A Formal Semantics for Concept Understanding Relying on

Description Logics

Farshad Badie

Department of Communication, Aalborg University, Aalborg, Denmark

Keywords: Concept Understanding, Conceptualisation, Terminology, Interpretation, Formal Semantics, Description

Logics, Ontologies.

Abstract: In this research, Description Logics (DLs) will be employed for logical description, logical characterisation,

logical modelling and ontological description of concept understanding in terminological systems. It’s

strongly believed that using a formal descriptive logic could support us in revealing logical assumptions

whose discovery may lead us to a better understanding of ‘concept understanding’. The Structure of

Observed Learning Outcomes (SOLO) model as an appropriate model of increasing complexity of humans’

understanding has supported the formal analysis.

1 INTRODUCTION

The central focus of this research is on concepts. My

point of departure is the special focus on the fact that

there is a general problem concerning the notion of

‘concept’, in linguistics, psychology, philosophy and

computer science. This research aims at providing a

logical description (and analysis) of the use of

‘concepts’ in terminological knowledge

representation systems, and, thus, I need to assume

concepts’ applications in order to be comprehensible

in the context and in my logical formalism. Taking

into consideration (Baader et al., 2010) and

(Rudolph, 2011), a concept might be correlated with

a distinct ‘entity’ or to/with its essential features,

characteristics and properties. Note that an entity’s

properties express its relationships with itself and

with other entities. Through the lens of Predicate

Logic, a concept might be considered to be

equivalent to a [unary] predicate. It shall be

emphasised that this remark is not about language,

but this is how concepts are perceived by logicians.

Accordingly, it could be claimed that predicates

could—logically—express the characteristics of

concepts in terminological systems. More

specifically, predicates assign characteristics,

features and properties of concepts into some

subjects. It’s believed that predicates may determine

the applications of logical descriptions. As all

logicians know, predicates play fundamental roles in

reasoning processes and in giving satisfying

conditions for definitions of [logical] truth. By

taking into consideration that ‘a predicate expresses

a condition that the entities referred to may satisfy,

in which case the resulting sentence will be true (see

(Blackburn, 2016))’, predicates can be applied in

expressing meanings within formal semantics.

Subsequently, the formal semantics could focus on

multiple conditions through definitions of truth (and

falsity). The central objective of formal semantics

can be said to be formalising and manipulating the

relationships between the signifiers of a description

and what the signifiers do [or have been designed to

do], see (Jackendoff, 1990; Gray et al., 1992;

Barsalou, 1999; Resnik, 1999).

As mentioned, the central focus of this research

is on concepts (and through the lens of Predicate

Logic). Concepts and their interrelationships will be

used to establish the basic terminology adopted in

the modelled domain regarding the hierarchical

structures. My logical descriptions will have a

special focus on my methodological assumption that

expresses that ‘human beings can find out that an

individual thing/phenomenon is an instance of a

formed concept, and, thus, their individual grasp of

that concept (in the form of their conceptions)

provide foundations for producing their own

conceptualisations’. This article will focus on

describing and characterising humans’ concept

understandings and will deal with a formal-semantic

model for figuring out the underlying logical

Badie F.

A Formal Semantics for Concept Understanding Relying on Description Logics.

DOI: 10.5220/0006113800420052

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 42-52

ISBN: 978-989-758-220-2

assumptions of ‘concept understanding’. The term

‘understanding’ will be observed from multiple

perspectives, and, subsequently, the expressiveness

of the semantic model’s descriptions will be

improved. The Structure of Observed Learning

Outcomes (SOLO) taxonomy is an appropriate

model of increasing complexity of humans’

understanding. SOLO as a descriptive model of

knowing and understanding can support my

formalism. Additionally, its taxonomical structure

could be expressed in the form of some logical

inclusions. In this research, the formal semantic

analysis of [concept] understanding is based on

Description Logics (DLs). I believe that DLs can

support me in proposing an understandable logical

description for clarifying concept understanding.

DLs as the profound formalism are used for

representing predicates and for formal reasoning

over them. They mainly focus on terminological

knowledge. It is of a terminological system’s

particular importance in providing a logical

formalism for knowledge representation systems,

and, also, for semantic representations and

ontologies (as formal and explicit specification of a

shared conceptualisation on the domains of interest),

see (Davies et al., 2003; Staab and Studer, 2009).

The main contributions of this research are: (i)

providing a formal semantics (relying on DLs) for

conceptual analysis of concept understanding, and

analysing a knowledge representation formalism for

expressing concept understanding, and (ii) designing

and formalising an ontology that provides a

structural representation of concept understanding

within the analysed semantic model.

2 DESCRIPTION LOGICS

First, I shall mention that (Baader et al., 2010) is my

main reference to Description Logics. Description

Logics (DLs) represent knowledge in terms of

individuals (objects, things), concepts (classes of

things), and roles (relationships between things).

Individuals correspond to constant symbols,

concepts to unary predicates, and roles to binary (or

any other n-ary) predicates and relations in Predicate

Logic. Reconsidering the predicate P in Predicate

Logic, we have [possibly specified] concept C in

DLs. There are two kinds of atomic symbols, which

are called atomic concepts and atomic roles. These

symbols are the elementary descriptions from which

we can inductively (by employing concept

constructors and role constructors) build the

specified descriptions. Considering N

, N

and N

the sets of atomic concepts, atomic roles and

individuals respectively, the ordered triple N

, N

 represents a signature. The set of main logical

symbols in ALC (Attributive Language with

Complements: the Prototypical DL, see (Schmidt-

Schauss and Smolka, 1991)) is: { Conjunction (⊓:

And), Disjunction (⊔: Or), Negation

(¬: Not),

Existential Restriction (∃: There exists ... ),

Universal Restriction (∀: For all ... ) }. We also

have Atomic Concepts (A), Top Concept (

⊤:

Everything) and Bottom Concept (

⊥: Nothing) in

ALC.

In order to define a formal semantics, we need to

apply terminological interpretations over our

signatures. More particularly, any [terminological]

interpretation consists of (i) a non-empty set ∆ (that

is the interpretation domain and consists of any

variable that occurs in any of the concept

descriptions), and (ii) an interpretation function .

(let me call it ‘interpreter’). The interpreter assigns

to every individual (like a) a ‘a

∈ ∆

’. Also, it

assigns to every atomic concept A (or every atomic

unary predicate) a set A

⊆ ∆

, and to every atomic

role P (or every atomic binary predicate) a binary

relation

⊆ ∆

× ∆

. Table 1 reports the syntax

and the semantics of ALC.

Table 1: The Prototypical Description Logic.

Syntax Semantics

A A

⊆ ∆

P P

⊆ ∆

× ∆

⊤ ∆

⊥ ∅

C ⊓ D (C ⊓ D)

= C

∩ D

C ⊔ D (C ⊔ D)

= C

∪ D

¬C (¬C)

= ∆

\ C

∃R. C { a | ∃b.(a,b) ∈ R

∧ b ∈ C

}

∀R. C { a | ∀b.(a,b) ∈ R

⊃ b ∈ C

}

A knowledge base in DLs usually consists of a

number of terminological axioms and world

descriptions (so-called: ‘assertions’), see Table 2.

The terminological interpretation I is called a model

of an axiom [or a model of a basic world

description], if, and only if, it can semantically

satisfy it, see Tables 2 and 3. In the following Tables

P is an atomic role, R and S are role descriptions, A

is an atomic concept, and C and D are concept

descriptions.

A Formal Semantics for Concept Understanding Relying on Description Logics

Table 2: Axioms and World Descriptions in DLs.

Name Syntax

Semantics

Concept Inclusion Axiom C ⊑ D

⊆ D

Role Inclusion Axiom

R ⊑ S R

⊆ S

Concept Equality Axiom

C ≡ D C

= D

Role Equality Axiom

R ≡ S R

= S

Concept Assertion

C(a) a

∈ C

Role Assertion

R(a, b) (a

, b

) ∈ R

Table 3: Inductive Concept Descriptions.

Over Concept Over Role

⊆ ∆

× ∆

⊥

= ∅ ⊥

= ∅

(¬C)

= ∆

\ C

(¬R)

= (∆

× ∆

) \ R

(C ⊓ D)

= C

∩ D

(R ⊓ S)

= R

∩ S

Let me start the logical analysis with two examples:

Example 1. Mary has verified that ‘there is a young

student’ and ‘there is a non-old student’ are

expressing the same matter. Her verification

between these two propositions is expressible in DLs

by:

∃hasStudent.Young ≡ ∃hasStudent.¬Old. It’s

realisable that Mary has assumed the axiom stating

that

Young and Old are two disjoint concepts, and, in

fact, the logical term ‘

Young ⊓ Old ⊑ ⊥’ has formed

a terminological axiom for Mary. It’s obvious that

Mary’s interpretation over (i)

Young ⊓ Old ⊑ ⊥

(meaning that

Young and Old are disjoint concepts)

and (ii)

Person ⊑ Young ⊔ Old (meaning that every

person is either young or old) has played crucial

roles here. In fact, Mary has interpreted, and,

respectively, understood that these two sentences

(‘there is a young student’ and ‘there is a non-old

student’) have the same meanings. More

specifically, Mary’s terminological interpretation

(over i and ii) has produced her understanding of an

equivalence between the concept descriptions

∃hasStudent.Young and ∃hasStudent.¬Old. We

can see that Mary’s interpretation has been restricted

(limited) to her understanding of disjointness of the

concept descriptions

∃hasStudent.Young and

∃hasStudent.¬Old. At this point I shall claim that

the concepts (concept descriptions) C and D are

logically and semantically equivalent, when ‘for all’

possible terminological interpretations like I, we

have:

= D

. In this example, if one person, say

John, does not assume the axioms stating that

‘

Young and Old are two disjoint concepts’ and

‘every person is either young or old’, then there will

not be an equivalence relation between

∃hasStudent.Young and ∃hasStudent.¬Old. Let me

conclude that Mary’s and John’s understandings are

dissimilar, because they have had different

terminological interpretations in their minds (and it

is because of their different conceptions and concept

formations). For example, regarding John’s

terminological interpretation, the proposition ‘there

is a middle-aged student’ could be added beside

‘there is a young student’ and ‘there is a non-old

student’. In fact, John could have the axiom

‘Person

⊑ Young ⊔ MiddleAged ⊔ Old (meaning that every

person is young or middle aged or old)’ in his mind.

Consequently, John by taking this axiom (based on

his own conception) into consideration doesn’t

understand ‘

∃hasStudent.Young’ and

‘∃hasStudent.¬Old’ as equivalent concept

descriptions.

Example 2. Mary has verified that ‘Anna has a child

who is a philosopher’ and ‘Anna has a child who is a

painter’ could be jointly expressed by ‘Anna has a

child who is a philosopher and painter’. Translated

into DLs we have her expression as followings

∃hasChild.Philosopher ⊓ ∃hasChild.Painter ≡

∃hasChild.(Philosopher ⊓ Painter). Suppose that

Anna has two children and one is a philosopher and

the other one is a painter. Then,

∃hasChild.(Philosopher ⊓ Painter) is not equivalent

∃hasChild.Philosopher ⊓ ∃hasChild.Painter.

Actually, Mary has not proposed a correct

description, and this is because of her inappropriate

terminological interpretation. Accordingly, her

understanding has followed her inappropriate

interpretation. In fact, she incorrectly (semantically:

False) has understood that the proposition ‘Anna has

a child who is a philosopher and painter’ expresses

the same matter. Reconsidering the proposed

formalism,

∃hasChild.Philosopher ⊓

∃hasChild.Painter

and ∃hasChild.(Philosopher ⊓

Painter) are not—semantically—the same and there

should not be an equivalence symbol between them.

Thus, Mary’s interpretation has not been

satisfactory. Subsequently, her understanding is not

satisfactory and appropriate.

3 A SEMANTIC MODEL FOR

CONCEPT UNDERSTANDING

In this section I clarify my logical conceptions of

‘concept understanding’. The term ‘understanding’

is very complicated and sensitive in psychology,

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

neuroscience, cognitive science, philosophy and

epistemology. There has not been any complete

model for describing this term, but there are some

proper models of (i) understanding of understanding

(see (Foerster, 2003)), (ii) understanding

representation (see (Peschl and Riegler, 1999;

Webb, 2009)), and (iii) specification of the

components of understanding (i.e., from the

cognition’s and from the affects’ perspectives), see

(Chaitin, 1987; Kintsch et al., 1990; di Pellegrino et

al., 1992; MacKay, 2003; Zwaan and Taylor, 2006;

Uithol et al., 2011; Uithol and Paulus, 2014). This

research—by analysing a formal semantics—focuses

on the junctions of ‘understanding of concept

understanding’ and ‘concept understanding

representation’ in terminological systems, and, more

specifically, it focuses on logical analysis of concept

understanding and its terminological representation.

3.1 Concept Understanding as a

Relation (and Function)

I shall claim that concept understanding—as a

relation—could relate ‘the characteristics and

attributes of a concept’ with ‘a description’. More

specifically, understanding is a function (mapping)

from a concept into some propositions (and

statements) which could be interpreted as ‘concept

descriptions’. In fact, the characteristics and

properties of a concept by means of the

understanding function become mapped into

concept descriptions. Let me be more specific:

A. A human being—by concept understanding—

attempts to map the significant characteristics of

concepts into some concept descriptions. For

example, ‘breathing’—as a biological and

psychological process—is a characteristic and trait

of all animals, and, thus,

breathing (that is a role) is

the characteristic of the concept

Animal. Then, (i)

knowing the fact that the individual

horse is an

instance of the concept

Animal (Formally:

Animal(horse)), and (ii) drawing the [concept

subsumption] inference

‘Horse ⊑ Animal’,

collectively lead us to knowing and to understanding

that ‘horses breathe’ (or equivalently: ‘horses do

breathing’). The role

breathing could be manifested

in the concept Breath. Therefore, (i) and (ii)

collectively lead us to expressing the concept

description ‘

Animal(horse) ⊓ ∃hasTrait.Breath’ for

the individual

horse (as an instance of the concept

Animal), and, respectively, for the concept Horse (as

a sub-concept of Animal).

B. A human being—by concept understanding—

attempts to map the concepts’ properties and their

interrelationships with themselves into some concept

descriptions. For example, the one who knows that

‘male horses breathe’, by taking the terminological

and assertional axioms {

Animal(horse), Horse ⊑

Animal, MaleHorse ⊑ Horse, FemaleHorse ⊑ Horse}

into consideration can know and understand that

‘female horses breathe’ as well.

C. A human being—by concept understanding—

attempts to map the concepts’ properties and their

relationships with other concepts into some concept

descriptions. For example, the one who knows that

‘horses breathe’ (and as described:

Animal(horse) ⊓

∃hasTrait.Breath

), could—respectively—know and

understand that the individual

rabbit (that is an

animal) breathes as well. So, she/he could express

that ‘rabbits breathe’, and, in fact,

Animal(rabbit) ⊓

∃hasTrait.Breath.

Conclusion. Relying on Predicate Logic [and on

DLs], the phenomenon of ‘concept understanding’

could be interpreted as a ‘binary predicate’ [and as a

‘role’] of human beings on expressing some concept

descriptions. Let me represent this role by

‘

understanding’.

3.2 Concept Understanding as a

Conceptualisation

The concept understanding could be interpreted to

be the limit/type of conceptualisation. Accordingly,

humans need to conceptualise concepts in order to

understand them. In (Badie, 2016a) and (Badie,

2016b), I have defined a ‘conceptualisation’ as “a

uniform specification of the separated

understandings”. In fact, any concept understanding

could be interpreted as a local manifestation of a

global conceptualisation. Additionally, human

beings’ grasps of concepts could provide proper

foundations for generating their own

conceptualisations. I shall claim that ‘concept

understanding’ could be acknowledged as a limited

type of humans’ concept constructions, when the

concept constructions are supported by their own

conceptualisations. Therefore,

conceptualising is a

role of human beings. This conclusion—relying on

DLs—could be represented by the ‘role inclusion (or

role subsumption)’

understanding ⊑ conceptualising.

In other words, ‘understanding a concept’ has been

acknowledged as the sub-role of ‘conceptualising

that concept’. On the other hand, it is not the case

that all conceptualisations are understandings.

A Formal Semantics for Concept Understanding Relying on Description Logics

fact, all the conceptualised concepts could not be

understood.

3.3 Concept Understanding as an

Interpretation-based Model

Generally, an interpretation is the act of elucidation,

explication and explanation, see (Simpson and

Weiner, 1989). According to (Honderich, 2005) and

through the lens of philosophy, “…in existential and

hermeneutic philosophy, ‘interpretation’ becomes

the most essential moment of human life: The

human being is characterized by having an

‘understanding’ of itself, the world, and others. This

understanding, to be sure, does not consist—as in

classical ontology or epistemology—in universal

features of universe or mind, but in subjective–

relative and historically situated interpretations of

the social. …”. Regarding (Blackburn, 2016) and

through the lens of logic, an ‘interpretation’ of a

logical system assigns meanings (or semantic

values) to the formulae and their elements. At this

point I shall take into consideration that the

phenomenon of ‘interpretation’ could have a

conjunction with the phenomenon of ‘terminological

interpretation’ in formal languages. More

specifically, the one who has engaged her/his

interpretations to explicate [and justify] what [and

why] she/he means by classifying a

thing/phenomenon as an instance of a concept, needs

to interpret the non-logical signifiers of different

concept descriptions within her/his linguistic

expressions.

Considering any set of non-logical symbols (that

have no logical consequences) in a terminology, a

terminological interpretation over humans’

languages could be described to be constructed

based on the tuple Interpretation Domain,

Interpretation Function. The interpretation domain

(or the universe of interpretation) might be called

‘universe of discourse’. As mentioned in previous

section, the interpretation domain must be non-

empty. This non-empty set forms the range of any

variable that occur in any of the concept descriptions

within linguistic expressions. It’s a fact that the

collection of the rules and the processes that manage

different terms and descriptions in linguistic

expressions, cannot have any meaning until the non-

logical signifiers and constructors are given

terminological interpretations. The interpretations

prepare humans for producing their personal

meaningful [and understandable] concept

descriptions. Hence, I have recognised all ‘concept

understandings’ as ‘concept interpretations’. This

conclusion—relying on DLs—could be represented

by the ‘role inclusion’

understanding ⊑ interpreting.

Therefore, ‘concept understanding’ has been

expressed as the sub-role of ‘concept interpreting’.

But, on the other hand, not all interpretations (of

concepts) imply understandings (of concepts).

Equivalently, it is not the case that all

interpretations are understandings. In other words,

all the interpreted concepts may not be understood.

Accordingly, considering any interpretation as a

function, ‘concept understanding’ is recognised as

an ‘interpretation function’.

From this point I apply the function UND (as a

limit of the interpretation function I)

in my

formalism. Then, C

UND

represents ‘Concept

Understanding’, where C stands for Concept.

Consequently, considering UND as a kind of

interpretation, there exists a tuple like D

understood

, where (i) D

represents the understanding

domain (that consists of the variables that occur in

any of the concept descriptions which are going to

be understood), and (ii) C

understood

is the understood

concept. C

understood

is achievable based on the

understanding function –

UND

. Relying on the

function –

UND

⊆ C

⊆ ∆

UND

⊆ ∆

It shall be stressed that D

UND

expresses

‘understanding all concepts belonging to the

understanding domain’. Note that –

UND

(that is a

function) can provide a model for a terminological

[and assertional] axiom. Therefore, the desired

model (i) is a restricted form of a terminological-

interpretation-based model, and (ii) can satisfy the

semantics of the terminological and assertional

axioms (read ‘UND ⊨ Axiom’: UND satisfies the

axiom), see Table 4. Consequently:

UND

⊆ C

⊆ ∆

& -

UND

: C → C

UND

Where: C

UND

⊆ D

UND

⊆ ∆

I shall emphasise that we are not able to conclude

that C

⊆ D

UND

. On the other hand, we certainly

know that , C

UND

⊆ ∆

(because C

UND

⊆ C

and C

⊆ ∆

). According to the analysed characteristics, the

UND understanding model in my terminology is

constructed based upon the tuple Understanding

Domain, Understanding Function as:

UND = D

UND

, -

UND

.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

Table 4: Understanding Model: Terminologies, World

Descriptions and their Semantics.

Name Description, Semantics

Understanding a

Concept Inclusion

UND ⊨ (C ⊑D) ⇒

UND

⊆ D

UND

Understanding a

Role Inclusion

UND ⊨ (R ⊑S) ⇒

UND

⊆ S

UND

Understanding a

Concept Equality

UND ⊨ (C ≡D) ⇒

UND

= D

UND

Understanding a

Role Equality

UND ⊨ (R ≡S) ⇒

UND

= S

UND

Understanding a

Concept Assertion

UND ⊨ C(a) ⇒

UND

∈ C

UND

Understanding a

Role Assertion

UND ⊨ R(a, b) ⇒

UND

, b

UND

) ∈ R

UND

Table 5 is based on Table 4 and itemises inductive

concept descriptions and their semantics as the

products of the understanding model.

Table 5: Understanding Inductive Concept Descriptions.

Model Satisfies the

Vocabulary

Semantics

UND ⊨ ⊤

⊤

UND

= ⊤

UND ⊨ ⊥ ⊥

UND

= ∅

UND ⊨ ¬R (¬R)

UND

= ⊤ \ R

UND

UND ⊨ ¬C (¬C )

UND

= D

UND

\ C

UND

UND ⊨ (R ⊓ S) (R ⊓ S)

UND

∩ S

UND

UND ⊨ (C ⊓ D) (C ⊓ D)

UND

= C

UND

∩ D

UND

3.4 Concept Understanding as a

Product of Functional Roles

How could we employ DLs in order to describe an

understanding function as a [functional] role of a

human being? Let me interpret functional roles

(features) as the roles that are existentially functions,

and, thus, they can express functional actions,

movements, procedures and manners of human

beings. Let N

be a set of functional roles and N

the set of role [descriptions]. Obviously: N

⊆ N

and informally, functional roles are some kinds of

roles.

Lemma. The UND understanding model is–

semantically—structured over:

a. the understanding domain (or D

b. the understanding function (or -

UND

), and

c. the set D

UND

(or equivalently, the effect of the

understanding function -

UND

on the Top

concept)

that represents understanding all

atomic concepts (everything) in the

understanding domain.

Analysis. The UND understanding model associates

with each atomic concept a subset of D

UND

, and

with each ordinary atomic role a binary relation over

UND

× D

UND

. Note that any functional role can be

recognised as a partial function. More specifically,

considering F = f

○ ··· ○ f

(F is a chain of

functional roles), the chain f

UND

○ ··· ○ f

UND

represents the composition of n partial

understanding functions. In fact, by employing

UND, any f

UND

—semantically—supports the

[overall] functional role F

UND

. Note that for all i in

(1,n), f

i+1

produces the input of f

. Therefore, the

understanding of f

i+1

(the output of f

i+1

) provides the

input of the understanding of f

. In particular, any

concept description could be understood over the

subsets of D

UND

. This characteristic is very useful

in making a strong linkage between the terms

‘understanding’ and ‘chain of functional roles’. It

supports my semantic model in scheming and

describing "the understanding as the product of a

chain of functional roles, where the functional roles

are the partial understanding functions". You will

see how it works.

3.5 Humans’ Functional Roles through

SOLO’s Levels

According to (Biggs and Collis, 2014), the Structure

of Observed Learning Outcomes (SOLO) taxonomy

is a proper model that can provide an organised

framework for representing different levels of

humans’ understandings. This model is concerned

with various complexities of understanding on its

different layers. According to SOLO and focusing

on humans’ levels of knowledge with regard to a

concept, we have:

 Pre-structured knowledge. Here humans’

knowledge of a concept is pre-structured (and is

the product of their pre-conceptions).

 Uni-structured knowledge. Humans have a

limited knowledge about a concept. They may

know one or few isolated fact(s) about a

concept.

 Multi-structured knowledge. They are getting to

know a few facts relevant for a concept, but

they are still unable to link and relate them

together.

 Related Knowledge. They have started to move

towards deeper levels of understanding of a

concept. Here they are able to link different

facts and to explain several conceptions of a

concept.

A Formal Semantics for Concept Understanding Relying on Description Logics

 Extended Abstracts. This is the most

complicated level. Humans are not only able to

link lots of related conceptions [of a concept]

together, but they can also link them to other

specified and complicated conceptions. Now

they are able to link multiple facts and

explanations in order to produce more

complicated extensions relevant for a concept.

Obviously, the extended abstracts are the products of

deeper comprehensions of related structures. Related

structures are the products of deeper

comprehensions of multi-structures. The multi-

structures are the products of deeper

comprehensions of uni-structures, and the uni-

structures are the products of deeper

comprehensions of pre-structures. Let me select a

process (as a sample of humans’ functional roles)

from any of the SOLO’s levels and formalise it.

According to SOLO, creation (with regard to an

understood concept) is an instance of ‘extended

abstracts’, justification (with regard to an understood

concept) is an instance of ‘related structures’,

description (with regard to an understood concept) is

an instance of ‘multi-structures’ and identification

(with regard to an understood concept) is an instance

of ‘uni-structures’. Therefore, Creation,

Justification, Description and Identification are four

processes which could be analysed as functions in

the model. Any of these functions can support a

functional role as a ‘partial understanding function’:

i. Creation has interrelatedness with

creatingOf

that is a functional role and extends the humans’

mental abstracts.

ii. Justification has interrelatedness with the

functional role

justifyingOf that relates the lower

structures.

iii. Description has correlation with the

functional role

describingOf that produces the multi-

structures.

iv. Identification has correlation with the

functional role

identifyingOf that generates the uni-

structures.

It shall be emphasised that

identifyingOf,

describingOf, justifyingOf and creatingOf are only

four examples of functional roles within SOLO’s

categories, and, in fact, the SOLO’s levels are not

limited to these functions. For example,

followingOf

and

namingOf are two other instances of uni-

structures,

combiningOf and enumeratingOf are two

other instances of multi-structures,

analysingOf and

arguingOf are two other instances of related

structures, and formulatingOf and theorisingOf are

two other instances of extended abstracts.

As mentioned, the functional roles

creatingOf,

justifyingOf, describingOf and identifyingOf represent

the equivalent roles of the creation, justification,

description and identification functions respectively.

Furthermore, these functions are the partial functions

of the understanding function. Obviously, the

understanding function (that is a process) could also

be considered to be equivalent to a functional role

understandingOf. Employing the ‘role inclusion’

axiom we have: (1)

creatingOf ⊑ understandingOf,

(2)

justifyingOf ⊑ understandingOf, (3) describingOf ⊑

understandingOf

, and (4) identifyingOf ⊑

understandingOf. Equivalently: (1) creation ⊆

understanding, (2) justification ⊆ understanding,

(3) description ⊆ understanding, and (4)

identification ⊆ understanding.

It shall be claimed that understandingOf—

conceptually and logically—supports ‘the

understanding function based on the analysed

understanding model (or UND)’. Similarly, we can

define CRN, JSN, DSN and IDN as sub-models of

UND for representing creation, justification,

description and identification respectively. Any of

these models can—semantically—satisfy the

terminologies and world descriptions in Table 4.

Accordingly—relying on inductive rules—they can

satisfy concept descriptions in Table 5.

Note that CRN (as a model) fulfils the desires of

UND better (and more satisfying) than JSN, DSN

and IDN. Considering D

as the understanding

domain, we have:

UND

⊆

CRN

⊆

JSN

⊆

DSN

⊆

IDN

More specifically:

 D

CRN

represents the model of creation over the

understanding domain. It consists of concepts

which are (or could be) ‘created’ by human

beings. Formally: C

CRN

∈ D

CRN

 D

JSN

represents the model of justification over

the understanding domain. It consists of

concepts which are (or could be) ‘justified’ by

human beings. Formally: C

JSN

∈ D

JSN

 D

DSN

represents the model of description over

the understanding domain. It consists of

concepts which are (or could be) ‘described’ by

human beings. Formally: C

DSN

∈ D

DSN

 D

IDN

represents the model of Identification

over the understanding domain. It consists of

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

concepts which are (or could be) ‘identified’ by

human beings. Formally: C

IDN

∈ D

IDN

Proposition. The terminological axioms and the

world descriptions (in Table 4) and inductive

concept descriptions (in Table 5) are all valid and

meaningful for CRN, JSN, DSN and IDN. Therefore,

inductive concept descriptions are also valid and

meaningful over the concatenation of the creation,

justification, description and identification functions

that have supported these terminological models.

Proposition. All satisfactions based on IDN are

already satisfied by DSN, JSN and CRN over

DSN

JSN

and D

CRN

respectively. Informally, if a human

being is able to describe, justify and create with

regard to her/his conception of a concept, so, she/he

is already capable of identifying that concept.

Furthermore, she/he might be able to identify

something else with regard to her/his conception of

that concept.

Formal Analysis. The semantics of the composite

function ‘creation (justification (description

(identification (C))))’—that is the product of the

chain of functional roles—supports the proposed

semantic model on D

UND

, which is the central

domain of the understanding (central part of the

understanding domain). Considering all the roles

relevant for the concept C, we have:

1. (∀R

.C)

CRN

{ a ∈ D

CRN

| ∀b.(a,b) ∈ R

CRN

→ b ∈ C

CRN

Therefore:

2. (∀R

.C)

JSN

{ a ∈ D

JSN

| ∀b.(a,b) ∈ R

JSN

→ b ∈ C

JSN

Therefore:

3. (∀R

.C)

DSN

{ a ∈ D

DSN

| ∀b.(a,b) ∈ R

DSN

→ b ∈ C

DSN

Therefore:

4. (∀R

.C)

IDN

{ a ∈ D

IDN

| ∀b.(a,b) ∈ R

IDN

→ b ∈ C

IDN

In the afore-itemised formalism R

, R

and R

stand for creatingOf, justifyingOf, describingOf and

identifyingOf respectively. Consequently, CRN, JSN,

DSN and IDN have been observed as roles of human

beings. Accordingly, it’s possible to represent the

chain of functional roles in the form of a collection

of implications as following:

(∀R

.C)

CRN

⇒ (∀R

.C)

JSN

⇒ (∀R

.C)

DSN

⇒

(∀R

.C)

IDN

It must be concluded that ‘any role based on a

conception of C’ to the left of any of arrows makes a

logical premise for ‘other roles based on conceptions

of C’ to the right of that arrow. It shall be stressed

that this is a very important terminological fact. The

concluded logical relationship represents a flow of

concept understanding from deeper layers to

shallower layers.

4 AN ONTOLOGY FOR

CONCEPT UNDERSTANDING

According to (Grimm et al., 2007; Staab and Studer,

2009), an ontology—from the philosophical point of

view—is described as studying the science of being

and existence. Ontologies must be capable of

demonstrating the structure of the reality of a

thing/phenomenon. They check multiple attributes,

particularities and properties that belong to a

thing/phenomenon because of its natural and

structural existence. An ontology—from another

perspective and through the lenses of information

and computer sciences—is described as an explicit

and formal specification of a shared

conceptualisation in a domain of interest. However,

in my opinion, there could be very strong

relationship between these two descriptions of

ontologies. In fact, ontologies in information

sciences attempt to mirror the things’/phenomena’s

structures in virtual and artificial systems. The

ontological descriptions in information sciences

tackle to provide appropriate logical and formal

descriptions of a phenomenon [and of its structure]

considering various concepts relevant for that

phenomenon. From this perspective, an ontology can

be schemed and demonstrated by semantic networks

and semantic representations. A semantic network is

a graph whose nodes represent concepts (e.g., unary

predicates) and whose arcs represent relations (e.g.,

binary/n-ary predicates) between the concepts.

Accordingly, semantic networks provide structural

representations of a thing/phenomenon. In Figure 1 I

have designed a semantic network as an ontology for

‘concept understanding’. This hierarchical semantic

representation, (1) specifies the conceptual

A Formal Semantics for Concept Understanding Relying on Description Logics

relationships between the most important ingredients

of this research, (2) demonstrates the logical

representation of concept understanding. It shows

how the proposed model attempts to represent

concept understanding. This semantic representation

can be interpreted as a specification of the shared

conceptualisation of concept understanding within

terminological systems. The proposed ontology can

be reformulated and formalised in ALC in the form

of a collection of fundamental terminologies as

following:

A Formal Ontology for Concept Understanding.

{

UnaryPredicate ⊑ Predicate, BinaryPredicate ⊑

Predicate, Concept ⊑ UnaryPredicate, Concept ⊑

∃hasInstance.Individual, BinaryPredicate ⊑

(∃hasNode.Individual ⊓ ∃hasNode.Individual), Role

⊑ BinaryPredicate, Relation ⊑ BinaryPredicate,

Function ⊑ Relation , Interpretation ⊑ Function,

Conceptualisation ⊑ Function, ConceptUnderstanding

⊑ Interpretation, ConceptUnderstanding ⊑

Conceptualisation, PartialFunction ⊑ Function,

FunctionalRole ⊑ Role, FunctionalRole ⊑

hasEquivalence.PartialFunction, FunctionalRole ⊑

Function, SubModel ⊑ Model, SemanticModel ⊑

Model, InterpretationSemanticModel ⊑

SemanticModel, UnderstandingSemanticModel ⊑

SemanticModel, UnderstandingSemanticSubModel ⊑

SubModel, UnderstandingSemanticSubModel ⊑

SemanticModel, InterpretationSemanticModel ⊑

∃hasSupport.Interpretation,

UnderstandingSemanticModel ⊑

∃hasSupport.InterpretationSemanticModel,

UnderstandingSemanticModel ⊑

∃hasSupport.UnderstandingSemanticSubModel,

UnderstandingSemanticSubModel ⊑

∃hasSupport.FunctionalRole

}

Figure 1: An Ontology for Concept Understanding.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

5 CONCLUSIONS

The readers of this article may ask “if the term

‘understanding’ in this research is related to the real

human beings, or if this research’s domain is only

information and computer sciences?” Actually,

that’s why I have employed Description Logics.

Under a plethora of names (among them

terminological systems and Concept Languages),

Description Logics (DLs) attempt to provide

descriptive knowledge representation formalisms

based on formal semantics to establish common

[conceptual and logical] grounds and

interrelationships between human beings and

machines. Description Logics supported me in

revealing some hidden conceptual assumptions that

could support me in having a better understanding of

‘concept understanding’. DLs—by considering

concepts as unary predicates and by applying

terminological interpretations over them—have

proposed a realisable logical description for

explaining the humans’ concept understanding. The

central contribution of the article has been providing

a formal semantics for logical analysis of concept

understanding. According to the logical analysis, a

background for terminological representation of

concept understanding has been expressed.

Consequently, a semantic representation [as an

ontology and a specification of the shared

conceptualisation of ‘concept understanding’] has

been designed and formalised.

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