Introduction
1. Motivation and Overview
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“She acknowledged it to be very fitting, that every little social commonwealth should dictate its own matters of discourse; and hoped, ere long, to become a not unworthy member of the one she was now transplanted into. With the prospect of spending at least two months at —, it was highly incumbent on her to clothe her imagination, her memory, and all her ideas in as much of — as possible.” Persuasion,
Chapter 6, (1818). Jane Austen According to the Scope & Purpose of the IEEE SUO project, “This standard will specify an upper ontology that will enable computers to utilize it for applications such as data interoperability, information search and retrieval, automated inferencing, and natural language processing.” Further, one of the three purposes of the project is “INTER-OPERABILITY: The standard will provide a basis for achieving Inter-Operability among various software and database applications.” The work presented here proposes a framework, the Information Flow Framework (IFF) to accomplish this goal of interoperability, as well as the goals related to automated reasoning and application areas. The IFF is manifested in the formalism of the IFF Foundation Ontology, which is designed to allow not only the interoperability among software and database applications, but also the semantic interoperability among various object-level ontologies themselves (Figure 1). As such, this work can be seen as complementary to the work being done in the IEEE SUO Working group on the SUMO. At this stage in the development of both the IFF and the SUMO, the major difference between the two works is that the SUMO effort is geared toward providing a single upper ontology, whereas the IFF is designed to support the semantic interoperability among various object-level ontologies. The IFF supports this interoperability by its architecture (Figure 1 and Section 2) and its use of a particular branch of mathematics known as category theory. A major reason that the IFF uses the architecture and formalisms that it does is to support modular ontology development. Modularity (constructing with standardized units or dimensions for flexibility and variety in use – Merriam-Webster) is essential (John Sowa) to good design decisions in every area of engineering, and most of all knowledge engineering and software engineering. In particular, modularity is important in the development, testing, maintenance, and use of ontologies. Development time will be shortened because separate groups would work on each module with little need for communication. When an ontology is maintained in modular fashion, errors and deficiencies can be traced to specific modules, thus limiting the scope of detailed error searching. With a modular design, it will be possible to make drastic changes to one module without the need to change other modules. Also with a modular design, by considering one module at a time the ontology as a whole will more easily be understood. With this understanding, the whole system can be better utilized. A monolithic ontology, on the other hand, would be very inefficient. In use, some parts would be very stable, well justified and shared across very broad communities, while other parts would not be adopted widely and would be quickly replaced (Austin Tate). A modular design will facilitate these consensual and evolutionary processes of renewal. The categorical approach of the IFF Foundation Ontology provides a framework for modular design via a structural metatheory of ontologies. Such a metatheory is a method for representing the structural relationships between ontologies. As will be quite evident in the IFF Model Theory Ontology, this structural metatheory is compatible with the representation of object-level ontologies in any suitable and appropriate ontology language. From the point of view (Robert Kent) that “every little social commonwealth should dictate its own matters of discourse,” would it not seem that the SUO should be structured in a modular fashion, as a loose confederation of participating ontologies, or perhaps better a league or commonwealth of ontologies? We now turn to the modular
architecture of the IFF Foundation Ontology. |
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“Philosophy cannot become scientifically
healthy without an immense technical vocabulary. We can hardly imagine our great-grandsons
turning over the leaves of this dictionary without amusement over the paucity
of words with which their grandsires attempted to handle metaphysics and
logic. Long before that day, it will have become indispensably requisite,
too, that each of these terms should be confined to a single meaning, which,
however broad, must be free from all vagueness. This will involve a
revolution in terminology; for in its present condition a philosophical
thought of any precision can seldom be expressed without lengthy
explanations.” Collected Papers 8:169. Charles Sanders Peirce |
2. Architecture
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The architecture of the IFF Foundation Ontology is based on the Information Flow Framework, which seeks to apply Information Flow to distributed logic, ontologies, and knowledge representation. The IFF takes a building-blocks approach towards the development of object-level ontological structure, This is a rather elaborate categorical approach that uses insights from the theory of distributed logic known as Information Flow (Barwise and Seligman 1997), and also uses ideas from the theory of Formal Concept Analysis initiated by Rudolf Wille (Ganter & Wille 1999). The IFF provides mechanisms for a principled foundation for a metalevel ontological framework – a framework for sharing ontologies, manipulating ontologies as objects, relating ontologies through morphisms, partitioning ontologies, composing ontologies via colimits, discussing ontological structure, noting dependencies between ontologies, declaring the use of other ontologies, etc. The IFF is primarily based upon the theory of Information Flow, which is centered on the notion of a classification. Information Flow is itself based upon the theory of the Chu construction of *-autonomous categories (Barr 1996), thus giving it a connection to concurrency and Linear Logic. The IFF is secondarily based upon the theory of Formal Concept Analysis, which is centered on the notion of a concept lattice. The IFF represents metalogic, and as such operates at the structural level of ontologies. In the IFF, there is a precise boundary between the metalevel and the object level. The architectural structure of the IFF Foundation Ontology is illustrated in Figure 1. Although the IFF Foundation Ontology is rather large, it is highly structured, representing the structural aspect of the SUO, its metalevel, in terms of meta-ontologies. The IFF Foundation Ontology is partitioned into three metalevels: top, upper and lower. These metalevels correspond to the set-theoretic distinction in foundations between (1) the generic, (2) the large and (3) the small (Mac Lane, 1971/1998).
From the set-theoretic perspective, the metalevel hierarchy of the IFF Foundation Ontology (Figure 2) corresponds to the hierarchy of (1) collections, (2) classes, and (3) sets. Table 1 locates particular kinds of collections and functions in the three-tiered framework of the IFF. In the IFF, all relations are strictly binary and all functions are strictly unary. Functions that are conceptually binary need to use pair notation for arguments. Table 1: Kinds of Specific Collections
Note, however, that the IFF Foundation Ontology is being developed from the perspective of category theory. From this perspective, the three IFF metalevels are distinguished by use. Each metalevel provides a language for the representation of the level below it – this is its raison d’etre. The top metalevel provides a language for representing the upper metalevel. The upper metalevel provides a language for representing and reasoning about the lower metalevel. The lower metalevel provides a language for representing and reasoning about the object level. Now, although one could apply the upper metalevel axiomatizations to object level information, a goal of the IFF is to apply these only to upper and lower metalevel information. If a similar axiomatization is needed for the object level, then such an axiomatization can be put in place at the metalevel. 1.
The top metalevel of the IFF Foundation Ontology is used to represent
and axiomatize the upper metalevel. The top metalevel does this by providing an
interface between the KIF logical language and the upper metalevel of the IFF
Foundation Ontology. The ability to express all of the mechanisms in the
upper metalevel needs a limited set of basic notions from mathematics:
general terminology for collections, relations, functions, limits, colimits,
and specific cases of these – restriction, abridgment, etc. ○
This ability is axiomatized in the IFF Basic KIF Ontology,
which contains rudimentary (fundamental) namespaces for collections,
functions, relations, limits and colimits. The IFF Basic KIF Ontology
provides an adequate foundation for representing ontologies in general and
for defining the other metalevel ontologies (Figure 1) in particular. All
upper metalevel ontologies import and use, either directly or indirectly, the
IFF Basic KIF Ontology. The IFF Basic KIF Ontology operates at the
set-theoretic level of generic collections, whereas Chris Menzel’s Basic Ontology
operates at the level of small sets (much of this will be subsumed by the IFF
Model Theory Ontology). 2.
The upper metalevel of the IFF Foundation Ontology is used to
represent and axiomatize the lower metalevel. The upper metalevel does
this by representing category theory, information flow and formal concept
analysis. The upper metalevel is partitioned into three sub-ontologies: the
Core Ontology, the Classification Ontology, and the Category Theory Ontology.
The upper metalevel is organized in order to be able to realize the
categorical principle (see below). In particular, the mechanism for
expressing composition and limits are axiomatized in the Core ontology, the
mechanisms for expressing categories, functors, natural transformations,
adjunctions are axiomatized in the Category Theory ontology, and the
mechanisms for combining ontologies in a lattice of theories is axiomatized
in the Classification ontology. ○
The IFF Core Ontology provides an anchor for the other two
ontologies at this level. It has a distinguished namespace for set-theoretic
classes and their functions. It also contains basic terminology and
axiomatics for large relations and endorelations. ○
The IFF Classification Ontology represents the theory of
distributed conceptual structures. This theory is concerned with the
distribution and conception of knowledge. It rests upon two related theories,
Information Flow and Formal Concept Analysis, which it seeks to unify. The
IFF Classification Ontology connects these two studies by representing the
basic theorem of Formal Concept Analysis as a categorical equivalence between
classifications and concept lattices at the level of functions, and the
categorical equivalence between bonds and complete adjoints at the level of
relations. With this, the IFF Classification Ontology accomplishes a
rapprochement between Information Flow and Formal Concept Analysis. ○
The IFF Category Theory Ontology represents in various
namespaces a baseline for the standard ideas of category theory. Much of the content
comes from well-known category-theoretic intuitions and some of the
semi-standard notation used in papers and textbooks. The IFF Category Theory
Ontology provides a framework for reasoning about metalogic, as represented
in the lower metalevel ontologies. Examples of provable statements are: “the
Classification namespace represents a category,” or “the IF classification
functor is left adjoint to the IF theory functor” (Kent 2000). The IFF
Category Theory Ontology encodes a KIF formalism for category theory
in one of its normal presentations. Other presentations, such as home-set or
arrows-only, may also have merit. The IFF Category Theory Ontology is based
upon the namespace for large graphs and graph morphisms that includes
horizontal multiplication of graphs and a theory of coherence – a category is
a monoid in the monoidal category of large graphs. 3.
The lower metalevel of the IFF Foundation Ontology is used to
represent and axiomatize the object metalevel. The lower metalevel is the
location for various modules that help represent the structural aspect of the
SUO using a categorical expression. That is to say, the lower metalevel
should follow as closely as possible the categorical principle: it should be
expressed at the level of classes and their functions, but using no
quantification, no logical connectives, and only the specific terminology
organized using composition, limits, and other categorical notions. This
should make inferencing highly efficient. The lower metalevel is used to the
build the objects and morphisms of ordinary categories such as Set, Classification, Hypergraph, … . It thus represents
these categories along with their associated functors, natural
transformations, and adjunctions. ○
The lower metalevel contains at present one module called the IFF
Model Theory Ontology consisting of about ten namespaces. This expression
of model theory, somewhat novel since it is based upon the theory of
Information Flow, is used to represent and axiomatize the object level. Use
of this ontology could be direct, but more likely will involve a
parse/translation from some “external” (to the IFF) ontological description
into the “internal” IFF semantic representation (see the expansion
of one of Adam Pease’s examples. ○
Other modules (namespaces or subontologies), in addition to the IFF
Model Theory Ontology, are perhaps possible at the lower metalevel. These
might include modules for categorical model theory, modules for modal, tense
and linear logic, modules for rough and fuzzy sets, modules for semiotics,
etc. 4.
The object level is where the content ontologies reside. These could
be very generic, such as a 4D ontology, or specific, such as ontologies for
government or higher education. It should be noted that the object level
satisfies a representation property similar to the categorical property
satisfied by the lower metalevel – the ontological language used is not KIF;
instead it is the terminology defined and axiomatized in the lower structural
level (terms such as ‘subtype’, ‘instance’, ‘expression’, ‘model’,
‘ontology’, ‘relation’, ‘entity’, ‘role’, etc.). The following principle is critically important in meta-ontology design. Categorical Principle: The design of any module
situated in the lower metalevel of the IFF Foundation Ontology should adhere
to the categorical property that its axiomatic representation is strictly
category-theoretic: ○
All axioms are expressed in terms of either terms from another lower
metalevel module or category-theoretic notions (upper metalevel notions),
such as classes, binary relations, order, the composition and identity class
functions, the pullback of diagrams of class functions, etc. ○
No axioms use explicit logical notation: No variables, quantification
(‘forall’, ‘exists’) or logical
connectives (‘and’, ‘or’, ‘not’, ‘=>’,‘<=>’) are used. ○
No axioms use terms from the IFF Basic KIF ontology, other than pair
bracketing ‘[-]’ or pair projection ‘(-
1)’, ‘(-
2)’. Here is some representative code in the function sub-namespace of the IFF Ontology of Small Sets (a part of the IFF Model Theory Ontology, which is under development). Notice how this code adheres to the Categorical Property. The lower metalevel terms being introduced in this fragment are: ‘function’, ‘source’, ‘target’ and ‘fn2rel’. The axiomatization uses both previously introduced lower-metalevel terms, and upper metalevel terms imported from the IFF Core Ontology. -
The term ‘set’ comes from the set namespace. -
The terms ‘rel$relation’, ‘rel$set1’ and ‘rel$set2’ come from the rel namespace. -
The terms ‘class’ and ‘subclass’ come from the SET namespace. -
The terms ‘function’, ‘source’, ‘target’, ‘restriction’, ‘composition’, ‘fn2rel’ and ‘injection’ come from the SET.FTN namespace. -
The term ‘tripling’ comes from the SET.LIM.PRD namespace. It uses no top metalevel terms, and it uses no logical variables, quantifiers or logical connectives.
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The
IFF Foundation Ontology consists of an adequate amount of set theory which,
on the one hand, is sufficiently flexible for the categorical inquiry
involved in the Information Flow Framework (IFF) but, on the other hand, is
sufficiently restrictive that IFF be consistent (does not produce
contradictions). The terminology of the IFF Foundation Ontology is
mathematical. As a result, mathematical aesthetics plays a large role in
development of the IFF Foundation Ontology, both as a stimulus for change and
as a guiding criterion in how to make those changes. Mathematical aesthetics
includes such things as simplicity, elegance and power of expression, etc.3. Language Levels, Role Types and Ontological Visibility
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At its core, the IFF Foundation Ontology is coded in the KIF logical language. As an ontological machine language, the Lisp-y style of logical expression of the KIF logical language is very useful, adequately expressive and conveniently terse. However, in the proposed IFF structure for the SUO the KIF logical language will not be used as the sine qua non for ontological expression. Instead, the IFF Foundation terminology is partitioned into language levels according to the following programming language analogy. 0.
The KIF logical language is like an ontological machine language. 1.
The IFF Basic KIF Ontology is like an ontological assembly
language. It provides a basic ontological apparatus for the SUO. The
purpose of this kind of terminology is to be an interface between the KIF
language and other ontological terminology, in general. Because of this
purpose, the IFF Basic KIF Ontology might also be called a bootstrap
ontology. 2.
The bulk of the IFF Foundation Ontology, as encoded in the of the
upper/lower metalevel namespaces, 3.
is like a high level programming language such as Lisp, Java,
ML, etc. The lower metalevel provides direct representation capabilities, and
the upper metalevel provides indirect structuring capabilities. 4.
The object level of the SUO is like various software applications,
such as word processors, browsers, spreadsheet software, databases, etc. The
object level provides the content of the SUO coming from the various domain
ontologies and philosophies with their own namespaces. The object-level ontologies
can be specified using the terminology axiomatized in the lower structural
level. A central goal of the IFF Foundation Ontology is to provide a principled mechanism for interoperability between the various modules at the object level. To support this goal, it is important that a complete distinction and an explicit boundary are kept between the object level and the metalevel. This fundamental partition must be obviously manifest in the SUO. Some ontologies at the object level may choose to represent and discuss metalevel concepts, but they would still be doing so at the object level. On the SUO email list and at the IJCAI-01 SUO Workshop, various comments and concerns have been expressed about intimidating users with category theory terminology. As John Sowa has remarked “the appropriate place for category theory is under the rug where no one will ever see it.” Since the IFF uses the categorical perspective, it makes heavy use of the various ideas and terminology of category theory. However, in the IFF Foundation Ontology this category theory terminology is invisible to the ordinary user – the user of object level ontologies. In the IFF approach the terminology that is visible to the ordinary user is that introduced in the object level ontologies, and the part of the IFF terminology visible to the object level content ontology developer is the terminology introduced in the lower metalevel. The latter is normal semantic terminology, but not category theory terminology. The IFF category theory terminology is introduced in the upper metalevel, and this is seen and used only at the lower metalevel, but not at the object level. This fact is represented by the dependencies in the IFF Architecture Diagram (Figure 1). In the above programming language analogy, the users of software applications should not see source code, but the programmers of those applications need to have it visible in front of them. By analogy, in the IFF approach, certain people working in certain roles will need to see and use category theory terminology, whereas others will want it to be invisible. This remains true, even when the lattice of theories is visible. These issues center on role types and ontological visibility of the IFF terminology and axiomatizations. As has been repeatedly stated, the IFF Foundation Ontology is partitioned into four levels: the object level, and the lower, upper and top metalevels. The visibility of these levels depends upon the role that people play with respect to the IFF. Two distinctions are important in a discussion of role types and visibility: ○ The contrast or
distinction between the designers of an ontology versus the users of that
ontology. Note that a designer must do some reasoning about the module being
developed. ○ The four levels of
terminology mentioned above. These two distinctions are combined into six
possible role types, of which we identify two pairs, ending up with four
possible distinct role types. The first two role types do not want to see
category theory terminology – they want it swept under the rug. However, the
second two role types need to use category theory terminology. |
Table 2: Role Types and Ontological Visibility
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4. The object level user A person in
this role (with respect to the IFF) sees only (4) object level terminology, that
is, they see only domain/middle/upper ontology terminology. Example
terminology: a. From a Movie domain
ontology: ‘movie’, ‘actor’, ‘rating’, ... b. From an upper object level
ontology: ‘3D’, ‘4D’, ‘object’, ‘event’, ‘continuant’, ‘occurrent’, ... |
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3. The object level module designer = the lower metalevel user A
person in this role (with respect to the IFF) sees both (4) object level and
(3) lower metalevel (for example, model theory, fuzzy/rough set, semiotic) terminology.
Example terminology: a. From the IFF Model Theory
Ontology: ‘instance’, ‘type’, ‘classification’, ‘subtype’, ‘disjoint’,
‘expression’, ‘arity’, ‘sentence’, ‘language’, ‘model’, ‘satisfies’,
‘1st-order interpretation’, ... b. From a fuzzy/rough set
module: ‘fuzzy-subset’, ‘upper-approximation’, ‘rough-set’, … |
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Here is the visibility
boundary for category theory terminology (John Sowa’s “rug”). |
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2. The lower metalevel module designer = the upper metalevel user A
person in this role (with respect to the IFF) sees both (3) lower metalevel
(model theory and other modules) and (2) upper metalevel (core,
classification and category theory) terminology. Example terminology: a. From the IFF Model Theory
Ontology: ‘instance’, ‘type’, ‘classification’, ‘expression’, ‘arity’,
‘sentence’, ‘language’, ‘model’, ‘satisfies’, ‘1st-order
interpretation’, ... b. From the IFF Core
Ontology: ‘class’, ‘function’, ‘source’, ‘target’, ‘relation’, ‘limit’,
‘colimit’, … c. From the IFF
Classification Ontology: ‘classification’, ‘concept-lattice’,
‘concept-morphism’, ... d. From the IFF Category
Theory Ontology: ‘category’, ‘functor’, ‘pullback’, ‘colimit’, ‘monad’, ... |
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1. The upper metalevel module designer A
person in this role (with respect to the IFF) sees both (2) the upper
metalevel (core, classification and category theory) and (1) the top
metalevel (KIF interface) terminology. This would include Robert Kent, as the
preliminary designer of the IFF upper metalevel and other category-theorists
that might help in the future. Example terminology: a. From the IFF Core
Ontology: ‘class’, ‘function’, ‘source’, ‘target’, ‘relation’, ‘limit’,
‘colimit’, … b. From the IFF
Classification Ontology: ‘classification’, ‘concept-lattice’,
‘concept-morphism’, ... c. From the IFF Category
Theory Ontology: ‘category’, ‘functor’, ‘pullback’, ‘colimit’, ‘monad’, ... From the IFF Basic KIF Ontology: ‘collection’, ‘function’, ‘relation’, ‘binary-product’, ‘tuple’, … |
4. Truth and Modularity
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The categorical approach of the IFF Foundation Ontology provides a framework for modular design via a structural metatheory of ontologies. Such a metatheory is a method for representing the structural relationships between ontologies. As will be quite evident in the IFF Model Theory Ontology, this structural metatheory is compatible with the representation of object-level ontologies in any suitable and appropriate ontology language. A categorical approach for modular design has previously been used in the Kestrel Institute’s SpecWare system. Specware uses the categorical notions of sheaves and colimits for realizing a modular approach to software specification. Colimits, which are mechanisms for the fusion (amalgamation) of diagrams within categories, can be constructed from primitives (indexed objects and morphisms in the diagram) using sums (coproducts) and/or quotients (coequalizers, pushouts). Colimits are already represented in the IFF Foundation Ontology: the Core Ontology has a representation for colimits in the quasicategory of set-theoretic classes and class functions; the Classification Ontology has a representation for colimits in the quasicategory of large classifications and infomorphisms, based upon the Core colimits; and the Category Theory Ontology has a representation for colimits in any large category. Extensions to the IFF Category Theory Ontology will be able represent systems such as Specware by adding a namespace for sheaves. A study by Michael Uschold has demonstrated the power of colimits as a composition mechanism, not only for software specifications but also for ontologies. The IFF Foundation Ontology advocates the use colimits for realizing a modular approach to ontology specification, not in terms of specifications and sheaves, but in terms of concept lattices and concept morphisms. John Sowa has proposed
the use of “an infinite lattice of all possible theories as the theoretical foundation
of the SUO.” He advocates that the SUO have as its goal “a framework that can
support an open-ended collection of ontological theories (potentially
infinite) organized in a lattice together with systematic metalevel
techniques for moving from one to another, for testing their adequacy for any
given problem, and for mixing, matching, combining, and transforming them to
whatever form is appropriate for whatever problem anyone is trying to solve.”
The IFF approach advocates that the IFF Foundation Ontology be used as this
framework, and proposes that Concept Lattice, the category of concept lattices, be used in the
SUO as a representation for John Sowa’s “lattice of all theories.” A very
detailed formalism for the category of concept lattices is incorporated in
the IFF Classification Ontology. The objects in this category are large
concept lattices, and each concept lattice has an underlying large
classification as its base and on which it is built. Large means that the
collections of instances, types and formal concepts are set-theoretic
classes, and not just small sets. Thus, this category can accommodate the
truth concept lattice for a 1st-order language, whose instance
collection is the class of all (model-theoretic) structures for that
language, and whose formal concepts have as their intents the (closed)
ontologies for that language. Amongst other things, the axiomatization of
large concept lattices in the IFF Classification Ontology answers Chris
Menzel’s foundational
concerns about a “reasonable class theory.” The morphisms in this
category are called concept morphisms. Each concept morphism has an
underlying infomorphism as its base and on which it is built. In addition,
each concept morphism has a pair of adjoint monotonic functions mapping
between the source and target concept lattices. Being adjoint, one function
preserves the joins and the other function preserves the meets of formal
concepts. In terms of the truth concept lattices, they preserve the joins and
meets of ontologies. The essential property of the category of concept
lattices (Kent, 2001) is that it is categorically equivalent to Classification, the underlying category
of classifications and infomorphisms, which is complete/cocomplete.
Therefore, by a basic result from category theory, the category of concept
lattices is also complete/cocomplete. This means that we can construct both
limits and colimits. However, the colimits are of most interest. A build process
that constructs colimits in the Concept Lattice quasicategory can be
described as follows. ○ [Base (start) step] Such a build process starts with the truth classifications (or their categorical equivalent, the truth concept lattices) and the interpretation infomorphisms between truth classifications associated with 1st-order interpretations (or their categorical equivalent, the interpretation concept morphisms). Other interesting infomorphisms between truth classifications may come from formalizing and axiomatizing the methodology that is used in various ontological merge operations. ○ [Recursion (iteration, induction) step] More interesting concept lattices are iteratively built by taking colimits (sums and/or quotients) of previously built large classifications (or their categorical equivalent, large concept lattices) and previously built large infomorphisms (or their categorical equivalent, large concept morphisms). Sums may involve taking the disjoint union of theories, whereas quotients may involve identifying certain pairs of sentences of theories and restricting the models accordingly. The notion of the truth classification of a 1st-order language is discussed as example 4.6 on page 71 of (Barwise and Seligman, 1997). Both this and the associated truth concept lattice, which is illustrated on slides 5 and 6 of the presentation, use the terminology of the IFF Model Theory Ontology and occur as examples within the IFF Classification Ontology. The truth classification of a 1st-order language L is the meta-classification, whose instances are L-structures (models), whose types are L-sentences, and whose classification relation is satisfaction. In the IFF approach, the concept lattice of the truth meta-classification functions is an appropriate base “lattice of ontological theories.” A formal concept in this lattice has an intent that is a closed theory (set of sentences) and an extent that is the collection of all models for that theory. The theory (intent) of the join or supremum of two concepts is the intersection of the theories (conceptual intents), and the theory (intent) of the meet or infimum of two concepts is the theory of the common models or the closure of the union of the theories (conceptual intents). It is important to note that this truth framework is a fibered structure. Which means that this is not a single lattice of theories but an infinite collection of such lattices, where each truth classification truth-classification(L) and each truth concept lattice truth-concept-lattice(L) is based upon (indexed by, parameterized by) a particular 1st-order language L. So a particular object level ontology – be it a domain ontology, a middle level ontology, or an upper level ontology – will be an element of a lattice based on its 1st-order language. Internally, the various ontologies (theories) are connected by generalization/specialization and sideways jumping. Externally, the lattices in the IFF framework can be connected by concept morphisms in numerous ways: by 1st-order interpretations, by colimit injections (for example, the FCA notions of apposition and subposition), or by concept morphisms defined by ontology merging, etc. The colimit operation can be viewed as an internalization operator, taking an (external) indexed collection of concept lattices and concept morphisms, and producing a single (internal) fused concept lattice. The notion of a 1st-order interpretation is discussed as example 4.11 on page 74 of (Barwise and Seligman [op. cit.]). 1st-order interpretations are axiomatized in the IFF Model Theory Ontology. A 1st-order interpretation I from one 1st-order language L1 to a second 1st-order language L |