Reconstructing a Logic from Tractatus: Wittgenstein’s Variables and Formulae

Abstract

It is and has been widely assumed, e.g., in Hintikka and Hintikka (1986), that the logical theory available from Wittgenstein’s Tractatus Logico-Philosophicus (Wittgenstein 1984) affords a foundation for (or is at least consistent with) the conventional logic represented in standard formulations of classical propositional, first-order predicate, and perhaps higher-order formal systems. The present article is a detailed attempt at a mathematical demonstration, or as much demonstration as the sources will allow, that this assumption is false by contemporary lights and according to a preferred account of argument validity. When Wittgenstein’s description of the forms of propositions or Sätze in the 5-numbered remarks and Remark 6 is given a close reconstruction, one sees that no Tractarian proposition is logically equivalent to a simple universally or existentially quantified formula of first-order predicate logic. Therefore, although Wittgenstein employs the sign ∀—or (x), in his notation—occasionally in explanations and illustrations, e.g., 4.0411 and 5.1311, when it comes to logic, the sign ∀ should receive in Tractatus a semantical treatment that is nonstandard. En route to that result, we show that the hierarchy of variables–and, hence, of propositions–defined at 5.501 incorporates the expressive power of (at least) finitary classical propositional logic. Also, when constructed over a first-order language for arithmetic, formulae corresponding to the specification of 5.501 (but salted with parameters) pick out all and only arithmetic sets of numbers. Consequently, Wittgenstein’s hierarchy of iterated N-propositions–as described in vide Remark 6–does not collapse: at any level k, one finds propositions at k + 1 or above that are not logically equivalent to any proposition formed at k or below.

The second-named author is responsible for all translations from Wittgenstein’s German.