Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T03:46:13.007Z Has data issue: false hasContentIssue false
  • English
  • Français

A MODAL TRANSLATION FOR DUAL-INTUITIONISTIC LOGIC

Published online by Cambridge University Press:  12 February 2016

YAROSLAV SHRAMKO
YAROSLAV SHRAMKO*
Affiliation:
Department of Philosophy Kryvyi Rih State Pedagogical University
*
*DEPARTMENT OF PHILOSOPHY KRYVYI RIH STATE PEDAGOGICAL UNIVERSITY KRYVYI RIH, 50086, UKRAINE E-mail: shramko@rocketmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct four binary consequence systems axiomatizing entailment relations between formulas of classical, intuitionistic, dual-intuitionistic and modal (S4) logics, respectively. It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula. An analogous modal translation of dual-intuitionistic formulas then consists of prefixing ◊ to every subformula of the translated formula. The philosophical importance of this result is briefly discussed.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 2 , June 2016 , pp. 251 - 265
Copyright
Copyright © Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Atiyah, M. (2008). Duality in mathematics and physics, In Conferències FME. Vol. V, Curs Riemann, 2007–2008, Barcelona: Facultat de Matemàtiques i Estadística, pp. 69–91.Google Scholar
Bellin, G. & Biasi, C. (2004). Towards a logic for pragmatics. Assertions and conjectures, Journal of Logic and Computation, 14(4), pp. 473–506.Google Scholar
Bocheński, J. M. & Menne, A. (1962). Grundriß der Logistik. Paderborn: Schöningh.Google Scholar
Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge: Cambridge University Press.Google Scholar
Crolard, T. (2001). Subtractive logic, Theoretical Computer Science, 254(1–2), pp. 151–185.Google Scholar
Curry, H. B. (1963). Foundations of Mathematical Logic. New York: McGraw-Hill.Google Scholar
Czermak, J. (1977). A remark on Gentzen’s calculus of sequents, Notre Dame Journal of Formal Logic, 18(3), pp. 471–474.CrossRefGoogle Scholar
Dunn, J. M. (1995). Positive modal logic, Studia Logica, 55, pp. 301–317.Google Scholar
Fitting, M. (1969). Intuitionistic Logic, Model Theory and Forcing, Amsterdam: North-Holland Pub. Co.Google Scholar
Fitting, M. (1970). An embedding of classical logic in S4, Journal of Symbolic Logic, 35, pp. 529–534.CrossRefGoogle Scholar
Gödel, K. (1933). Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums 4, pp. 39–40. Reprinted, with English translation, in Gödel, K. (1986). Collected Works. I: Publications 1929–1936, S. Feferman et al., editors. Oxford: Oxford University Press, pp. 300–303.Google Scholar
Goodman, N. D. (1981). The logic of contradiction, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 27, pp. 119–126.Google Scholar
Goré, R. (2000). Dual intuitionistic logic revisited, In Dyckhoff, R., editor. Proceedings Tableaux 2000 (Lecture Notes in Artificial Intelligence 1847), Springer Verlag, Berlin, pp. 252–267.Google Scholar
Goré, R. & Postniece, L. (2010). Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic, Journal of Logic and Computation, 20, pp. 233–260.Google Scholar
Heyting, A. (1956). Intuitionism: An Introduction. Amsterdam: North-Holland.Google Scholar
Humberstone, L. (2005). Contrariety and subcontrariety: The anatomy of negation (with special reference to an example of J.-Y. Béziau), Theoria, 71, pp. 241–262.Google Scholar
Kripke, S. (1965). Semantical analysis of intuitionistic logic I, Formal Systems and Recursive Functions, Crossley, J. N. and Dummett, M. A., editors. Amsterdam: North-Holland, pp. 92–130.Google Scholar
Lewis, D. (1982). Logic for equivocators, Noûs, 16, pp. 431–441.Google Scholar
Łukowski, P. (1996). Modal interpretation of Heyting-Brouwer logic, Bulletin of the Section of Logic, 25, pp. 80–83.Google Scholar
Miller, D. (2005). Out of Error: Further Essays on Critical Rationalism. Aldershot and Brookfield, Ashgate.Google Scholar
Nelson, D. (1949). Constructible falsity, Journal of Symbolic Logic, 14, pp. 16–26.Google Scholar
Novikov, P. S. (1977). Constructive Mathematical Logic from the Classical Point of View (in Russian), Moscow: Nauka.Google Scholar
Pinto, L. & Uustalu, T. (2010). Relating sequent calculi for bi-intuitionistic propositional logic, In van Bakel, S., Berardi, S., Berger, U., editors. Proceedings Third International Workshop on Classical Logic and Computation, EPTCS, 47, pp. 57–72.Google Scholar
Popper, K. (1962). Conjectures and Refutations: The Growth of Scientific Knowledge, New York, London: Basic Books.Google Scholar
Pozza, C. D. & Garola, C. (1995). A pragmatic interpretation of intuitionistic propositional logic, Erkenntnis, 43, pp. 81–109.Google Scholar
Přenosil, A. (2014). A duality for distributive unimodal logic, In Goré, R., Kooi, B., Kurucz, A., editors. Advances in Modal Logic, vol. 10, London: College Publications, pp. 423–438.Google Scholar
Raatikainen, P. (2004). Conceptions of truth in intuitionism, History and Philosophy of Logic, 25, pp. 131–145.Google Scholar
Rauszer, C. (1974). Semi-Boolean algebras and their applications to intuitionistic logic with dual operations, Fundamenta Mathematicae, 83, pp. 219–249.Google Scholar
Routley, R. (1979). Dialectical logic, semantics and metamathematics, Erkenntnis, 14, pp. 301–331.Google Scholar
Schroeder-Heister, P. (2009). Schluß und Umkehrschluß: ein Beitrag zur Definitionstheorie, In Gethmann, C. F., editor. Deutsches Jahrbuch Philosophie 02. Lebenswelt und Wissenschaft, Felix Meiner Verlag, Hamburg, pp. 1065–1092.Google Scholar
Shramko, Y. (2005). Dual intuitionistic logic and a variety of negations: the logic of scientific research, Studia Logica, 80, pp. 347–367.Google Scholar
Shramko, Y. (2012). What is a genuine intuitionistic notion of falsity?, Logic and Logical Philosophy, 21, pp. 3–23.Google Scholar
Skolem, T. (1919). Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. In Skrifter utgit av Videnskabsselskapet i Kristiania, vol. 3, 1919; reprinted in: Skolem, T. (1970). Selected Works in Logic, J. E. Fenstad, editor. Oslo, Bergen, Tromsö: Universitetforlaget, pp. 67–101.Google Scholar
Tranchini, L. (2012). Natural deduction for dual-intuitionistic logic, Studia Logica, 100, pp. 631–648.Google Scholar
Troelstra, A. & van Dalen, D. (1988). Constructivism in Mathematics, Volume I, Amsterdam: North-Holland.Google Scholar
Turquette, A. R. (1948). Review: Gr.C. Moisil, Logique Modale, Journal of Symbolic Logic, 13, pp. 162–163.Google Scholar
Urbas, I. (1996). Dual-intuitionistic logic, Notre Dame Journal of Formal Logic, 37, pp. 440–451.Google Scholar
Wansing, H. (2008). Constructive negation, implication, and co-implication, Journal of Applied Non-Classical Logics, 18, pp. 341–364.CrossRefGoogle Scholar
Wansing, H. (2010). Proofs, disproofs, and their duals, In Beklemishev, L., Goranko, V. and Shehtman, V., editors. Advances in Modal Logic, v. 8, College Publications, pp. 483–505.Google Scholar
Wansing, H. (2013). Falsification, natural deduction, and bi-intuitionistic logic, Journal of Logic and Computation, doi: 10.1093/logcom/ext035.Google Scholar
Wolter, F. (1998). On logics with coimplication, Journal of Philosophical Logic, 27, pp. 353–387.Google Scholar