Cut-elimination and quantification in canonical systems

Anna Zamansky*, Arnon Avron

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

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
Volume82
Issue number1
DOIs
StatePublished - Feb 2006

Funding

FundersFunder number
Israel Academy of Sciences and Humanities

    Keywords

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

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