A variable is the sign of a formal concept.

Wittgenstein now extends the discussion to formal concepts in the same sense as he discussed formal properties. He introduced the expression 'formal concepts' to distinguish between them and concepts proper. (They are confused in the whole of traditional logic.) The sign for the object itself shows us that something falls under a formal concept. A proposition cannot tell us that. (A name obviously signifies an object, a numeral signifies a number, etc.)

Functions can express concepts proper but not formal concepts. This is because functions do not express formal properties. Instead, all symbols whose meaning falls under a formal concept express the formal property that is a characteristic feature of that concept. In this way, a propositional variable in which this characteristic feature alone is constant expresses the formal concept. The propositional variable signifies the formal concept, and its values signify the objects that fall under it.

A variable is the sign of a formal concept, because it displays a constant form that all its values possess, and that form can be regarded as a formal property of those values. Thus the variable name 'x' is actually the sign for the pseudo-concept object of consideration.

Wherever the word 'object' ('thing', 'matter' etc.) is used properly to denote a formal concept, it will become a variable name in conceptual notation. For example, in the proposition, 'There are 2 objects which. . .', 2 objects is expressed by ' (∃x,y) ... ' and not otherwise.

Wherever objects is used as a proper concept, pseudo-propositions result that make no sense. So one cannot say, for example, 'There are objects', as one might say, 'There are books', and it is just as impossible to say, 'There are 100 objects'.

It is absurd to speak of the total number of objects. The same applies to the words 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). Expressions such as '1 is a number', 'There is only one zero', and all similar ones make no sense. (It also makes no more sense to say, 'There is only one 1', as to say, '2 + 2 at 3 o'clock equals 4'.)

A formal concept is a given once an object falls under it. One cannot, therefore, introduce both a formal concept as well as objects belonging to it as primitive ideas at the same time. So one cannot introduce both the concept of a function and specific functions as primitive ideas (as Russell does), nor, for that matter, the concept of a number along with particular numbers.

If we want to express the general proposition 'b is a successor of a' in conceptual notation, then we need to express the general term of a series of forms. We can do this by giving its first term and the general form of the operation that produces the next term out of the proposition that precedes it, like this:

aRb,
(∃x):aRx.xRb,
(∃x,y):aRx.xRy.yRb,
...

We can only express the general term of a series of forms using a variable, because the concept 'term of that series of forms' is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the one above is incorrect; it contains a vicious circle.)

To ask whether a formal concept exists makes no sense, for no proposition can give an answer. (So, for example, the question, 'Are there unanalysable subject-predicate propositions?' cannot be asked.)

Logical forms are not enumerable. For this reason, there are no preeminent numbers in logic, and no possibility of philosophical monism or dualism, etc.