Conceptual Modeling is a discipline of great relevance to several areas in Computer Science. In a series of papers [1,2,3] we have been using the General Ontological Language (GOL) and its underlying upper level ontology, proposed in [4,5], to evaluate the ontological correctness of conceptual models and to develop guidelines for how the constructs of a modeling language (UML) should be used in conceptual modeling. In this paper, we focus on the modeling metaconcepts of classifiers and objects from an ontological point of view. We use a philosophically and psychologically well-founded theory of universals to propose a UML profile for Ontology Representation and Conceptual Modeling. The formal semantics of the proposed modeling elements is presented in a language of modal logics with quantification restricted to Sortal universals.
Conceptual Modeling is regarded as a discipline whose importance spreads throughout several areas in the realm of Computer Science (e.g. Software Engineering, Information Systems Design, Domain Engineering, Database Design, Requirements Engineering, and Knowledge Engineering, among others). Its main objective is concerned with identifying, analyzing and describing the essential concepts and constraints of a universe of discourse with the help of a (diagrammatic) modeling language that is based on a small set of basic meta-concepts (forming a metamodel). Ontological modeling, on the other hand, is concerned with capturing the relevant entities of a domain in an ontology of that domain using an ontology specification language that is based on a small set of basic, domain-independent ontological categories (forming an upper level ontology). While conceptual modeling languages are evaluated on the basis of their successful use in the practice domain information modeling, ontology specification languages and their underlying upper level ontologies have to be rooted in principled philosophical theories about what kinds of things exist and what their basic relationships with each other are.
The Unified Modeling Language (UML) is a language initially proposed as a unification of several
different visual notations and modeling techniques used for systems design [
We believe, however, that defining constructs of a conceptual modeling only in terms of its mathematical semantics, although essential, it is not sufficient to make it a suitable ontology representation language. The position defended here is that, in order to model reality, a conceptual modeling language should be founded on formal upper-level ontologies. In other words, it should have both, formal and ontological semantics.
In a series of papers we have been employing the General Ontological Language (GOL) and its
underlying upper level ontology, proposed in [
modeling constructs. In [
Although classifier modeling constructs are fundamental in conceptual modeling (being present in all major conceptual modeling languages) there is still a deficiency of methodological support for helping the user of the language deciding how to model the elements of a given domain. In practice, a set of primitives is often used to model distinctions in different types of classifiers (Type, Role, State, Mixin, among others). However, the choice of how the elements that denote universal properties in a domain (viz. Person, Student, Red Thing, Physical Thing, Deceased Person, Customer) should be modeled is often made in ad hoc manner. Likewise, it is the judgment of what are the admissible relations among these modeling elements.
This paper proposes a philosophically and psychologically well-founded theory of universals (section 2). This theory is further used to generate a typology of UML classifiers together with a set of methodological guidelines that governs its use (section 3). Additionally, we provide a formal characterization of the types of universals proposed in section 2 in a language of modal logics with restricted quantification (section 4). Finally, section 6 elaborates on some conclusions and future work.
In [
In other words, if we have the patterns (exactly) five X… and the Y which is Z…, only the substitution of X,Y,Z by CNs will produce sentences which are grammatical. To see that, we can try the substitution by the adjective Red in the sentence (i): (exactly) five red were in the kitchen last night. A request to ‘count the red in this room’ cannot receive a definite answer: Should a red shirt be counted as one or should the shirt, the two sleeves, and two pockets be counted separately so that we have five reds? The problem in this case is not that one would not know how to finish the counting but that one would not know how to start since arbitrarily many subparts of a red thing are still red.
The explanation for this feature unique of CNs lies on the function that determinates (demonstratives and quantifiers) play in noun phrases, which is to determine a certain range on individuals. Both reference and quantification requires that the thing (or things) which are referred or which form the domain of quantification are determinate individuals, i.e., their conditions for individuation and identity must be determinate. In other words, if it is not determinate how to count Xs or how to identify X that is the same as Y, the sentences in the patterns (i) and (ii) do not express determinate propositions, i.e. propositions with definite truth values.
The distinction between the grammatical categories of CNs and arbitrary general terms can be
explained in terms of the ontological categories of Sortal and Characterizing universals [
In [
The statement that we can only make identity and quantification statements in relation to a Sortal
amounts to one of the best-supported theories in the philosophy of language, namely, that the identity
of an individual can only be traced in connection with a Sortal Universal, which provides a principle of
individuation and identity to the particulars it collects [
As argued by Kripke [
It is not the case that two incompatible principles of identity could apply to the same individual x,
otherwise x would not be a viable entity (determinate particular) [
As a consequence of the uniqueness principle we define a second postulate: Postulate 2: An Object in a conceptual model of the domain cannot instantiate more than one CMClass representing an ultimate Substance Sortal.
In the example above, the sortal Person is the unique substance sortal that defines the validity of
the claim that Mick Jagger is the same as Mike Philip or, in other words, that Mike Philip persists
through changes in height, weight, age, residence, etc… as the same individual. Person can only be the
sortal that supports the proper name Mick Jagger in all possible situations because it applies necessarily
to the individual referred by the proper name, i.e. instances of Person cannot cease to be so without
ceasing to exist. As a consequence, the extension of a substance sortal is world invariant. This
metaproperty of classifiers is named Modal Constancy [
Sortals such as Boy and Adult Man in the example above, but also Student, Employee, Caterpillar
and Butterfly, Philosopher, Writer, Alive and Deceased, which possibly apply to a continuant during a
certain phase of its existence, are named phased-sortal in [
Contrary to substance sortals, phased-sortals apply to individuals contingently and, thus, do not
enjoy modal constancy. For example, for an individual John instance of Student, we can easily imagine
John moving in an out of the Student type, while being the same individual, i.e. without loosing his
identity. Moreover, for every instance x of Student in a world w, there is another world w’ in which x is
not an instance of Student. This meta-property of classifiers is name anti-rigid in [
By considering how these different universals stand w.r.t rigidity another postulate can be derived: Postulate 3: A CM-Class representing a rigid classifier cannot be a subclass a CM-Class representing an anti-rigid classifier
If PS is a phased-sortal and S is the substance sortal specialized by PS, there is a specialization
condition φ such that x is a PS iff x is a S that satisfies φ [
Phases constitute possible stages in the history of a substance sortal. Examples are: (a) Alive and Deceased: as possible stages of a Person; (b) Catterpillar and Butterfly of a Lepidopteran; (c) Town and Metropolis of a City; (d) Boy, Male Teenager and Adult Male of a Male Person. Classifiers representing phases constitute a partition of the substance sortal they specialize. For example, if <Alive,Deceased> is a phase-partition of a sustance sortal Person then for every world w, every Person x is either and instance of Alive or of Deceased but not of both. Moreover, if x is an instance of Alive in world w then there is world w’ such that x is not an instance of Alive in w’, which in this case, implies that x is an instance of Deceased in w’.
Contrary to phases, roles do not necessarily form a partition of substance sortals. Moreover, they differ from phases in terms of the specialization condition φ. For a phase P, φ represents a condition that depends solely on intrinsic properties of P. For instance, one might say that if Mick Jagger is a Living Person then he is a Person who has the property of being alive or, if Spot is a Puppy then it is a Dog which has the property of being less than a year old. For a role R, conversely, φ depends on extrinsic (relational) properties of R. For example, one might say that John is a Student then John is a Person who is enrolled in some educational institution or that, if Peter is a Customer then Peter is a Person who buys a Product y from a Supplier z. In other words, an entity plays a role in a certain context, demarcated by its relation with other entities.
Although Frege argued at length that “one cannot count without knowing what to count”, in artificial logical languages inspired by him, natural language general terms such as CNs, adjectives and verbs are treated uniformly as predicates. For instance, if we want to represent the sentence “there are tall men”, in the fregean approach of classical logic we would write ∃x Man(x) ∧ Tall(x). This reading puts the count noun Man (which denotes a Sortal) on an equal logical footing with the predicate Tall. Moreover, in this formula, the variable x is interpreted into a “supposedly” universal kind Thing. So, the natural language reading of the formula should be “there are things which have the property of being a man and the property of being tall”. Since, by postulate 1, all individuals must be instances of a substance sortal we must conclude that Thing is a unique universal ultimate sortal which is able to supply a principle of identity for all elements that we consider in our universe of discourse. Moreover, by postulate 2, this principle of identity must be unique. Can that be the case?
In [20], Hirsch argues that concepts such as Thing, (Entity, Element, among others) are dispersive,
i.e. they cover many concepts with different principles of identity. For instance, in the extension of
Thing we might encounter an individual x which is a cow and an individual y which is a watch. Since
the principles of identity for Cows and Watches are not the same we conclude that Thing cannot supply
a principle of identity for its instances. Otherwise, x and y would obey incompatible principles of
identity and, thus, would not be determinate individuals. Therefore, as defended in [
The Unified Modeling Language (UML) has built in extension mechanisms that allow one to
modify the language elements to suite certain modeling needs. A coherent set of such extensions,
defined accordingly to a specific purpose or domain, constitutes a UML profile [
A Stereotype is a lightweight extension mechanism that allows one to specialize UML modeling elements by defining additional constraints and sometimes a different graphical notation, so that they behave in some aspects as if they were instances of elements defined in new virtual metamodel. Stereotypes are also used to indicate difference in meaning or usage between modeling elements with a similar structure.
In [
«kind»
A «subkind» A
A «phase»
A «role»
A «category»
A «roleMixin» A
A kind represents a substance sortal, i.e. rigid, externally independent universals that supply a principle of identity for its instances.
Examples could be instances of Natural Kinds (such as Person, Dog, Tree) and artifacts (Chair, Car, Television).
A subkind is a rigid, externally independent restriction of a kind which carries the principle of identity supplied by the kind. An example could be the subkind MalePerson of the kind Person. In general, the stereotype «subkind» can be omitted in conceptual models without loss of clarity.
It represents the phased-sortals phase, i.e. anti-rigid and externally independent universals defined as part of a partition of a kind. For instance, the partition {Catterpillar, Butterfly} of the kind Lepdopterum.
It represents a phased-sortal role, i.e. antirigid and externally dependent universal. For instance, the role student played by instance of the kind Person.
It represents a rigid and externally independent non-sortal, a dispersive universal that aggregates essential properties which are common to different kinds. For example, the category RationalEntity as a generalization of Person and IntelligentAgent.
It represents an anti-rigid and externally A role mixin cannot have direct instances dependent non-sortal, a dispersive universal and must be depicted as an abstract class. that aggregates properties which are common A supertype of a role mixin cannot be a to different roles. It includes formal roles member of {« kind », « subkind », such as whole/part and initiatior/ responder. « phase », « role »}. Let X be a class stereotyped as « roleMixin » and r be an association representing X’s restriction condition. Then, #X.r ≥ 1
Every object in conceptual model using this profile must be an instance of a Kind, directly or indirectly (postulate 1). Moreover, it cannot be an instance of more than one ultimate Kind (postulate 2). A supertype of a kind cannot be a member of {« subkind », « phase », « role », « roleMixin »} A sypertype of a subkind cannot be a member of {« phase », « role », « roleMixin »} The phases {P1…Pn} that form a partition of a Kind K are defined in UML as a disjoint and complete generalization set. The kind K is always depicted as an abstract class.
Roles and Phases are anti-rigid universals and cannot appear in a conceptual model as a superclass of a Kind (postulate 3). Moreover: Let X be a class stereotyped as « role » and r be an association representing X’s restriction condition. Then, #X.r ≥ 1 A category cannot have direct instances and must be depicted as an abstract class. A supertype of a category cannot be a member of {« kind », « subkind », « phase », « role », « roleMixin »} «mixin»
A
The stereotype «mixin» represents properties
which are essential to some of its instances
and accidental to others (a meta-property
named semi-rigidity in [
A mixin cannot have direct instances and must be depicted as an abstract class. A supertype of a mixin cannot be a member of {« kind », « subkind », « phase », « role », « roleMixin »}
In this section we provide a formal characterization of the notions discussed in section 2 by using a
language L of quantified modal logics with identity. A model M in this language is a structure
<W,D,δ> where W is a non-empty set of worlds, D is a non-empty domain of objects and δ is an
interpretation function assigning values to the non-logical constants of the language. The domain D of
quantification is that of possibilia, which includes all possible entities independent of their actual
existence. Therefore we shall quantify over a constant domain in all possible worlds. Moreover, all
worlds are equally accessible and therefore we omit the accessibility relation from the model structure.
As a result we have a language that differs from the simplest language of quantified modal logic (QS5)
[
(i) (∀S,x) A (ii) (∃S,x) A which can be read as for every instance of S A holds and there is an instance of S such that A holds, respectively.
In this article, (i) and (ii) are meta-linguistic abbreviations to the formulas (∀x S(x) → A) and (∃x
S(x) ∧ A), respectively, i.e., they conform to the Fregean analysis of restricted quantification. However,
the primitive objects of quantification (elements of D) are continuants and as proposed in [
Second, individual constants of the language represent proper names of individuals (continuants)
and, therefore, the interpretation function δ defined as
(iii) δ(c,w) ∈ D, in which c is an individual constant
(iv) δ(S,w) ⊆ D, in which S is a sort
(v) δ(Pn,w) ⊆ Dn, in which P is a n-ary predicate
must obey the following constraint: for all w,w’ ∈ W, δ(c,w) = δ(c,w’), i.e. the interpretation of an
individual constant c (proper name) is world invariant. This amount to Kripke’s thesis that proper
names are rigid designators [
The quantification restricted in this way makes explicit what is only implicit in standard predicate logics. As previously discussed, suppose we want to state the following proposition : (a) There are red tasty apples. In classical predicate logic we would write down a logical formula such as (b) ∃x (apple(x) ∧ tasty(x) ∧ red(x)). In an ontological reading, (b) states that “there are things which are red, tasty and apple”. The theory proposed section 2 rejects that we can conceptually grasp an individual under a general concept such as Thing or Entity or, what is almost the same, that a logic (or conceptual modeling language) should presupposed the notion of a bare particular. Moreover, it states that only a sortal (e.g. Apple) can carry a principle of identity for the individuals it collects, a property which is absent in attributions such as Red and Tasty. For this reason, a logical system when as used to represent a formalization of conceptual models, should not presupposed that the representations of natural general terms such as Apple, Tasty and Red stand in the same logical footing. For this reason, (a) should be represented as (∃Apple,x) (tasty(x) ∧ red(x)) in which the sortal binding the variable x it is the one responsible for carrying its principle of identity.
Let F and G be two arbitrary universals such that F is specializes G. As a consequence we have that
1. □ (∀F,x G(x)) if G is a rigid universal then
2. □(∀G,x □G(x)) or in other words, for all w,w’ ∈ W we have that δ(G,w) = δ(G,w’)
For instance, Figure 1 depicts an example with the kind Person and its subkind Man. In this case we have the following instantiations of (1) and (2):
□ (∀Man,x Person(x)) □(∀Man,x □Man(x)) □(∀Person,x □Person(x)) In fact, in this example, the subkinds Man and Woman form a partition of the kind Person. In general, if <U1…Un> is a partition of a universal U then we have that
3. □(∀U,x U1(x) ⊕…⊕ Un(x)) and, in this specific case, □(∀Person,x Man(x) ⊕ Woman(x)).
In the same figure 1, another partition is present, namely, the phase-partition Child, Adolescent, Adult of the kind Person. Phases are always defined as a partition and, thus, formula (3) always hold for a phase-partition <K1…Kn>of a substance sortal S. Besides that, for all Ka,Kb ∈ <K1…Kn> such that a ≠ b we have that
4. □(∀Ka,x ◊Kb(x)) in the example of figure 1, □(∀Child,x ◊Adolescent(x)) □(∀Child,x ◊Adult(x)) □(∀Adolescent,x ◊Child(x)) □(∀Adolescent,x ◊Adult(x)) □(∀Adult,x ◊Child(x)) □(∀Adult,x ◊Adolescent(x)) Formula (4) implies
5. □(∀Ki,x ◊¬Ki(x))
which is a more general statement of anti-rigidity and, hence, applies to all phased-sortals including
roles. In figure 2, Student represents a role played by instances of the kind Person. As previously
mentioned, roles differ from phases w.r.t. their specialization conditions. In figure 1, the association
enrollment φenrollment ⊆ Student × School represents a extrinsic property that must necessary apply to all
instances of Student. In general, we can state the following: Let R be a role that specializes a sortal S
(named its allowed type) and let φ be a relation representing the restriction condition for R, such that φ
⊆ R × T, where T represents a type on which R is externally dependent [
6. □(∀R,x ∃T,y φ(x,y)) in the case of figure 1
□(∀Student,x ◊¬Student(x)) □(∀Student,x ∃School,y φ(x,y))
Finally, we can show why the postulate 3 (section 2) must be reinforced in conceptual models. To see that is the case suppose there is a rigid classifier G which specializes an anti-rigid classifier F. Let {a,b,c,d} and {a,b} be the extension of F and G in world w, respectively. By (5), there is a world w’ in which a ∈ δ(F,w) is not in δ(F,w’) (a ∉ δ(F,w’)) . By (2), however, δ(G,w) = δ(G,w’) and, by (1), δ(G,w’) ⊆ δ(F,w’), ergo, a ∈ δ(F,w’) which is a contradiction. We have therefore shown that it is not the case that a rigid classifier could specialize an anti-rigid one.
«kind» Person
Man Woman «kind» Person «role» Student enrollment 1
School «phase» Child
«phase» Adolescent «phase»
Adult
The development of a well-grounded, axiomatized upper level ontology is an important step towards the definition of real-world semantics for conceptual modeling diagrammatic languages. In this paper, we use a philosophically and psychologically well-founded theory of universals to address the problem of classifiers in conceptual modeling.
This theory is further used in the definition of a UML profile for Ontology Representation and Conceptual Modeling. The profile comprises of a set of stereotypes representing distinctions on types of classifiers proposed by the theory (e.g., Kind, Role, Phase, Category, Mixin) as well as a set of constraints on the possible relations to be established between these elements (representing the postulates of the theory).
A formalization of the theory is provided in a language of first-order modal logics with
quantification restricted to Sortal universals. This formalization shall be extended in a future paper in
which the difference between Sortals and arbitrary general terms will be emphasized. In particular, we
intend to use separated intentional properties (in the spirit of Gupta’s logic of Common Nouns [
We believe that these results contribute to the task of defining ontological foundations and principled engineering tools for the discipline of conceptual modeling.
Acknowledgements Part of the work presented here has been developed during an extended visit of the first author to the ISTC/LOA (Trento, Italy) and in collaboration with the OntoMed research group (Leipzig, Germany). We would like to thank Nicola Guarino, Heinrich Herre, Claudio Masolo, Laure Vieu, Alessandro Oltramari, Stefano Borgo, Emanuele Botazzi, and Roberta Ferrario for fruitful discussions and for providing valuable input to the issues of this article.