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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.
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