The algebraic significance of weak excluded middle laws

Pub Date : 2022-01-29 DOI:10.1002/malq.202100046

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 $K$ 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 $K$ has a greatest proper $K$ -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 $KC$ . We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of $S 4$ has a global consequence relation with a WEML iff it extends $S 4.2$ , while every axiomatic extension of $R^{t}$ with an IL has a WEML.

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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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