Frege never explicitly advocated a doctrine of types like Russell, but a naive type theory can be found in his concept-script in his 1893 Grundgesetze der Arithmetik. Frege did not endorse a semantic account of typing, unlike most type theorists who use different types for booleans, natural numbers, products, etc. His concept-script is confined to a syntactic approach where object terms are identified with closed terms but the behavior of function terms alternates between that of closed terms of a function type and open terms. My aim in this paper is to rehabilitate Frege as a key figure in the history of type theory with an exegetical account of the developments anticipated in his concept-script from the perspective of modern type theory.

中文翻译：

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弗雷格的类型理论

弗雷格从未明确提倡像罗素这样的类型学说，但可以在他 1893 年的 Grundgesetze der Arithmetik 中的概念脚本中找到一种朴素的类型理论。弗雷格不认可类型的语义解释，这与大多数类型理论家使用不同类型的布尔值、自然数、乘积等不同。他的概念脚本仅限于句法方法，其中对象术语与封闭术语标识，但函数项在函数类型的闭项和开项之间交替。我在本文中的目的是通过从现代类型论的角度对他的概念脚本中预期的发展进行解释性说明，使弗雷格重新成为类型论历史上的关键人物。