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Categoricity for inferential $ω$-logic and $L_{ω_1,ω}$

John T. Baldwin, Constantin C. Brîncuş

TL;DR

The paper develops two ω-rule–based extensions of logic to achieve categoricity without relying on full second-order commitments. It introduces the inferential $I-\omega$-rule, proving categoricity for $PA^{\omega}$ and $Q^{\omega}$, and a two-sorted $G-\omega$-logic that ties complete $L_{\omega_1,\omega}$ sentences to pseudo-elementary theories, establishing structural equivalence and corollaries about non-generative models. Philosophically, it confronts the doxological challenge and circularity through a cognitive-modeling framework that emphasizes concept formation and framework distinctions. The results illuminate how weaker, inferentially defined logics can determine truth-conditions and structure while avoiding onerous arithmetical commitments, suggesting a unified view bridging model theory and inferentialism with broad foundations implications.

Abstract

This paper provides two extensions of first order logic by `$ω$-rules'. In each case we characterize the countable structures whose theory in the logic is categorical (has a unique model). In the one-sorted inferential $ω$-logic, both Robinson's system $Q$ and Peano Arithmetic become categorical. In the two-sorted generalized $ω$-logic we show each complete $L_{ω_1,ω}$ sentence defines the same class of structures as a first-order theory with the appropriate $G-ω$-rule. These logics are much weaker than second order logic and we argue that they do not appeal to the arithmetical concepts that the categoricity theorems themselves aim to secure. The results depend on proving that the inferential rules for the logics are categorical, i.e. they uniquely determine certain truth-conditions for the logical connectives and quantifiers. We provide an extensive answer to the doxological challenge (on referential determinacy) proposed in \cite{ButtonWalshbook} and we develop a philosophical view of mathematics -which we call {\em cognitive modelism}- according to which classical mathematics is best understood as a complex process of constructing and developing a distinctive class of concepts, rather than merely describing a fixed pre-existing realm of structures. KEYWORDS: categoricity, inferentialism, first-order logic, first-order theories, $ω$-rules, $L_{ω_1,ω}$.

Categoricity for inferential $ω$-logic and $L_{ω_1,ω}$

TL;DR

The paper develops two ω-rule–based extensions of logic to achieve categoricity without relying on full second-order commitments. It introduces the inferential -rule, proving categoricity for and , and a two-sorted -logic that ties complete sentences to pseudo-elementary theories, establishing structural equivalence and corollaries about non-generative models. Philosophically, it confronts the doxological challenge and circularity through a cognitive-modeling framework that emphasizes concept formation and framework distinctions. The results illuminate how weaker, inferentially defined logics can determine truth-conditions and structure while avoiding onerous arithmetical commitments, suggesting a unified view bridging model theory and inferentialism with broad foundations implications.

Abstract

This paper provides two extensions of first order logic by `-rules'. In each case we characterize the countable structures whose theory in the logic is categorical (has a unique model). In the one-sorted inferential -logic, both Robinson's system and Peano Arithmetic become categorical. In the two-sorted generalized -logic we show each complete sentence defines the same class of structures as a first-order theory with the appropriate -rule. These logics are much weaker than second order logic and we argue that they do not appeal to the arithmetical concepts that the categoricity theorems themselves aim to secure. The results depend on proving that the inferential rules for the logics are categorical, i.e. they uniquely determine certain truth-conditions for the logical connectives and quantifiers. We provide an extensive answer to the doxological challenge (on referential determinacy) proposed in \cite{ButtonWalshbook} and we develop a philosophical view of mathematics -which we call {\em cognitive modelism}- according to which classical mathematics is best understood as a complex process of constructing and developing a distinctive class of concepts, rather than merely describing a fixed pre-existing realm of structures. KEYWORDS: categoricity, inferentialism, first-order logic, first-order theories, -rules, .
Paper Structure (13 sections, 10 theorems, 22 equations)

This paper contains 13 sections, 10 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Rules, Valuations and Logical Inferentialism
  3. Interludio: Tarski and Carnap on the $\omega$-rule
  4. Tarski on the $\omega$-rule and Logical Consequence
  5. Carnap's Use of the $\omega$-rule in Substitutional Settings
  6. The Inferential $\omega$-rule and the Categoricity of $PA^{\omega}$
  7. Categorical structures in $L_{\omega_1 \omega}$ with an Inferential $G-\omega$-rule
  8. Structures, Models, and the Modelists Doxological Challenge
  9. The Modelists and the Doxological Challenge
  10. The Doxological Challenge and Carnap's Linguistic Frameworks
  11. Cognitive Modelism
  12. Against the Circularity Argument
  13. Conclusion.

Key Result

Lemma 2.3

Each permissible valuation $v$ gives a $\tau$-structure for the vocabulary $\tau$ whose symbols appear in the domain of $v$. All permissible valuations on a $\tau$-structure $M$ give the same values to the quantifier free sentences of $\tau$.

Theorems & Definitions (54)

  • Definition 2.1: Vocabularies and structures
  • Definition 2.2: Permissible Valuations
  • Lemma 2.3
  • Remark 2.5: Scholia on Definition \ref{['langdef']}.2
  • Definition 2.6: Varieties of Constants
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 44 more