It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church's simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of first-level function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege's theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege's Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names.

翻译：人们常认为，弗雷格在《算术基本法则》中提出的函数层级理论，预示了支撑丘奇简单类型理论的类型层级。这种观点大致认为，弗雷格在《算术基本法则》的说明性语言中预设了简单类型理论意义上的函数类型。然而，这一视角难以处理二元函数名称，也难以将函数视为不完全实体。我提出并辩护一种对《算术基本法则》中第一级函数名称的替代性解释：将其理解为简单类型理论中的开放项，而非函数类型的封闭项。这一解释虽然仍是非历史的，但能更忠实地以类型理论近似弗雷格的层级理论，并可以自然地扩展以容纳第二级函数。其可行性基于两个关键观察：弗雷格的罗马标记本质上类似于开放项，且弗雷格缺乏区分罗马标记与函数名称的明确标准。