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

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. 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.