Similarly the series of propositions “aRb”,
“(∃x):aRx.xRb”,
“(∃x,y):aRx.xRy.yRb”, etc.
(If b stands in one of these relations to a, I call b a successor of a.)
4.126
In the sense in which we speak of formal properties we can now speak also of formal concepts.
(I introduce this expression in order to make clear the confusion of formal concepts with proper concepts which runs through the whole of the old logic.)
That anything falls under a formal concept as an object belonging to it, cannot be expressed by a proposition. But it is shown in the symbol for the object itself. (The name shows that it signifies an object, the numerical sign that it signifies a number, etc.)
Formal concepts, cannot, like proper concepts, be presented by a function.
For their characteristics, the formal properties, are not expressed by the functions.
The expression of a formal property is a feature of certain symbols.
The sign that signifies the characteristics of a formal concept is, therefore, a characteristic feature of all symbols, whose meanings fall under the concept.
The expression of the formal concept is therefore a propositional variable in which only this characteristic feature is constant.
4.127
The propositional variable signifies the formal concept, and its values signify the objects which fall under this concept.
4.1271
Every variable is the sign of a formal concept.
For every variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values.
4.1272
So the variable name “x” is the proper sign of the pseudo-concept object.
Wherever the word “object” (“thing,” “entity,” etc.) is rightly used, it is expressed in logical symbolism by the variable name.
For example in the proposition “there are two objects which …”, by “(∃x,y) …”.
Wherever it is used otherwise, i.e. as a proper concept word, there arise senseless pseudo-propositions.
So one cannot, e.g. say “There are objects” as one says “There are books.” Nor “There are 100 objects” or “There are ℵ0 objects.”
And it is senseless to speak of the number of all objects.