Lambek elegantly characterized part of natural language. As is well-known, his substructural logic L, and its non-associative version NL, handle basic function/argument composition well, but not scope taking and syntactic displacement—at least, not in their full generality. In previous work, I propose $$\text {NL}_\lambda $$NLλ, which is NL supplemented with a single structural inference rule (“abstraction”). Abstraction closely resembles the traditional linguistic rule of quantifier raising, and characterizes both semantic scope taking and syntactic displacement. Due to the unconventional form of the abstraction inference, there has been some doubt that $$\text {NL}_\lambda $$NLλ should count at a legitimate substructural logic. This paper argues that $$\text {NL}_\lambda $$NLλ is perfectly well-behaved. In particular, it enjoys cut elimination and an interpolation result. In addition, perhaps surprisingly, it is decidable. Finally, I prove that it is sound and complete with respect to the usual class of relational frames.

中文翻译：

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$$\hbox {NL}_\lambda $$ NL λ 作为范围和运动的逻辑

Lambek 优雅地表征了自然语言的一部分。众所周知，他的子结构逻辑 L 及其非关联版本 NL 可以很好地处理基本的函数/参数组合，但不能很好地处理范围获取和句法置换——至少，不能完全通用。在之前的工作中，我提出了 $$\text {NL}_\lambda $$NLλ，它是 NL 补充了单个结构推理规则（“抽象”）。抽象与量词提升的传统语言规则非常相似，具有语义范围获取和句法置换的特征。由于抽象推理的非常规形式，有人怀疑 $$\text {NL}_\lambda $$NLλ 是否应该算作合法的子结构逻辑。这篇论文认为 $$\text {NL}_\lambda $$NLλ 是完美的。特别是，它享受削减消除和插值结果。此外，也许令人惊讶的是，它是可判定的。最后，我证明它相对于通常的关系框架类是合理和完整的。