The characterizing properties of a proof-theoretical presentation of a given logic may hang on the choice of proof formalism, on the shape of the logical rules and of the sequents manipulated by a given proof system, on the underlying notion of consequence, and even on the expressiveness of its linguistic resources and on the logical framework into which it is embedded. Standard (one-dimensional) logics determined by (non-deterministic) logical matrices are known to be axiomatizable by analytic and possibly finite proof systems as soon as they turn out to satisfy a certain constraint of sufficient expressiveness. In this paper we introduce a recipe for cooking up a two-dimensional logical matrix (or B-matrix) by the combination of two (possibly partial) non-deterministic logical matrices. We will show that such a combination may result in B-matrices satisfying the property of sufficient expressiveness, even when the input matrices are not sufficiently expressive in isolation, and we will use this result to show that one-dimensional logics that are not finitely axiomatizable may inhabit finitely axiomatizable two-dimensional logics, becoming, thus, finitely axiomatizable by the addition of an extra dimension. We will illustrate the said construction using a well-known logic of formal inconsistency called mCi. We will first prove that this logic is not finitely axiomatizable by a one-dimensional (generalized) Hilbert-style system. Then, taking advantage of a known 5-valued non-deterministic logical matrix for this logic, we will combine it with another one, conveniently chosen so as to give rise to a B-matrix that is axiomatized by a two-dimensional Hilbert-style system that is both finite and analytic.
翻译:逻辑的证明理论表达方式的特性可能取决于证据形式主义的选择、逻辑规则的形状和由给定证据系统操纵的序列的形状、后果的基本概念,甚至取决于其语言资源的外表性,以及它嵌入的逻辑框架。据知(非确定性)逻辑矩阵所决定的标准(一维)逻辑的特性,即使输入矩阵在孤立性上不够清晰,而且可能也是有限的证明系统,只要它们能满足足够直观的某种限制。在本文中,我们引入一种配方,用两种(可能是局部的)非确定性语言资源及其嵌入的逻辑框架来烹调一种二维逻辑矩阵(或B矩阵)的形状。我们将表明,这种结合可能导致B-矩阵满足充分表达性特性,即使输入矩阵在孤立性上不够充分清晰化,我们也会利用这一结果来显示,对于一个不固定的直观的直观的直观逻辑的直径直径直的逻辑, 将使得这个直径的逻辑的直径直的二维值(或B矩阵的直径的直径直的逻辑进进化), 将使得这个直的逻辑的逻辑的逻辑进化系统的二维进化成为一个进化的进化的进化的进化的进化的进化的进化的进化, 。