Can anyone explain to a non-logician how Tractatus 3.333 refutes (or fails to refute) Russell's Paradox? Please explain his use of symbols!


Wittgenstein's understanding of Russell's Theory of Type is very shallow and muddle-headed, but I'll make an effort to interpret what he says.

Wittgenstein thinks that a function's prototype determines what kind of argument it can take - this is exactly how some computer language, like C, define functions. Since the function is defined this way, if it is used otherwise, the resultant meaning is undefined:

F(x) is defined as a function that takes individuals as arguments.

G(U(x)) is defined as a function that takes first order function as arguments, where 1st-order function are those who takes only individuals as arguments.

Since F(x) can only take individuals as arguments, F(F(x))'s meaning is undefined.

Suppose there is such a thing as F(F(x)), the outer function takes first-order function as arguments, the inner function takes individual as arguments, therefore they are actually different functions using the same symbol.

∃ means there exists. (∃Φ):F(Φ(u)).Φ(u)=Fu means there exists a 1st-order function Φ such that Φ satisfies the 2nd-order function F(Φ(u)) and Φ(u) = Fu. Wittenstein tries to show that F(Φ(u)) and Fu are different functions having in common only the letter F.

Ignoring Wittgenstein's misuse of symbolism, his argument can be simplified like this: I designated this basket for apples only, therefore you must not put baskets in this basket, and if you see a basket inside another basket, the outer basket must be a totally different type of basket. Basically, Wittgenstein just demonstrated how the Theory of Types works, but failed to explain the underlying principles and concerns that gave rise to the Theory of Types.

Whitehead and Russell, in Principia Mathematica, invoked the vicious circle principle to dispel Russell paradox. The vicious circle principle states that a totality must not contain itself as a constituent because the totality cannot be determinate until each of its constituent is determinate; if one of the constituent is the totality itself, then the totality is indeterminate.

The vicious circle principle determines why a function must not take itself as an argument.

A function Fŷ -please notice the hat - denotes a totality. Suppose there are only two individuals in the world, Socrates and the earth, then the functions "ŷ is a man" denotes the totality {Socrates is a man, the earth is a man}. It follows that the meaning of Fŷ presupposes - or depends on - its values.

Fy - no hat here - denotes one of Fŷ's values when y is substantiated, i.e., either "socrates is a man" or "the earth is a man", but ambiguous. It follows that propositions of the form Fy must not involve Fŷ because Fŷ is indeterminate until Fy is determinate and, if Fy depends on Fŷ, Fy is indeterminate. Therefore, F(Fŷ) has no meanings.

  • 2
    A more precise and charitable simplification: I designated this basket for apples only, therefore nothing but apples fits; if you place a basket inside my basket, then you have no right to nevertheless claim that it is a basket for apples only. Rather than being "muddle-headed" and "shallow", I think this is a beautifully concise way of solving the problem linguistically, which is what Wittgenstein was justly famous for. – Rhyme Dec 6 '19 at 8:39

A similar question was posted on SE and answer, which I've reproduced here:

Wittgenstein is alluding to how Russell himself solved the Paradox - the theory of ramified types. He alludes to this in:

3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).

And he reformulates as

3.333 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.

A functional sign is simply the sign of the function; the function being what the sign signifies. He expands what he means by this:

If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer function F and the inner function F must have different meanings;

for the inner has the form g(fx), the outer the form h(g(fx)).

That is F(F(fx)) is different from F(F(fx)) because in the expression they signify different things, that is they have different meanings or precisely functions; and only the sign 'F' is common to both, as he affirms:

Common to both functions is only the letter “F”, which by itself signifies nothing.

and by

This is at once clear, if instead of “F(F(u))” we write “There exists g : F(gu). gu = Fu”.

Herewith Russell’s paradox vanishes.

This resolution is also discussed in the paper appended by user4894, Wittgensteins Tractatus 3.333 and Russells Paradox by Urmas Sutrop:

On the other hand, Ostrow points that there is no paradox. “In ‘F(F(fx))’, the first ‘F’ and the second will not have the same meaning, since, to use Russellian terminology, the first ‘F’ ranges over propositional functions of type n, while the second ranges over functions of type n + 1” (Ostrow 2002: 66-67). In this case the inner and the outer functions play different roles and the common letter F denoting both functions is not confusing at all. This is very promising approach, but unfortunately the Wittgenstein’s formula “(∃φ) : F(φu) . φu = Fu” is not discussed at all in this paper.

Ostrow, Matthew B. (2002) Wittgenstein’s Tractatus: a dialectical interpretation.


I am somewhat out of depths, it must be said, but I find Russell's solution more like a sticking plaster and prefer the solution given by his colleague George Spencer Brown. Russell himself seemed to like it albeit he didn't seem to quite understand its significance.

The reason would be that the paradox arises in metaphysics, where R's solution is no help but GSB's solution works. The theory of types/classes seems like a technical fudge to me, but perhaps this only is because I'm not a mathematician.

Pardon me if this is off-topic.

  • I think this should be a comment rather than an answer, since it doesn't actually address the OP's specific concern. That said, do you have a reference re: GSB? – Noah Schweber Sep 22 '19 at 14:09
  • @NoahSchweber - You're quite right. I was shooting the breeze, which I shouldn't do. If you google his name and 'Laws of Form' this should produce lots of links. There are some good general essays on him worth a browse by Robert Robertson. (president of the US Jungian Society). . – user20253 Sep 23 '19 at 11:45

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