Canonical calculi with (n,k)-ary quantifiers

Abstract
Propositional canonical Gentzen-type systems, introduced in [2], 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. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends 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 show that the following statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination. We also show that coherence is not a necessary condition for standard cut-elimination, and then characterize a subclass of canonical systems for which this property does hold
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