The logic of distributive bilattices
The logic of distributive bilattices
复制标题
分配双格的逻辑
作者：
Félix Bou;U. Rivieccio
Ginsberg（1988），Comput.内部：265-316）作为一个统一的框架，在人工智能推理，是代数结构，证明在许多领域的有用。近年来，Arieli和Avron（1996，J. Logic Lang. Inform.，5，25-63）发展了一种基于一类双格矩阵的逻辑系统，称为逻辑双格，并为它提供了一种Gentzen式演算。这种逻辑本质上是著名的Belnap-Dunn四值逻辑到双格标准语言的扩展。我们的目的是从抽象代数逻辑的角度研究Arieli和Avron的逻辑。我们引入了一个希尔伯特式公理化，以调查该逻辑的代数模型的属性，证明每个公式可以减少到一个等价的范式，我们的公理化是完整的w.r.t. Arieli和Avron的语义。这样，我们就可以根据AAL的标准对这种逻辑进行分类。例如，我们表明，它是非原代数和非自扩张的。我们还刻画了它的Tarski同余和它的约化广义模型的代数约化类，在AAL的一般理论中，它通常被认为是代数逻辑的代数对应物。这个类是由最小的非平凡双格生成的簇，它严格包含在逻辑双格的代数约简类中。另一方面，我们证明了我们的逻辑的约化模型的代数约化类严格包含在其约化广义模型的代数约化类中。得到的另一个有趣的结果是，正如一些著名逻辑的无蕴涵片段所发生的那样，我们可以将我们的逻辑与一个根岑演算联系起来，它在以下意义上是代数化的：
Bilattices, introduced by Ginsberg (1988, Comput. Intell., 265–316) as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron (1996, J. Logic Lang. Inform., 5, 25–63) developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic (AAL). We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of