Sequent Systems for Negative Modalities
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.
KeywordsNegative modalities sequent systems cut-admissibility Analyticity
Mathematics Subject Classification03B53 03B60 03B45 03F05
Open access funding provided by Max Planck Society. The authors acknowledge partial support by CNPq, by The Israel Science Foundation (Grant No. 817-15), by the Marie Curie project GeTFun (PIRSES-GA-2012-318986) funded by EU-FP7, and by the Humboldt Foundation. They also take the chance to thank Elaine Pimentel, Heinrich Wansing, Alessandra Palmigiano, Dorota Leszczyńska-Jasion, and two anonymous referees for their comments on an earlier incarnation of this manuscript.
- 1.Avron, A., Ciabattoni, A., Zamansky, A.: Canonical calculi: invertibility, axiom expansion and (non)-determinism. In: Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science—Theory and Applications, CSR’09, pp. 26–37. Springer, Berlin (2009)Google Scholar
- 3.Avron, A., Zamansky, A.: A paraconsistent view on \(B\) and \(S5\). In: Beklemishev, L., Demri, S., Máté, A. (eds.) Advances in Modal Logic, vol. 11, pp. 21–37. College Publications, London (2016)Google Scholar
- 13.Gentzen, G.: Untersuchungen über das logische Schließen I, 1934 An English translation appears in ‘The Collected Works of Gerhard Gentzen, edited by E. Szabo, North-Holland (1969)Google Scholar
- 14.Greco, G., Ma, M., Palmigiano, A., Tzimoulis, A., Zhao, Z.: Unified correspondence as a proof-theoretic tool. J. Log. Comput. exw022 (2016). doi: 10.1093/logcom/exw022
- 17.Humberstone, L.: Sentence connectives in formal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, 2015th edn. Fall, Cambridge (2015)Google Scholar
- 19.Jaśkowski, S.: A propositional calculus for inconsistent deductive systems (in Polish). Studia Societatis Scientiarum Torunensis, Sectio A, 5, 57–77 (1948). Translated into English in Studia Logica, 24, 143–157 (1967), and in Logic and Logical Philosophy 7, 35–56 (1999)Google Scholar
- 22.Lahav, O., Marcos, J., Zohar, Y.: It ain’t necessarily so: basic sequent systems for negative modalities. In: Beklemishev, L., Demri, S., Máté, A. (eds.) Advances in Modal Logic, vol. 11, pp. 449–468. College Publications, London (2016)Google Scholar
- 23.Lahav, O., Zohar, Y.: SAT-based decision procedure for analytic pure sequent calculi. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) Automated Reasoning, Volume 8562 of Lecture Notes in Computer Science, pp. 76–90. Springer, Berlin (2014)Google Scholar
- 25.Marcos, J.: Modality and paraconsistency. In: Bilkova, M., Behounek, L. (eds.) The Logica Yearbook 2004, pp. 213–222. Filosofia, Vaudreuil-Dorion (2005)Google Scholar
- 28.Nelson, D.: Negation and separation of concepts in constructive systems. In: Heyting, A. (ed.) Constructivity in Mathematics, Studies in Logic and the Foundations of Mathematics, pp. 208–225. North-Holland, Amsterdam (1959)Google Scholar
- 29.Onishi, T.: Substructural negations. Australas. J. Log. 12(4), 177–203 (2015) Article no. 1Google Scholar
- 32.Rauszer, C.: An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Dissertationes Mathematicae, vol. CLXVII. Instytut Matematyczny Polskiej Akademii Nauk, Warsaw (1980)Google Scholar
- 34.Ripley, D.W.: Negation in Natural Language. Ph.D. thesis, University of North Carolina at Chapel Hill (2009)Google Scholar
- 38.Vakarelov, D.: Consistency, completeness and negation. In: Priest, G., Sylvan, R., Norman, J. (eds.) Paraconsistent Logic: Essays on the Inconsistent, pp. 328–363. Philosophia Verlag, Munchen (1989)Google Scholar
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.