Canonical calculi with (n, k)-ary quantifiers

Arnon Avron, Anna Zamansky

Research output: Contribution to journalArticlepeer-review


Propositional canonical Gentzen-type systems, introduced in 2001 by Avron and Lev, 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. A constructive coherence criterion for the non-triviality of such systems was defined and it was shown that a system of this kind admits cutelimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). In 2005 Zamansky and Avron extended 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, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k ∈ {0, 1} and for a canonical calculus G show that it is coherent precisely when it has a strongly characteristic 2Nmatrix, which in turn is equivalent to admitting strong cut-elimination.

Original languageEnglish
Article number2
JournalLogical Methods in Computer Science
Issue number3
StatePublished - 6 Aug 2008
Externally publishedYes

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (grant No 809/06).


  • Automated deduction
  • Cut elimination
  • Gentzen-type systems
  • Many-valued logic
  • Non-deterministic matrices
  • Proof theory
  • Quantifiers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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