Abstract
The Interrogative Model of Inquiry (IMI) and Dynamic Epistemic Logics (DELs) are two central paradigms in formal epistemology. This paper is motivated by the observation of a significant complementarity between them: on the one hand, the IMI provides a framework for investigating inquiry represented as an idealized game between an Inquirer and Nature, along with an account of the interaction between questions and inferences in information-seeking processes, but is lacking a formulation in the multi-agent case; on the other hand, DELs model various operations of information change in multi-agent systems, but the field is lacking a proper integration of question and inference dynamics, along with an application to the investigation of inquiry processes. The goal of this paper is to integrate the two paradigms in such a way as to combine their respective insights. To this end, we develop a formal system called DEL\(_\mathrm{IMI }\) which aims to represent the interaction between question and inference dynamics in inquiry—as described by the IMI—in a multi-agent setting, and this in such a way as to enable an investigation of inquiry games with multi-agent dimensions. The DEL\(_\mathrm{IMI }\) system is designed to represent the possible moves of such inquiry games through three types of epistemic actions: agents addressing questions to Nature, agents addressing questions to other agents, agents drawing logical inferences. We then show how the resulting framework can be used to formally define multi-agent inquiry games. We conclude by evaluating the interest of the DEL\(_\mathrm{IMI }\) system for the IMI and DELs paradigms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For an overview of the different operations of information change that have been studied within the DELs paradigm, we refer the reader to van Benthem (2011).
We will refer to the former as social inquiry games, and to the latter as higher-order inquiry games.
Inferences or deductive moves in the IMI are usually represented by Hintikka as tableau construction steps in the tableau method (D’Agostino 1999). A formalization of the IMI along this line can be found in (Hintikka et al. 1999). Since this is a standard representation of inferences, we will not discuss it further in the following and will rather focus on the IMI representation of questions.
In this last reference, Hintikka has proposed an important evolution of his theory of questions based on what he calls the ‘second-generation’ epistemic logic which introduces, and makes use of, the notion of independence. In this paper, we will only be concerned with the original formulation of the theory based on the ‘first-generation’, or the classical version of, epistemic logic.
A propositional question \(Q = (\gamma _1,\ldots ,\gamma _k)\) does not require that the answers to \(Q\) be formulas from the language of propositional logic. In the framework of interrogative logic (Hintikka et al. 1999), answers to propositional questions can be first-order formulas. In this paper, we will consider propositional questions where answers can be formulas from the language of multi-agent epistemic logic.
For a review of possible solutions to deal with logical omniscience, we refer the reader to Halpern and Pucella (2011).
See the PhD thesis of Velázquez-Quesada (2011) for a full report on this line of research.
The other boolean connectives \(\vee \), \(\rightarrow \) and \(\leftrightarrow \) are defined in the usual way.
The awareness modalities correspond to the ones introduced by Fagin and Halpern (1988) in the so-called awareness logic. In this paper, our use of the awareness modalities is purely instrumental in defining the notion of explicit knowledge. In particular, we do not aim to analyze any intuitive or informal notion of what it means for an agent to be “aware of” a given statement. We use the term ‘awareness’ and the expression ‘aware of’ only for being consistent with the literature on awareness logic. Thus, awareness of \(\varphi \) for an agent does not imply any particular epistemic attitude towards \(\varphi \) on the part of the agent.
Throughout this paper, we make the common assumption that the indistinguishability relation for any agent \(a \in \mathsf {Ag}\) is an equivalence relation.
Several options are available for defining explicit knowledge from implicit knowledge and awareness. For instance, Fagin and Halpern (1988) propose the alternative \(\mathsf {Ex}_a \varphi \,\,{:=}\,\, K_a \varphi \wedge A_a \varphi \). For a discussion of this issue see Sect. 3 of van Benthem and Velázquez-Quesada (2010). These choices are independent of the main concerns of this paper, and the DEL\(_\mathrm{IMI }\) system can perfectly be developed in the same fashion based on different definitions of explicit knowledge.
At this stage, different sets of assumptions can be adopted in the DEL\(_\mathrm{IMI }\) system, reflecting different modelling choices. Such sets of assumptions should then be adapted to the specific situation one is interested to represent. In order to maintain a certain level of generality and flexibility for DEL\(_\mathrm{IMI }\), we are here restricting ourselves to a set of minimal assumptions necessary to model properly the requirements (i)–(iii).
This property is called ‘weak introspection’ in van Benthem and Velázquez-Quesada (2010).
One might be worried that this assumption is defined in a circular way as it makes use of the notion of valid principle on our intended class of models while this assumption will be part of the definition of our intended class of models. A way out of this difficulty is to understand the notion of valid principle as given syntactically by the logic \(\mathsf {EL}_\mathsf {IMI}\) to be defined in the next section. Given the soundness and completeness theorem for the logic \(\mathsf {EL}_\mathsf {IMI}\) we will provide, one can realize a posteriori that the set of formulas \(\mathsf {EL}_\mathsf {IMI}\) corresponds indeed to the set of semantically valid principles on our intended class of models.
More precisely, the condition that “\(\varphi \in \fancyscript{L}_s\) is a valid principle” should be understood as “\(\varphi \in \mathsf {EL}_\mathsf {IMI}\)”, where \(\mathsf {EL}_\mathsf {IMI}\) will be defined in the next section.
This assumption follows Hintikka’s theory of questions, as receiving one true answer to a propositional question \(Q\) suffices to bring the questioner into an epistemic state satisfying the desideratum of \(Q\).
In the following, by the term ‘IMI epistemic model’ we will refer to models of this class.
The completeness via canonical model method is presented in Chap. 4 of Blackburn et al. (2002).
In the remaining of this paper, we will use the following abbreviations: we will write \(Q_O\) for ‘\((\gamma _1,\ldots ,\gamma _k)\)’, \(Q_A\) for ‘\((\varphi _1,\ldots ,\varphi _n)\)’ and \(I\) for ‘\(\{\psi _1,\ldots ,\psi _m\} \hookrightarrow \psi _c\)’.
Such an operation has been suggested in van Benthem and Velázquez-Quesada (2010) under the name ‘explicit seeing operation’.
In this paper, we will refer to this type of games—in which goals and actions are inherently epistemic—as informational games.
If \(X\) is a set of IMI epistemic models, we denote by \(X / \cong \) the quotient set of \(X\) by \(\cong \), i.e., the set of equivalence classes induced by \(\cong \) on \(X\).
This restriction is necessary for the representation of answering moves we adopt here. It does not, however, impact the modelling power of the framework, as our primary interest is to represent questions and inferences concerning factual and higher-order information.
This goal formula is exactly the desideratum of question \(Q\) according to Hintikka’s theory of questions. Notice that having the notion of desideratum as specifying the epistemic state in which the agent wants to be brought in is in direct line with Hintikka’s conception of the desiderata of questions. This is therefore the natural place to integrate and represent the notion of desideratum in our approach.
In the notation introduced previously, this means respectively that \(\mathsf {Ag} = \{a,b,c\}\) and \(\mathsf {P} = \{p,q\}\).
In DEL\(_\mathrm{IMI }\), the awareness sets of the different agents contain all the valid principles on the class of models \(\mathbf {DEL_{IMI}}\). In the figures of this section, we only represent in the awareness sets the formulas which do not correspond to valid principles.
For a discussion of this issue, see (van Benthem and Velázquez-Quesada (2010), p. 9).
Notice that the non-persistence of explicit knowledge in situations where higher-order information plays a role in inquiry might constitute an important challenge to the proof-theoretic approach of interrogative logic (Hintikka et al. 1999), as one will have to provide additional mechanisms to represent possible changes in previously acquired explicit knowledge.
For a discussion of the social dimensions of scientific knowledge and inquiry from the point of view of the philosophy of science see (Longino 2013).
References
Ågotnes, T., van Benthem, J., van Ditmarsch, H., & Minica, S. (2011). Question–answer games. Journal of Applied Non-Classical Logics, 21(3–4), 265–288.
Ågotnes, T., & van Ditmarsch, H. (2011). What will they say? Public announcement games. Synthese, 179(1), 57–85.
Artemov, S. (2008). The logic of justification. The Review of Symbolic Logic, 1(4), 477–513.
Blackburn, P., De Rijke, M., & Venema, Y. (2002). Modal logic. Cambridge: Cambridge University Press.
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40(1), 55–94.
Ciardelli, I., & Roelofsen, F. (2013). Inquisitive dynamic epistemic logic. Synthese. doi:10.1007/s11229-014-0404-7.
D’Agostino, M. (1999). Tableau methods for classical propositional logic. In M. D’Agostino, D. Gabbay, R. Haehnle, & J. Posegga (Eds.), Handbook of tableau methods (pp. 45–123). Dordrecht: Kluwer.
Duc, H. (1997). Reasoning about rational, but not logically omniscient, agents. Journal of Logic and Computation, 7(5), 633–648.
Enqvist, S. (2009). Interrogative belief revision in modal logic. Journal of Philosophical Logic, 38(5), 527–548.
Fagin, R., & Halpern, J. (1988). Belief, awareness, and limited reasoning. Artificial Intelligence, 34(1), 39–76.
Genot, E. (2009). The game of inquiry: The interrogative approach to inquiry and belief revision theory. Synthese, 171(2), 271–289.
Groenendijk, J., & Roelofsen, F. (2009). Inquisitive semantics and pragmatics. In J. M. Larrazabal & L. Zubeldia (Eds.), Meaning, content, and argument: Proceedings of the ILCLI international workshop on semantics, pragmatics, and rhetoric. Sebastian: University of the Basque Country Press.
Groenendijk, J., & Stokhof, M. (1984). Studies on the semantics of questions and the pragmatics of answers. PhD thesis, Universiteit van 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.
Grossi, D., & Velázquez-Quesada, F. (2009). Twelve angry men: A study on the fine-grain of announcements. In X. He, J. Horty, & E. Pacuit (Eds.), Logic, rationality, and interaction. LNCS (Vol. 5834, pp. 147–160). New York: Springer.
Halonen, I., & Hintikka, J. (2005). Toward a theory of the process of explanation. Synthese, 143(1), 5–61.
Halpern, J. Y., & Pucella, R. (2011). Dealing with logical omniscience: Expressiveness and pragmatics. Artificial Intelligence, 175(1), 220–235.
Hintikka, J. (1976). The semantics of questions and the questions of semantics: Case studies in the interrelations of logic, semantics, and syntax. Acta Philosophica Fennica, 28(4).
Hintikka, J. (1985). A spectrum of logics of questioning in recent developments in dialogue logics. Philosophica, 35(1), 135–150.
Hintikka, J. (1988). What is the logic of experimental inquiry? Synthese, 74(2), 173–190.
Hintikka, J. (1992). The interrogative model of inquiry as a general theory of argumentation. Communication and Cognition, 25(2–3), 221–242.
Hintikka, J. (1999). Inquiry as inquiry: A logic of scientific discovery, Jaakko Hintikka selected papers (Vol. 5). Dordrecht: Kluwer.
Hintikka, J. (2007). Socratic epistemology: Explorations of knowledge-seeking by questioning. Cambridge: Cambridge University Press.
Hintikka, J., Halonen, I., & Mutanen, A. (1999). Interrogative logic as a general theory of reasoning. In Inquiry as inquiry: A logic of scientific discovery (pp. 47–90). Dordrecht: Kluwer.
Jago, M. (2009). Epistemic logic for rule-based agents. Journal of Logic, Language and Information, 18(1), 131–158.
Longino, H. (2013). The social dimensions of scientific knowledge. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/spr2013/entries/scientific-knowledge-social/.
Peliš, M., & Majer, O. (2009). Logic of questions from the viewpoint of dynamic epistemic logic. In M. Peliš (Ed.), The Logica Yearbook (2009) (pp. 157–172). London: College Publications.
Peliš, M., & Majer, O. (2011). Logic of questions and public announcements. In N. Bezhanishvili, S. Löbner, K. Schwabe, & L. Spada (Eds.), Eighth international Tbilisi symposium on logic, language and computation (2009). Lecture notes in computer science (pp. 145–157). Berlin: Springer.
Pucella, R. (2006). Deductive algorithmic knowledge. Journal of Logic and Computation, 16(2), 287–309.
Roelofsen, F. (2013). Algebraic foundations for the semantic treatment of inquisitive content. Synthese, 190(1), 79–102.
van Benthem, J. (2008). Merging observation and access in dynamic logic. Journal of Logic Studies, 1(1), 1–17.
van Benthem, J. (2011). Logical dynamics of information and interaction. Cambridge: Cambridge University Press.
van Benthem, J. (2013). Logic in games. Cambridge, MA: MIT Press.
van Benthem, J., & Minică, Ş. (2012). Toward a dynamic logic of questions. Journal of Philosophical Logic, 41(4), 633–669.
van Benthem, J., & Velázquez-Quesada, F. (2010). The dynamics of awareness. Synthese, 177, 5–27.
van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic Epistemic Logic. Synthese Library (Vol. 337). Heidelberg: Springer.
Velázquez-Quesada, F. (2009). Inference and update. Synthese (Knowledge, Rationality and Action), 169(2), 283–300.
Velázquez-Quesada, F. R. (2010). Dynamic epistemic logic for implicit and explicit beliefs. In O. Boissier, A. E. F. Seghrouchni, S. Hassas, & N. Maudet (Eds.), MALLOW 2010, CEUR workshop proceedings (Vol. 627), Lyon, France. http://ceur-ws.org/Vol-627.
Velázquez-Quesada, F. (2011). Small steps in dynamics of information. PhD Thesis, Institute for Logic, Language and Computation (ILLC), Universiteit van Amsterdam (UvA), Amsterdam.
Wiśniewski, A. (2001). Questions and inferences. Logique & Analyse, 173–175, 5–43.
Acknowledgments
I am very grateful to Johan van Benthem for his many advices and comments on this paper. I have benefited from discussions of earlier versions of this work with Can Başkent, Emmanuel Genot, Eric Pacuit, Ştefan Minică, Gabriel Sandu and Fernando Velázquez-Quesada. Finally, I am indebted to two anonymous reviewers for suggesting substantial improvements in the orientation and motivations of an earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamami, Y. The interrogative model of inquiry meets dynamic epistemic logics. Synthese 192, 1609–1642 (2015). https://doi.org/10.1007/s11229-014-0460-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0460-z