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We can analyze the complex objects of everyday speech into simpler parts by means of definitions. For instance, if I don't understand what you mean by "sesquipedalian," I can ask you to define it in terms of a number of simpler words. Ultimately, simple symbols—the symbols for names—cannot be further defined: they are fully analyzed. The meaning of a name in a fully analyzed proposition is the object it refers to (3.203).
Names themselves cannot be defined, so we cannot say what they mean. Instead, we must show what they mean by using what Wittgenstein calls "elucidations": by using the name in a proposition, we can approach a clearer understanding of what it means.
Wittgenstein accepts Frege's and Russell's view of a proposition as a function of the expressions contained in it (3.318). For instance, "the hat is on the table" is just one value of the variable function "the x is on the y." Expressions such as "hat" or "table" then fill out these functions, giving them a sense; the expressions themselves are meaningless outside the context of a proposition (3.314).
In a perfect sign-language, the signs we use to express propositions and the variables in them should be crystal clear, so there is never any confusion of meaning (3.325). In natural language, this is never the case, so that the same sign is often used in different ways. For instance, we sometimes use the word "is" as a copula ("John is tall"), sometimes as a sign for identity ("Paris is the capital of France"), and sometimes to signify existence ("There is an even prime number") (3.323).
A sign has no meaning independent of its use. Wittgenstein criticizes Russell's Theory of Types because it endows signs themselves with meaning (3.331). The Theory of Types can be disposed of simply by recognizing that a proposition that makes a statement about itself is being used in two different ways, and so it cannot be the same proposition. The "F" in "F(fx)" and the first "F" in "F(F(fx))" range over different kinds of variables, so saying that the two Fs have the same meaning is as confused as saying that the "is" of identity and the "is" of existence have the same meaning (3.333).
The propositions "the hat is on the table" and "the book is on the shelf" are identical in all their essential features, and vary only in the accidental fact of what values are given to their variables. While hats and tables, books and shelves, are not essential to the proposition, the fact that they are the kinds of values that fill out the proposition tells us about the proposition's essential features. That is, we learn what kinds of things can be substituted into the proposition "the x is on the y" and thus learn what place that proposition holds in logical space. In turn, we can learn about the structure of logical space by observing the places different propositions can hold in it (3.42). Thus, while propositions such as "the hat is on the table" tell us nothing general, we can infer a great deal about the general structure of logical space by observing the structure of these ordinary propositions (3.3421).
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