×

Questions and dependency in intuitionistic logic. (English) Zbl 1453.03025

This paper shows how the inquisitive logic and dependence logic which allow for a logical analysis of questions and dependencies between propositions can be developed on the intuitionistic basis. As a result, the intuitionistic inquisitive logic (InqI) is introduced, which deals not only with intuitionistic statements, but also with questions and formulas that express dependencies. To this effect, the authors develop a kind of Kripke-semantics for intuitionistic logic based on the notion of support at a team, rather than on the notion of truth at a possible world. Namely, having a standard intuitionistic Kripke model \(M = \langle W, R, V \rangle\), a team in \(M\) is defined as a set of worlds \(t \subseteq W\). Moreover, a team \(t^\prime\) is an extension of a team \(t\) iff \(t \subseteq R[t]\), where \(R[t] : = \bigcup_{w \in t} R[w] \) (\(R[w] = \{ w^\prime : wRw^\prime \}\)). Then one defines the intuitionistic notion of support with respect to a team in a Kripke model, so that, e.g., an atomic proposition \(p\) is supported by a team \(t\) in \(M\) iff \(p\) is true at every world \(w\) from this team: \(M, t \models p \Leftrightarrow V(w, p) = 1\) for all \(w \in t\). This definition is then naturally extended to compound formulas. To deal with questions one enriches the standard intuitionistic language with a new connective ‘inquisitive disjunction’ (\(\scriptstyle\mathbb{V}\)), where \(\varphi \:{\scriptstyle\mathbb{V}}\: \psi\) is regarded as a question whether \(\varphi\) or \(\psi\). The support condition for inquisitive disjunction is then as follows: \(M, t \models \varphi \:{\scriptstyle\mathbb{V}}\: \psi \Leftrightarrow M, t \models \varphi \mbox{ or } M, t \models \psi \). It turns out that a question \(\mu\) determines another question \(\nu\) in a team \(t\) of a model \(M\) iff the team \(t\) supports the implication \(\mu \rightarrow\nu\). The authors introduce the notion of entailment between formulas of InqI and construct a natural deduction system which is obtained from the respective system for classical inquisitive logic by simply dropping the double negation elimination rule. This system is sound and complete with respect to the proposed semantics. Thus, the authors conclude, “the only difference between the classical and the intuitionistic version of inquisitive logic lies in the underlying logic of statements, while the relation between statements and questions is the same in both cases”.

MSC:

03B65 Logic of natural languages
03B60 Other nonclassical logic
03B20 Subsystems of classical logic (including intuitionistic logic)
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid Link

References:

[1] Abramsky, S., and J. Väänänen, “From IF to BI: A tale of dependence and separation,” Synthese, vol. 167 (2009), pp. 207-30. · Zbl 1175.03016
[2] Armstrong, W. W., “Dependency structures of data base relationships,” pp. 580-83 in Information Processing 74 (Stockholm, 1974), edited by J. L. Rosenfeld, North-Holland, Amsterdam, 1974. · Zbl 0296.68038
[3] Chagrov, A., and M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, New York, 1997. · Zbl 0871.03007
[4] Ciardelli, I., “Modalities in the realm of questions: Axiomatizing inquisitive epistemic logic,” pp. 94-113 in Advances in Modal Logic, Vol. 10, edited by R. Goré, B. Kooi, and A. Kurucz, College Publications, London, 2014. · Zbl 1385.03006
[5] Ciardelli, I., “Dependency as question entailment,” pp. 129-81 in Dependence Logic: Theory and Applications, edited by S. Abramsky, J. Kontinen, J. Väänänen, and H. Vollmer, Birkhäuser/Springer, Cham, 2016. · Zbl 1429.03112
[6] Ciardelli, I., “Questions as information types,” Synthese, vol. 195 (2018), pp. 321-65. · Zbl 1455.03038
[7] Ciardelli, I., “Inquisitive semantics and intermediate logics,” master’s dissertation, University of Amsterdam, Amsterdam, 2009, https://www.illc.uva.nl/Research/Publications/Reports/MoL-2009-11.text.pdf.
[8] Ciardelli, I., “Questions in logic,” Ph.D. dissertation, University of Amsterdam, Amsterdam, 2016. · Zbl 1390.03027
[9] Ciardelli, I., J. Groenendijk, and F. Roelofsen, “On the semantics and logic of declaratives and interrogatives,” Synthese, vol. 192 (2015), pp. 1689-728. · Zbl 1357.03064
[10] Ciardelli, I., and M. Otto, “Bisimulation in inquisitive modal logic,” pp. 151-66 in Theoretical Aspects of Rationality and Knowledge, edited by J. Lang, vol. 251 of Electronic Proceedings in Theoretical Computer Science (EPTCS), EPTCS, n.p., 2017.
[11] Ciardelli, I., and F. Roelofsen, “Inquisitive logic,” Journal of Philosophical Logic, vol. 40 (2011), pp. 55-94. · Zbl 1214.03019
[12] Ciardelli, I., and F. Roelofsen, “Inquisitive dynamic epistemic logic,” Synthese, vol. 192 (2015), pp. 1643-87. · Zbl 1357.03046
[13] Ebbing, J., L. Hella, A. Meier, J.-S. Müller, J. Virtema, and H. Vollmer, “Extended modal dependence logic,” pp. 126-37 in Logic, Language, Information, and Computation, edited by L. Libkin, U. Kohlenbach, and R. de Queiroz, vol. 8071 of Lecture Notes in Computer Science, Springer, Heidelberg, 2013. · Zbl 1394.03047
[14] Frittella, S., G. Greco, A. Palmigiano, and F. Yang, “A multi-type calculus for inquisitive logic,” pp. 215-33 in Logic, Language, Information, and Computation, edited by J. Väänänen, A. Hirvonen, and R. de Queiroz, vol. 9803 of Lecture Notes in Computer Science, Springer, Berlin, 2016. · Zbl 1429.03128
[15] Galliani, P., “Inclusion and exclusion dependencies in team semantics: On some logics of imperfect information,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 68-84. · Zbl 1250.03047
[16] Galliani, P., and L. Hella, “Inclusion logic and fixed point logic,” pp. 281-95 in Computer Science Logic 2013, edited by R. Ronchi della Rocca, vol. 23 of Leibniz International Proceedings in Informatics, Schloss Dagstuhl, Wadern, 2013. · Zbl 1356.03071
[17] Grädel, E., and J. Väänänen, “Dependence and independence,” Studia Logica, vol. 101 (2013), pp. 399-410. · Zbl 1272.03125
[18] Kontinen, J., J.-S. Müller, H. Schnoor, and H. Vollmer, “A van Benthem theorem for modal team semantics,” pp. 277-91 in Twenty-fourth EACSL Annual Conference on Computer Science Logic (CSL 2015), edited by S. Kreutzer, vol. 41 of Leibniz International Proceedings in Informatics, Schloss Dagstuhl, Wadern, 2015. · Zbl 1373.03024
[19] Pitts, A. M., “On an interpretation of second-order quantification in first-order intuitionistic propositional logic,” Journal of Symbolic Logic, vol. 57 (1992), pp. 33-52. · Zbl 0763.03009
[20] Punčochář, V., “Weak negation in inquisitive semantics,” Journal of Logic, Language, and Information, vol. 24 (2015), pp. 323-55. · Zbl 1369.03108
[21] Punčochář, V., “A generalization of inquisitive semantics,” Journal of Philosophical Logic, vol. 45 (2016), pp. 399-428. · Zbl 1415.03036
[22] Punčochář, V., “Algebras of information states,” Journal and Logic and Computation, vol. 27 (2017), pp. 1643-75. · Zbl 1444.03072
[23] Punčochář, V., “Substructural inquisitive logics,” Review of Symbolic Logic, vol. 12 (2019), pp. 296-330. · Zbl 07063893
[24] Sano, K., and J. Virtema, “Characterizing frame definability in team semantics via the universal modality,” pp. 140-55 in Logic, Language, Information, and Computation, edited by V. de Paiva, R. de Queiroz, L. S. Moss, D. Leivant, and A. G. de Oliveira, vol. 9160 of Lecture Notes in Computer Science, Springer, Heidelberg, 2015. · Zbl 1465.03063
[25] Sano, K., and J. Virtema, “Characterizing relative frame definability in team semantics via the universal modality,” pp. 392-409 in Logic, Language, Information, and Computation, edited by J. Väänänen, A. Hirvonen, and R. de Queiroz, vol. 9803 of Lecture Notes in Computer Science, Springer, Berlin, 2016. · Zbl 1478.03041
[26] Väänänen, J., Dependence Logic: A New Approach to Independence Friendly Logic, vol. 70 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2007. · Zbl 1117.03037
[27] Väänänen, J., “Modal dependence logic,” pp. 237-54 in New Perspectives on Games and Interaction, edited by K. Apt and R. van Rooij, vol. 4 of Texts in Logic and Games, Amsterdam University Press, Amsterdam, 2008. · Zbl 1377.03011
[28] Yang, F., “Expressing second-order sentences in intuitionistic dependence logic,” Studia Logica, vol. 101 (2013), pp. 323-42. · Zbl 1272.03126
[29] Yang, F., “Modal dependence logics: Axiomatizations and model-theoretic properties,” Logic Journal of the IGPL, vol. 25 (2017), pp. 773-805.
[30] Yang, F., “On extensions and variants of dependence logic: A study of intuitionistic connectives in the team semantics setting,” Ph.D. dissertation, University of Helsinki, Helsinki, 2014.
[31] Yang, F., and J. Väänänen, “Propositional logics of dependence,” Annals of Pure and Applied Logic, vol. 167 (2016), pp. 557-89. · Zbl 1355.03021
[32] Yang, F., and J. Väänänen, “Propositional team logics,” Annals of Pure and Applied Logic, vol. 168 (2017), pp. 1406-41. · Zbl 1422.03058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.