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In the final part of his introduction to the Tractatus Logico-philosophicus, Russell provides a possible solution to the problem of the impossibility of self-reference of logic:

There is one purely logical problem in regard to which these difficulties are peculiarly acute. I mean the problem of generality. In the theory of generality it is necessary to consider all propositions of the form fx where fx is a given propositional functions. This belongs to the part of logic which can be expressed, according to Mr Wittgenstein's system. But the totality of possible values of x which might seem to be involved in the totality of propositions of the form fx is not admitted by Mr Wittgenstein among the things that can be spoken of, for this is no other than the totality of things in the world, and thus involves the attempt to conceive the world as a whole; ``the feeling of the world as a bounded whole is the mystical''; hence the totality of the values of x is mystical (6.45). This is expressly argued when Mr Wittgenstein denies that we can make propositions as to how many things there are in the world, as for example, that there are more than three.

Russell's solution:

These difficulties suggest to my mind some such possibility as this: that every language has, as Mr Wittgenstein says, a structure concerning which in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit. Mr Wittgenstein would of course reply that his whole theory is applicable unchanged to the totality of such languages.

Is it correct to say that the problem Russell is focusing on is that of the impossibility, for logic, of representing the logical form? Could you explain to me more specifically in which sense the solution provided does solve the problem I quoted?

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    It seems that Russell is alluding to the hierarchy of language-metalanguage (and logic-metalogic) that was absent from Principia Mathematica. This hierarchy is at the core of Tarski's truth definition. – Mauro ALLEGRANZA Mar 22 '16 at 17:00
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    I agree with @MauroALLEGRANZA, what Russell appears to be getting at here is the object-language / meta-language distinction but going on to say that we could treat the meta-language in turn as an object-language requiring yet another meta-language, and so on. I think Russell is critiquing Wittgenstein's take on logic, not logic as it was then generally conceived. As to what exact charge he is laying at Wittgenstein's feet I am unsure but I think you're intuition is pointing in the right direction. – igravious May 20 '16 at 2:41
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The issues of generality, self-reference and a hierarchy of languages would likely not be a solution to Wittgenstein's mysticism. As Russell suggests (Tractatus, page 23)

Wittgenstein would of course reply that his whole theory is applicable unchanged to the totality of such languages.

G. E. M. Anscombe illustrates Wittgenstein's position by quoting 6.52 of the Tratatus:

We feel that even if all possible scientific questions be answered, the problems of life have still not been touched at all. Of course there is then no question left, and just this is the answer.

Here is her interpretation: (page 170)

This comment can be taken in two ways: First, Wittgenstein might be saying - and this is what Professor Ayer, for example, would make of his remarks - that people who have wanted to say what the meaning of life consisted in have had nothing in them but a lot of nonsense. This cannot be the right interpretation; for he speaks of people 'to whom the meaning of life has become clear'. But he says of them that they have not been able to say it. Now such people have not failed for want of trying; they have usually said a great deal. He means that they have failed to state what they wished to state; that it was never possible to state it as it is possible to state indifferent truth.

If her interpretation is correction Russell's solution of a hierarchy of languages would not help, nor is the reason the result of logical self-reference such as that presented by Russell's paradox.


Anscombe, G. E. M. (1971). An introduction to Wittgenstein's Tractatus. St. Augustine's Press.

Wittgenstein, L. (1922). Tractatus logico-philosophicus. Retrieved on May 11, 2019 from Internet Archive at https://archive.org/details/in.ernet.dli.2015.221720/page/n188

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