One of the most common scenarios of handling incomplete information occurs in relational databases. They describe incomplete knowledge with three truth values, using Kleene's logic for propositional formulae and a rather peculiar extension to predicate calculus. This design by a committee from several decades ago is now part of the standard adopted by vendors of database management systems. But is it really the right way to handle incompleteness in propositional and predicate logics?

Our goal is to answer this question. Using an epistemic approach, we first characterize possible levels of partial knowledge about propositions, which leads to six truth values. We impose rationality conditions on the semantics of the connectives of the propositional logic, and prove that Kleene's logic is the maximal sublogic to which the standard optimization rules apply, thereby justifying this design choice. For extensions to predicate logic, however, we show that the additional truth values are not necessary: every many-valued extension of first-order logic over databases with incomplete information represented by null values is no more powerful than the usual two-valued logic with the standard Boolean interpretation of the connectives. We use this observation to analyze the logic underlying SQL query evaluation, and conclude that the many-valued extension for handling incompleteness does not add any expressiveness to it.

处理不完整信息的最常见场景之一发生在关系数据库中。他们用三个真值描述不完整的知识，使用 Kleene 的命题公式逻辑和对谓词演算的相当奇特的扩展。几十年前一个委员会的这种设计现在是数据库管理系统供应商采用的标准的一部分。但这真的是处理命题和谓词逻辑中不完备性的正确方法吗？

我们的目标是回答这个问题。使用认知方法，我们首先表征关于命题的部分知识的可能水平，这导致了六个真值。我们对命题逻辑的连接词的语义强加合理性条件，并证明 Kleene 逻辑是标准优化规则适用的最大子逻辑，从而证明这种设计选择是合理的。然而，对于谓词逻辑的扩展，我们证明了额外的真值不是必需的：一阶逻辑对具有由空值表示的不完整信息的数据库的每个多值扩展并不比通常的二值逻辑更强大连接词的标准布尔解释。我们用这个观察来分析底层 SQL 查询评估的逻辑，