The problem was discussed under this name by Bertrand Russell, but can be traced back to Plato.
In Plato's Sophist, the simplest kind of sentence consists of just a proper name and a universal term (i.e. a predicate).
If, as Plato thinks,[1] the predicate is the name of some universal concept or form, how do we explain how the sentence comes to be true or false?
Set membership is a formal representation of the relation between the two parts of the proposition, and there are certain philosophical problems connected with this, as Frege realised when he investigated the distinction between concept and object.
This is a dogma that even Frege's most faithful followers found difficult to swallow.
The difficulty was discussed in detail in The Principles of Mathematics by Russell, who saw no resolution.