Formal Properties and Relations.

In a certain sense, we can assign formal properties to objects and matters of fact as we discuss them. This applies to structural properties of facts as well. In the same sense, we can also discuss formal relations and structural relations.

Also, Wittgenstein prefers to say 'internal property' instead of structural property in case of a fact and say 'internal' instead of structural in case of a relation. He introduces these expressions to clarify the very widespread confusion between internal relations and relations proper (external relations).

A property is internal if it is unthinkable that its object should not possess it. Different shades of blue relate, eo ipso, as lighter to darker. Because it is unthinkable that these two particular objects should not relate in this way, the relation is internal. (Note the shift in the use of 'object of consideration.' It corresponds to both 'property' and 'relation.')

Internal properties and internal relations cannot be declared in assertions. Rather, they can simply be seen in those sentences that describe the relevant circumstances and deal with the relevant objects. An internal property of a fact can also be called a feature of that fact (in the sense in which we speak of facial features, for example).

That a situation has an internal property, becomes apparent in the proposition that describes it as an internal property of the proposition. It is not expressed by means of another proposition. It makes no sense to assert that a proposition has a formal property or not. One cannot distinguish forms from one another by saying that one has this property and another that property because doing that presupposes that it makes sense to ascribe either property to either form.

That situations have an internal relation to each other, expresses itself in language as an internal relation between the sentences describing them. This resolves the controversy of whether all relations are internal or external.

Wittgenstein calls a series that is ordered by an internal relation a series of forms. The series of natural numbers is not ordered by an external relation but by an internal relation. The same is true of the series of propositions

»aRb«,
»(∃x):aRx.xRb«,
»(∃x,y):aRx.xRy.yRb«,
and so forth.

(If b stands in one of these relations to a, b is a successor of a.)