The notion of normality has gained significant attention in recent works in epistemology. Goodman and Salow (Philosophical Studies, 175, 183–196. 2018, 2021, Philosophical Review, 132(1), 89–145. 2023) introduced an abstract class of structures, namely normality structures, and a framework based on them that can support a diverse range of normality-based epistemic theories. In this work, we provide a formal study of the normality structures introduced by Goodman and Salow (2021). There are two general motivations for such a study: first, to explore the structures as mathematical entities, and second, to study them as semantic structures appropriate for reasoning about epistemic and doxastic concepts. To pursue these aims, we propose two distinct languages: a modal language and a conditional one. We then develop the logic of normality structures in each language and establish their soundness and completeness with respect to normality structures. Within the second language we propose, not only can we define certain epistemic modalities that satisfy the desired conditions, but we can also potentially provide the logic for a more generalized class of frames, suitable for a more abstract notion of normality that relaxes some of the more controversial conditions imposed by Goodman and Salow. Finally, we briefly demonstrate how these formal tools can contribute to ongoing debates in formal epistemology.
