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

Abstract

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.

This is the second half of the split article (arXiv:2602.02854v1). The first half is now the updated version of the initial full article. The first paper (Categoricity for an inferential $\omega$-logic and in $L_{{\omega }_1,\omega }$) contains the technical results while the present one provides a philosophical discussion of these results