逻辑滤子的方程可定义性
Equational definitions of logical filters
AI总结：
本文提出紧滤子的方程可定义性（EDCF）及其局部、参数化层级，并给出各变体的语义刻画，表明其可适用于许多不可代数化或分类表现不佳的逻辑。
摘要：
一个有穷命题逻辑可以通过两种不同方式获得代数解读：将公式翻译为方程、逻辑规则翻译为拟方程，或者将逻辑规则直接翻译为方程。前一种代数解释类型已得到广泛研究，并构成代数化理论的基础。本文将系统发展后一种代数解释的理论。更确切地说，我们考察该性质的一种语义形式，称之为紧滤子的方程可定义性（EDCF）。与已被充分研究的演绎—分离定理（DDT）变体层级相平行，该性质也具有局部、参数化以及参数化局部变体。我们的主要结果以类似于现有 DDT 刻画的精神，给出了 EDCF 每一种变体的语义刻画。尽管 EDCF 层级与 DDT 层级在可代数化逻辑上是一致的，但 EDCF 的部分吸引力来自这样一个事实：即便在代数逻辑中其他现有分类方面表现不佳的逻辑，也常常享有 EDCF。
英文摘要：
A finitary propositional logic can be given an algebraic reading in two different ways: by translating formulas into equations and logical rules into quasi-equations, or by translating logical rules directly into equations. The former type of algebraic interpretation has been extensively studied and underlies the theory of algebraization. Here we shall develop a systematic theory of the latter type of algebraic interpretation. More precisely, we consider a semantic form of this property which we call the equational definability of compact filters (EDCF). Paralleling the well-studied hierarchy of variants of the deduction--detachment theorem (DDT), this property also comes in local, parametrized, and parametrized local variants. Our main results give a semantic characterization of each of these variants of the EDCF in a spirit similar to the existing characterizations of the DDT. While the EDCF hierarchy and the DDT hierarchy coincide for algebraizable logics, part of the interest of the EDCF stems from the fact it is often enjoyed even by logics which are not well-behaved in terms of other existing classifications in algebraic logic.
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