# zbMATH — the first resource for mathematics

Strong noncontingency: on the modal logics of an operator expressively weaker than necessity. (English) Zbl 1441.03019
The author provides a detailed study of propositional modal logic with unary strong noncontingency operator. It has intuitive meaning to the effect that if a formula with this operator is true, then it is necessarily true, and if it is false, then it is necessarily false. It may be applied as a formal way of expressing the content of question-embedding verbs like “knowing whether” or “knowing who”, as analysed by Hintikka. For example “$$a$$ knows whether $$p$$” may be understood as “if $$p$$, then $$a$$ knows that, and if not-$$p$$, then $$a$$ knows that not-$$p$$”, and such understanding can be formalised by means of the new operator. However, the paper is not concerned much with linguistic or philosophical motivations and applications. It is a formal study of logical properties of the new operator.
This new operator is compared with respect to its expressive power with operators of standard necessity, noncontingency and essence/accident. In particular, it is shown that logic of strong noncontingency is less expressive than standard logic of necessity, incomparable with the logic of noncontingency, and expressively equivalent to the logic of essence and accident. In contrast to necessity, the new operator is rather weak tool for frame definability; although symmetry is definable, the properties of seriality, reflexivity, transitivity and Euclideaness are not definable. Two sections (4, 5) are devoted to characterization results based on suitable notion of bisimulation. It is shown that this logic is a bisimulation-invariant fragment of standard modal logic and of the first-order logic. Axiomatization and completeness results for the basic logic and some extensions occupy Section 6 and 7. Finally a comparison with Humberstone’s work on the logic of agreement operator is provided.
The paper is interesting and readable. Despite the philosophical motivations beyond the new operator, it is adressed rather to formal logicians interested in technical development of propositional modal logic.
##### MSC:
 03B45 Modal logic (including the logic of norms) 03B42 Logics of knowledge and belief (including belief change)
Full Text:
##### References:
 [1] Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2001. · Zbl 0988.03006 [2] Cresswell, M. J., “Necessity and contingency,” Studia Logica, vol. 47 (1988), pp. 145-49. · Zbl 0666.03015 [3] Fan, J., and H. van Ditmarsch, “Neighborhood contingency logic,” pp. 88-99 in Logic and Its Applications, edited by M. Banerjee and S. Krishna, vol. 8923 of Lecture Notes in Computer Science, Springer, Berlin, 2015. · Zbl 1304.03049 [4] Fan, J., Y. Wang, and H. van Ditmarsch, “Almost necessary,” pp. 178-96 in Advances in Modal Logic, Vol. 10, edited by R. Goré, B. Kooi, and A. Kurucz, College Publications, London, 2014. · Zbl 1385.03017 [5] Fan, J., Y. Wang, and H. van Ditmarsch, “Contingency and knowing whether,” The Review of Symbolic Logic, vol. 8 (2015), pp. 75-107. · Zbl 1375.03023 [6] Hintikka, J., Knowledge and Belief, Cornell University Press, Ithaca, 1962. · Zbl 1384.03102 [7] Hintikka, J., “The semantics of questions and the questions of semantics: Case studies in the interrelations of logic, syntax, and semantics,” Acta Philosophica Fennica, vol. 28 (1976). · Zbl 0392.03004 [8] Humberstone, L., “The logic of non-contingency,” Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 214-29. · Zbl 0833.03004 [9] Humberstone, L., “The modal logic of agreement and noncontingency,” Notre Dame Journal of Formal Logic, vol. 43 (2002), pp. 95-127. · Zbl 1046.03008 [10] Karttunen, L., “Syntax and semantics of questions,” Linguistics and Philosophy, vol. 1 (1977), pp. 3-44. · Zbl 0342.02002 [11] Kuhn, S., “Minimal non-contingency logic,” Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 230-34. · Zbl 0833.03005 [12] Marcos, J., “Logics of essence and accident,” Bulletin of the Section of Logic, vol. 34 (2005), pp. 43-56. · Zbl 1117.03305 [13] Montgomery, H., and R. Routley, “Contingency and non-contingency bases for normal modal logics,” Logique et Analyse, Nouvelle Séries, vol. 9 (1966), pp. 318-28. · Zbl 0294.02008 [14] Steinsvold, C., “Completeness for various logics of essence and accident,” Bulletin of the Section of Logic, vol. 37 (2008), pp. 93-101. · Zbl 1286.03073 [15] Steinsvold, C., “A note on logics of ignorance and borders,” Notre Dame Journal of Formal Logic, vol. 49 (2008), pp. 385-92. · Zbl 1180.03017 [16] Steinsvold, C., “Being wrong: Logics for false belief,” Notre Dame Journal of Formal Logic, vol. 52 (2011), pp. 245-53. · Zbl 1252.03035 [17] van der Hoek, W., and A. Lomuscio, “A logic for ignorance,” Electronic Notes in Theoretical Computer Science, vol. 85 (2004), pp. 117-33. · Zbl 1270.03042 [18] van Ditmarsch, H., J. Fan, W. van der Hoek, and P. Iliev, “Some exponential lower bounds on formula-size in modal logic,” pp. 139-57 in Advances in Modal Logic, Vol. 10, edited by R. Goré, B. Kooi, and A. Kurucz, College Publications, London, 2014. · Zbl 1385.03029 [19] Wang, L., and O. Tai, “Skeptical conclusions,” Erkenntnis, vol. 72 (2010), pp. 177-204. [20] Zolin, E. E., “Completeness and definability in the logic of noncontingency,” Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 533-47. · Zbl 0989.03019
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.