Canonical calculi with (n,k)-ary quantifiers

TL;DR

This paper extends the theory of canonical Gentzen-type systems with (n,k)-ary quantifiers, providing a constructive coherence criterion and linking coherence to cut-elimination via non-deterministic matrices.

Contribution

It generalizes the characterization of canonical systems to (n,k)-ary quantifiers and establishes a connection between coherence, 2Nmatrices, and cut-elimination.

Findings

Coherence criterion remains constructive for (n,k)-ary quantifiers.

A canonical calculus with k∈{0,1} is coherent iff it has a strongly characteristic 2Nmatrix.

Coherence is equivalent to admitting strong cut-elimination.

Abstract

Propositional canonical Gentzen-type systems, introduced in 2001 by Avron and Lev, 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. A constructive coherence criterion for the non-triviality of such systems was defined and it was shown 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). In 2005 Zamansky and Avron extended 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 for a canonical calculus G show that it is coherent precisely when it has a strongly characteristic 2Nmatrix, which in turn is equivalent to admitting strong cut-elimination.