4.1272 Thus the variable name 'x' is the proper sign for the pseudo-concept
object. Wherever the word 'object' ('thing', etc.) is correctly used, it is
expressed in conceptual notation by a variable name. For example, in the
proposition, 'There are 2 objects which. . .', it is expressed by ' (dx,y)
... '. Wherever it is used in a different way, that is as a proper concept-
word, nonsensical pseudo-propositions are the result. 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', or, 'There are !0
objects'. And it is nonsensical 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). '1 is a number', 'There is only one zero', and all similar
expressions are nonsensical. (It is just as nonsensical to say, 'There is
only one 1', as it would be to say, '2 + 2 at 3 o'clock equals 4'.)
4.12721 A formal concept is given immediately any object falling under it
is given. It is not possible, therefore, to introduce as primitive ideas
objects belonging to a formal concept and the formal concept itself. So it
is impossible, for example, to introduce as primitive ideas both the
concept of a function and specific functions, as Russell does; or the
concept of a number and particular numbers.
4.1273 If we want to express in conceptual notation the general
proposition, 'b is a successor of a', then we require an expression for the
general term of the series of forms 'aRb', '(d : c) : aRx . xRb', '(d x,y)
: aRx . xRy . yRb', ... , In order to express the general term of a series
of forms, we must use 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.) We can
determine the general term of a series of forms by giving its first term
and the general form of the operation that produces the next term out of
the proposition that precedes it.
4.1274 To ask whether a formal concept exists is nonsensical. For no
proposition can be the answer to such a question. (So, for example, the
question, 'Are there unanalysable subject-predicate propositions?' cannot
be asked.)
4.128 Logical forms are without number. Hence there are no preeminent
numbers in logic, and hence there is no possibility of philosophical monism
or dualism, etc.
4.2 The sense of a proposition is its agreement and disagreement with
possibilities of existence and non-existence of states of affairs. 4.21 The
simplest kind of proposition, an elementary proposition, asserts the
existence of a state of affairs.
4.211 It is a sign of a proposition's being elementary that there can be no
elementary proposition contradicting it.