Abstract
We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.
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References
Baaz, M., and R. Iemhoff, ‘Skolemization in intuitionistic logic’, Submitted, 2005.
Beckmann, A., and N. Preining, ‘Linear Kripke Frames and Gödel Logics’, Submitted, 2005.
Beeson, M., Foundations of Constructive Mathematics, Springer, Berlin, 1985.
Corsi, G., ‘A cut-free calculus for Dummett's LC quantified’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35 (1989), 289–301.
Corsi, G., ‘A logic characterized by the class of connected models with nested domains’, Studia Logica 48, No. 1 (1989), 15–22.
Corsi, G., ‘Completeness theorem for Dummett's LC quantified and some of its extensions’, Studia Logica 51, No. 2 (1992), 317–335.
Gentzen, G., ‘Untersuchungenüber das logische Schliessen.’ I., Math. Z., vol. 39 (1934), 176–210.
Gentzen, G., ‘Untersuchungenüber das logische Schliessen.’ II., Math. Z., vol. 39 (1934), 405–431.
Heyting, A., ‘Die formalen regeln der intuitionistische Mathematik’ II, Sitzungsberichte der Preussischen Akademie von Wissenschaften. Physikalisch-mathematische Klasse, 1930, 57–71.
Iemhoff, R., ‘A note on linear Kripke models’, Journal of Logic and Computation, 2005, to appear.
Mints, G. E., ‘Axiomatization of a Skolem function in intuitionistic logic’, in Faller M. et al., (eds.), Formalizing the dynamics of information, CSLI Lect. Notes 91, 2000, 105–114.
Preining, N., Complete Recursive Axiomatizability of Godel Logics, PhD-thesis, Technical University Vienna, 2003.
Scott, D. S., ‘Identity and existence in intuitionistic logic’, in Fourman et al., (eds.), Applications of sheaves, Proc. Res. Symp. Durham 1977, Lect. Notes Math. 753, 1979, 660–696.
Takano, M., ‘Another proof of the strong completeness of the intuitionistic fuzzy logic’, Tsukuba J. Math. 11, No. l, 1987, 101–105.
Takeuti, G. and M. Titani, ‘Intuitionistic fuzzy logic and intuitionistic fuzzy set theory’, Journal of Symbolic Logic 49 (1984), 851–866.
Troelstra, A. S., and D. Van Dalen, Constructivism in Mathematics, vol. I North-Holland, 1988.
Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science 43, Cambridge University Press, 1996.
Unterhalt, M., Kripke-Semantik mit partieller Existenz, PhD-thesis, University of Münster, 1986.
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Baaz, M., Iemhoff, R. Gentzen Calculi for the Existence Predicate. Stud Logica 82, 7–23 (2006). https://doi.org/10.1007/s11225-006-6603-6
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DOI: https://doi.org/10.1007/s11225-006-6603-6