Abstract
An (n,k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical Gentzen-type systems with (n,k)-ary quantifiers are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of an (n,k)-ary quantifier is introduced. The semantics of such systems for the case of k ∈ {0,1} are provided in [16] using two-valued non-deterministic matrices (2Nmatrices). A constructive syntactic coherence criterion for the existence of a 2Nmatrix for which a canonical system is strongly sound and complete, is formulated there. In this paper we extend these results from the case of k ∈ {0,1} to the general case of k ≥ 0. We show that the interpretation of quantifiers in the framework of Nmatrices is not sufficient for the case of k > 1 and introduce generalized Nmatrices which allow for a more complex treatment of quantifiers. Then we show that (i) a canonical calculus G is coherent iff there is a 2GNmatrix, for which G is strongly sound and complete, and (ii) any coherent canonical calculus admits cut-elimination.
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Avron, A., Zamansky, A. (2007). Generalized Non-deterministic Matrices and (n,k)-ary Quantifiers. In: Artemov, S.N., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2007. Lecture Notes in Computer Science, vol 4514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72734-7_3
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DOI: https://doi.org/10.1007/978-3-540-72734-7_3
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