How did Gottlob Frege treat sentences as having the truth-values True or False?

Gottlob Frege treated sentences as having truth-values—True or False—by analyzing them through the lens of mathematical logic and functional application.1

In his framework, Frege distinguished between "saturated" expressions (sentences and singular terms) and "unsaturated" expressions (predicates). He viewed predicates as functions, analogous to mathematical functions. For example, when a predicate like "...is a lecturer" is applied to a name like "John," the resulting sentence, "John is a lecturer," functions as an expression whose value is determined to be True if the condition is met, and False otherwise.1

Furthermore, Frege treated sentential connectives, such as "and" or "not," as functions that produce new truth-values when applied to sentences as arguments. In this logical analysis, the truth-value of a complex sentence is determined by the outputs of these predicate-functions. While both the referent and the sense of an expression contribute to the truth or falsehood of a sentence, the truth-value itself serves as the ultimate referent of the sentence in Frege's system.12