Academic journal article Trames

Wittgenstein's Tractatus 3.333 and Russell's Paradox

Academic journal article Trames

Wittgenstein's Tractatus 3.333 and Russell's Paradox

Article excerpt

Nobody will wish to assert of the class of men that it is a man.

Gottlob Frege

This is at once clear, if instead of "F(F(u))" we write "([there exists][phi]) : F([phi]u) x [phi]u = Fu".

Ludwig Wittgenstein

1. Introduction

Russell's paradox is still topical: it will not go away (Moorcroft 1993). It seems that Moorcroft has rightly pointed out that 'the impact of the Paradox is not so much a problem that must be solved in some way but the abandonment or modification of a principle that was taken to be intuitive and obvious' (1993: 101). The paradox arises by considering the class of all classes which are not members of themselves. Such a class appears to be a member of itself if and only if it is not a member of itself. Ludwig Wittgenstein thought that Russell's paradox vanishes in his 'Tractatus logico-philosophicus' (prop 3.333). (1) In this paper Russell's paradox (contradiction) will be mentioned many times. To be clear, I present here a version of Russell's paradox which Bertrand Russell drafted at a mature age:

Some classes are members of themselves, some are not, the class of all classes is a class, the class of not-teapots is a not-teapot. Consider the class of all the classes not members of themselves; if it is a member of itself, it is not a member of itself; if it is not, it is (see Garciadiego 1992: xiii).

It is interesting that Post-Tractarian Wittgenstein utters '"the class of cats is not a cat."--How do you know?' and indicates that 'there is a language game with this sentence'. Hence he holds this opinion about the Russellian contradiction which 'would stand like a monument (with a Janus head)' (Wittgenstein 1956, 182e and 131e). (2)

But to our analysis it is important that young Wittgenstein declares that he has solved Russell's paradox--'Herewith Russell's paradox vanishes' (1922, prop. 3.333(4)). He presents his solution of Russell's paradox in a prima facie simple formula

"([there exists][phi]) : F([phi]u) x [phi]u = Fu"

(Wittgenstein 1922, prop. 3.333(4)).

There is an intension in this proposition that the formula solves the paradox better than Bertrand Russell in his theory of logical types. It is curious that Wittgenstein's solution of Russell's paradox is disregarded both by the Russellians and most Wittgensteinians.

At first, Bertrand Russell himself ignored this solution of his paradox. When he, for the first time, read the manuscript of Wittgenstein's 'Tractatus' he sent Ludwig Wittgenstein a letter and queries about the text of the 'Tractatus' on a separate sheet (McGuinnes and von Wright 1990, 107-108. Letter #7 from Bertrand Russell to Ludwig Wittgenstein from 13 August 1919). Here and after this letter, Bertrand Russell did not pay any attention to Wittgenstein's solution of his contradiction. It is typical that Wittgenstein's solution of this paradox is not discussed in the special studies on Russellian contradiction (Quine 1963, Garciadiego 1992, Link 2004) or in the comparison of Whitehead and Russell's 'Principia' and Wittgenstein's 'Tractatus' (e.g. Rao 1998).

Russell discovered his paradox in 1901, originally formulating it in terms of predicates rather than in terms of sets (similar antinomy was discovered by Cesare-Bura Forti already in 1897). Next year Russell wrote to Gottlob Frege of his paradox which showed that the axioms Frege was using to formalize his logic were inconsistent (cf. Frege 1976: 211 ff.) As a result Frege immediately added an appendix to the second volume of his 'Grundgesetze der Arithmetik' (1903) which discussed Russell's contradiction. Russell himself first disputed his paradox in depth in an appendix to his 'Principles of mathematics' (1903) and he developed the idea in his theory of types. The naive set theory created at the end of the 19th century by Georg Cantor and used by Frege leads to a contradiction. Ivor Grattan Guinnes describes Russell's paradox as a true paradox, a double contradiction, not another neo-Hegelian puzzle to be resolved by synthesis (2000: 311). …

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