonymous reviewers, as well as Editor-in-Chief Wiebe van der Hoek, for very useful feedback. We are also grateful to Alexandru Baltag, Johan van Benthem, Jan van Eijck, Jeroen Groenendijk, Yacin Hamami, Sonja Smets, Matthijs Westera, and especially to Yanjing Wang for helpful discussion of the ideas presented here and closely related topics. Financial support from the Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged.
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Ciardelli, I.A., Roelofsen, F. Inquisitive dynamic epistemic logic. Synthese 192, 1643–1687 (2015). https://doi.org/10.1007/s11229-014-0404-7
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DOI: https://doi.org/10.1007/s11229-014-0404-7