Carnapian Frameworks and Categoricity of Arithmetic via Inferential $\omega$-logics

We provided in \cite{BaldwinBrincusI} extensions of first order logic by modified inferential definitions of the classical $\omega$-rule in $1$ or $2$ sorts. These logics are categorical in the inferential sense. Arithmetic has a unique countable model in each case, e.g. first order PA is categorical in our first logic. The 2-sorted case interprets $L_{\omega_1,\omega}$. In this paper, we discuss two philosophical problems raised by Button and Walsh \cite{ButtonWalshbook} concerting the identification of a unique isomorphism class. First, we argue that the doxological challenge (on referential determinacy) gets a clear answer if placed in an appropriate (Carnapian) linguistic framework and is meaningless otherwise. To clarify this approach, we address Button-Walsh's dismissal of concepts-modelism by developing the notion of {\em cognitive modelism}, according to which classical mathematics is a complex process of constructing and developing a distinctive class of concepts. Second, we argue that the inferential $\omega$-logics, that are much weaker than second order logic, do not appeal to the arithmetical concepts that the categoricity theorems proved within these logics aim to secure.