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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).
Frege and Russell recognized that the subject-predicate form of grammar masks the underlying logical form of a proposition. Rather than read sentences as being composed of subjects and predicates, they read sentences as being composed of functions and variable placeholders. Consequently, they analyze "all horses are mammals" as a function of a variable, x: "For all x, if x is a horse, then x is a mammal." This kind of analysis has many advantages: it makes quantifier logic possible ("there exists an x such that..." or "for all x..."), it can dissect sentences that refer to non-existent things (e.g. "The present king of France is bald"), and it gives a general form to all propositions that permits a wide range of deduction and inference.
Any given function will allow for a number of different variables. For instance, there are many values of x that will satisfy the function "x is a horse." Frege would talk about the "extension" of the "is a horse" concept as all values of x that satisfy "x is a horse," i.e. all horses. Russell would talk about sets, or classes, of things, all of which satisfy a certain function, so that there is, for instance, the "set of all horses" (which is the set of all objects that satisfy the function "x is a horse"), the "set of all prime numbers," the "set of all vegetables beginning with 'R,'" and so on. We can also talk about sets of sets, such as the "set of all sets with two members" (which Russell would use as a definition for the number two), or the "set of all sets that have at least one member that begins with the letter 'A,'" and so on. And if there can be sets of sets, Russell inferred, it must also be possible for a set to contain itself. For instance, the "set of all sets that begin with the letter 'S'" is itself a set that begins with the letter "S," and so must be a member of itself. We can then imagine that there must be a "set of all sets that contain themselves as members," and also a "set of all sets that do not contain themselves as members."
Does the "set of all sets that do not contain themselves as members" contain itself as a member? Careful reflection will reveal that if it does contain itself as a member, then it cannot contain itself as a member; and if it doesn't contain itself as a member, then it must contain itself as a member. This odd contradiction is called Russell's Paradox, and since it can be derived from the basic laws of logic as Frege and Russell understood them, it casts a shadow of doubt over all their achievements in logical analysis.
Russell's Theory of Types is his answer to this paradox. According to Russell, there are different orders of sets, so that first order sets could contain only objects as members, second order sets could contain objects and first order sets, and so on. Accordingly, a symbolism would be required to differentiate the symbols for objects from the symbols for first order sets, and so on.
Wittgenstein claims the Theory of Types is unnecessary if we recognize that the meaning of a sign is made apparent by its use in propositions. He claims that signs can only signify within the context of a proposition, and that their meaning becomes apparent in the way they are used in a proposition. That "hat" and "table" are possible values of the function "the x is on the y" and "two" and "purple" are not tells us something about the meaning of these words. Accordingly, a function cannot be used to talk about itself (a set cannot contain itself), since that would be to give it two different uses. Since meaning is determined by use, two functions being used in different ways cannot possibly have the same meaning, i.e. be the same function.
In other words, Wittgenstein is claiming that the sense of a proposition is entirely internal to the proposition itself: the elements of a proposition are related only to one another, and not to anything external to them. This contradicts Frege, who argued that every proposition has an external meaning, namely its truth- value.
Commentators on Wittgenstein's later philosophy (best represented by the Philosophical Investigations) talk about his emphasis on the importance of use in determining meaning. It is worth noting that this emphasis is present in Wittgenstein's early philosophy as well.