Mathematical Connections

 

The Model Theory of Information Flow

The 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 Model Theory of Onto Logic

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 L-models and L-expressions.


IFF Language Components

Each categorical object language, model or ontology is explicitly represented in the IFF language.

Language

A ontological language L is a hypergraph; that is, a triple L = ent(L), rel(L),  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)).

Text Box: Language
o	Entity type declarations
o	Relation type declarations, with attached signature

 


Model

A ontological model A = ent(A), rel(A),  is either a hypergraph of classifications or a classification of hypergraphs. It is a two-dimensional 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)).


 


Ontology

An ontology O 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.


 

 

 


Connecting Onto Logic with Information Flow

 

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