Equational definitions of logical filters

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2025-05-28 DOI:10.1016/j.apal.2025.103617

Michele Pra Baldi , Adam Přenosil

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引用次数: 0

Abstract

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, whereas little systematic attention has been paid to the latter type. We investigate a semantic form of the latter type of algebraic interpretation, 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. The main results of this paper characterize 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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逻辑滤波器的等式定义

有限命题逻辑可以用两种不同的方式进行代数解读：将公式转化为方程，将逻辑规则转化为准方程，或者将逻辑规则直接转化为方程。前一种类型的代数解释已经被广泛研究，是代数化理论的基础，而后一种类型的代数解释却很少得到系统的关注。我们研究了后一种代数解释的语义形式，我们称之为紧滤波器的等式可定义性（EDCF）。与演绎分离定理（DDT）的变体层次结构相似，这个性质也存在于局部、参数化和参数化的局部变体中。本文的主要结果以类似于DDT现有特征的精神描述了EDCF的每一个变体。虽然EDCF层次结构和DDT层次结构在可代数逻辑中是一致的，但EDCF的部分兴趣源于这样一个事实，即即使在代数逻辑中其他现有分类方面表现不好的逻辑也经常使用它。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.