Conceptualization

A Novel Intensional-based Model

Islam Ali and Hamada Ghenniwa

Department of Electrical and Computer Engineering, Western University, London, Ontario, Canada

Keywords: Conceptualization, Extensional Logic, Intensional Logic, Extensionalization.

Abstract: A formal treatment of conceptualization is essential and a fundamental aspect of knowledge representation,

Ontologies and information engineering. Several approaches have been proposed based on extensional logic

and extensional reduction model. However, in this paper we highlight several limitations of their

applicability for modelling conceptualizations in dynamic and open environments, due to several strong

assumptions that are not adequate for dynamic and open environments. To this end we argue that intension

based model is a natural and adequate model. We present a model based on the theory of Properties

Relations and Propositions. This description takes the concepts and relations as primitives and, as such,

irreducible. The proposed description is then extended to describe the world in more details by capturing the

properties of the domain concepts.

1 INTRODUCTION

Ontology is a very active topic in the knowledge

formalization, information sharing, and artificial

intelligence communities. The progress in artificial

intelligence, knowledge engineering, the semantic

web, information sharing, information integration

and P2P systems made the development of

ontologies essential for information systems.

Especially in dynamic systems and open

environment, ontology plays a very important role in

facilitating the interaction and collaboration between

several agents. Ontology specifies a

conceptualization which is essential for the

formalization of knowledge (Genesereth and

Nilsson, 1987). A conceptualization is defined as

“an abstract model that consists of the relevant

concepts and the conceptual relations that exist in a

certain domain” (Xue, 2010). This definition

emphasizes the intensional nature of a

conceptualization.

(Gruber, 1993) defined a conceptualization as

"the objects, concepts, and other entities that are

presumed to exist in some area of interest and the

relations that hold amongst them" (Gruber, 1993).

This definition reflects the extensional account of

the conceptualization described in (Genesereth and

Nilsson, 1987). (Guarino, Oberle and Staab, 2009)

argued that a conceptualization is about concepts.

And as such, the conceptualization should not

change unless the meanings do change (Guarino and

Giaretta, 1995). And so, (Guarino and Giaretta,

1995) defined a conceptualization as "an intensional

semantic structure that encodes the implicit roles

constraining the structure of a piece of reality". This

definition also shows that conceptualization is of an

intensional nature. As will be shown later,

extensional logic cannot describe intensional

contexts. And this is why an intensional notation is

required for the task of describing a

conceptualization.

An extensional reduction notation for describing

a conceptualization is proposed (Guarino and

Giaretta, 1995), (Guarino, 1998), and (Guarino et al.,

2009). This model followed the possible world

approach (Anderson, 1984) for intensional logic.

The extensional reduction model is more adequate

than the extensional model as it deals with

conceptual relations as opposed to extensional

relations in the extensional model. There are,

however, several formal and intuitive concerns about

the possible world approach that reduces the

intensional entities to extensional ones (Bealer,

1993), (Bealer, 1998a). It is also noticed that

extensional reduction model is appropriate for

describing systems in which the set of existing

entities is not allowed to change while the relations

between them may change. However, the

257

Ali I. and Ghenniwa H..

Conceptualization - A Novel Intensional-based Model.

DOI: 10.5220/0004145202570264

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2012), pages 257-264

ISBN: 978-989-8565-30-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

extensional reduction model does not adequately

describe information systems or dynamic systems in

which new entities are allowed to enter and/or leave

the world. It also describes the domain in terms of

extensional entities rather than concepts. The

extensional reduction notation also treats the

concepts as relations, which is found to be

inappropriate and unintuitive. For these reason the

need for an intensional-based notation for describing

a conceptualization arises.

In this work, two different approaches for

describing a conceptualization are discussed and

analyzed. These approaches are fundamentally

different as they belong to different classes of logic.

The PRP theory (Bealer, 1979) for intensional logic

is then discussed. An intensional model for the

conceptualization, based on the PRP theory, is

proposed. This intensional model avoids the

limitations of the extensional and the extensional

reduction notations. The proposed notation is also

extended to support a more fine-grained description

of a conceptualization in which the properties of the

domain concepts are captured.

The rest of the paper is organized as follows:

section 2 briefly explains the assumptions,

applicability, and limitations of the extensional logic

and the intensional logic. Section 3 then discusses

the extensional model for a conceptualization. Then,

Section 4 describes the extensional reduction

notation that is based on the possible world

approach. The critical points in these two

approaches are discussed in section 5 which sheds

some light on the PRP theory (Bealer, 1997) and

proposes the intensional model for the

conceptualization. In the same section, both course-

grained and fine-grained intensional-based

descriptions for the conceptualization are proposed.

Finally, section 6 concludes the work.

2 EXTENSIONAL LOGIC AND

INTENSIONAL LOGIC

This section explains briefly the extensional logic

and the intensional logic as applied to modelling a

conceptualization. We will start by a simple example

(Fitting, 2007): If someone tells you that the

Morning star is the Evening star, this changes your

knowledge. This is because, now you know that the

Morning star and the Evening star are equal.

However, even though the two signs ("Morning star"

and "Evening star") designate the same object, they

do not have the same meaning. In this sense,

meanings are the intensions, and things they

designate are the extensions. A context that cares

only about extensions is called an extensional

context. On the other hand, if the context cares about

the meanings, it is an intensional context (Fitting,

2007).

One of the major differences that help

distinguishing between the intensional and

extensional contexts is the applicability of

substitutivity (Bealer, 1982). In other words, a

context in which substitutivity does not apply can be

recognized as an intensional context. However, for

extensional contexts, the substitutivity of equivalents

always holds. The following argument (Bealer 1982)

explains the failure of the principle of substitutivity

in the intensional contexts.

x believes that everything runs.

Everything runs if and only if everything walks.

.'. x believes that everything walks

It is obvious that the above argument is intuitively

invalid. This is because the substitutivity is used in

an intensional context in which it does not apply.

Sentences like; “It is known that…”, “It is believed

that …”, “It is said that…”, “It is necessary that…”

are typical intensional contexts (Fitting, 2007). For a

computer scientist, expressing the belief of an agent

or the knowledge of an information system follows

the same role. That is why the belief of an agent and

the knowledge of an information system are

intensional matters.

Intensional systems are those in which

intensional features can be represented (Fitting,

2007). These are the systems that cannot be

described in extensional logic. In order to describe

such systems, several theories for intensional entities

were proposed. Some of these theories included

some reduction and some others adopted a non-

reductionist view. Those theories, which

incorporated reduction, reduce the intensional

entities to extensional entities (Bealer, 1998). An

example of such category of theories is the possible

world approach (Anderson, 1984) and (Lewis,

1986). When used for describing a

conceptualization, the reductionist approaches

assume that the world has fixed set of entities. As

such, these approaches are applicable if one is

interested in describing a static system with a fixed

set of entities in which the relations between objects

are allowed to change. These approaches, however,

are not adequate for describing information systems

or dynamic systems in which entities or agents can

enter and/or leave the system at any time.

The non-reductionist approaches, however, take

the intensional entities such as concepts, relations,

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and properties, at face-values, i.e. as real irreducible

entities. An example of theories of this category is

the theory of Properties, Relations, and Propositions

(Bealer, 1997), (Bealer, 1982), and (Bealer, 1983).

Modelling the conceptualization using this class of

logic is more adequate for dynamic systems and

open environment. It allows for the description of

intensional contexts such as belief and knowledge. It

also accounts for changes in the world as long the

concepts and the meanings do not change.

3 EXTENSIONAL MODEL

The extensional model is based on the extensional

logic. And as such it describes the conceptualization

in terms of declarative sentences and ordinary

relations. According to this model, a

conceptualization is formally defined as a triple E

, F

, R

) consisting of a universe of discourse, a

functional basis set for that universe, a relational

basis set (Genesereth and Nilsson, 1987). The

universe of discourse D

is a set of all entities, or

what is called extensions, in the domain. A function

maps an entity e

D

to another entity e

D

based

on an interrelation between the two entities. The set

of functions that are emphasized in the

conceptualization is referred to as the functional

basis set F

. And finally, the relational basis set R

the set of all extensional relations that hold between

the elements of D

The following example (Genesereth and Nilsson,

1987) explains the extensional model of

conceptualization:

Consider the blocks world that has only one concept

(block). And consider a specific instance of this

world in which there are five blocks arranged as

shown in Figure 1.

Figure 1: Five blocks on a table example (Genesereth and

Nilsson, 1987).

In this example, it is assumed that there is only one

function in this domain that is relevant to the

conceptualization. This function is called Hat, and it

maps each item to its hat (the item that lies directly

above it). It is also assumed that there are four

different relations that are relevant to this

conceptualization. These relations are On, Above,

Clear, and Table. The conceptualization for this

world, according to the extensional description, is

= (D

, F

, R

) where D

= {a, b, c, d, e}, F

{hat

}, and R

= {on

, above

, clear

, table

}. The

members of both (F

and R

) are ordered tuples on

the elements of D

. In that sense, on

= {(a, b), (b,

c), (d, e)}, above

= {(a, b), (a, c), (b, c), (d, e)},

clear

= {a, d}, and table= {c, e}.

In the previous example, the extensions, in the

snapshot of the blocks world shown in Figure 1,

were described using the extensional notation. It

should be noticed, however, that the extensional

logic cannot describe intensional matters. And this is

because extensional logic substitutes equivalent

entities based on their extensions. And this does not

apply for intensional contexts.

4 EXTENSIONAL REDUCTION

MODEL

The fact that “an agent, or an information system,

for simplicity, believes something about the world”

cannot, adequately, be described using extensional

logic. This is because it is an intensional context.

And as such, describing such contexts using

extensional logic might result in unintuitive

arguments. (Guarino and Giaretta, 1995) also

pointed out that “the extensional notation of

conceptualization is only useful if one is interested

in an isolated snapshot of the world”. For instance, if

a different arrangement of blocks is considered, as

shown in Figure 2, the corresponding

conceptualization, according to the extensional

notation, will be different.

Figure 2: A different configuration for the five blocks

(Guarino and Giaretta, 1995).

It is argued (Guarino and Giaretta, 1995),

(Guarino, 1998) and (Guarino et al., 2009) that, the

conceptualization should focus on the meaning

instead of a particular state of the world. And so, the

conceptualization should not change when the

arrangement of the blocks, in the blocks world,

changes. In order to capture such intuition, the

possible world theory (Anderson, 1984) is adopted

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259

as a basis for describing the conceptualization

(Guarino and Giaretta, 1995), (Guarino, 1998), and

(Guarino et al., 2009). This theory reduces the

intensional entities to extensional entities, i.e.

extensional functions or sets (Bealer, 1998).

Using the possible world reduction, (Guarino

and Giaretta, 1995) formally described the

conceptualization as a triple E

= (D

, W

, R

)

(Guarino et al., 2009). In this model, D

is a domain

of objects, W

is a set of possible worlds, and R

a set of conceptual relations. According to this

model, a conceptual relation of arity n on D

is a

function from the set of possible world W

to the set







of all possible n-ary relations on D

. It is also

worth mentioning that, in this model the concepts

are treated as relations, or functions, from W





(Guarino et al., 2009).

Referring to the blocks world example shown in

Figure 1, the conceptualization for the blocks world

based on the extensional reduction model is E

er1

, W

er1

, R

er1

) where: D

er1

= {a, b, c, d, e}, W

er1

, w

, …} the set of possible worlds, i.e.,

the set of all possible configurations of the members

of D

, and R

= {Block

, Clear

, Table

, On

Above

} is the set of relations from W

























} respectively.

In order to show that the extensional reduction

model has an advantage over the extensional model,

the configuration in Figure 2 will be described

according to the two models. The conceptualization

for the world shown in Figure 2, according to the

extensional model, is E

= (D

, F

, R

) where D

{a, b, c, d, e}, F

= {hat

}, and R

= {on

, above

clear

, table

}. The members of the two sets, F and

R, are ordered tuples on the elements of D. In that

sense, on

= {(a, b), (c, d), (d, e)}, above

= {(a, b),

(c, d), (c, e), (d, e)}, clear

= {a, c}, and table

= {b,

e}. Here it is noticed that D

, however, R

≠R

and in turn E

≠E

. On the other hand, the

configuration in Figure 2, described using the

extensional reduction model, is E

er2

= (D

er2

, W

er2

). And based on the possible world reduction, it

can be shown that D

er1

er2

. This is obvious since

the entities in the world have not changed, i.e. the

five blocks in both Figure 1 and Figure 2. Since W

is the set of all possible configurations of the

elements of D

, and since D

er1

er2

, it can also be

shown that W

er1

= W

er2

. And finally since R

is a set

of relations from W

to2







. It is also obvious that,

er1

and R

er2

are equivalent. And in turns, E

er1

= E

er2

as one would expect.

5 INTENSIONAL MODEL

It is clear that the extensional reduction, or the

possible world approach, is more expressive as

compared to the extensional model. As discussed in

the previous section, different arrangements of the

same entities will not result in different

conceptualization. This is because the meanings of

the relations between them do not change. However,

for several reasons, this model needs to be further

revisited, especially in the context of knowledge

formalization, information systems, information

integration, and open environments.

There are several formal and intuitive concerns

about the possible world reduction (Bealer, 1993),

(Bealer, 1998a). First and foremost is that, it is a

reduction that reduces the intensional entities to

extensional entities. Further discussions about the

possible world reduction can also be found in

(Adams, 1974) and (Jubien, 1988) (as cited in

Bealer, 1993). Bealer proposed a non-reductionist

formulation for intensional logic that is compatible

with actualism as opposed to possibilism. The theory

of Properties Relations and Propositions (Bealer,

1997) and (Bealer, 1982), and (Bealer, 1993) takes

properties, relations and propositions as real

irreducible intensional entities.

Before the formal description is proposed, some

important definitions will be discussed first. We will

start with the definition of a concept. “Cognitive

scientists generally agree that a concept is a mental

representation that picks out a set of entities, or a

category. That is, concepts refer, and what they refer

to are categories” (Medin and Rips, 2005). In other

words, the term concept denotes a general, abstract,

idea of a category. A particular, is a concrete entity

that exists in space and time as opposed to a concept.

This does not mean that every instance of a category

is exactly the same. But, only that from some

perspective they are treated equivalently based on

something they have in common. The relation

between a concept and particular will be referred to

as abstraction. So a concept is created by keeping

the characteristics that are common between several

particulars while abstracting away the

characteristics that are uncommon.

A conceptualization is also defined as an abstract

model that consists of the relevant concepts and the

conceptual relations that exist in a certain domain

(Xue, 2010). Again this definition emphasizes the

fact that the conceptualization is about concepts and

meanings. And so, the conceptualization should

remain the same even when the state of the world is

changed or a particular is introduced to the world.

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This assumes that the new particular that is

introduced to the system is an extension of a concept

that is already captured in the conceptualization. It is

only the introduction of a new concept or conceptual

relation that should result in different

conceptualization.

Figure 3: The blocks world with 6 entities instead of 5.

In order for this point to be clear, Figure 3 shows

another example of the blocks world in which

another block f is introduced to the world. Let us

assume that the conceptualization for the

configuration shown in Figure 3, based on the

extensional model, is E

. As discussed before, it is

quite evident that E

, E

, and E

are different. This

is partially taken care of in the extensional reduction

model that is based on the possible world approach.

The conceptualization for the configuration in

Figure 3, based on the extensional reduction model,

will be referred to as E

er3

= (D

er3

, W

er3

, R

er3

). As

mentioned earlier, E

er1

and E

er2

are equivalent.

However, D

er3

= {a, b, c, d, e, f}, and that means

er1

er2

≠D

er3

. Since W

is defined as all possible

configurations of elements of the domain D

, then

er1

er2

≠W

er3

. Because R

is a set of relations

from W

to 2







, it is easy to show that

er1

er3

≠R

er3

and in turn E

er1

er2

≠E

er3

Since the extensional reduction model is based

on the possible world reduction, the so called

conceptual relations are, in fact, extensional

relations between the set of possible worlds and the

set of extensions in the domain. It is also clear that,

the introduction of a new extension to the world

changes the conceptualization. According to the

intensional model, introducing a new particular,

which corresponds to a concept that is already

captured in the system, should not change the

conceptualization.

Based on the above discussion, an intensional

model that accounts for the instantiation, or

extensionalization, is required. Being an intensional

model, it should take the relations as intensional

entities rather than reducing them to extensional

functions. The intensional model should also capture

the concepts (based on the observation of the

particulars) instead of capturing the particulars

themselves. Especially in the context of information

systems, inserting a record in the database, for

instance, can be considered as sort of instantiation.

And this should not affect the conceptualization.

It is also worth mentioning that, the extensional

reduction model treats the concepts as relations and

mixes them with the set of relations. We find this

inappropriate and unintuitive for the purpose of this

work. This is because, the concepts are abstractions

of entities that exist in certain time and space while

the relations are abstractions of the interrelations

between these entities. Even though both concepts

and relations are intensional entities, they are

different in nature.

Another observation is that, the relations in the

possible world approach are separated from the

domain. And even though this is not wrong for

describing the conceptualization, we adopt the view

that the relations are intensional entities and should

be taken as primitive, irreducible, entities (Bealer,

1997) and (Bealer, 1982). And as such, it is more

adequate to treat the intensional relations as part of

the domain. In that sense, both the set of concepts

and the set of conceptual relations will be members

of the domain.

Finally, it is also important that the model can be

expanded in order to describe the world in more

details. An example of this would be a model that

describes the properties of the concepts as will be

shown later.

Motivated by the above observations, a new

intensional model for describing the

conceptualization is proposed. This model is based

on Bealer’s intensional logic (Bealer, 1997) (Bealer,

1983). The following section will shed some light on

Bealer’s intensional logic. Then the proposed model

will be explained. The proposed model will then be

extended to describe the properties assigned to the

concepts.

5.1 Intensional Algebra

This section briefly explains the Intensional

Algebraic structure according to the theory of

Properties Relations and Propositions for intensional

logic (Bealer, 1979). For more details about the

theory of PRP, we refer the reader to (Bealer, 1979)

(Bealer, 1982), (Bealer, 1983). The theory of PRP is

a non-reductionist intensional formalization for

intensional logic. This formalization takes the

properties, the relations, and the propositions as real

irreducible entities instead of reducing them to

extensional entities. According to the theory of PRP,

an intensional algebra is a structure (D, J, K)

consisting of a domain D, a set of logical operations

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261

J, and a set of possible extensionalization functions

K (Bealer, 1979) and (Bealer, 1998b). The domain D

divides into subdomains that include the intensional

entities of the domain. The set of logical operations

includes, but not limited to, conjunction, negation,

singular predication, existential generalization, and

so on. And the possible extensionalization functions

assign extensions to relevant items in the domain.

Extensionalization can be defined as the process of

keeping the abstraction distinct and maintaining the

relationship between the abstractions and observed

facts (Aparaso, 2010). In other words,

extensionalization is the connection between reality

and the conception of the observer.

5.2 Intensional Description of

Conceptualization

As mentioned earlier, the conceptualization is

defined as abstract model that consists of the

relevant concepts and relations that exist in certain

domain (Xue, 2010). This definition will be revised

as “an abstraction that consists of the relevant

concepts and relations that exist in certain domain”.

We purposely take off the word model from the

definition because it might imply the use of formal

language, or the lead to the illusion of being

something physical. In order to intensionally

describe conceptualization, an intensional structure,

based on the theory of PRP, is used. This structure is

formally explained below and various advantages of

the new model are discussed.

Figure 4: The relation between the conceptualization and

the reality.

According to the intensional notation, a

conceptualization, is described as a tuple E

= (D, K)

in which D is a domain and K is a set of

extensionalization functions. The domain D, in

turns, consists of the set of concepts C and the set of

conceptual relations R, written as D= (C, R). The set

of concepts C captures abstracts to all relevant

entities in the world. And the set of relations R can

be further decomposed into binary relations R

ternary relations R

, and so on. The members of the

set of extensionalization functions K assign entities

of the reality to the corresponding concepts and

conceptual relations in the conceptualization. Figure

4 explains how an extensionalization function relates

elements of the reality to both concepts and

intensional relations in the conceptualization.

Figure 4 shows how the particulars are related to

the conceptualization through the extensionalization

function. Note that, the predicate Sit (Cat, Mat) does

not describe certain instances of the concepts Cat or

Mat. Rather it intensionally means that entities

corresponding to the concept Cat can be described

as Sitting on any entity that can be referred to as a

Mat. And as such the conceptualization

corresponding to the world in Figure 4 can be

described as E

= (D

, K

). In that case, D

= ({Cat,

Mat}, {Sit (Cat, Mat)}).

Figure 5: Two cats sitting on a mat.

The question now is, what changes to reality

should affect the conceptualization? Or in other

words, when should the conceptualization change?

In order to answer this question Figure 5 and Figure

6 are closely examined. In Figure 5, one can see two

cats sitting on a mat. Is the conceptualization that

describes the world in Figure 5 different from the

one that describes the world in Figure 4? In order to

answer this question we need to answer the

following questions first:

Did the world change? If yes then:

a. Were extensions of new concepts introduced to

the world? If yes, then:

i. Are these concepts relevant to our

conceptualization?

b. Were extensions of new relations

introduced to the world? If yes, then:

i. Are these intensional relations relevant to

our conceptualization?

By looking at Figure 5; the answer to the first

question is YES. This is because another cat is now

sitting on the mat. However, since the concept that is

already captured in E

, this should not change the

conceptualization. This is because the introduction

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of a new cat does not change the meaning of the

concept cat. Now let us examine the relations

between relevant concepts in Figure 5. There seem

to be a relation between the two cats, as one of them

is beside the other. Now, if this relation is relevant to

our conceptualization, this will be perceived as a

binary relation on the concept Cat, i.e.

SidebySide(Cat, Cat). However, if this relation is

irrelevant to our conceptualization, it will be

abstracted out and the conceptualization Ei4 will be

able to describe the Cat/Mat world in Figure 5. And

as such, our conceptualization captures the facts that,

there can be cats, and there can be mats, and cats can

sit on mats. No matter how many cats, how many

mats, and how many cats are sitting on mats, this

should not affect the conceptualization.

Figure 6: The relation between the conceptualization and

the reality.

By examining Figure 6 and trying to answer the

same questions above, one can observe that the

world has changed. This change adds both an

extension of a new concept, Dog, and extensions of

new conceptual relations, i.e. SidebySide (Dog, Cat)

and Sit (Dog, Mat). The next question would be, is

the concept Dog relevant to our conceptualization?

If the answer is No, then the concept Dog will be

abstracted out and the conceptualization won't be

affected. However, if the concept Dog is relevant to

our conceptualization then the conceptualization

should change in order to account for that new

concept. In a similar way, we will need to answer

the question about the conceptual relations and

whether or not they are relevant to our

conceptualization.

5.3 Advantages of the Intensional

Model

As discussed earlier, the extensional reduction

model is more appropriate for describing the

conceptualization as compared to the extensional

model. The extensional reduction model, however, is

suitable for describing static systems in which the

configuration of the system may change, without

introducing new entities. For the sake of describing

information systems, or dynamic system that exist in

open environment, the extensional reduction model

may not be a good candidate. For this reason and for

several reasons, mentioned above, the need for an

intensional-based model is quite evident.

The intensional model has further improved the

description of conceptualization so that it describes

the relations as real irreducible entities instead of

reducing them to extensional functions. It also deals

with concepts rather than extensional entities or

objects. Moreover, the intensional relation separates

the concepts from the relations as they are different

in nature. This is different from the extensional

reduction model which treats the concepts as

relations. Furthermore, since the intensional model

treats the intensional relations as primitive entities,

they are considered to be part of the domain. It is

also worth mentioning that the use of the singular

term in the intensional logic (Bealer, 1987) avoids

higher order syntax for intensional logic (Majkic,

2009). And finally, the proposed intensional

description of conceptualization is easy to expand so

that it describes more details about the world. In the

next section, it will be shown how the intensional

model will be expanded to describe the properties of

the domain concepts.

5.4 Fine Grained Description

Recall in section 5.2, the intensional model of

conceptualization describes the conceptualization as

a tuple (D

, K

). In this description, D is composed of

subdomains containing both the concepts C, and the

conceptual relations R. This model can further be

extended to describe not only the relations between

concepts, but also the properties of the concepts. A

particular instance of a property is referred to as an

abstract particular or a trope (Bacon, 2008).

Following the PRP theory (Bealer, 1997) and

(Bealer, 1998), the properties are taken as primitive

entities and considered as part of the domain. The

values assigned to the properties can be thought of

as concepts. However these concepts are not of

direct relevance to the conceptualization. And as

such, these concepts are going to be called extrinsic

concepts Ce. On the other hands, the concepts that

are of direct relevance to the domain will be referred

to as intrinsic concepts C

The expanded model of the conceptualization is

defined as a tuple E

= (D

, K

). In this model, the

domain consists of four members D

= (C

, C

, R, P).

These four members represent intrinsic concepts,

extrinsic concepts, relations, and properties. An

example of a property in Figure 4 would be the color

of the cat. Assuming that a Cat can have one of

Conceptualization-ANovelIntensional-basedModel

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several colors (Black, White, Grey, or Brown), these

colors are considered extrinsic concepts in our

conceptualization. The fact that cats can have the

grey color will be described by the property Color

(Cat, Grey). This should not be confused with

asserting certain fact about a certain entity in the

world. However, this should be understood as an

intensional property that can be read as “extensions

of the concept Cat can be attributed as having a

Grey Color”.

As shown in this section, the intensional

deception of conceptualization can be expanded.

This allows scalability and gives more control on the

description of a system. The ability to expand the

intensional description is used here to describe the

properties of the domain concepts. However, we

expect this property to offer flexibility in describing

even more details about the system.

6 CONCLUSIONS

In this work, extensional and extensional reduction

models for describing a conceptualization are

critically discussed and analysed. It was shown that,

while the extensional description is suitable for

describing a certain state of the world, the

extensional reduction description is appropriate for

describing a static world. For information systems,

multi-agent systems, and in general, any dynamic

system in which entities can enter and leave the

system, it is shown that there is a need for an

intensional description of the conceptualization. An

intensional model for describing a conceptualization

is proposed. This model is based on the PRP theory

for intensional logic. The advantages of the

intensional description are discussed. And, both

course-grained and fine-grained descriptions for the

conceptualization are provided.

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