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Canonical Gentzen-Type Calculi with (n,k)-ary Quantifiers

  • Anna Zamansky
  • Arnon Avron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)

Abstract

Propositional canonical Gentzen-type systems, introduced in [1], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [1] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [14] extends these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n,k)-ary quantifiers, which bind k distinct variables and connect n formulas. We show that the coherence criterion remains constructive for such systems, and that for the case of k∈{0,1}: (i) a canonical system is coherent iff it has a strongly characteristic 2Nmatrix, and (ii) if a canonical system is coherent, then it admits cut-elimination.

Keywords

Function Symbol Predicate Symbol Canonical System Introduction Rule Syllogistic Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anna Zamansky
    • 1
  • Arnon Avron
    • 1
  1. 1.Tel Aviv UniversityIsrael

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