Density Elimination and Rational Completeness for First-Order Logics

  • Agata Ciabattoni
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4514)

Abstract

Density elimination by substitutions is introduced as a uniform method for removing applications of the Takeuti-Titani density rule from proofs in first-order hypersequent calculi. For a large class of calculi, density elimination by this method is guaranteed by known sufficient conditions for cut-elimination. Moreover, adding the density rule to any axiomatic extension of a simple first-order logic gives a logic that is rational complete; i.e., complete with respect to linearly and densely ordered algebras: a precursor to showing that it is a fuzzy logic (complete for algebras with a real unit interval lattice reduct). Hence the sufficient conditions for cut-elimination guarantee rational completeness for a large class of first-order substructural logics.

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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Agata Ciabattoni
    • 1
  • George Metcalfe
    • 2
  1. 1.Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 WienAustria
  2. 2.Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240USA

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