Carnapian Frameworks and Categoricity of Arithmetic via Inferential $ω$-logics
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918470728286208 |
|---|---|
| author | Baldwin, John T. Brîncuş, Constantin C. |
| author_facet | Baldwin, John T. Brîncuş, Constantin C. |
| contents | We provided in \cite{BaldwinBrincusI} extensions of first order logic by modified inferential definitions of the classical $ω$-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 $L_{ω_1,ω}$. 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 $ω$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Carnapian Frameworks and Categoricity of Arithmetic via Inferential $ω$-logics Baldwin, John T. Brîncuş, Constantin C. Logic 03A05, 03C75, 03B10, 03B72 We provided in \cite{BaldwinBrincusI} extensions of first order logic by modified inferential definitions of the classical $ω$-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 $L_{ω_1,ω}$. 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 $ω$-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. |
| title | Carnapian Frameworks and Categoricity of Arithmetic via Inferential $ω$-logics |
| topic | Logic 03A05, 03C75, 03B10, 03B72 |
| url | https://arxiv.org/abs/2604.24943 |