On strong standard completeness in some MTL \(_\Delta \) expansions

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• 作者：Amanda Vidal ; Félix Bou ; Francesc Esteva ; Lluís Godo
• 关键词：Mathematical fuzzy logic ; Left ; continuous t ; norms ; Monoidal t ; norm logic ; Infinitary rules ; Standard completeness
• 刊名：Soft Computing
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：21
• 期：1
• 页码：125-147
• 全文大小：
• 刊物类别：Engineering
• 刊物主题：Computational Intelligence; Artificial Intelligence (incl. Robotics); Mathematical Logic and Foundations; Control, Robotics, Mechatronics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1433-7479
• 卷排序：21

文摘

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard \(\mathsf {BL}\)-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm \(*\), find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system \(\mathsf {L}_*^\infty \), expanding the language with \(\Delta \) and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti–Titani density rule. Then we show that \(\mathsf {L}_*^\infty \) is indeed strongly complete with respect to the standard algebra . Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions.