On strong standard completeness in some MTL$$_\Delta $$Δ expansions
On strong standard completeness in some MTL$$_\Delta $$Δ expansions
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关于某些 MTL$$_Delta $$Δ 扩展中的强标准完整性
DOI：
10.1007/s00500-016-2338-0
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发表时间：
2017
期刊：
影响因子：
4.1
通讯作者：
L. Godo
中科院分区：
文献类型：
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作者：
Amanda Vidal;Félix Bou;F. Esteva;L. Godo
摘要在本文中，受Franco Montagna先前关于标准$$\mathsf {BL}$$ bl -代数的无穷公理化的工作的启发，我们着重于以下问题的统一方法：给定一个左连续t-范数$$*$$ *，找到一个相对于标准代数强完备的公理系统（可能有无穷规则）
这个系统将是基于一元t范数逻辑的扩展。首先，我们引入了一个无限公理系统$$\mathsf {L}_*^\infty $$ L∗∞，用$$\Delta $$ Δ和数个真值常数扩展了语言，并且只有一个无限推理规则，该规则的灵感来自Takeuti-Titani密度规则。然后证明$$\mathsf {L}_*^\infty $$ L *∞对于标准代数确实是强完备的。此外，该方法被推广到使用附加算子对这些逻辑的展开进行公理化，这些算子的预期语义在[0,1]上满足一些正则性条件。
AbstractIn this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard $$\mathsf {BL}$$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 $$\mathsf {L}_*^\infty $$L∗∞, expanding the language with $$\Delta $$Δ and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti–Titani density rule. Then we show that $$\mathsf {L}_*^\infty $$L∗∞ 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.