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Higher-Order Illative Combinatory Logic

  • Łukasz Czajka (a1)

Abstract

We show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

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[1] Barendregt, Henk, Bunder, Martinw, and Dekkers, Wil, Systems of illative combinatory logic complete for first-order propositional and predicate calculus, this Journal, vol. 58 (1993), no. 3, pp. 769788.
[2] Bunder, Martin W. and Dekkers, Wil, Pure type systems with more liberal rules, this Journal, vol. 66 (2001), no. 4, pp. 15611580.
[3] Bunder, Martin W. and Dekkers, Wil, Equivalences between pure type systems and systems of illative combinatory logic, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 2, pp. 181205.
[4] Curry, Haskell B., Feys, Robert, and Craig, William, Combinatory logic, vol. 1, North-Holland, 1958.
[5] Czajka, Łukasz, A semantic approach to illative combinatory logic, Computer Science Logic 2011 (Bezem, Marc, editor), LIPIcs, vol. 12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011, pp. 174188.
[6] Dekkers, Wil, Bunder, Martin W., and Barendregt, Henk, Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic, this Journal, vol. 63 (1998), no. 3, pp. 869890.
[7] Dekkers, Wil, Bunder, Martin W., and Barendregt, Henk, Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus, Archive for Mathematical Logic, vol. 37 (1998), no. 5-6, pp. 327341.
[8] Seldin, Jonathan P., The logic of Church and Curry, Logic from Russell to Church (Gabbay, Dov M. and Woods, John, editors), Handbook of the History of Logic, vol. 5, North-Holland, 2009, pp. 819873.

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