Abstract
This is a review, with historical and critical comments, of a paper by I. E. Orlov from 1928, which gives the oldest known axiomatization of the implication-negation fragment of the relevant logic R. Orlov's paper also foreshadows the modal translation of systems with an intuitionistic negation into S4-type extensions of systems with a classical, involutive, negation. Orlov introduces the modal postulates of S4 before Becker, Lewis and Gödel. Orlov's work, which seems to be nearly completely ignored, is related to the contemporancous work on the axiomatization of intuitionistic logic.
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References
Ackermann W., 1925, Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit,Mathematische Annalen 93, pp. 1–36.
Anderson A. R. and Belnap N. D. Jr, 1975,Entailment: The Logic of Relevance and Necessity, vol. I, Princeton University Press, Princeton.
Becker O., 1930, Zur Logik der Modalitäten,Jahrbuch für Philosophie und phänomenologische Forschung 11, pp. 497–548.
Brouwer L. E. J., 1925, Zur Begründung der intuitionistischen Mathematik, I,Mathematische Annalen 93, pp. 244–257 (reprinted in Brouwer, 1975).
Brouwer L. E. J., 1925a, Intuitionistische Zerlegung mathematischer Grundbegriffe,Jahresbericht der Deutschen Mathematiker-Vereinigung 33, pp. 251–256 (reprinted in Brouwer, 1975).
Brouwer L. E. J., 1975,Collected Works, vol. I: Philosophy and Foundations of Mathematics, North-Holland, Amsterdam.
Chagrov, A. and Zakharyashchev, M., 1990, Modal companions of intermediate logics: A survey (manuscript).
Church A., 1936, A bibliography of symbolic logic,The Journal of Symbolic Logic 1, pp. 121–218.
Church A., 1951, The weak theory of implication, in: A. Menne et al. eds.Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften, Alber, Munich, pp. 22–37 (abstract: The weak positive implicational calculus,The Journal of Symbolic Logic 16 (1951), p. 238).
Došen K., 1986, Modal translations and intuitionistic double negation,Logique et Analyse (N.S.)29, pp. 81–94.
Došen K., 1988, Sequent-systems and groupoid models, I and II,Studia Logica 47, pp. 353–385,48 (1989), pp. 41–65,49 (1990), p. 614.
Došen K., 1990, Modal translations of Heyting and Peano arithmetic,Publications de l'Institut Mathématique (N.S.)47(61), pp. 13–23.
Došen K., 1992, Modal translations in substructural logics,Journal of Philosophical Logic 21, pp. 283–336.
Dunn J. M., 1986, Relevance logic and entailment, in: D. Gabbay and F. Guenthner eds,Handbook of Philosophical Logic, vol. III: Alternatives to Classical Logic D. Reidel, Dordrecht, pp. 117–224.
Gentzen G., 1935, Untersuchungen über das logische Schließen,Mathematische Zeitschrift 39, pp. 176–210, 405–431 (English translation in:The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam. 1969).
Girard J.-Y., 1987, Linear logic,Theoretical Computer Science 50, pp. 1–102.
Glivenko V. I., 1928, Sur la logique de M. Brouwer,Académic Royale de Belgique, Bulletin de la Classe des Sciences (5e série)14, pp. 225–228.
Glivenko V. I., 1929, Sur quelques points de la Logique de M. Brouwer,Académie Royale de Belgique, Bulletin de la Classe des Sciences (5e série)15, pp. 183–188.
Gödel K., 1933, Eine Interpretation des intuitionistischen Aussagenkalküls,Ergebnisse eines mathematischen Kolloquiums 4, pp. 39–40 (reprinted with English translation in K. Gödel,Collected Works, vol. I: Publications 1929–1936, Oxford University Press, New York, 1986).
Heyting, A., 1930, Die formalen Regeln der intuitionistischen Logik,Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 42–56.
Heyting A., 1930a, Sur la logique intuitioniste,Académie Royale de Belgique, Bulletin de la Classe des Sciences (5e série)16, pp. 957–963.
Heyting A., 1971,Intuitionism: An Introduction, third revised edition, North-Holland, Amsterdam (first edition 1956).
Heyting A., 1978, History of the foundation of mathematics,Nieuw Archief voor Wiskunde (3)26, pp. 1–21.
Hilbert D., 1923, Die logischen Grundlagen der Mathematik,Mathematische Annalen 88, pp. 151–165 (reprinted in: D. Hilbert,Gesammelte Abhandlungen, vol. 3, Springer, Berlin, 1935).
Hughes G. F. and Cresswell M. J., 1968,An Introduction to Modal Logic, Methuen, London.
Khinchin A. I., 1928, Objection à une note de MM. Barzin et Errera,Académie Royale de Belgique, Bulletin de la Classe des Sciences (5c série)14, pp. 223–224.
Kolmogorov A. N., 1925, On the principle of excluded middle (in Russian),Matematicheski\>i Sbornik 32, pp. 646–667 (English translation in: J. van Heijenoort ed.,From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967).
Kuratowski K., 1992, Sur l'opération Ā de l'Analysis Situs,Fundamenta Mathematicae 3, 182–199.
Lewis C. I. and Langford C. H., 1932,Symbolic Logic, Century, New York.
Moh Shaw-Kwei, 1950, The deduction theorems and two new logical systems,Methodos 2, pp. 56–75.
Orlov I. E., 1928, The calculus of compatibility of propositions (in Russian),Matematicheskii Sbornik 35, pp. 263–286.
Routley R., Meyer R. K. et al., 1982,Relevant Logics and Their Rivals, pt I: The Basic Philosophical and Semantical Theory, Ridgeview, Atascadero.
Troelstra A. S., 1990, On the early history of intuitionistic logic, in: P. P. Petkov ed.,Mathematical Logic, Plenum, New York, pp. 3–17.
van Stigt W. P., 1990,Brouwer's Intuitionism. North-Holland, Amsterdam.
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Došen, K. The first axiomatization of relevant logic. J Philos Logic 21, 339–356 (1992). https://doi.org/10.1007/BF00260740
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DOI: https://doi.org/10.1007/BF00260740