Notre Dame Journal of Formal Logic

Basic Predicate Calculus

Wim Ruitenburg


We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.

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Notre Dame J. Formal Logic Volume 39, Number 1 (1998), 18-46.

First available in Project Euclid: 7 December 2002

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Zentralblatt MATH identifier

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.) 03D35: Undecidability and degrees of sets of sentences 03F30: First-order arithmetic and fragments


Ruitenburg, Wim. Basic Predicate Calculus. Notre Dame J. Formal Logic 39 (1998), no. 1, 18--46. doi:10.1305/ndjfl/1039293019.

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  • [1] Ardeshir, M., and W. Ruitenburg, Basic Propositional Calculus I, TR 418, Marquette University, Milwaukee, 1995. Forthcoming in Mathematical Logic Quarterly.
  • [2] Ardeshir, M., Aspects of Basic Logic, Ph.D. thesis, Marquette University, Milwaukee, 1995.
  • [3] Heyting, A., Die formalen Regeln der intuitionistischen Logik, Verlag der Akademie der Wissenschaften, Berlin, 1930.
  • [4] Heyting, A., Die formalen Regeln der intuitionistischen Mathematik, Verlag der Akademie der Wissenschaften, Berlin, 1930.
  • [5] Ruitenburg, W., ``Constructive logic and the paradoxes,'' Modern Logic, vol. 1 (1991), pp. 271--301.
  • [6] Ruitenburg, W., ``Basic logic and Fregean set theory,'' pp. 121--42 in Dirk van Dalen Festschrift, Quaestiones Infinitae, vol. 5, edited by H. Barendregt, M. Bezem, and J. W. Klop, Utrecht University, Utrecht, 1993.
  • [7] Scott, D., ``Identity and existence in intuitionistic logic,'' pp. 660--96 in Applications of Sheaves, Lecture Notes in Mathematics, 753, edited by M. P. Fourman, C. J. Mulvey, and D. S. Scott, Springer-Verlag, Berlin, 1979.
  • [8] Visser, A., ``A propositional logic with explicit fixed points,'' Studia Logica, vol. 40 (1981), pp. 155--75.