The ERA of FOLE: Superstructure

This paper discusses the representation of ontologies in the first-order logical environment {\ttfamily FOLE}. An ontology defines the primitives with which to model the knowledge resources for a community of discourse. These primitives consist of classes, relationships and properties. An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. This paper continues the discussion of the representation and interpretation of ontologies in the first-order logical environment {\ttfamily FOLE}. The formalism and semantics of (many-sorted) first-order logic can be developed in both a \emph{classification form} and an \emph{interpretation form}. Two papers, ``The {\ttfamily ERA} of {\ttfamily FOLE}: Foundation'', defining the concept of a structure, and the current paper, defining the concept of a sound logic, represent the \emph{classification form}, corresponding to ideas discussed in the ``Information Flow Framework''. Two papers, ``The {\ttfamily FOLE} Table'', defining the concept of a relational table, and ``The {\ttfamily FOLE} Database'', defining the concept of a relational database, represent the \emph{interpretation form}, expanding on material found in the paper ``Database Semantics''. Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form incorporates the relational data model of Codd. A fifth paper ``{\ttfamily FOLE} Equivalence'' proves that the classification form is equivalent to the interpretation form. In general, the {\ttfamily FOLE} representation uses a conceptual structures approach, that is completely compatible with the theory of institutions, formal concept analysis and information flow.