  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&#8712;{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.
