Wittgenstein's Tractatus Logico-Philosophicus & Russell's Paradox

"No proposition can make a statement about itself, because a propositional sign cannot be contained in itself," Wittgenstein says at 3.332 of Tractatus Logico-Philosophicus (the decimal figures hereafter refers to the numbers in the Tractatus). He also "disposes" Russell's paradox at 3.333.

Let us consider the following proposition p:

This statement is false.

Clearly, if p is true, then from the content of p, p is false. On the contrary, if p is false, then what p says is true, so that p is true. It is the liar paradox.

At first glance, the problem of the liar paradox is due to self-reference of the proposition. Let us consider a more general version of the paradox. Let P and Q denote the following proposition respectively:

Q is true; P is false.

We see that the paradoxes arise from the actually meaningless definitions of the objects (that is, the propositions) concerned. To define an object, we must use already well-defined terms. A self-referential proposition is meaningless because it is defined by itself. For the last version of the paradox, both P and Q are defined by what are being defined! Let us return to the self-referential liar paradox and rewrite p as

p is false.

Consider a function F(q):

q is false.

Then F(p) is the proposition:

p is false.

Besides, F(F(p)) is the proposition:

F(p) is false.

Here, as mentioned in 3.333, p, F(p) and F(F(p)) are in fact different propositions, and they should not be signified by the same sign such as p (as what liar paradox does), as states in 3.325:

In order to avoid such errors we must make use of a sign-language that excludes them by not using the same sign for different symbols and by not using in a superficially similar way signs that have different modes of signification: that is to say, a sign-language that is governed by logical grammar--by logical syntax. (The conceptual notation of Frege and Russell is such a language, though, it is true, it fails to exclude all mistakes.)

Wittgenstein does not mention the liar paradox but that of Russell. Russell's paradox is a set-theoretical paradox that asks whether the set R defined by

R = {x: x is not an element of x}

is an element of itself. If we use f(x) to denote the proposition

x is not an element of x,

then

R = {x: f(x)}.

However, is the definition of R meaningful? If it was, it had to be able to tell if any given object was an element of R. In particular, could it tell if the object R (if it could be defined) itself was an element? In other words, is f(R) true (or false)? Actually, if R was well-defined, f(R) could be rewritten as f({x: f(x)}), which was also a meaningless self-referential proposition (covered by the definition)!

Russell's solution to his paradox was his theory of types. He arranged all propositions into a hierarchy. The lowest level of the hierarchy consisted of propositions about individuals, not sets. The next lowest level consisted of propositions about sets of individuals. The next lowest level consisted of propositions about sets of sets of individuals, and so on. Thereby the definition (one of these "valid" propositions) of any set only referred to objects of the same "type" (at the same level).

It seems that Russell's theory of types is just an ad hoc solution to the problem of self-reference of his paradox (3.332). However, we see that the rules in his theory are not just syntactic, but semantic - he had to mention the meaning of signs when establishing the rules for them (3.331). On the other hand, Wittgenstein provided a syntactic solution to these paradoxes. More precisely, Wittgenstein did not provide any solution (because a solution is more or less ad hoc), but it is just that there is no such paradox in the system of his Tractatus.