The N-Operator and the Elimination of the Identity Sign in Ludwig Wittgenstein's Tractatus Logico-Philosophicus

The correct logical notation proposed by Ludwig Wittgenstein in his Tractatus Logico-Philosophicus consists of two provisions: the complete elimination of the sign of identity (the identity or non-identity of objects should be expressed through the identity or non-identity of signs (5.53)) and the postulated ability to obtain any first-order logic formulas through the application of the N-operator. The first position rejects the definition of identity given by Leibniz (adopted also by Russell in Principia Mathematica) and in the Tractatus is justified in a certain way (the author does not consider Wittgenstein's arguments). The second position is introduced without any argument. Regarding the rules for eliminating the identity sign and applying the N-operator, Wittgenstein limited himself to examples. The aim of this review is to summarize the main recent results of various attempts to present the correct logical notation proposed by Wittgenstein in a systematic way. Disputes over the fundamental possibility or impossibility of the systematic use of logic without an identity sign with varying degrees of success have been going on since the publication of the Tractatus (the reaction of Ramsay and Russell and the consequences for the Principia Mathematica project are well known). Recent works by Juliet Floyd and Gregory Landini have returned interest to the problem). But significant success in understanding Wittgenstein's intentions has been linked to recent results by Kai Wehmeier and especially by Tim Lampert. Regarding the definition of the rules for using the N-operator, substantial success was reached by Brian Rogers and Kai Wehmeier. The first part of this article discusses the rules and the algorithm for translating first-order logic formulas through the N-operator proposed by Rogers and Wehmeier (the well-known debate between Robert Fogelin and Peter Geach regarding the N-operator is also briefly considered). The second part sets forth the algorithm proposed by Wehmeier for translating standard formulas of first-order logic into a language without an identity sign. In the third part, the author considers another similar algorithm (given by Lampert) and the corresponding criticism of the translation rules of Rogers and Wehmeier. In the final part of this article, additional arguments in favor of Lampert's position are formulated.

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