Abstract
Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction- and cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of \({\square\,A}\) from A.
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References
Avron A. (1991) Simple consequence relations. Information and Computation 92: 105–139
Barcan R. (1946) The deduction theorem in a functional calculus of first order based on strict implication. The Journal of Symbolic Logic 11: 115–118
Barcan Marcus R. (1953) Strict implication, deducibility and the deduction theorem. The Journal of Symbolic Logic 18: 234–236
Basin D., Matthews S., Viganò L. (1998) Natural deduction for non-classical logics. Studia Logica 60: 119–160
Chagrov A., Zakharyaschev M. (1997) Modal logic. Clarendon Press, Oxford
Chellas B. (1980) Modal logic: An introduction. Cambridge University Press, Cambridge
Fagin R., Halpern J., Moses Y., Vardi M. (1995) Reasoning about knowledge. MIT Press, Cambridge
Fagin R., Halpern J., Vardi M. (1992) What is an inference rule?. The Journal of Symbolic Logic 57: 1018–1045
Feys R. (1965) Modal logics. In: Dopp J. (ed) Collection de Logique Mathématique, Série B, IV. E. Nauwelaerts Éditeur, Paris
Fitting M. (2007) Modal proof theory. In: Blackburn P., van Benthem J., Wolter F. (eds) Handbook of modal logic. Elsevier, Amsterdam, pp 85–138
Ganguli S., Nerode A. (2004) Effective completeness theorems for modal logic. Annals of Pure and Applied Logic 128: 141–195
Gentzen, G. (1934–1935). Untersuchungen über das logische Schliessen [Investigations into logical deduction]. Mathematische Zeitschrift, 39, 176–210, 405–431. Translated in Gentzen, G. (1969). The collected papers of Gerhard Gentzen (pp. 68–131) (M. Szabo, Ed.). North-Holland.
Goldblatt R. (1992) Logics of time and computation (2nd ed.). CSLI Publications, Stanford
Herbrand, J. (1930). Recherches sur la theorie de la demonstration. Ph.D. thesis, University of Paris. English translation in W. Goldfarb (Ed.). (1971). Jacques Herbrand: Logical writings. Harvard University Press.
Hilbert D. (1923) Die logischen Grundlagen der Mathematik. Mathematische Annalen 88: 151–165
Hilbert D., Bernays P. (1934) Grundlagen der Mathematik I. Springer, Berlin
Hughes, G., & Cresswell, M. (1968). An introduction to modal logic, Methuen and Co., London.
Hughes G., Cresswell M. (1996) A new introduction to modal logic. Routledge, London
Kleene S. (1952) Introduction to metamathematics. Van Nostrand, New York
Kripke S. (1959) A completeness theorem in modal logic. The Journal of Symbolic Logic 24: 1–14
Lemmon W. (1957) New foundations for Lewis modal systems. The Journal of Symbolic Logic 22: 176–186
Lesniewski S. (1929) Grundzüge eines neuen System der Grundlagen der Mathematik. Fundamenta Mathematicae 14: 1–81
Lewis C., Langford C. (1932) Symbolic logic. The Century Co, New York
Mints, G. (1992). Lewis’ systems and system T (1965–1973). In Selected papers in proof theory (pp. 221–294). Bibliopolis North-Holland (Russian original 1974).
Montague R., Henkin L. (1956) On the definition of formal deduction. The Journal of Symbolic Logic 21: 129–136
Negri S. (2005) Proof analysis in modal logic. Journal of Philosophical Logic 34: 507–544
Negri S. (2009) Kripke completeness revisited. In: Primiero G., Rahman S. (eds) Acts of knowledge: History, philosophy and logic. College Publications, London, pp 233–266
Negri S., von Plato J. (2001) Structural proof theory. Cambridge University Press, Cambridge
Porte, J. (1982). Fifty years of deduction theorems. In J. Stern (Ed.), Proceedings of the Herbrand symposium, Logic Colloquium ’81 (pp. 243–250). North-Holland.
Satre T. (1972) Natural deduction rules for modal logic. Notre Dame Journal of Formal Logic 13: 461–475
Smorynski, C. (1984). Modal logic and self-reference. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. II, pp. 441–495). Reidel. Reprinted in Handbook of philosophical logic (Vol. 11). Kluwer (2002).
Sundholm, G. (1983). Systems of deduction. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic. (Vol. I, pp. 133–188). Dordrecht: Reidel.
Sundholm G. (2002) Varieties of consequence. In: Jacquette D. (ed) A companion to philosophical logic. Blackwell, Oxford, pp 241–255
Tarski, A. (1930). Über einige fundamentale Begriffe der Metamathematik [Some fundamental concepts of metamathematics], Comptes Rendus de Séances de la Société des Sciences et des Lettres de Varsovie, Classe III (Vol. 23, pp. 22–29). English translation by J. Woodger in Tarski, A. (1956). Logic, semantics, metamathematics, papers from 1923 to 1938 (pp. 30–37). Oxford: Oxford University Press.
Troelstra A., Schwichtenberg H. (2000) Basic proof theory (2nd ed.). Cambridge University Press, Cambridge
von Plato J. (2009) Gentzen’s logic. In: Gabbay D., Woods J. (eds) Handbook of the history of logic. Elsevier, Amsterdam, pp 667–721
Zeman J. (1967) The deduction theorem in S4, S4.2, and S5. Notre Dame Journal of Formal Logic 8: 56–60
Zeman J. (1973) Modal logic: The Lewis-modal systems. Oxford University Press, Oxford
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Hakli, R., Negri, S. Does the deduction theorem fail for modal logic?. Synthese 187, 849–867 (2012). https://doi.org/10.1007/s11229-011-9905-9
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DOI: https://doi.org/10.1007/s11229-011-9905-9