Generalized Non-deterministic Matrices and (n,k)-ary Quantifiers

Abstract

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.

A Triple Correspondence in Canonical Calculi: Strong Cut-Elimination, Coherence, and Non-deterministic Semantics

“…In the following two subsections, we briefly reproduce the relevant definitions from [4,3] of canonical rules with (n, k)-ary quantifiers and of the framework of non-deterministic matrices. Note the important addition of the definition of full canonical systems, which include the rule of substitution.…”

“…We use the simplified representation language from [4,3] for a schematic representation of canonical rules.…”

“…Our main semantic tool in this paper are generalized non-deterministic matrices introduced in [3], which are a generalization of non-deterministic structures used in [2,17,4].…”

“…The second problem is that even in case of k = 1, coherence is not a necessary condition for standard cut-elimination, and so the triple correspondence seems to be lost. Now the first problem was solved in [3] by introducing generalized two-valued Nmatrices (2GNmatrices), where a more complex approach to quantification is used. However, the problem of finding an appropriate form of cut-elimination for which coherence is a necessary condition, and reestablishing the triple correspondence was explicitly left open in [4] and then also in [3].…”

“…Now the first problem was solved in [3] by introducing generalized two-valued Nmatrices (2GNmatrices), where a more complex approach to quantification is used. However, the problem of finding an appropriate form of cut-elimination for which coherence is a necessary condition, and reestablishing the triple correspondence was explicitly left open in [4] and then also in [3]. In this paper we provide a full solution to the second problem 4 .…”