Mathematical Connections 

The Model Theory of Information FlowThe basic element of information flow is the classification. Classifications and their morphisms comprise the mathematical context called Classification. A theory is a type set and a set of sequents known as constraints. Theories and their morphisms comprise the mathematical context called Theory. A classification and a theory with shared type set form a logic. Logics and their morphisms comprise the mathematical context called Logic. The mathematical contexts Classification, Theory and Logic are categories. These contexts are connected by a pair of inverse passages: every classification A has an associated theory Th(A), namely all constraints satisfied by all instances; every theory T has an associated classification Cla(T) consisting of all formal instances (binary partitions of types). The invertible pair of passages Th and Cla is an adjunction. This adjunction lifts by factorization to an a set of inverse passages between Logic and Theory. 

Log 
Cla 
Th 
th 
Log 
Logic 
Classification 
Theory 
The basic element of onto logic is the model. Models and their morphisms comprise the mathematical context called Model. The type aspect of models is a hypergraph known as an ontological language. Languages and their morphisms comprise the mathematical context called Language. The important truth (meta)classification has models as instances, expressions as types and satisfaction as incidence relation. An ontology is just a set of expressions. Models and ontologies are connected by the derivation operators in the truth concept lattice. Language morphisms defined ontology morphisms. Ontologies and their morphisms comprise the mathematical context called Ontology. The mathematical contexts Language, Model and Ontology are categories. Model and Ontology are both indexed by Language – each model has an underlying type hypergraph and each ontology (ontological theory) is defined over a component language. An ontological language L determines a category of models Model(L). Satisfaction is the classification relation in the truth classification between Lmodels and Lexpressions.
Each categorical object – language, model or ontology – is explicitly represented in the IFF language.
A ontological language L is a hypergraph; that is, a triple L = áent(L), rel(L), ¶ñ consisting of
o a set of entity types ent(L),
o a set of relation types rel(L), and
o a signature map ¶ : rel(L) ® tuple(ent(L)).
A ontological model A = áent(A), rel(A), ¶ñ is either a hypergraph of classifications or a classification of hypergraphs. It is a twodimensional structure consisting of
o a hypergraph of instances inst(A),
o a hypergraph of types typ(A),
o an entity classification ent(A),
o a relation classification rel(A), and
o a tuple projection ¶ : rel(A) ⇉ tuple(ent(L)).
An ontology O Í expr(L) is a set of expressions of a language L. The expressions in O are called constraints. Two subsets of constraints are distinguished, forming entity and relation theories.
Onto Logic 
Information Flow 

Language 
Type set 

Model

Classification 
Logic 
Ontology 
Theory 

Ontologic Interpretation 
Terminology 
Please send questions, comments and suggestions
about this page to: Robert E. Kent rekent@ontologos.org
Copyright © 2000 TOC (The Ontology
Consortium). All rights reserved. Revised: July 2000