The algebraic significance of weak excluded middle laws

IF 0.4 4区 数学 Q4 LOGIC
Tomáš Lávička, Tommaso Moraschini, James G. Raftery
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引用次数: 2

Abstract

For (finitary) deductive systems, we formulate a signature-independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety K $\mathsf {K}$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of K $\mathsf {K}$ has a greatest proper K $\mathsf {K}$ -congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends KC $\mathsf {KC}$ . We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of S 4 $\mathsf {S4}$ has a global consequence relation with a WEML iff it extends S 4 . 2 $\mathsf {S4.2}$ , while every axiomatic extension of R t $\mathsf {R^t}$ with an IL has a WEML.

弱排除中间律的代数意义
对于(有限)演绎系统,我们给出了弱排除中间律(WEML)的一个与签名无关的抽象,强化了现有的不一致引理(IL)的一般概念。特别有趣的是拟变量K $\mathsf {K}$对演绎系统进行代数化的情况。我们证明,在这种情况下,如果_有一个WEML(在一般意义上),则K $\mathsf {K}$中每一个相对子直接不可约的元素都有一个最大固有K $\mathsf {K}$ -同余;如果∧有不一致引理,则反之成立。该结果以适当的形式推广到所有的原代数逻辑。如果超直觉逻辑扩展了KC $\mathsf {KC}$,则具有WEML。我们描述了正常模态逻辑和相关逻辑的IL和WEML。S4 $\mathsf {S4}$的普通扩展如果扩展S4,则与WEML具有全局推论关系。2 $\mathsf {S4.2}$,而rt $\mathsf {R^t}$的每一个具有IL的公理扩展都有一个WEML。
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来源期刊
CiteScore
0.60
自引率
0.00%
发文量
49
审稿时长
>12 weeks
期刊介绍: Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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