The syntax of many-valued relations

Abstract

In this paper we show how many-valued relations syntactically can be formulated using powertype constructors. This in turn enables to describe the syntax of generalized relations in the starting point sense where the category sets and relations is isomorphic to the Kleisli category of the powerset monad over the category of sets. We can then generalize to work over monoidal closed categories, and thereby description logic, formal concepts and rough sets can be viewed as depending on that powertype constructor, and within a setting of many-valued λ-calculus. In order to achieve this, we will adopt a threelevel arrangement of signatures [4], and demonstrate the benefits of using it. Bivalent and untyped relational adaptations typically appear in terminology and ontology, and we will illuminate this situation concerning classifications in health. Extensions to multivalent and typed nomenclatures provides an enrichment that is beneficial in practical use of health classifications and nomenclatures.