Notre Dame Journal of Formal Logic

Syntactic Preservation Theorems for Intuitionistic Predicate Logic

Jonathan Fleischmann

Abstract

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich Theorem, which characterizes the sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies n ( = Π n 0 ) and n ( = Σ n 0 ) , and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic.

Article information

Source
Notre Dame J. Formal Logic, Volume 51, Number 2 (2010), 225-245.

Dates
First available in Project Euclid: 11 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1276284784

Digital Object Identifier
doi:10.1215/00294527-2010-014

Mathematical Reviews number (MathSciNet)
MR2667934

Zentralblatt MATH identifier
1254.03016

Subjects
Primary: 03B20: Subsystems of classical logic (including intuitionistic logic) 03C40: Interpolation, preservation, definability 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
Secondary: 03F55: Intuitionistic mathematics

Keywords
Kripke models intuitionistic predicate logic preservation theorems formula hierarchies Keisler Sandwich Theorem

Citation

Fleischmann, Jonathan. Syntactic Preservation Theorems for Intuitionistic Predicate Logic. Notre Dame J. Formal Logic 51 (2010), no. 2, 225--245. doi:10.1215/00294527-2010-014. https://projecteuclid.org/euclid.ndjfl/1276284784


Export citation

References

  • [1] Andréka, H., J. van Benthem, and I. Németi, "Back and forth between modal logic and classical logic", Logic Journal of the IGPL, vol. 3 (1995), pp. 685--720.
  • [2] Bagheri, S. M., and M. Moniri, "Some results on Kripke models over an arbitrary fixed frame", Mathematical Logic Quarterly, vol. 49 (2003), pp. 479--84.
  • [3] Burr, W., "Fragments of Heyting arithmetic", The Journal of Symbolic Logic, vol. 65 (2000), pp. 1223--40.
  • [4] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
  • [5] van Dalen, D., Logic and Structure, 4th edition, Universitext. Springer-Verlag, Berlin, 2004.
  • [6] Ellison, B., J. Fleischmann, D. McGinn, and W. Ruitenburg, "Kripke submodels and universal sentences", Mathematical Logic Quarterly, vol. 53 (2007), pp. 311--20.
  • [7] Ellison, B., J. Fleischmann, D. McGinn, and W. Ruitenburg, "Quantifier elimination for a class of intuitionistic theories", Notre Dame Journal of Formal Logic, vol. 49 (2008), pp. 281--93.
  • [8] Hinman, P. G., Fundamentals of Mathematical Logic, A. K. Peters Ltd., Wellesley, 2005.
  • [9] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
  • [10] Keisler, H. J., "Theory of models with generalized atomic formulas", The Journal of Symbolic Logic, vol. 25 (1960), pp. 1--26.
  • [11] Połacik, T., "Partially-elementary extension Kripke models: A characterization and applications", Logic Journal of the IGPL, vol. 14 (2006), pp. 73--86.
  • [12] Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics: An Introduction. Vol. I, vol. 121 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1988.
  • [13] Visser, A., "Submodels of Kripke models", Archive for Mathematical Logic, vol. 40 (2001), pp. 277--95.
  • [14] Visser, A., J. van Benthem, D. de Jongh, and G. R. Renardel de Lavalette, "NNIL, A study in intuitionistic propositional logic", pp. 289--326 in Modal Logic and Process Algebra (Amsterdam, 1994), edited by A. Ponse, M. de Rijke, and Y. Venema, vol. 53 of CSLI Lecture Notes, CSLI Publications, Stanford, 1995.