Cut-Elimination and Quantification in Canonical Systems

Studia Logica 82 (1):157-176 (2006)
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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.

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Arnon Avron
Tel Aviv University

References found in this work

Mathematical logic.Joseph R. Shoenfield - 1967 - Reading, Mass.,: Addison-Wesley.
Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - New York: Cambridge University Press. Edited by Jan Von Plato.
Investigations into Logical Deduction.Gerhard Gentzen - 1964 - American Philosophical Quarterly 1 (4):288 - 306.
Investigations into Logical Deduction.Gerhard Gentzen, M. E. Szabo & Paul Bernays - 1970 - Journal of Symbolic Logic 35 (1):144-145.

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