Canonical Propositional Gentzen-Type Systems

Canonical propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules which have the subformula property, introduce exactly one occurrence of a connective in their conclusion, and no other occurrence of any connective is mentioned anywhere else in their formulation. We provide a constructive coherence criterion for the non-triviality of such systems, and show that a system of this kind admits cut elimination iff it is coherent. We show also that the semantics of such systems is provided by non-deterministic two-valued matrices (2-Nmatrices). 2-Nmatrices form a natural generalization of the classical two-valued matrix, and every coherent canonical system is sound and complete for one of them. Conversely, with any 2-Nmatrix it is possible to associate a coherent canonical Gentzen-type system which has for each connective at most one introduction rule for each side, and is sound and complete for that 2-Nmatrix. We show also that every coherent canonical Gentzen-type system either defines a fragment of the classical two-valued logic, or a logic which has no finite characteristic matrix.