Frege and Russell introduced generality in association with logical product or logical sum. This made it difficult to understand the propositions ((∃x). fx) and ((x).fx), which include both ideas.
What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants. The generality-sign appears as an argument. But if objects are given, then that already gives us all objects. If elementary propositions are given, then all elementary propositions are thereby given.
It is incorrect to render the proposition '(∃x) . fx' in the words, 'fx is possible ' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition. Rather we determine which applies by an expression's being a tautology, a proposition that makes sense, or a contradiction. The precedent which we cite must lie in the symbol itself.
One can describe the world completely by means of fully generalized propositions, which means that one need not begin by assigning a name to any particular object. To then arrive at a conventional expression, one must simply follow the expression: " There is one and only one x such that ... " with: "And this x is a".
A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in ((∃x, φ).φx) we have to mention 'φ' and 'x' separately. They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is characteristic of a composite symbol to have something in common with other symbols.
The truth or falsity of every proposition, after all, changes the general construction of the world. And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more elementary proposition i true.)