The Tractatus 6.123:

6.123 Clearly the laws of logic cannot in their turn be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough, since it is not applied to itself.)

This looks a lot like "No theory can prove its own consistency" of Gödel's 2nd incompleteness theorem.

Is that what Wittgenstein meant, or did he mean something else?


Not on this at least. Wittgenstein is alluding to Frege on logical syntax. From Tractatus:"Frege says that any legitimately constructed proposition must have a sense. I say that any proposition is legitimately constructed". Laws of syntax are similar in form to ethical laws: thou shalt not (form such and such sentences). Wittgenstein's response in both cases is "and what if I do?" Frege attempted to justify syntactic rules ("laws of logic") by appealing to other laws of logic, so did Russell with his type theory to avoid contradictions of self-reference. Wittgenstein rejects this idea as "clearly" wrong.

Bearn explains:"If the combination of signs is nonsense then we don't need a law to tell us that we should not combine the signs in this way. What we need is a logical syntax that makes logical structure, which is already there, clear. Logic must take care of itself... According to Wittgenstein, the so-called laws of logic are built into unspeakable structure of logical space... They are manifest in the fact that some combinations of signs and not others make sense. Russell misconstrued the task of logic as the installation of rules obedience to which would keep our propositions within the realm of sense. The theory of types was of precisely this nature. Moralists make the same error. They attempt to construct a set of moral rules obedience to which will give our lives meaning." In other words, Wittgenstein favors unified logic that fuses language, meta-language, meta-meta-language, etc., and has a single set of "logical laws" that function synthetically, not formally, "one is enough". As Russell pointed out in another context, such a view would make mathematical logic "impossible".

Here is Friedman's characterization of the Tractatus from Logical Truth and Analyticity in Carnap's "Logical Syntax of Language" (p.85) more generally:"For Wittgenstein, there can be only one language the single interconnected system of propositions within which everything that can be said must ultimately find a place; and there is no way to get "outside" this system so as to state or describe its logical structure: there can be no syntactic metalanguage. Hence logic and all "formal concepts" must remain ineffable in the Tractatus... Of course, the Tractatus is itself quite clear on the restricted scope of its conception of logic and mathematics in comparison with Frege's (and Russell's) conception. Wittgenstein's response to this difficulty is also all too clear: so much the worse for classical mathematics and set theory".

As for Wittgenstein's relation to Gödel's incompleteness, his "notorious paragraph" shows, depending on one's point of view, that he either misunderstood it, or concluded that it has nothing philosophically valuable to say. Either way he would not have been anticipating it. Essentially, he rejects a premise of Gödel's proofs, that "truth" can be interpreted as distinct from "provability", which renders completeness issues moot:"Just as we ask: “‘provable’ in what system?”, so we must also ask: “‘true’ in what system?” ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system... If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up.".

  • "so did Russell with his type theory to avoid contradictions of self-reference. Wittgnstein rejects this idea as "clearly" wrong." --> But isn't that exactly what Godel did, Proving that Russell couldn't escape self reference no matter how hard he tried? Jun 26 '15 at 2:25
  • 1
    @Alexander S King Russell created type theory to avoid paradoxes in Frege's logic, Gödel kept it (the original incompleteness proof was in the formalism of Principia), and put object/meta language distinction on top of it. He then showed that with that in place the kind of reasoning that used to lead to paradoxes produces a non-trivial result about the type theory instead. Wittgenstein moved in the opposite direction: instead of formalizing logic and getting paradoxes, and then formalizing it some more to avoid them, he discards the initial formalization.
    – Conifold
    Jun 29 '15 at 1:55
  • 1
    @Alexander S King Wittgenstein rejects what is common to Russell and Gödel, not inches from one to the other, and he definitely rejects Gödel's ideas ahead of time. In his "logic" completeness is ineffable and consistency proofs are redundant. I added a link to Friedman comparing positions of Russell, Gödel and Wittgenstein. There is no literal self-reference in Gödel's sentence, it is only "self-referring" when viewed externally, type theory successfully avoids self-reference, its problem is that it makes most of mathematics intractable along the way.
    – Conifold
    Jun 29 '15 at 2:02
  • The link regarding Friedman's characterization of Tractatus doesn't work.
    – user13627
    Dec 20 '16 at 4:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.