In his 'Tractatus logico-philosophicus', Ludwig Wittgenstein declares that he has solved Russell's paradox. He presents the solution in a prima facie simple formula "(∃φ) : F(φu) . φu = Fu". This solution is disregarded both by the Russellians and most Wittgensteinians. In this paper I try to read the above formula and translate it into a class membership language (CML). I show that this formula contradicts Russell's logic. I investigate and compare the different forms of this formula in Wittgenstein's manuscripts and in the different editions of the 'Tractatus'. I also take a look at the different interpretations of this formula in literature. My position is that Wittgenstein created his own transcendental logic that can show self-referential (reflexive) sentences without reflexivity. TRAMES, 2009, 13(63/58), 2, 179–197 WITTGENSTEIN’S TRACTATUS 3.333 AND RUSSELL’S PARADOX Urmas Sutrop Institute of the Estonian Language, Tallinn, and the University of Tartu Abstract. In his ‘Tractatus logico-philosophicus’, Ludwig Wittgenstein declares that he has solved Russell’s paradox. He presents the solution in a prima facie simple formula “(∃φ) : F(φu) . φu = Fu”. This solution is disregarded both by the Russellians and most Wittgensteinians. In this paper I try to read the above formula and translate it into a class membership language (CML). I show that this formula contradicts Russell’s logic. I investigate and compare the different forms of this formula in Wittgenstein’s manuscripts and in the different editions of the ‘Tractatus’. I also take a look at the different interpretations of this formula in literature. My position is that Wittgenstein created his own transcendental logic that can show self-referential (reflexive) sentences without reflexivity. DOI: 10.3176/tr.2009.2.06 Keywords: Ludwig Wittgenstein, Bertrand Russell, paradox, logic, history of logic, history of philosophy 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 “(∃φ) : F(φu) . φu = Fu”. Ludwig Wittgenstein Urmas Sutrop 180 1. Letter #7 from Bertrand Russell to Ludwig Wittgenstei n from 13 August 1919). Here and after this letter, Bertrand Russell did not pay a ny attention to Wittgen stein’s solution of his contradiction. It is typical that Witt genstein’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). I ask how this formula corresponds to or contradicts the rules of Russell’s logic, that is, is this formula significant in Russellian sense? Then I take a look at some misprints in the different editions of ‘Tractatus’. To find out the correct formula I try to follow the historical genesis of this formula. After that I will discuss some interpretations of Wittgenstein’s solution of Russell’s paradox and show that Wittgenstein created a new transcendental logic that is free from reflexivity. Finally I will take a look at some consequences of the solution. 3 One can read on the Russell’s paradox, possible solutions, and how Russell discovered Graf Hoensbroech (1939: 355) holds that Russell’s paradox rests on the figure “φφ”, in which “φ” is to be a variable function and the juxtaposition of the two “φ”s is to express the relation “predicate of”. He thinks that Russell’s paradox does not imply that there cannot be propositions in the form ‘ φφ’ in the object-language, but only in the meta-language (see Langford 1939: 132). Considering these remarks I try to analyze one of such formulas as ‘ φ(φ)’– Wittgenstein’s solution of Russell’s paradox. For that I will try to open a door and trespass upon a self-reflexive field of knowledge. James Conant (2000: 212-213), for example, holds that the whole point of §§3.3-3.344 of the Tractatus is that the identity of the object referred to by a name is only fixed by the use of the name in a set of significant [sinnvolle] propositions. Of course, Conant expresses the critique of Russellian doctrines in Wittgenstein’s ‘Tractatus’ in the quoted passage. His position is similar to those who argue that saying and showing are different or the (visually) same sign can show different objects or propositions. In a historical study of Wittgenstein’s solution of paradoxes, his solution of Russell’s paradox is only quoted (Dumitriu 1974: 233). 5 Wilhelm Vossenkuhl (2000) points out that we should make a distinction between Saying and Showing. One can say something through sentences, i.e. through language and one can only show what is unsayable through sentences. Wittgenstein’s solution is not philosophical or linguistic but logical. A symbol is allowed to express only what it can express. The Russell paradox is a linguistic one. It belongs to the realm of Saying, but Wittgenstein’s solution is logical, it belongs to the realm of Showing. Russell’s paradox (his Theory of Types) contains self-reference and for that reason is circular. Wittgenstein’s solution in his formula “(∃φ ) : F(φu) . φu = Fu” contains no self-reference (reflexivity). The 5 Russell holds another position and accepts th e identity of (visually) different signs: ἐφ(ε) and ἐψ(ε) are identical when φ x and ψx are equivalent for all values of x (Russell to Frege on 10.7.1902; Frege 1976: 220). Wittgenstein’s Tractatus 3.333 and Russell’s paradox 193 symbols show the same sentence that lies behind Russell’s paradox, but without saying it. Wittgenstein’s logic is not a th eory but a reflexion of the world. His logic is transcendental (see, e.g. Wittgenstein 1973a: 169, # 6.13). Nevertheless, we have an access to what is showed by Wittgenstein’s transcendental logic.