Abstract: Russell strived to defend the view that arithmetic is nothing but logic by conceiving of the natural numbers as the finite cardinalities. The received view is that Russell’s logicist project has failed owing to a reduction of arithmetic to logic along these lines presupposing the axiom of infinity’s logicality. At the end of the day, even Russell rejected the view that the axiom of infinity was a truth of logic.
In this talk I will begin by offering a glance at a result of my own according to which arithmetic is interpretable in a very weak higher-order modal logic without presupposing the axiom of infinity’s truth (much less its logicality). I’ll then put forward a novel form of neoRussellian logicism which gets support from this result, and offer some further reasons in its favor. I’ll conclude by comparing the neoRussellian logicism proposed with other, seemingly related, philosophies of arithmetic.