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

TL;DR

This paper explores Carnapian frameworks and inferential $\omega$-logics to establish categoricity of arithmetic, addressing philosophical issues about mathematical reference and concept development.

Contribution

It introduces inferential $\omega$-logics that are categorical and interprets $L_{\omega_1,\omega}$ in a novel logical framework, clarifying philosophical debates.

Findings

Arithmetic has a unique countable model in the proposed logics.

Inferential $\omega$-logics are weaker than second order logic but still achieve categoricity.

A philosophical framework clarifies the doxological challenge and concept-modelism in mathematics.

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.