ON THE INTEGRATION OF KNOWLEDGE IN A PROPOSITIONAL
LOGICAL LAYER
Sebastian Bab
Technische Universit¨at Berlin, Fakult¨at IV, Sekretariat FR 3-2, Franklinstr. 28/29, 10587 Berlin, Germany
Keywords:
Propositional knowledge, Knowledge integration, Logic of integration, Propositional logics.
Abstract:
In general knowledge is complex which means that it is not an isolated entity, but a phenomenon which is
coming (or can be derived) from different sources of knowledge. The position of the present paper is that
it can be highly beneficial to study the integration of knowledge coming from different knowledge sources
as an explicit propositional logical layer in knowledge engineering. Here the term of proposition is to be
understood in the understanding of Frege as the inherent sense of a formal expression. We discuss a certain
family of propositional logics – the so-called ∈
T
-logics – which allow for an explicit interpretation of formulas
as propositions. We argue that the family of ∈
T
-logics and their models offer a very expressive logical setting
which is able to realize a certain scenario of the integration of knowledge from different knowledge sources.
1 MOTIVATION
The initial observation on which the position of the
present paper is based is that in the general case
knowledge is complex and thus not appearing in an
isolated form, but in a form of combination and inte-
gration of information entities coming from different
sources. The thesis of this paper is that an explicit
study of the propositional logical conditions of com-
binations and integrations of knowledge can be highly
beneficial for achieving a sense (in the Fregean under-
standing of the term) related form of integration and
to reproduce natural ways of combining and integrat-
ing knowledge adequately.
Most approaches to the integration of knowl-
edge and information in today’s existing information
systems include the use of certain ontological con-
cepts. An integration of heterogeneous information
or knowledge entities is realized by breaking down
the entities to the smallest concepts of the domain
of discourse given by the ontology. The integration
then takes place on this level of the ontology by the
identification of similarities or relationships between
the single parts of the information entities (compare
for example with the ideas of ontology-based data
integration). The uses of ontologies include, be-
yond others, single ontology and multiple ontology
approaches. In the first case one single ontology is
used for the description of concepts, while in the sec-
ond case a combination of multiple ontologies is used
which requires for the definition of adequate map-
pings between the involved ontologies (see (Wache
et al., 2001)). It is one important observation for the
idea of the present paper that such a definition of map-
pings between ontologies must include certain logical
considerations to guarantee a consistent cooperation
of the involved ontologies.
It is the position of this paper that the described
ontology based approaches of breaking down entities
to their smallest concepts does not reflect the natural
way in which integrations of knowledge take place.
Furthermore ontologies are very fixed and thus rather
inflexible concepts which do not allow for sudden
context dependent changes of the meanings of certain
concepts. However, such context dependent changes
in the conceptions of things are very natural in the
human way of knowledge exchanges, negotiations, or
discussions.
1
The position of this paper is that due to
that naturality it is of interest and necessity to study
ways for formal describing and reasoning
2
about in-
tegrations of knowledge which first of all take the in-
herent senses of entities and their relationships on a
semantical level into account.
In computer science one can observe two con-
ceptions of integration which can be characterized as
bottom-up or top-down approaches. The first case
1
Compare for example with the explanations on concep-
tions of Mahr in (Mahr, 2010a).
2
Compare with (Davis et al., 1993, Role 3).
299
Bab S..
ON THE INTEGRATION OF KNOWLEDGE IN A PROPOSITIONAL LOGICAL LAYER.
DOI: 10.5220/0003688902990303
In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2011), pages 299-303
ISBN: 978-989-8425-80-5
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

corresponds to the described approach of ontologies
where one defines the domain of discourse first as a
fine granulated basis for the understanding of entities
and where integration of heterogeneous entities origi-
nates from this fine granulated basis. The second case
of top-down approaches to integration is more close
to the human thinkings, as it takes the relationships
between complex concepts of different entities into
account first, and only afterwards breaks down the re-
lationships to more granular units.
If one attempts to integrate entities of a different
form from a logical point of view one does not have
to specify the nature of every single information unit
in the first place, but is more interested in the logical
characteristics of the integration, that means the rela-
tionships between complex concepts. The similarity
to the concept-driven thinking of the humans gives a
justification for an additional logical consideration of
the integration of knowledge in knowledge engineer-
ing.
Consider for example the development of a new
car model. Here the variety of knowledge ranges from
aspects of the molecular nature of the tires of the car
to questions of the included sound system. Thus the
variety of knowledge is so huge and divers and pro-
vided with multiple different languages and logical
principles that it cannot be adequately covered in one
view. It is an obvious observationthat in the industrial
development an integration of the different views of
the developers on the car does not start from breaking
down their concepts to the smallest bits of informa-
tion, but in fact by integrating their views on a logical
level of concepts. Although a multiple ontology ap-
proach to this example can be successful and as well
must include logical considerations in the definitions
of the mappings between the ontologies, these logical
considerations are not explicitely given and it cannot
be reasoned about the logical characteristics of the in-
tegration of concepts of the several views. Thus there
is a need for a logical layer in the field of knowledge
integration.
In this paper we discuss a formal treatment of
knowledge integration in such a logical layer which
describes integration over different sources. As men-
tioned before such a logical layer must take the in-
herent senses of entities (or concepts) explicitely into
account. To do this we understand the term of
knowledge as propositional knowledge in the sense of
Searle (see (Searle, 2008)). Here the notion of propo-
sition can be defined as follows
3
:
A proposition is the inherent sense of a formal
expression. Propositions can be either true or false.
3
For detailed essays on the notion of propositions see for
example (Mahr, 2010b) and (Robering, 2011).
Here the notion of sense is used in the understand-
ing of Frege as the inherent idea of a formal expres-
sion (see for example (Frege, 2008; Frege, 2007)). In
the following sections we will define a scenario of
integration of knowledge and information which can
be seen as a setting allowing for the studying of cer-
tain kinds of integrations. We discuss a certain fam-
ily of propositional logics, the family of ∈
T
-logics.
In the understanding of the present paper a proposi-
tional logic is not to be understood in the classical way
where expressions are directly evaluated to truth val-
ues, but as a logic in which formulas are explicitely
interpreted as propositions and where these proposi-
tions are explicitely available as entities in the seman-
tics of the logics.
It is the position of the present paper that the fam-
ily of ∈
T
-logics and their models offer a very expres-
sive logical tool which can be used to realize the sce-
nario and thus a wide range of information and knowl-
edge integrations. Here the integration over differ-
ent sources of knowledge is realized by a represen-
tation of the knowledge sources as models of certain
propositional logics. These models are integrated into
an ∈
T
-logic which acts as a meta-level language for
the description of and reasoning about the underly-
ing sources of knowledge, their relationships and their
integration. More practical questions like for exam-
ple questions of a representation of concrete existing
sources of knowledge in certain propositional logics
shall not be dealt with in the present paper.
2 SCENARIO OF INTEGRATION
OF KNOWLEDGE AND
INFORMATION
Part of the position of this paper is that any real
knowledge integration can be adequately comprised
by the following scenario of integration which is a
generalization and advancement of the work of (Mahr
and Bab, 2005). The goal of the following scenario
is not to give a definition of the notion of integration
in knowledge engineering, but to state a formal set-
ting which allows for the studying and reproduction
of the results of knowledge integrations in a logical
fundament.
For a studying of knowledge integrations in a log-
ical setting two different cases are conceivable which
both shall be represented in our scenario. The first
is to assume that we have given a complex integrated
knowledge object and that we logically describe and
reason about the integration of the knowledge sources
which has taken place and which led to the integrated
KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development
300

object. The second is to assume to have given a cer-
tain set of knowledge sources on an object, but not
the integrated knowledge object itself. The task in
this second case is not only to describe an already ex-
isting integration of different sources of knowledge,
but to actually perform the integration in the logical
setting. In this case questions of consistency arise
which have to be dealt with in the integration, while
in the first case with a given integrated knowledge ob-
ject the different knowledge sources on the object can
be considered as consistent as there already has taken
place a successful integration of the different sources
of knowledge.
The scenario of integration is defined as follows:
Scenario of Integration of Knowledge and
Information. Given different views V
1
, . . . ,
V
n
on a complex knowledge object A. Each
view V
i
represents a source of knowledge for
the object A. Each view can be identified by a
model of a certain propositional logic. Each of
these models includes a set of propositions by
explicitely interpreting any formula of the log-
ics as true or false propositions. Each of these
sets of propositions represents the knowledge
of the corresponding view on the object A.
The goal of integration is to define a meta-
level propositional logic which offers means
of expression to describe and reason about that
certain set of propositions which is represent-
ing the integrated knowledge of the single sets
of propositions given by the single models of
the views on the object A.
It is the position of the present paper that the pro-
posed scenario of integration of knowledge and in-
formation is adequate to cover most of the concepts
of integration in knowledge engineering. An explicit
interpretation of the sources of knowledge on a com-
plex object A as certain sets of propositions and the
construction of new propositions stating any relation-
ships between the single propositions results in such
a set of propositions which represents the whole inte-
grated knowledge on the object A. The construction
of a propositional logic which allows for the descrip-
tion of and reasoning about the resulting set of propo-
sitions has the advantage that even complex concepts
(represented in certain complex formulas) do not have
to be broken down to their atomic parts, but do them-
selves denote certain propositions in the models of the
propositional logic.
We will state a possible realization of the scenario
of integration of knowledge and information based on
the concepts of propositional logics and ∈
T
-logics in
Section 4.
3 A SHORT OUTLINE OF
∈
T
-LOGICS
In the following section we want to discuss a possible
realization of the scenario given in the previous sec-
tion using the logics of the family of ∈
T
-logics. Every
∈
T
-logic has a modeltheoretic semantics where the
semantic entities are given by propositions. To get a
better understanding of the ideas of ∈
T
-logics we will
now give a short outline of the most important works
in this field.
∈
T
-logics are propositional logics offering means
for formulating self-referential and explicit truth
statements while preserving a total truth predicate and
thus being free from antinomies. The first logic in this
field is the classical ∈
T
-logic by Str¨ater (see (Str¨ater,
1992)) which forms a theory of truth and proposi-
tions in the context of the re-construction of natu-
ral language semantics by means of self-referential
structures (see for example (Mahr et al., 1990; Mahr,
1993; Bab et al., 2008)). The expressions of classical
∈
T
-logic are built over propositional variables, con-
stants, classical connectives, means for the quantifi-
cation over propositional variables, as well as means
for propositional equivalence and predicates for truth
and falsity.
The works of Str¨ater were picked up by Zeitz in
(Zeitz, 2000) where he extends Str¨aters concept in
the way that every ∈
T
-logic can extend an arbitrary
underlying object-level logic (represented in a cer-
tain abstract form) whose components become con-
stants in the corresponding ∈
T
-logic. In this context
Zeitz’ ∈
T
-logics can be interpreted as a logic theory
for the meta-level reasoning over propositions origi-
nating from underlying arbitrary logics.
In (Bab, 2007) Bab generalized and newly inter-
preted the work of Zeitz in the way that his class of
∈
µ
-logics subsumes the concepts of Zeitz ∈
T
-logics,
but furthermore allows for the integration of modali-
ties from arbitrary modal logics as means of expres-
sion in a Kripke-like semantics. Thus, until this point,
∈
µ
-logics are the most expressive and general logics
in the field of ∈
T
-logics allowing for the reasoning
about situation dependent propositions. Moreover,
Bab gave a new interpretation of ∈
T
-logics as a theory
of propositions.
Str¨ater, Zeitz, and Bab stated different model ex-
istence theorems which prove that all ∈
T
-logics are
free from antinomies despite their total truth predi-
cates and their ability to model self-referential sen-
tences and impredicative quantification. ∈
T
-logics
have been proven to be a suitable concept for truth
and reference, because they avoid antinomies which
necessarily appear with logics having total truth pred-
ON THE INTEGRATION OF KNOWLEDGE IN A PROPOSITIONAL LOGICAL LAYER
301

icates and at the same time allow for representa-
tions of decidable relations and computable functions
(see (Tarski, 1935)). Furthermore ∈
T
-logics allow
for an implicit representation of antinomies like the
liar paradox over unsatisfiable propositional equa-
tions (see for example (Str¨ater, 1992; Zeitz, 2000;
Bab, 2007)).
The theory of intensional models of ∈
T
-logics was
widely extended by the work of (Bab and Wieczorek,
2010) which shows that ∈
T
-logics offer a wide range
of different intensional models. Furthermorethere ex-
ist sound and complete calculi for the ∈
T
-logics by
Str¨ater and Zeitz.
The interpretation of formulas as propositions in
∈
T
-logics is achieved by a sense function which is
part of every model of every ∈
T
-logic. Here the in-
terpretation of formulas as propositions is not arbi-
trary, but must comply with certain natural rules and
concepts of logic. The models of an ∈
T
-logic define
the range of propositions which can be denoted by
formulas. However, the interpretation of formulas as
propositions is free in the sense, that it can differ from
one model to the other. Thus ∈
T
-logics offer logical
concepts which allow for the studying of the inherent
senses of formulas in different contexts. By using the
situation dependency of, for example, the class of ∈
µ
-
logics, one gets a rich logical tool for a formal treat-
ment of integrations.
Another logic in the family of ∈
T
-logics, but of
a different nature, was defined by Lewitzka in (Le-
witzka, 2009). His ∈
I
-logic offers a non-Fregean in-
tuitionistic logic with a truth predicate and a falsity
predicate as intuitionistic negation. ∈
I
-logic can be
seen as an extension of the works of Zeitz, but with-
out using the concept of quantification.
4 DESCRIBING AND
REASONING ABOUT
KNOWLEDGE INTEGRATIONS
IN ∈
T
-LOGICS
In the following we want to state a possible realiza-
tion of the scenario given in Section 2. It is the posi-
tion of the present paper that the family of ∈
T
-logics
and their models offer a very expressive logical set-
ting which is able to realize the scenario and thus
a wide range of information and knowledge integra-
tions. The proposed realization of the scenario can be
achieved in the following way:
1. In a first step the different views on the object A
must be represented by models of certain propo-
sitional logics which offer sets of true and false
propositions describing the propositional knowl-
edge which can be said about A from the point of
the corresponding views. These model construc-
tions can be accomplished by models of certain
∈
T
-logics of appropriate expressive power.
2. In a second step the propositions of the single
views have to be integrated to form an overall set
of propositions representing the integrated knowl-
edge on A. This integration of propositions can
take place in a certain model of an ∈
T
-logic acting
as an integration logic in which single proposi-
tions are combined to more complex propositions
according to logical connectives. The ∈
T
-logic
here acts as the integrator performing the integra-
tion depending on certain specifications of the as-
pired method of integration.
In this step inconsistencies between the single
views become directly obvious if there exists for
example a proposition which occurs to be a true
proposition in the one view and a false proposi-
tion in the other. In this case we have a proposition
which is true in the context of one view and false
in the context of another view which cannot be
the case in any consistent integration process. In
the following we assume that the different sources
of knowledge resp. their representation as sets of
propositions of the corresponding views are con-
sistent.
3. The goal of integration can then be accomplished
in a third step by defining a certain ∈
T
-logic
which offers means of expression to state and to
reason about any of the propositions of the inte-
grated knowledge on A.
When looking at the realization one could argue
that the integration logic of the second step would be
sufficient to act as the one logic which meets the goal
of integration as stated in the scenario of integration
of knowledge and information. In the sense of this pa-
per, however, there is an important difference between
the logic which performs the integration and a logic
which can describe or reason about the integrated
range of propositions from a meta-level perspective.
A logic of the latter sense can represent an integration
in a much more general and common way which is
due to the observation that in certain integration situ-
ations there is a need to regard an integrated object in
a language which is completely independent from the
languages of the single views to be integrated.
The existence of sound and complete calculi for
the ∈
T
-logics by Str¨ater and Zeitz and the proofs of
the existence of certain intensional models for every
∈
T
-logic can be seen as an indicator that the proposed
realization of the scenario using the family of ∈
T
-
KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development
302

logics is a reasonably approach to an explicit logical
handling of integrations.
5 CONCLUSIONS AND FURTHER
WORK
In the present paper we argued that a study of integra-
tions of knowledge from different knowledge sources
in an explicit propositional logical layer is a natural
approach which can be highly beneficial to enrich the
studies of the integration of knowledge in knowledge
engineering.
The proposed scenario of integration of knowl-
edge and information is not limited to the field of
knowledge engineering, but can be taken as a basis for
a general understanding of integrations in computer
science. The family of ∈
T
-logics allows for the meta-
level reasoning about integrations like knowledge in-
tegration by integrating the sets of propositions which
represent the states of affair of the underlying differ-
ent sources of knowledge. The explicit interpretation
of formulas as propositions in the ∈
T
-logic layer thus
allows for an integration which explicitely takes the
senses of the objects and the logical characteristics of
the integration, that means the relationships between
complex concepts, into account. The wide range of
different models of ∈
T
-logics allows for representing
and comparing arbitrary different ideas of integration
as any model includes its own set of denotable propo-
sitions and its own sense function which interprets the
formulas as propositions.
The presented ideas of this work can be seen as a
basis for a treatment of knowledge integrations which
extends the established regarded concepts by an ex-
plicit propositional logical layer. However, the pro-
posed ideas of a scenario of integration and its realiza-
tion in propositional logics like ∈
T
-logics does have
to be elaborated in detail, which is part of ongoing
and soon to be published work of the author. Beyond
that studies on a theoretical logical framework which
can be seen as the main attention of the work of the
author more questions emerge from a more practical
point of view. These include questions of the practical
representation of specific knowledge sources as mod-
els of certain propositional logics or the elaboration
of a real application in form of a case study.
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