Infinity and Generality in the Tractatus

Abstract

There has been a good deal of controversy over the expressive completeness of Wittgenstein’s operator N, presented in the Tractatus. James Connelly gives an explication of operator N in an attempt to dispel charges against its expressive capacities as the sole operator in Wittgenstein’s proposed logical system. Connelly argues that a proper appreciation of infinity as it was then understood by Wittgenstein is fundamental to the exculpation of N. I evaluate Connelly’s discussion and demonstration of actual as opposed to potential infinity. I then raise objections regarding Connelly’s conflation of actual infinity and finitude, and his introduction and use of notational variables within propositional expressions. I present mathematical induction and recursion as alternative methods for containing infinity. I consider Wittgenstein’s differentiation of logical form and logical structure, from which he attempts to, but cannot, justify semantic rule-formation based on syntactic rule-formation. I conclude that N is expressively and functionally incomplete.