This topic refers to the course Model-theoretic inferentialism and categoricity (Brîncus). Here is a description and a bibliography for addressing it.
J. Hintikka, "Is there completeness in mathematics after Gödel?", in Philosophical topics 17:2, 69-90, pp. 69-73, 77-79 [Hintikka distinguishes between categoricity, deductive and semantic completeness, and argues, among other things, that using a semantically incomplete logical system may be more useful for the aims of the working mathematician. Someone may present Hintikka's view from pp. 77-79 and try to give an inferentialist answer to it]
J. Garson, What logic means: from proof theory to model-theoretic semantics, Cambridge University Press, 2013 [pp. 1-24] [This is a technical book on model-theoretic semantics inferentialism. Garson introduces three kinds of models for reading off the meanings of the logical terms from the rules of inference (deductive, local and global). Someone may present the way in which the natural deduction rules fix the classical meanings of the logical terms if we use local models (pp. 38-89). The pages 1-24 are excellent for an introduction to logical inferentialism]
J. Warren, Shadows of syntax. Revitalizing logical and mathematical conventionalism, Oxford University Press, 2020 [pp. 78-86, 261-271] [The book argues tha tlogical inferentialism provides us with an explanation of the fact that logical and arithmetical truths are by-products of the syntax of language. Someone may present Warren's argument for accepting the ω-rule (pp. 263-270). Likewise, someone may discuss Warren's argument that open-ended ω-rule provides us with categoricity (pp. 270-271)]
Additional bibliography
R. Carnap, Formalization of logic, Harvard University Press, 1943 [pp. 3-6, 94-96, 135-147] [This is a technical book on the non-normal interpretations of propositional and first-order logic. The selected pages, however, offer a general understanding of Carnap's project of attaining a full formalization, i.e., a categorical formalization, of classical logic]
J. Murzi & B. Topey, "Categoricity by convention", in Philosophical studies 178, pp. 3391-3420 [pp. 3392-3397] [This is an article on Carnap's categoricity problem. The authors provide a naturalist inferentialist solution to this problem. The paper may be read for a better understanding of (Carnap 1943) and (Garson 2013)]