Categorization and association are fundamental for our understanding of the world, for which reason notions such as category, individual and relation are of major importance, for applied and formal ontology as well as for conceptual modeling. Several of the available foundational ontologies consider those notions at the meta level (only) and focus on individuals as their overall domain. This notwithstanding, we see a need for establishing well-founded ontological theories of categories and relations within the field of applied ontology, not at least as a foundation for conceptual modeling. In this paper we report on our work on an ontology of categories and relations (and cognate notions), abbreviated CR, which is intended to serve as a module for foundational ontologies. The approach has its background in the development of the General Formal Ontology (GFO) as well as in ontological semantics, a novel account of establishing semantics for representation or modeling languages on the basis of ontologies. The conceptual foundations of the ontology are introduced and we present initial axiomatization efforts.
Ontologies continue to be on the advance in many areas. Creating ontologies means to
analyze a certain domain of interest in terms of the conceptual constituents by means of
which the ontology is going to be formed. The major basic constituents of most
ontologies today are categories and relations.3 This is not surprising, as categorization /
classification4 and association are fundamental for our understanding of the world.
Accor1Correspondence Address: Augustusplatz 10, 04109 Leipzig, Germany
Email: frank.loebe@informatik.uni-leipzig.de
2Parts of this submission are based on sections of [
4We use the word ‘categorization’ as a synonym of ‘classification’ herein. However, we employ only ‘category’ for a broadly conceived notion of instantiable entities. ‘Class’ is reserved for several notions that dingly, they rank equally important in formal and applied ontology. We see it as a (if not the) central task of ontological analysis of any subject to address the questions of “Which (categories of) entities are there?” and “How are those entities interrelated?”.
Basically the same questions – we claim – are likewise present, at least implicitly,
in almost all modeling enterprises. They are thus of similarly central nature for
conceptual modeling, in particular, and for modeling in computer and information sciences in
general. There the notions of category and relation frequently appear together among
the very basic concepts of modeling approaches – of course, with varying terminology.
For example, John Mylopoulos [
Categories and relations are further two highly generic / abstract notions, which leads
us to top-level ontologies (TLOs) as well as to meta ontologies. TLOs are ontologies
concerned with notions of very high generality, typically aiming at entities that appear
in a large number of domain-specific areas. A meta ontology should provide notions that
serve to characterize the constituents of an ontology and their interplay, which frequently
also relates to language constructs that are employed in representing ontologies. The
preceding considerations that categories and relations account for basic constituents of
ontologies may thus suggest a meta-ontological nature of themselves, cf. also [1, sect.
2.3.3] and [
The overall goal of our work is to establish an ontology of categories and relations,
which involves further cognate notions resulting from the analysis. There is a direct
connection to the field of top-level ontologies, on the one hand, and to meta-ontological and
semantic considerations, on the other hand. First of all, the project has emerged in the
much wider context of constructing the General Formal Ontology (GFO) [
5These two words are used interchangeably herein, albeit in-depth philosophical analyses may allow for distinguishing differing understandings. The term logical individual, however, leads to formal logic and Tarskian model theory, where it refers to a member of the universe of a formal model / interpretation structure.
6Surely, limitations based on the current state of science and knowledge apply.
nation with further TLOs and (2) as an ontological foundation for, primarily, meta
modeling. The latter point has yet another intimate connection, namely to a novel semantic
account that we propose in [
In this paper we report on preliminary work in progress towards an ontology of categories and relations, named / abbreviated CR. The conceptual foundations of the ontology are elaborated in sect. 2. Sect. 3 first summarizes a partial formalization which utilizes description logic / OWL2, which covers primarily a taxonomic fragment of the ontology. Secondly, a short preview on an axiomatization in first-order logic (FOL) is included. A discussion, also comprising some pointers to related work, is given in sect. 4 before the paper concludes with sect. 5.
Before expounding the view of categories and relations as it constitutes CR, let us highlight some aspects that have an impact on developing the ontology. First of all, for both notions there is a large amount of material available in philosophy, cognitive science, computer and information sciences as well as in further areas that aims at defining or characterizing these notions. The amount becomes vast if we broaden the scope to literature which uses one or both notions or which is otherwise closely linked to them or just to aspects that one could consider in order to develop an ontology of categories and relations. Surveying even a fragment of this body of work deserves its own publication and cannot be accomplished herein. Therefore we present the subsequent theory of categories and relations without delivering a prior state of the art, while we shall mention connections to a very focused set of approaches in sect. 4.
That variety of literature comes along with an actual variety of differing views and
understandings of categories and relations, e.g., cf. the analysis in [
Another guiding aspect is the desideratum of establishing a theory capable of
selfreflection / self-analysis. More precisely, categorizing and interrelating the constituents
of CR should not require additional notions that are not already covered by CR. That
desideratum results from the more influential aim to eventually utilize the ontology in
7It is not in any form closer related to the Ontological Semantics by Sergei Nirenburg, Victor Raskin and
others [
As a final note, we admit that the phrase “ontology of categories and relations” may cause expectations to find an in-depth classification or taxonomy of kinds of categories, relations or further constituents of the theory. To counteract such expectation, let us state already that this is not the case, although occasionally we draw inspiration from having special cases in mind. But first of all, herein we are interested in the notions (which we deem relevant) in themselves and an axiomatic characterization of them. A limited taxonomy emerges on that basis nevertheless, yet substantial extensions, e.g. analyzing different kinds of categories, remain future tasks.
In the domain “surrounding” the term ‘category’, we first identify four principal notions, namely entity, category, individual, and instantiation, depicted together with major interdependencies in Fig. 1.9 As a kind of minimal consensus for categories we merely require that they are subject to being instantiated10, hence to the instantiation / categorization relation in a specific way. This relation, in Fig. 1 shown as the UML association named by the double colon :: (as in the FOL formalization of CR), links entities to the categories that they instantiate. In this connection an entity is called an instance11 of a category. Remembering the intent to constrain categories as weakly as possible, no assumptions are made regarding the degree of generality of categories. Insofar they can be very specific, e.g., (for another ontology than CR) as determined by the phrase “female lion that lived in South Africa between 1994 and 1999”, or they can be highly abstract categories as in top-level ontologies.
Indeed, entity is the category that is instantiated by anything as soon as that
anything exists. Existence is thereby understood in the broadest conceivable sense, following
GFO’s recourse to, a.o., Roman Ingarden and different modes of existence [
8Alternatively, [
9We employ class diagrams of the Unified Modeling Language version 2 (UML2) [
10Note here already that “being subject to” does neither imply that every category is instantiated, nor even that it can possibly be instantiated, because neither non-instantiated categories are to be excluded, nor such that cannot be instantiated at all, like ‘round square’.
11The UML rolenames (following the terminology as in [18, p. 576 ff.], although ‘association end name’ is official for UML2) can better or only be explained after some parts of sect. 2.3. Besides the mnemonic parts of the role names, it may suffice here that the suffix ‘RL’ stands for ‘role’ itself and the difference between instance as just introduced and the instance role becomes clear in the section below. {disjoint, complete}
Thus categories cannot only be instantiated, but they are instances themselves. Not at least, any category instantiates the category Category (indicating a technical CR name by this font), which may be in conflict with one or another TLO. In line with many TLOs, however, CR includes (also) individuals / particulars, i.e., entities that are instances of categories, but are not subject to being instantiated themselves. Entity is partitioned into categories and individuals, which entails that every entity instantiates either Category or Individual. In contrast, it is not enforced that categories must have any instances.
Considering instantiation anew, the above paragraphs imply that CR yields no
stratified nor well-founded account of instantiation. Without further restrictions, there may
be instantiation chains of arbitrary length, as well as instantiation cycles. For example,
we subscribe to the facts that Entity instantiates Entity (cycle of length 1), and that Entity
instantiates Category and Category instantiates Entity (length 2). This is not too
uncommon in applied ontology and possibly even more accepted in knowledge representation
and the Semantic Web, cf. e.g. the notion of Resource in the Resource Description
Framework (RDF) [2, ch. 2–3]. Without room for going into details (cf. [1, sect. 2.4.2.3]
for a bit more), this affects relationships between CR and set theory, for instance, if the
notion of the extension of a category is considered, defined as the set of all and exactly
the instances of this category. Then, relying on a wellfounded set theory, not every
category may be equipped with a proper extension, or one adopts an alternative set-theoretic
approach, such as non-wellfounded set theory [
A related and here final aspect is that of comprehension axioms for categories, which
may easily lead to inconsistency of the theory, e.g., in analogy to Russell’s paradox [
As already indicated in sect. 2.1, the part of CR on relations is more specifically determined than the deliberately weak theory of categories. The basic view on relations combines a positionalist account12 with one of individualized properties. More precisely, relations in CR are categories which are instantiated by individuals, called relators, that are composed of roles and that have the power of relating / mediating between / gluing together arbitrary entities. Relators establish the relation that they instantiate between the entities / arguments / relata that they connect. At the categorial level, one may consider different kinds of relator categories, but relations are special insofar that they capture the connecting character of their instances among the entities related.
12See [22, esp. sect. 3.2] for a condensed exposition of that stance within conceptual modeling, with pointers to treatments in philosophy.
RelatorC
RoleC Relation 1..*
baseEIsA
To discuss a concrete example and at the same time to demonstrate a case that is commonly seen as a formal relation (which holds immediately as soon as the relata are given), we consider in the domain of natural numbers that 1 is less than 2, commonly written 1 < 2. Analyzing this fact in CR, we can first distinguish four entities, namely the numbers 1 and 2, the less-than relation <, and a particular relator r< instantiating < and mediating particularly between 1 and 2. With just that relator available, it is not clear in which – potentially distinct – ways 1 and 2 participate in the relation.
This is accounted for by roles (and so-called base role categories at the level of categories / relations), of which there is one per argument. Roles are individuals that can be seen as parts of relators, although within CR we name that relation role of, symbolized by ⊸ in the FOL formalization, or has role, denoted by ⟜, in inverse reading.13 In the example, r< is composed of two roles, say, qg and ql. However, (grasping) the “semantics” / “nature” of the two regarding < relies on specific role categories that they instantiate. Indeed, the overall semantics of < depends on determining role categories such that relators composed of roles of those categories adequately reflect in which way each single argument participates in the relator (and thereby / loosely speaking, “in the relation”). We say in such cases that a role category is a base role category for a relation. For < let us name its base role categories less and greater.
Another connection is required to / from a role w.r.t. the corresponding argument of the relator, e.g., from ql to 1. This connection is called plays / plays role / fills role, with the FOL predicate symbol ↝ (or ↜, if read reversely). To conclude our example coherently, we need 1 ↝ ql and 2 ↝ qg . Overall, the role analysis of 1 < 2 yields the subsequent formula, where all # denote constant symbols with the same intended meaning of categories or relations as the predicate symbols without the prescript, thus reifying the latter (cf. also sect. 3).
r< :: #< ^ ql :: #less ^ qg :: #greater ^ 1 ↝ ql ^ ql ⊸ r < ^ r< ⟜ qg ^ qg ↜ 2
Meanwhile, not only are the three fundamental relations for CR introduced, namely instantiation, role-playing and role-having, but the bottom layer of Fig. 2 is already cove13Importantly, this inversion is merely one of expressing the relation in a linear language, while ‘role of’ and ‘has role’ refer to the very same relation, from a positionalist stance. Cf. also [23, esp. sect. 4.1] on this aspect. red by the preceding paragraphs. Even base role categories are mentioned above, yet they deserve further consideration. They capture the different ways / kinds in which entities may be involved in the specific relation, contributing to the comprehensibility of a relation. Their name originates from the notion of a role base for a relation, which comprises not only the base role categories, but possibly further constraints, cf. [1, sect. 2.4.3.3].
Here we restrict ourselves to considering the additional categories BasePlayerC and BasePlayerAbleC in Fig. 2, which should be less common (even under different names), but are likewise informative w.r.t. relations as well as useful in expressing certain connections. All three kinds of categories result from a trichotomy derivable from understanding role notions. Take ‘instance’ as an example, already introduced above, and observe that the role name near Entity in Fig. 1 is instanceRL, where RL abbreviates ‘role’. First, ‘instance’ may be said to be a role / base role category itself (and thus an appropriate UML rolename). If CR is applied to instantiation itself, indeed, there is an instance role within any instantiation relator, which is played by the entity that instantiates something in virtue of that relator. Secondly, in r< :: #< we would say that r< is the instance. From this point of view ‘instance’ is rather a category of (primarily) non-role entities that actually stand in the instantiation relation (to other entities). This categorization is one from the point of view of the relation. For that reason we use instanceRL for the role category, reserving ‘instance’ for the players of that role / base role category. Thirdly and associated with common domain and range specifications for relations, there is the question of which entities are at all subject to playing a role of a relation. That question can typically be given an answer from the point of view of the relation or the role. For example, all entities that can be instances are ‘categorizable’. Everything that can be instantiated is ‘instantiable’. These are cases of BasePlayerAbleC. Note that those are always roleor relation-based categories, in contrast to categories of role- and relation-independent entities. Claiming that Instantiable is equivalent to Category then yields that all and only categories are subject to being Instantiated, i.e., to playing the InstantiatedRL. Table 1 accounts for key components of the role bases of the three fundamental CR relations.
Relation
BaseRoleC
BasePlayerC
BasePlayerAbleC
Category 3. Towards formalizing the theory CR Our primary aim in formalizing CR is to develop its axiomatization in first-order logic (FOL). Beforehand, we started with a partial axiomatization in description logic (DL). For the purposes of this paper, all formalizations are solely considered under standard set-theoretic semantics.
The informal exposition of CR involves a number of14 categories in intertwined ways.
Therefore we focused on the taxonomy first, limited to DL and by utilizing Protégé [
The corresponding theory in DL is presented in [1, App. B, p. 281–283], such that
we do not include specific DL axioms here. Moreover, this version of the ontology is
available as an OWL2 file, see [
Still, some observations about this version of CR appear to be of interest. First of all, the reasoner succeeded in classifying the ontology file without reporting inconsistency of the overall file, and without detecting incoherent, i.e., unsatisfiable DL concepts. A larger fragment of the resulting classified taxonomy of CR is depicted in Fig. 3. It still includes some auxiliary notions that turned out useful in the course of the axiomatization. In particular, we expect that the categories Relating (orange; the union of Role and Relator) and NonRelating (blue; the complement of Relating relative to Entity / OWL Thing) are widely usable, e.g., as NonRelating is silently assumed in many cases of referring to Individual. Two major partitionings of Entity are (1) that into categories and individuals as in Fig. 1 and (2) that into non-relating entities, roles and relators.
14Indeed, not all categories discussed in [1, sect. 2.4 and 6.1] or contained in the OWL file cr-dl.owl
[
Entity Individual
Relating Role
Relator NonRelator NonRole InstanceRL InstantiatedRL
PlayerRL PlayedRL RoleRL ContextRL
BasePlayerAbleC
BasePlayerC
NonRelating Category
RoleCategory RelatorCategory
BaseRoleC
Relation
Figure 3. Taxonomy fragment of selected CR categories, as classified by HermiT [
Overall, Fig. 3 corresponds well to our expectations. Yet there is one result that was not immediately obvious from the DL axioms, because it grows out of the domain and range constraints of CR relations and some axioms about the notions introduced as role bases of those relations, cf. Table 1. In summary, the reasoner concludes the subsequent (extensional) equivalences among CR categories. Each line corresponds to one of those nodes in Fig. 3 labeled in bold face. Although this high number of equivalent categories may seem awkward for an ontology in general, inspection shows that all of them are (in the light of CR) reasonable cases of intensionally distinct, but coextensional categories.
The axiomatization in first-order logic (FOL) is still in a preliminary stage, first of all. It follows the ideas of ontological semantics [1, ch. 4–5], which implies some methodological specialties. Nevertheless, all formulas specified in this section can properly be considered under classical set-theoretic semantics. We consider three binary relations of CR as fundamental relations, which are the only ones that are solely axiomatically characterized. These relations are instantiation (::), role playing ( ) and role having (⊸), ↝ where those symbols are binary predicate symbols that are used in infix notation.
Every other CR notion is first reified by a logical constant symbol and thus subject to the domain of discourse.15 Moreover, predicate symbols are typically introduced for convenience, but all of them must be “explained” by being equipped with a so-called predication definition, intended to fix the “semantics” of atomic formulas with that predicate. Notationally, like in sect. 2.3, we adopt the convention to let reifying constants be preceded by # . For example, if Ent is a unary predicate symbol intended to denote the CR category Entity, then #Ent is a logical constant for which there is the same intent. Pred ( ) denotes all predicates of / in the argument, Const ( ) all constant symbols, ar ( ) the arity of its argument.
Subsequently we specify a very limited, selective preview on the FOL axiomatization over the signature CR. Formulas that are / were intended as axioms, but are already shown to be entailed by others, are kept in the list of “axioms” / formulas, yet numbered with prefix C (with references to entailing formulas in their description), whereas the numbering of non-entailed and thus proper axioms has prefix A, that of definitions D.
D1. P (x) $df x :: #P j P 2 Pred ( CR); ar (P ) = 1
(uniform predication definitions for all unary predicates) D2. Ent(x) $df 9y : y = x (x is an entity :iff x exists) 15Introducing a constant symbol and thus reification applies likewise to the fundamental relations.
A1. #P 6= #Q j #P; #Q 2 Const ( CR) (any two constants refer to distinct entities) A2. 9x : P (x) j P 2 Pred ( CR); ar (P ) = 1 (all CR categories are instantiated) C1. Cat(#P ) j P 2 Pred ( CR); ar (P ) = 1
(categorization of constants as categories [A2, D1, A4]) C2. Ent(x) (everything is an entity [D2]) A3. :9x : Role(x) ^ Relator(x) (role and relator are disjoint categories) A4. x :: y ! Ent(x) ^ Cat(y) (instantiation relates entities with categories) A5. x :: y ^ Relation(y) ! Relator(x) (instantiation of relations yields relators) A6. x :: y ^ RoleCat(y) ! Role(x) (instantiation of role categories yields roles) A7. x ↝ y ! Ent(x) ^ Role(y) (plays relates entities with roles) A8. x ⊸ y ! Role(x) ^ Relator(y) (role-of relates roles with relators) A9. Relator(x) ! 9y : Relation(y) ^ x :: y (every relator instantiates 1 relation) A10. x ⊸ y ^ x ⊸ z ! y = z (the relator of a role is uniquely determined) C3. x ⊸ y ! (:9u u ⊸ x) ^ (:9v v ⊸ y) (role-of is a single-step relation [A3]) C4. (9y : y :: x) ! Cat(x) (if an entity is instantiated, it is a category [A4]) C5. Ent(x) $ 9y : Cat(y) ^ x :: y
(every entity instantiates a category [C2, D1 for Ent, A4])
Admittedly, this is just a minimal selection of axioms, primarily demonstrating some axioms resulting from the reification aspect and then focusing on constraints of the fundamental relations. One can observe that some assumptions on the reifying constants have already effects on those relations. More of that can already be found in [1, sect. 6.1.2], e.g., where most of the common properties of orderings turn out not to be satisfied by instantiation, based on few axioms, a.o., having entity as a universal category.
The full theory is still under development. This goes in tandem with working on a consistency proof for the FOL theory.
With CR we aim at a theory of categories and relations that is well-suited as a foundation for the semantics of languages as well as for other ontologies. The latter aspect concerns meta-ontological capabilities of CR, i.e., CR notions lend themselves to an ontological analysis of the constituents of ontologies. That aim of serving as a foundation for other ontologies is as broad as to cover arbitrary ontologies, including CR itself. It is therefore pursued by proposing a self-contained / self-analytic theory. As may be clear from the sections above, all notions of CR are themselves considered to be in the intended domain of discourse of the theory, such that, for example, all CR constituents can be categorized by CR categories. Moreover, literally all relations of CR are subject to the same analytic account based on relators composed of roles. This applies even to the three fundamental binary relations, cf. the discussion of that aspect in [1, sect. 2.4.4.3 and near the end of sect. 6.1.2]. A price to be paid for this is that CR is subject to infinite regresses, which lead not only to infinite (logical) models of the full theory, but which are often considered as harmful, if not necessary to be avoided. Unfortunately, we lack the space herein to argue in detail against necessarily seeing a problem with the particular regresses that occur, on philosophical, but also logical grounds. At least, we accept them in favor of the – for us – very attractive feature that CR does not rely on any additional assumptions nor on any kinds of entities that are considered solely external to the theory or that are declared to be not analyzable.
A few words remain to be devoted to related work. Focusing on applied ontology
and existing ontologies first, we mention the Unified Foundational Ontology (UFO) [
In the domain of conceptual modeling, we see two closely related lines of research.
Admittedly, only recently having become much more clearly aware of the abundance of
modeling and meta modeling approaches involving a positionalist account of relations
(and thus roles), see e.g. [22, esp. sect. 3.2], one may consider to what extent CR might
yield novel contributions conceptually, for instance, remembering the discussion of base
role, base player and base playerable categories, and / or whether it may serve as a
foundation for ontology-driven conceptual (meta) modeling, e.g., in the sense of [
In this paper we argue for the striking importance of well-developed accounts of
categories and relations and the remaining need to establish formal ontological theories for this
subject matter. We are in the process of working on one candidate theory, abbreviated as
CR. Primarily its conceptual approach is elaborated herein to some extent, following [
Not surprisingly, we consider CR as a promising approach, while that largely remains a claim herein. Our next steps comprise of completing the first-order axiomatization and its meta-theoretic analysis. Afterwards or possibly starting in parallel, it will be instructive to apply and relate CR not only to itself, but to other ontologies as well as in the foundation of conceptual modeling.