Abstract
We introduce a Gentzen-style modal predicate logic and prove the cut-elimination theorem for it. This sequent calculus of cut-free proofs is chosen as a proxy to develop the proof-theory of the logics introduced in [14, 15, 4]. We present syntactic proofs for all the metatheoretical results that were proved model-theoretically in loc. cit. and moreover prove that the form of weak reflection proved in these papers is as strong as possible.
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References
Avron A.: ‘On modal systems having arithmetical interpretations’. J. of Symb. Logic 49(3), 935–942 (1984)
Enderton H.B.: A mathematical introduction to logic. Academic Press, New York (1972)
Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls, vol. 4, Ergebnisse eines mathematischen Kolloquiums, 1933, pp. 39–40.
Kibedi F., Tourlakis G.: ‘A modal extension of weak generalisation predicate logic’. Logic Journal of the IGPL 14(4), 591–621 (2006)
Kleene S.C.: Introduction to metamathematics. Noordhoff, Groningen (1952)
Leivant D.: ‘On the proof theory of the modal logic for arithmetic provability’. Journal of Symbolic Logic 46(3), 531–538 (1981)
Mendelson E.: Introduction to mathematical logic, 3rd edition. Wadsworth & Brooks, Monterey California (1987)
Sambin G., Valentini S.: ‘A Modal Sequent Calculus for a Fragment of Arithmetic’. Studia Logica 39(2/3), 245–256 (1980)
Sambin G., Valentini S.: ‘The Modal Logic of Provability. The Sequential Approach’. Journal of Philosophical Logic 11(3), 311–342 (1982)
Schütte K.: Proof Theory. Springer-Verlag, New York (1977)
Shoenfield J.R.: Mathematical Logic. Addison-Wesley, Reading, Massachusetts (1967)
Takeuti G.: Proof Theory. North-Holland, New York (1975)
Tourlakis G.: Lectures in Logic and Set Theory; Volume 1: Mathematical Logic. Cambridge University Press, Cambridge (2003)
Tourlakis G., Kibedi F.: ‘A modal extension of first order classical logic–Part I’. BSL 32(4), 165–178 (2003)
Tourlakis G., Kibedi F.: ‘A modal extension of first order classical logic–Part II’. BSL 33(1), 1–10 (2004)
Valentini S.: ‘The Modal Logic of Provability: Cut-Elimination’. Journal of Philosophical Logic 12(4), 471–476 (1983)
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Schwartz, Y., Tourlakis, G. On the Proof-Theory of two Formalisations of Modal First-Order Logic. Stud Logica 96, 349–373 (2010). https://doi.org/10.1007/s11225-010-9294-y
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DOI: https://doi.org/10.1007/s11225-010-9294-y