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While propositions can depict all of reality, they cannot depict its logical form (4.12). A proposition can only depict what is external to it, so in order to depict logical form it would have to do so from a perspective outside of logical space. Rather than depict logical form, a proposition shows it (it shares its logical form with the reality it depicts), and "what can be shown, cannot be said" (4.1212).
At 4.122, Wittgenstein introduces the notion of formal, or internal, properties, those properties that show themselves (as opposed to being spoken about) in a proposition. These properties define the logical structure of propositions, facts, and objects. Propositions have the same internal properties as the facts they depict (4.124).
A formal concept defines the formal properties of an object, state of affairs, or fact. Formal concepts are to be sharply distinguished from concepts proper (4.126): while a concept proper can be expressed as a function and can feature in propositions, a formal concept cannot be spoken about at all. An example of a concept proper is "x is a horse"; an example of a formal concept is "x is a number." We cannot say that x is a number: that it is a number shows itself. Any attempt to use a formal concept in a proposition (e.g. "two is a number," "purple is a color") will result in a nonsensical pseudo-proposition (4.1272).
Contrary to Frege and Russell, Wittgenstein asserts that formal concepts are not represented in logical notation as sets or functions (4.1272) and that they cannot be introduced in the same way that objects are (4.12721). That is, we can say "there is an x, such that
" but we cannot say, "there is an object, such that
." Rather than express them as sets or functions (e.g. express "x is an object" as the function O(x)), Wittgenstein suggests that formal concepts are expressed as variables (4.1271). The variable x in the proposition "x is a horse" signifies an object, because it holds the place of an object in that proposition: the "x" in that proposition can stand for any object. We cannot talk about objects directly as functions, but we can show their existence in our use of variables.
That we can say nothing about formal concepts also implies that we cannot talk about the number of formal concepts, or ask what kinds of formal concepts there are. Propositions can only talk about objects and states of affairs, and there are no objects or states of affairs corresponding to formal concepts.
Wittgenstein's introduces formal concepts to clarify a distinction that he feels was ignored by Frege and Russell. In analyzing the logical properties of language, Frege made a fundamental distinction between objects and concepts. In a proposition like "the president of America is a Texan," "the president of America" is an object (it denotes a specific thing in the world to which we can ascribe properties) and "Texan" is a concept (it is a category into which one or more object may fall). We could say that "the president of America" is a value of the function "is a Texan." More generally, in any proposition of the form "x is a y," "x" will represent an object and "y" will represent a concept.
Frege ran into trouble with this distinction when he tried to talk about logical properties themselves. How do we talk about, say, "the concept of a horse"? We can say things about it, ascribe properties to it, so it must be an object. We would have to say, "'the concept of a horse' is an object." And, since the "x" in "x is a y" must always be an object, that would mean there is no true proposition of the form "x is a concept."
Frege and Russell both developed and employed powerful logical machinery that helps our understanding of language and philosophy. However, this logical machinery foundered when it was turned upon itself. Russell's paradox unearths similar problems with self-referentiality in talking about sets that are members of themselves. How can logic talk about itself without falling into paradox? And if it cannot, how can we have confidence its reliability?
Wittgenstein's sharp distinction between what can be said and what can be shown is essentially attempts to extricate logic from these difficulties. At 4.0312 he claims: "My fundamental idea is that the 'logical constants' are not representatives; that there can be no representatives of the logic of facts." Another way of putting this "fundamental idea" is to say that we cannot talk about logic: logic is something that shows itself.
The distinction between formal concepts and concepts proper is supposed to highlight this difference. According to Wittgenstein, there is a fundamental difference between "x is a horse" and "x is a concept." Only grammar leads us to think the two are equivalent. Wittgenstein wants to tell us that only the former of those two propositions has a sense. If Frege is right, and any y in a proposition of the form, "x is a y" is a concept, then the fact that y is a concept shows itself from the place it holds in the proposition. And, as Wittgenstein says at 4.1212, "what can be shown, cannot be said." Any attempt to say something of the form "x is a concept" is an attempt to say what can only be shown, and the result is not a proposition, but just plain nonsense.
This might seem a bit harsh. Surely, "two is a number," "purple is a color," or other statements that describe the formal properties of objects or concepts have sense. I can certainly understand what you mean when you say, "two is a number." Wittgenstein's point is that you only think you understand, and that this illusion of understanding is a result of your being misled by a familiar grammatical structure. The test, according to Wittgenstein, of whether a proposition has sense or not is to ask what possible situation in the world it represents. There is no possible situation corresponding to "two is a number," and so Wittgenstein concludes that it is nonsense.
At the end of the Tractatus, Wittgenstein will deal with the self- referentiality of his own propositions in a mystifying way. In his discussion of formal concepts, we can see he is already pulling the rug out from under himself. If, as he claims, we cannot say anything about formal properties and concepts, what are we to make of the propositions of the Tractatus itself? Almost every single one of them makes some mention of objects, facts, states of affairs, or some such. The difficult question of how we are supposed to consider these propositions will be addressed in the commentary on the final section of the text.