In this paper we introduce a language of first-order hybrid logic in which function symbols are interpreted by partial functions and prove a number of completeness results. Syntactically, our language builds on the basic propositional hybrid language, has a primitive unary predicate symbol $\mathsf{DEN}$ which tests whether a term denotes or not, and permits satisfaction operators to rigidify predicate and function symbols. Semantically, our system is actualist, allows terms to be undefined, and has no truth-value gaps. But should we follow Fitting and Mendelsohn and rule out domain elements not belonging to any world, or should we tolerate them? Here we explore both options. As we shall see, while the choice makes no difference when it comes to validity, it has consequences for richer logics.

在本文中，我们介绍了一种一阶混合逻辑语言，其中函数符号由偏函数解释，并证明了一些完备性结果。在句法上，我们的语言建立在基本的命题混合语言之上，有一个原始的一元谓词符号$\mathsf{书房}$它测试一个术语是否表示，并允许满足运算符刚性化谓词和函数符号。从语义上讲，我们的系统是现实主义的，允许未定义的术语，并且没有真值差距。但是我们应该遵循 Fitting 和 Mendelsohn 并排除不属于任何世界的域元素，还是应该容忍它们？在这里，我们探讨了这两种选择。正如我们将看到的，虽然选择在有效性方面没有区别，但它对更丰富的逻辑有影响。