Boolean-valued models for first-order languages generalize two-valued models, in that the value range is allowed to be any complete Boolean algebra instead of just the Boolean algebra 2. Boolean-valued models are interesting in multiple aspects: philosophical, logical, and mathematical. The primary goal of this paper is to extend a number of critical model-theoretic notions and to generalize a number of important model-theoretic results based on these notions to Boolean-valued models. For instance, we will investigate (first-order) Boolean valuations, which are natural generalizations of (first-order) theories, and prove that Boolean-valued models are sound and complete with respect to Boolean valuations. With the help of Boolean valuations, we will also discuss the Löwenheim-Skolem theorems on Boolean-valued models.

一阶语言的布尔值模型概括了二值模型，因为值范围允许是任何完整的布尔代数，而不仅仅是布尔代数 2。布尔值模型在多个方面都很有趣：哲学、逻辑、和数学。本文的主要目标是扩展一些关键的模型理论概念，并将基于这些概念的一些重要的模型理论结果推广到布尔值模型。例如，我们将研究（一阶）布尔估值，这是（一阶）理论的自然推广，并证明布尔估值模型对于布尔估值是合理且完整的。在布尔值的帮助下，我们还将讨论布尔值模型上的 Löwenheim-Skolem 定理。