The Ontological Square is a four-categorial scheme that is obtained by crossing two formal distinctions which underpin conceptual modelling languages and top-level ontologies alike: that between types (or universals) and tokens (or particulars) on the one hand, and that between characters (or features) and their bearers (or substrates) on the other hand. Thus the Ontological Square consists of particular substrates, called substances, and universal substrates, called kinds, as well as particular characters, called modes or moments, and universal characters, called attributes. In this article, I try to elucidate the basic ontological assumptions underlying this four-category scheme and I propose a calculus of many-sorted second-order logic that is meant to capture these intuitions by an enrichment of standard atomic logical form. A first-order semantics can be designed for such a Logic of the Ontological Square, with respect to which the latter's soundness can be established.