Cut-elimination and quantification in canonical systems

Anna Zamansky, Arnon Avron

Research output: Contribution to journalArticlepeer-review


Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a "canonical system" to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.

Original languageEnglish
Pages (from-to)157-176
Number of pages20
JournalStudia Logica
Issue number1
StatePublished - Feb 2006
Externally publishedYes

Bibliographical note

Funding Information:
This research was supported by THE ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Sciences and Humanities.


  • Canonical systems
  • Cut elimination
  • Non-deterministic matrices
  • Proof theory

ASJC Scopus subject areas

  • Logic
  • History and Philosophy of Science


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