Π2-rule systems and inductive classes of Gödel algebras

IF 0.6 2区 数学 Q2 LOGIC
Rodrigo Nicolau Almeida
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引用次数: 0

Abstract

In this paper we present a general theory of Π2-rules for systems of intuitionistic and modal logic. We introduce the notions of Π2-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many Π2-rule systems extending LC=IPC+(pq)(qp), and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in LC: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all Π2-rules which are admissible are derivable, and (2) show that the problem of admissibility of Π2-rules over LC is decidable.
Gödel代数的Π2-rule系统和归纳类
本文给出了直觉逻辑和模态逻辑系统的一般理论Π2-rules。我们引入Π2-rule系统和归纳类的概念,并提供模型完备性定理和代数完备性定理,作为我们的基本工具。作为一般理论的例证,我们从结构理论和逻辑的角度分析了Gödel代数的归纳类的结构。我们展示了不同于其他经过充分研究的设置(如逻辑,或单结论规则系统),有连续体许多Π2-rule系统扩展LC=IPC+(p→q)∨(q→p),并展示了我们的方法如何允许对著名的Takeuti-Titani规则的可接受性进行简单证明。我们的最终结果涉及LC中的一般可容许性问题:(1)我们给出了归纳完备类的完全分类,即所有可容许的Π2-rules都是可导的;(2)证明了LC上Π2-rules的可容许性问题是可判定的。
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来源期刊
CiteScore
1.40
自引率
12.50%
发文量
78
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.
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