Logic in the Tractatus

Review of Symbolic Logic 10 (1):1-50 (2017)

Authors
Max Weiss
Boston University
Abstract
I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.
Keywords Logic  Induction  Consequence  Wittgenstein  Russell  Frege  Tarski  Definability  Infinitary logic
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DOI 10.1017/s1755020316000472
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References found in this work BETA

Tractatus logico-philosophicus.Ludwig Wittgenstein - 1922 - Filosoficky Casopis 52:336-341.
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Towards a Theory of Part.Kit Fine - 2010 - Journal of Philosophy 107 (11):559-589.
[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.

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