We investigate a first-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specifically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.
Kai F. Wehmeier. "Wittgensteinian Predicate Logic." Notre Dame J. Formal Logic 45 (1) 1 - 11, 2004. https://doi.org/10.1305/ndjfl/1094155275