Categoricity for an inferential $\omega$-logic and in $L_{\omega_1,\omega}$

TL;DR

This paper extends first-order logic with $$-rules, characterizing structures where theories are categorical, and shows how certain logics define the same classes of structures as first-order theories with specific rules.

Contribution

It introduces two extensions of first-order logic with $$-rules, demonstrating their categoricity properties and equivalences to first-order theories with $G-$-rules.

Findings

Robinson's system Q and Peano Arithmetic become categorical in the one-sorted inferential $$-logic.

Each complete $L_{,}$ sentence defines the same class of structures as a first-order theory with a $G-$-rule.

The inferential rules for these logics are shown to be categorical, determining truth-conditions uniquely.

Abstract

This paper provides two extensions of first order logic by `$\omega$-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 $\omega$-logic, both Robinson's system $Q$ and Peano Arithmetic become categorical. In the two-sorted generalized $\omega$-logic we show each complete $L_{\omega_1,\omega}$ sentence defines the same class of structures as a first-order theory with the appropriate $G-\omega$-rule. 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.