4.1212
What can be shown cannot be said.

4.1213
Now we understand our feeling that we are in possession of the right logical conception, if only all is right in our symbolism.


4.122
We can speak in a certain sense of formal properties of objects and atomic facts, or of properties of the structure of facts, and in the same sense of formal relations and relations of structures.

(Instead of property of the structure I also say "internal property"; instead of relation of structures "internal relation".

I introduce these expressions in order to show the reason for the confusion, very widespread among philosophers, between internal relations and proper (external) relations.)

The holding of such internal properties and relations cannot, however, be asserted by propositions, but it shows itself in the propositions, which present the facts and treat of the objects in question.

4.1221
An internal property of a fact we also call a feature of this fact. (In the sense in which we speak of facial features.)

4.123
A property is internal if it is unthinkable that its object does not possess it.

(This bright blue color and that stand in the internal relation of bright and darker eo ipso. It is unthinkable that these two objects should not stand in this relation.)

(Here to the shifting use of the words "property" and "relation" there corresponds the shifting use of the word "object".)

4.124
The existence of an internal property of a possible state of affairs is not expressed by a proposition, but it expresses itself in the proposition which presents that state of affairs, by an intern al property of this proposition.

It would be as senseless to ascribe a formal property to a proposition as to deny it the formal property.

4.1241
One cannot distinguish forms from one another by saying that one has this property, the other that: for this assumes that there is a sense in asserting either property of either form.

4.125
The existence of an internal relation between possible states of affairs expresses itself in language by an internal relation between the propositions presenting them.


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 ₀ objects".

And it is senseless to speak of the number of all objects.

The same holds of the words "Complex", "Fact", "Function", "Number", etc.

They all signify formal concepts and are presented in logical symbolism by variables, not by functions or classes (as Frege and Russell thought).

Expressions like "1 is a number", "there is only one number nought", and all like them are senseless.

(It is as senseless to say, "there is only one 1" as it would be to say: 2?+?2 is at 3 o'clock equal to 4.)

4.1273
If we want to express in logical symbolism the general proposition "b is a successor of a" we need for this an expression for the general term of the formal series: aRb, ( x)?:?aRx?.?xRb, ( x,?y)?:?aRx?.?xRy?.?yRb,?.?.?. The general term of a formal series can only be expressed by a variable, for the concept symbolized by "term of this formal series" is a formal concept. (This Frege and Russell overlooked; the way in which they express general propositions like the above is, therefore, false; it contains a vicious circle.)

We can determine the general term of the formal series by giving its first term and the general form of the operation, which generates the following term out of the preceding proposition.

4.1274
The question about the existence of a formal concept is senseless. For no proposition can answer such a question.

(For example, one cannot ask: "Are there unanalyzable subject-predicate propositions?")

4.128
The logical forms are anumerical.

Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.