Strong Cut-Elimination, Coherence, and Non-deterministic Semantics
An (n, k)-ary quantiﬁer is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantiﬁers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantiﬁer is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantiﬁers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix. In addition, we deﬁne simple calculi, an important subclass of canonical calculi, for which coherence is equivalent to the weaker, standard cut-elimination property.
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