Abstract
We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.
Similar content being viewed by others
References
Crolard T.: Subtractive logic. Theoretical Computer Science 254, 151–185 (2001)
Czemark J.: A remark on Gentzen’s calculus of sequents. Notre Dame Journal of Formal Logic 18(3), 471–474 (1977)
de Campos Sanz, W., Falsity preservation. CLE e-print serie of Campinas University, Brazil 8, 2008, 2.
Dummett, M., What is a theory of meaning? (II). In G. Evans, and J. McDowell, (eds.), Truth and Meaning, Oxford University Press, 1976.
Dummett M.: The Logical Basis of Metaphysics. Duckworth, London. (1991)
Gentzen G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210 (1935)
Goodman N.: logic of contradiction. Zeitschrift für Logik und Grundlagen der Mathematik 27, 119–126 (1981)
Goré, R., Dual intuitionistic logic revisited. In R. Dyckhoff, (ed.), TABLEAUX00: Automated Reasoning with Analytic Tableaux and Related Methods, Springer, 2000.
Kamide N.: A note on dual-intuitionistic logic. Mathematical Logic Quarterly 49(5), 519–524 (2003)
Lopez-Escobar E. G. K.: Refutability and elementary number theory. Indagationes Mathematicae 34, 362–374 (1972)
Miller, D., Out of Error: Further Essays on Critical Rationalism. Ashgate, 2006.
Popper, K., What is dialectic? In Conjectures and Refutations: the Growth of Scientific Knowledge. Routledge & Kegan Paul, London, 1963, pp. 312–335.
Prawitz D.: Natural Deduction. A proof-theoretical study. Almqvist & Wiksell, Stockholm (1965)
Prawitz, D., Ideas and results in proof theory. In J. E. Fenstad, (ed.), Proceedings of the Second Scandinavian Logic Symposium, North Holland, Amsterdam, 1971, pp. 235-308.
Rauszer C.: A formalization of propositional calculus of H-B logic. Studia Logica 33(1), 23–34 (1974)
Rauszer C.: Semi-boolean algebras and their applications to intuitionistic logic with dual operations. Fundamaenta Mathematicae 83, 219–249 (1974)
Schroeder-Heister P.: Validity concepts in proof-theoretic semantics. Synthese 148, 525–571 (2006)
Schroeder-Heister, P., Schluß und Umkehrschluß: ein Beitrag zur Definitionstheorie. In C. F. Gethmann, (ed.), Akten des XXI Deutschen Kongresses f¨ur Philosophie (Essen, 15.-19.9.2008), Deutsches Jahrbuch Philosophie, Band 3, Felix Meiner Verlag, Hamburg, 2009.
Shramko Y.: Dual intuitionistic logic and a variety of negations: The logic of scientific research. Studia Logica 80, 347–367 (2005)
Tranchini, L., Refutation: a proof-theoretic account. In C. Marletti, (ed.), First Pisa Colloquium in Logic, Language and Epistemology, ETS, Pisa, 2010.
Troelstra, A. S., H. Schwichtemberg, Basic Proof Theory, Cambridge University Press, 1996.
Urbas I.: Dual-intuitionistic logic. Notre Dame Journal of Formal Logic 37(3), 440–451 (1996)
Uustalu, T., A note on anti-intuitionistic logic. Abstract presented at the Nordic Workshop on Programming Theory (NWPT’97), Tallinn, Estonia, 1997.
Wansing H.: Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics 18, 341–364 (2008)
Wolter F.: On logics with coimplication. Journal of Philosophical Logic 27, 353–387 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Heinrich Wansing
Rights and permissions
About this article
Cite this article
Tranchini, L. Natural Deduction for Dual-intuitionistic Logic. Stud Logica 100, 631–648 (2012). https://doi.org/10.1007/s11225-012-9417-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-012-9417-8