Rodrigo Sabadin Ferreira

Wittgenstein afirma no Tractatus que a teoria das classes é supérflua na Matemática e que isso está relacionado ao fato de que a generalidade exigida pela Matemática não é “acidental” (TLP 6.031). O objetivo deste texto é elucidar essa afirmação chamando a atenção para o que, seguindo Gregory Landini, tomaremos como uma forma de Logicismo compartilhada por Frege e Russell. Esta forma de Logicismo tem dois princípios básicos, a saber: o uso de uma teoria lógica cujas variáveis estruturadas incorporam o que hoje chamamos de teoria dos tipos simples e a análise de atribuições numéricas em termos de afirmações sobre conceitos ou atributos que empregam conceitos de ordem superior exatamente análogos aos chamados quantificadores ‘numericamente definidos’. Argumentamos que é esse arcabouço teórico que Wittgenstein está rejeitando em 6.031 e não apenas o uso da Teoria dos Conjuntos na Matemática. Também é defendido que a noção de classe ainda tem um papel essencial a desempenhar no Tractatus.AbstractWittgenstein claims in the Tractatus that the theory of classes is superfluous in Mathematics and that this is related to the fact that the generality required by Mathematics is not “accidental” (TLP 6.031). My aim in this paper is to elucidate this claim by calling attention to what, following Gregory Landini, I refer to as a form of Logicism shared by Frege and Russell. This form of Logicism has two main tenets, namely:the use of a logical theory whose structured variables embody what we nowadays call the simple theory of types and the analysis of number ascriptions in terms of assertions about concepts or attributes which employ higher-order concepts exactly analogous to so-called ‘numerically definite’ quantifiers. It is argued that it is this shared theoretical framework that Wittgenstein is rejecting in 6.031 and not just the use of Set Theory in Mathematics. It is also argued that the notion of class still has an essential role to play in the Tractatus.

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results.