The purpose of this paper is threefold. First, I visit the Fogelin–Geach-dispute, criticizeMiller's interpretation of the Geachian notationN(x:N(fx)) and conclude that Fogelin's argumentagainst the expressive completeness of the Tractariansystem of logic is unacceptable and that the adoptionof the Geachian notation N(x:fx) would not violate TLP5.32. Second, I prove that a system of quantificationtheory with finite domains and with N as the solefundamental operation is expressively complete. Lastly, I argue that the Tractarian system is apredicate-eliminated many-sorted theory (withoutidentity) with finite domains and with N as the solefundamental operation, and thus is expressivelycomplete.