1
Semantic Web Ontology and Natural Language
from the Logical Point of View
Marie Duží
Department of Computer science
VSB – Technical university of Ostrava
17. Listopadu 15
708 33 Ostrava – Poruba
E – mail: marie.duzi@vsb.cz
Abstract
The development of Semantic Web ontology languages in the last decade can be
characterised as “from markup languages to metadata”, i.e., a bottom up approach. The
current ontology languages are first briefly described and characterised from the logical
point of view. Generally, these languages are based on the firstorder predicate logic (FOL)
enriched with ad hoc higherorder constructs wherever needed. FOL is a mathematical logic,
and its language is stenography for mathematics with nice mathematical properties.
However, its expressive power is not rich enough to render the semantics of natural
languages, in which web users need to communicate. We argue that in the Semantic Web a
rich expressive language with transparent semantics is needed, in order to build up metadata
in the conceptual level of the Semantic Web architecture, to formally analyse natural
language and to conceptually analyse the content of the Web. A powerful logical tool of
transparent intensional logic (TIL) is described, which provides a logicalsemantic
framework for a finegrained knowledge representation and conceptual analysis of natural
language. TIL is based on a rich ontology of entities organised in an infinite ramified
hierarchy of types. The conceptual and terminological role of TIL in a multiagent world is
described, and we show that such a system can serve as a unifying logical framework both for
natural language representation and for an adequate knowledge representation. Concluding
we define the notion of inferable knowledge and show that the proposed logic accommodates
philosophical desiderata that should be met in a multiagent world of the Semantic Web.
1. Introduction
The Web was proposed as a tool for representing relationships between named objects,
drawing together knowledge from scattered systems into a common framework [2]. The main
aim of the Semantic Web initiative is to develop the current Web towards the original
proposal. W3C’s Semantic Web Activity develops standards and technologies, which are
designed to help machines to understand more information on the Web [12, 24]. The word
“semantic” in the context of Semantic Web is said to mean “machineprocessible” [4]. The
main idea is of having the data on the Web defined and linked in such a way that it can be
used for more effective information retrieval, knowledge discovery, automation, integration
and reuse of information across various applications, organisations and communities.
To meet these goals, the World Wide Web Consortium (W3C) has defined a layer
model for the Semantic Web (Figure 1), and knowledge representation and ontology
languages are being developed. The Semantic Web is vitally dependent on a formal meaning
assigned to the constructs of its languages. For Semantic Web languages to work well
together their formal meanings must employ a common view (or thesis) of representation
2
[13], otherwise it will not be possible to reconcile documents written in different languages. A
common underpinning is especially important for the Semantic Web as it is envisioned to
contain several languages, as in Tim BernersLee's “layer cake” diagram (Figure 1) first
presented at XML 2000 [3]. The diagram depicts a Semantic Web Architecture in which
languages of increasing power are layered one on top of the other. Unfortunately, the
relationships between adjacent layers are not specified, either with respect to the syntax or
semantics. Naturally but unfortunately, the model is being gradually realised in a bottom up
way; languages in particular layers come into being in a rapid but ad hoc way, without a deep
logical insight. Thus the languages often lack an exact semantics and ontological definitions
of particular entities; in the syntactic constructs, particular abstract levels are mixed together.
Figure 1 Semantic Web architecture
In philosophy, the notion of ontology has been understood as covering the ‘science of
being’, raising questions like “what is, what could be, or cannot be there, in the world, what
we can talk about”. In informatics, the notion ‘ontology’ is nowadays understood in many
distinct ways: formalisation, conceptual analysis, hierarchical classification,
conceptualisation, etc. In general, ‘ontology’ can be conceived of as a conceptual analysis of a
given universe of discourse, i.e., of what (which entities) we talk about, and how, by means of
which concepts we capture these entities. However, we are going to warn against confusing
the two levels: the level of using entities⎯ concepts and/or functions, and the level of
mentioning them. Such confusion is a source of never ending discrepancies and
misunderstandings. At the same time, the analyses have to take into account the fact that we
operate in a multiagent world.
Logical forms rendering the basic stock of explicit knowledge of a particular agent
serve as the base, from which logical consequences can be derived by an inference machine
so that to obtain the inferable stock of knowledge [8]. The more finegrained the analysis is,
the more accurate inferences the machine can perform. In the ideal case, the inference
machine can in principle derive just the logical consequences of the base: it does not over
infer (derive something that does not follow), nor underinfer (not being able to derive
something that does follow).
We show that the current stateofarts is far from the ideal case. Web ontology
languages are based on the 1
st
order predicate logic (FOL). Though FOL has become
stenography of mathematics, it is not expressive enough when used at the natural language
Trust level
Digital signature, annotation
Logical and inference level
Rulebased systems, SWRL, ML,
Ontology level
OWL, WordNet, RosettaNet, …
Metadata level
RDF, RDFS, SKIF, …
Structure level
XML, XLink, XML Schema, …
Internet level
Unicode, URI, …
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area. The obvious disadvantage of the FOL approach is that treating higherorder properties
and relations like individuals conceals the ontological structure of the universe, and
knowledge representation is not comprehensive. Moreover, when representing knowledge,
paradox of omniscience is inevitable: this is a critical defect with respect to a multiagent
system, which may lead to inconsistencies and chaotic behaviour of the system. For
applications where even the full power of FOL is not adequate, it would be natural to extend
the framework to higherorder logic (HOL). A general objection against using HOL logic is
its computational intractability. However, HOL formulas are relatively well understood, and
reasoning systems for HOLs do already exist, e.g., HOL [10] and Isabelle [21]. Though the
Web languages have been enriched by a few constructs exceeding the power of FOL, these
additional constructs are usually not well defined and understood. Moreover, particular
languages are neither syntactically nor semantically compatible. The W3C efforts at
standardization resulted in accepting the Resource Description Framework (RDF) language as
the Web recommendation. However, this situation is far from satisfactory. Quoting from
Horrocks and Schneider [13]: “The thesis of representation underlying RDF and RDFS is
particularly troublesome in this regard, as it has several unusual aspects, both semantic and
syntactic. A morestandard thesis of representation would result in the ability to reuse existing
results and tools in the Semantic Web.”
In this paper we provide a brief overview of the relevant portions of an expressive
logical system of Transparent Intensional Logic (TIL) from the point of view of the ‘multi
agent web world’. We concentrate on the two features of TIL which make it possible to
consequently distinguish between using and mentioning entities, namely the explicit
intensionalisation and temporalisation, and the rich ontology of entities organised into a two
dimensional infinite hierarchy of types. TIL provides a logicosemantic framework for a fine
grained logical analysis of natural language and an adequate representation of knowledge
possessed by autonomous agents who are (less or more) intelligent, but not omniscient.
A common objection against such a rich system, namely that it is too complicated and
computationally intractable, is in our opinion rather irrelevant: formal knowledge
specification in TIL is semantically transparent and comprehensible, with all the semantically
salient features explicitly present. In the Semantic Web we need such a highly expressive
language so that to first know what is there, and only afterwards to try deriving the
consequences. The fact that for higherorder logics such as TIL there is no semantically
complete system is not important. Only when knowing “what is there” we are able to derive
some consequences. Moreover, the TIL framework might at least serve as a both semantic and
terminological standard for the development of new languages, as an ideal to which we
should aim at.
The paper is organised as follows. First, in Chapter 2 we recapitulate the current state
of Web ontology languages from the logical point of view. We show here that the Web
ontology languages are mostly based on the firstorder predicate logic approach, which is far
from being a satisfactory state. In Chapter 3, an expressive system of the transparent
intensional logic (TIL) is introduced. After an informal introduction we provide particular
precise definitions. The method of logical analysis of language expressions is described
together with characterising important conceptual (analytical) relations, and the TIL method
of knowledge representation is introduced. In Chapter 4, the modern procedural theory of
concepts is described (for details see [18], [19], [5]), and important analytical relations and
properties of concepts are defined. Finally, in Chapter 5 the notion of inferable knowledge is
defined, and computational semantics is described in more details. Concluding Chapter 6
recapitulates the conceptual and terminological role of TIL in building multiagent systems in
the ‘InternetWeb Age’.
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2. Web Ontology Languages
The lowest levels of the Semantic Web are based on markup languages. By means of
schemas and ontology classifications, types of resources and types of interrelationships
between the resources can be specified. At the bottom level, the Unicode and URI layers
make sure that international character sets are used and provide means for identifying the
resources in the Semantic Web. At the very core level of the Web, there is the XML layer
with linking, style and transformation, namespace and schema definitions, which forms the
basis for the Semantic Web definitions. On the metadata level we can find RDF and RDF
Schema for describing resources with URI addresses and for defining vocabularies that can be
referred to by URI addresses, respectively. The ontology level supports the evolution of a
shared semantic specification and conceptualization of different application domains. The
ontology level is based on the OWL recommended by W3C, which is based on the 1
st
order
Description logic [1] framework. Based on common logic, the SKIF language accommodates
some higherorder constructs. At the logical level, simple inferences based on ontologies can
be drawn. As far as we know, the only ontology language supporting inferences at this level is
a Semantic Web Rule Language (SWRL) combining OWL and RuleML [14]. At the highest
level, the focus is on trust, i.e., how to guarantee the reliability of the data obtained from the
Web. Currently, a common research project CoLogNet II (Network of Excellence in
Computational Logic II) of fourteen European universities, lead by the Free University of
BozenBolzano, Italy, is prepared. The goal of this project can be characterised as the
development of a powerful Webinference machine based on a highly expressive logical
semantics.
2.1 Web Ontology languages from the logical Point of view
According to Horrocks and PatelSchneider [13] building ontologies consists in a hierarchical
description of important concepts in a domain, along with descriptions of properties of the
instances of each concept and relations between them. Current ontological languages
correspond roughly in their expressive power to the firstorder predicate logic (FOL), with
some higherorder ad hoc extensions. None of them makes it possible to express modalities
(what is necessary and what is contingent), distinguish between analytical and empirical
concepts, and handle higherorder concepts; perhaps only languages based on the Description
logic framework partly meet these goals. Concepts of nary relations are unreasonably
modelled as properties. True, each nary relation can be expressed by n unary relations
(properties): for instance the fact that G.W. Bush ordered to attack Iraq can be modelled by
two facts, namely that G.W. Bush has the property of ordering the attack to Iraq, and Iraq has
the property of being attacked by G.W. Bush’s order, but such a representation is not
comprehensive, and the equivalence of the two statements is concealed.
The basis of a particular way of providing meaning for metadata is embodied in the
model theory for RDF. RDF has unusual aspects that make its use as the foundation of
representation in the Semantic Web difficult at best. In particular, RDF has a very limited
collection of syntactic constructs, and these are treated in a very uniform manner in the
semantics of RDF. The data model of the RDF includes three basic elements. Resources are
anything with an URI address. Properties specify attributes and/or (binary) relations between
resources and an object used to describe resources. Statements of the form ‘subject, predicate,
object’ associate a resource and a specific value of its property. The RDF thesis requires that
no other syntactic constructs than the RDF triples are to be used and that the uniform semantic
treatment of syntactic constructs cannot be changed only augmented [13]. In RDFS we can
specify classes and properties of individuals, constraints on properties, and the relation of
subsumption (subclass, subproperty). It is not possible, for instance, to specify properties of
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properties, e.g., that the relation (property) is functional or transitive. Neither is it possible to
define classes by means of properties of individuals that belong to the class.
Recognition of the limitations of RDFS led to the development of new Web ontology
languages such as OIL, DAMLONT and DAML+OIL [23, 25]. These are used as the basis of
a new W3C Web ontology language called the OWL. As a second language for the Semantic
Web, OWL has been developed as an extension of RDFS. OWL (like DAML+OIL) uses the
same syntax as RDF (and RDFS) to represent ontologies, the two languages are syntactically
compatible. However, the semantic layering of the two languages is more problematical. The
difficulty stems from the fact that OWL (like DAML+OIL) is largely based on the
Description Logic [1], the semantics of which would normally be given by a classical first
order model theory in which individuals are interpreted as elements of some domain (a set),
classes are interpreted as subsets of the domain and properties are interpreted as binary
relations on the domain. The semantics of RDFS, on the other hand, are given by a non
standard model theory, where individuals, classes and properties are all elements in the
domain. Properties are further interpreted as having extensions which are binary relations on
the domain, and class extensions are only implicitly defined by the extension of the rdf:type
property. Moreover, RDFS supports reflection on its own syntax: interpretation of classes and
properties can be extended by statements in the language. Thus language layering is much
more complex, because different layers subscribe to these two different approaches.
The third group of ontology languages lies somewhere between the FOL framework
and RDFS. The group of relatively new languages include SKIF and Common Logic [11].
The SKIF syntax is compatible with functional language LISP, but in principle it is FOL
syntax. These languages have like RDFS a non standard model theory, with predicates being
interpreted as individuals, i.e., elements of a domain. Classes are however treated as subsets
of the domain, and their redefinition in the language syntax is not allowed.
Thus from the logical point of view, the ontological languages can be divided into
three groups: the FOL approach, the SKIF approach, and the RDF approach.
a) The FOL approach (DAML+OIL, OWL) is closely connected to the rather expressive
Description Logic (DL): Languages of this group talk about individuals that are elements of a
domain. The individuals are members of subclasses of the domain, and can be related to other
individuals (or data values) by means of properties (nary relations are called properties in
Web ontologies, for they are decomposed into n properties). The universe of discourse is
divided into two disjoint sorts: the object domain of individuals and the data value domain of
numbers. Thus the interpretation function assigns elements of the object domain to individual
constants, elements of data value domain to value constants, and subclasses of the data
domain to data types. Further, object and data predicates (or properties) are distinguished, the
former being interpreted as a subset of the Cartesian product of object domain, the latter a
subset of the Cartesian product of value domain. DL is relatively rather rich, though being an
FOL language. It makes it possible to distinguish intensional knowledge (knowledge on the
analytically necessary relations between concepts) and extensional knowledge (of contingent
facts).
The knowledge base of DL is divided into the socalled Tbox (according to
terminology or taxonomy) and Abox (according to contingent attributes of objects). Tbox
contains verbal definitions, i.e., a new concept can be defined composing known concepts.
For instance, a woman can be defined: WOMAN = PERSON & SEXFEMALE, and a
mother: MOTHER = WOMAN & ∃child(HASCHILDchild). Thus the fact that, e.g., mother
is a woman is analytical (necessary) true. In the Tbox there are also specifications of
necessary properties of concepts and relations between concepts: the property satisfiability
(corresponding to a nonempty concept), the relation of subsumption (intensionally contained
concepts), equivalence and disjointness (incompatibility). Thus, e.g., that a bachelor is not
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married is analytically (necessarily) true proposition. On the other hand, the fact that, e.g., Mr.
Jones is a bachelor is a contingent unnecessary fact. Such contingent properties (attributes) of
objects are recorded in Aboxes.
b) The SKIF approach: SKIF languages are syntactically compatible with LISP, i.e., the FOL
syntax is extended with the possibility to mention properties and use variables ranging over
properties. For instance, we can specify that John and Peter have a common property:
∃p . p(John) & p(Peter). The property they have in common can be, e.g., that they both love
their wives. We can also specify that the property P is true of John, and the P has the property
Q: P(John) & Q(P). If P is being honest and Q is being eligible, the sentence can be read as
that John is honest, which is eligible. The interpretation structure is a triple
<D, ext, V>, where D is the universe, V is the function that maps predicates, variables and
constants to the elements of D, and ext is the function that maps D into sets of ntuples of
elements of D. SKIF does not reduce the arity of predicates.
c) The RDF approach: These languages originally did not have a model theoretic semantics,
which led to many discrepancies. The RDF syntax consists of the socalled triples – subject,
predicate and object, where only binary predicates are allowed. This causes serious problems
concerning compatibility with more expressive languages. RDF(S) has become a Web
ontological recommendation defined by W3C, and its usage is world spread. The question is
whether it is a good decision. A classical FOL approach would be better, or even its standard
extension to HOL would be more suitable for ontologies. Formalisation in HOL is much more
natural and comprehensive, the universe of discourse is not a flat set of ‘individuals’, but
properties and relations can be naturally talked about as well, which is much more apt for
representation of ontologies.
Ontologies will play a pivotal role in the Semantic Web by providing a source of
shared and precisely defined concepts of entities that can be used in metadata. The degree of
formality employed in capturing these concepts can be quite variable, ranging from natural
language to logical formalisms, but increased formality and regularity clearly facilitates
machine understanding. We argue that conceptual formalisation should be meaning driven,
based on natural language. Formal language of FOL is not a natural language; it is a language
of non interpreted formulas that enable us to talk about individuals, expressing their properties
and relations between individuals. We cannot talk about properties of (individuals, properties,
functions, relations, properties of concepts, generally of higherorder objects), unless they are
simple members of the universe. Thus inference machine based on FOL can underinfer, or
paradoxically overinfer.
Here is an example: The office of the President of USA is certainly not an individual.
It can be occupied by individuals, but the holder of the office and the office itself are two
completely distinct things. The office (Church’s individual concept) necessarily has some
requisites (like being occupied by at most one individual), which no its occupant has. In FOL,
however, we have to treat the office as an individual. Paradoxes like the following arise:
• John Kerry wanted to become the President of USA.
• The President of USA knows that John Kerry wanted to become the President of USA.
• George W. Bush is the President of USA.
–––––––––––––––––––––––––––––––––––––
Hence what?
Systems based on FOL do not handle such a realistic situation in an adequate way.
Analysing (as it should be) ‘is’ in the third premise as the identity of individuals, we obtain
the obviously nonvalid (senseless) conclusion:
• George W. Bush knows that John Kerry wanted to become George W. Bush.
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True, this shortcoming is usually overcome in FOL approach by introducing a special binary
predicate ‘holds’. The third premise would then be translated into:
• (individual) George W. Bush holds (another individual) the President of USA.
But then the inference machine underinfers, because it does not make it possible to infer a
valid consequence of the above premises, namely that
• George W. Bush knows that John Kerry wanted to become the President of USA.
Another shortcoming of FOL approach is the impossibility to handle (contingently)
‘nondenoting terms’ like the President of Czech Republic in January 2003, or the King of
USA. In other words, translating the sentence
The President of CR does not exist
into an FOL language, we obtain a paradox of existence (that being an individual, it exists,
but actually it does not exist): ¬∃x (x = Pres(CR)), but from (x = Pres(CR)) by existential
generalisation we immediately derive that ∃x (x = Pres(CR)).
A theory formalising reasoning of intelligent agents has to be able to talk about the
objects of agents’ attitudes, to quantify over them, to express iterated attitudes and/or self
referential statements, like agent a knows that an agent b knows that (he) believes that P,
which in FOL leads to inconsistencies.
Such theories should also make it possible to express the distinction between
analytical and empirical concepts (what is necessary and what is just contingent), to express
empty concepts, to talk about concepts and, last but not least, to express nary relationsin
intension between any entities (not only individuals) of our ontology. While this is beyond the
expressive power of FOL, many richer logical systems with nonstandard operators are
proposed: modal, epistemic, intensional, temporal, nonmonotonic, paraconsistent, etc. These
logics can be characterized as theories with ‘syntactically driven axiomatization’. They
provide ad hoc axioms and rules that define a set of models, each logic partly solving
particular problem. Ontology language should be, however, universal, highly expressive, with
transparent semantics and meaning driven axiomatisation. We have such a system at hand:
the system of transparent intensional logic (TIL).
3. Transparent Intensional Logic (TIL)
3.1 Hierarchy of Types
TIL is a logic that does not use any non standard operators; in this sense it is classical.
However, its expressive power is very high: formalisation of meaning is comprehensive, with
transparent semantics, closed to natural language. Notation is an adjusted objectual version of
Church’s typed λcalculus, where all the semantically salient features are explicitly present.
The entities we can talk about are in TIL organised into twodimensional hierarchy of types.
This enables us to logically handle structured meanings as higherorder, hyperintensional
abstract objects, thus avoiding inconsistency problems stemming from the need to mention
these objects within the theory itself. Hyperintensionally individuated structured meanings
are procedures, structured from the algorithmic point of view, known as TIL constructions.
Due to typing, any object of any order can be safely, not only used, but also mentioned within
the theory.
On the ground level of the typehierarchy, there are settheoretical entities
unstructured from the algorithmic point of view belonging to a type of order 1. Given a so
called epistemic base of atomic types (οtruth values, ιindividuals, τtime points (or real
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numbers), ωpossible worlds), mereological complexity is increased by an induction rule of
forming partial functions: where α, β
1
,…,β
n
are types of order 1, the set of partial mappings
from β
1
×…× β
n
to α, denoted (α β
1
…β
n
), is a type of order 1 as well.
TIL is an openended system. The above epistemic base {ο, ι, τ, ω} was chosen,
because it is apt for naturallanguage analysis, but in the case of mathematics a (partially)
distinct base would be appropriate; for instance, the base consisting of natural numbers, of
type ν, and truthvalues. Derived types would then be defined over {ν, ο}.
A collection of constructions that construct entities of order 1, denoted by *
1
, serves as
a base for the induction rule: any collection of partial functions (α β
1
…β
n
) involving *
1
in
their domain or range is a type of order 2. Constructions belonging to a type *
2
that identify
entities of order 1 or 2, and partial functions involving such constructions, belong to a type of
order 3. And so on, ad infinitum.
Example: Binary mathematical functions like adding (+), dividing (:) are mappings of type
(τττ). The set of prime numbers is a mapping of type (ον)⎯ the characteristic function that
associates each natural number with a truthvalue: True, in case the number is a prime, False
otherwise.
3.2 Constructions
Constructions are structured from the algorithmic point of view; they are procedures
consisting of instructions specifying the way of arriving at lowerlevel (lessstructured)
entities. Since constructions are abstract, extralinguistic entities, they are reachable only via a
verbal definition. The ‘language of constructions’ is a modified version of the typed λ
calculus, where Montaguelike λterms denote, not the functions constructed, but the
constructions themselves. The modification is extensive. Church’s λterms form part of his
simple type theory, whereas our λterms belong to a ramified type theory [22].
Constructions qua procedures operate on input objects (of any type, even higherorder
constructions) and yield as output objects of any type. One should not conflate using
constructions as constituents of composed constructions and mentioning constructions that
enter as input objects into composed constructions, so we have to strictly distinguish between
using and mentioning constructions. The latter is, in principle, achieved by using atomic
constructions. A construction is atomic if it is a procedure that does not contain as a
constituent any other construction but itself. There are two atomic constructions that supply
objects (of any type) on which complex constructions operate: variables and trivialisations.
Variables are constructions that construct an object dependently on valuation: they v
construct. Variables can range over any type. If c is a variable ranging over constructions of
order 1 (type *
1
), then c belongs to *
2
, the type of order 3, and constructs a construction of
order 1 belonging to *
1
: the type of order 2. When X is an object of any type, the trivialisation
of X, denoted
0
X, constructs X without the mediation of any other construction.
0
X is the
atomic concept of X: it is the primitive, nonperspectival mode of presentation of X.
There are two compound constructions, which consist of other constituents:
composition and closure. Composition is the procedure of applying a function f to an
argument A, i.e., the instruction to apply f to A to obtain the value (if any) of f at A. Closure is
the procedure of constructing a function by abstracting over variables, i.e., the instruction to
do so. Finally, higherorder constructions can be used twice over as constituents of composed
constructions. This is achieved by a fifth construction called double execution (
2
C). For
instance, if a variable c belongs to *
2
, the type of order 3, and constructs a construction of
order 1 belonging to *
1
, the type of order 2,
2
c constructs an entity belonging to a type of
order 1.
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3.3 Definitions
Definition 1 (Construction)
i) Variables x, y, z, … construct objects of the respective types dependently on valuations v;
they vconstruct.
ii) Trivialisation: Where X is an object whatsoever (an extension, an intension or a
construction),
0
X constructs X.
iii) Closure: If x
1
, x
2
, …,x
n
are pairwise distinct variables that vconstruct entities of types α
1
,
α
2
, …, α
n
, respectively, and Y is a construction that vconstructs an entity of type β, then
[λx
1
…x
n
Y] is a construction called closure, which vconstructs a partial function of type
(β α
1
…α
n
) mapping α
1
×…× α
n
to β.
iv) Composition: If X vconstructs a function f of a type (β α
1
…α
n
), and Y
1
,…,Y
n
vconstruct
entities A
1
, …, A
n
of types α
1
,…,α
n
, respectively, then the composition [X Y
1
… Y
n
] v
constructs the value (an entity, if any, of type β) of the (partial) function f on the argument
〈A
1
, …, A
n
〉. Otherwise the composition [X Y
1
… Y
n
] does not vconstruct anything: it is
vimproper.
v) Double execution: If X is a construction of order n, n ≥ 2, that vconstructs a construction
X’ (of order n–1), then
2
X vconstructs the entity vconstructed by X’. Otherwise the
double execution
2
X is vimproper.
vi) Nothing is a construction, unless it so follows from i) through v).
Definition 2 (Ramified hierarchy)
Let B be a base, i.e. a collection of pairwise disjoint, nonempty sets.
T
1
(types of order 1)
i) Every member of B is an elementary type of order 1 over B.
ii) Let α, β
1
, ..., β
m
(m > 0) be types of order 1 over B. Then the collection (α β
1
... β
m
) of all
mary (total and partial) mappings from β
1
× ... × β
n
into α is a functional type of order 1
over B.
iii) Nothing is a type of order 1 over B unless it so follows from i) and ii).
C
n
(constructions of order n)
i) Let x be a variable ranging over a type of order n. Then x is a construction of order n
over B.
ii) Let X be a member of a type of order n. Then
0
X,
2
X are constructions of order n over B.
iii) Let X, X
1
, ..., X
m
(m > 0) be constructions of order n over B. Then [X X
1
...X
m
] is a
construction of order n over B.
iv) Let x
1
, ..., x
m
, X (m > 0) be constructions of order n over B. Then [λx
1
...x
m
X] is a
construction of order n over B.
T
n+1
(types of order n + 1)
Let ∗
n
be the collection of all construction of order n over B.
i) ∗
n
and every type of order n are types of order n + 1.
ii) If m > 0, and α, β
1
,...,β
m
are types of order n + 1 over B, then (α β
1
... β
m
) (see T
1
ii)) is a
type of order n + 1 over B.
iii) Nothing is a type of order n + 1 over B unless it so follows from i), ii).
Examples (a) The function +, defined on natural numbers (of type ν), is not a construction. It
is a mapping of type (ν νν), i.e., a set of triples, the first two members of which are natural
numbers, while the third member is their sum. The simplest construction of this mapping is
0
+. (b) The composition [
0
+ x
0
1] vconstructs the successor of any number x. (c) The closure
λx [
0
+ x
0
1] constructs the successor function. (d) The composition of this closure with
0
5, i.e.,
10
[λx [
0
+ x
0
1]
0
5], constructs the number 6. (e) The composition [
0
: x
0
0] does not vconstruct
anything for any valuation of x; it is vimproper. (f) The closure λx [
0
: x
0
0] is not improper, as
it constructs something, even though it is only a degenerate function, viz. one undefined at all
its arguments.
The constructions
0
+, [
0
+ x
0
1], λx [
0
+ x
0
1], [λx [
0
+ x
0
1]
0
5], [
0
: x
0
0], λx [
0
: x
0
0], all
mentioned above, are members of *
1
. When IMP is a set of vimproper constructions of order
1, i.e., when IMP is an object of type (ο*
1
), the composition [
0
IMP
0
[
0
: x
0
0]] is a member of
type *
2
, and it
constructs the truthvalue True. The constituent
0
[
0
: x
0
0] of this composition (a
member of type *
2
) is an atomic proper construction that constructs [
0
: x
0
0], a member of *
1
.
It is atomic, because the construction [
0
: x
0
0] is not used here as a constituent but only
mentioned as an input object. For further details, see [6], [22].
If ARITHUN is a set of arithmetic unary functions, then the composition
[
0
ARITHUN
2
c] vconstructs True if c v–constructs [λx [
0
+ x
0
1]]. The double execution
2
c v
constructs what is vconstructed by [λx [
0
+ x
0
1]], i.e., the arithmetic successor function.
Notational conventions An object A of the type α is called an αobject, denoted A/α. That a
construction C constructs an αobject will be denoted C → α. We use infix notation without
trivialisation for truthvalue connectives ∧ (conjunction), ∨ (disjunction), ⊃ (implication), for
an identity sign = and for binary number relations ≥, <, >, ≤.
Thus, for instance, we can write:
c → *
1
, c / *
2
,
2
c → (ττ),
2
c / *
3
,
ARITHUN / (ττ),
0
ARITHUN / *
1
, [
0
ARITHUN
2
c] → ο,
[
0
ARITHUN
2
c] / *
3
, x → τ, x / *
1
,
0
+ / *
1
,
0
+ → (τττ),
0
1 / *
1
,
0
1 → τ, [λx [
0
+ x
0
1]] / *
1
,
[λx [
0
+ x
0
1]] → (ττ).
Definition 3 (α)intension, (α)extension
(α)intensions are members of a type (αω), i.e., functions from possible worlds to the
arbitrary type α. (α)extensions are members of the type α, where α is not equal to (βω) for
any β, i.e., extensions are not functions from possible worlds.
Remark Intensions are frequently functions of the type ((ατ)ω), i.e., functions from possible
worlds to chronologies of the type α (in symbols: α
τω
), where a chronology is a function of
type (ατ).
Examples of intensions
• being happy is a property of individuals / (οι)
τω
.
• The president of the Czech Republic is an individual office (‘individual concept’) / ι
τω
.
• That Charles is happy is a proposition / ο
τω
.
• Knowing is an attitude of an individual to a construction, i.e., a relation that is a higher
order intension / (ο ι ∗
n
)
τω
.
3.4 Logical Analysis
We adhere to the constraint on naturallanguage analysis dictated by the principle of subject
matter: an admissible analysis of an expression E is a construction C such that C uses, as its
constituents, constructions of just those objects that E mentions, i.e., the objects denoted by
subexpressions of E (for details, see [16, 17]). Any such analysis is an adequate analysis of
E, the best one relative to a conceptual system determined by a set of atomic constructions
[19]. The principle is central to our general threestep method of logical analysis of language:
(i) Typetheoretical analysis Assign types to the objects mentioned, i.e., only those that are
denoted by subexpressions of E, and do not omit any semantically selfcontained sub
expression of E, i.e., use all of them.
11
(ii) Synthesis Compose constructions of these objects so as to construct the object D denoted
by E.
(iii) Type checking Use the assigned types for control so as to check whether the various
types are compatible and, furthermore, produce the right type of object in the manner
prescribed by the analysis.
A construction of an intension is usually of the form λwλt X, w → ω, t → τ. If C is a
construction of an intension Int, the composition [[C w] t] — the intensional descent of Int to
its extension (if any) at w,t — will be abbreviated C
wt
.
Example of analysis We are going to analyse the sentence,
“The President of USA is G.W.Bush”.
(i’) Presidentof / (ιι)
τω
—(an empirical function that dependently on the states of affairs
assigns an individual to an individual), USA / ι — (for the sake of simplicity), the
President of USA / ι
τω
— an individual office, G.W.Bush / ι, = / (ο ιι) — the identity
of individuals. The whole sentence denotes a proposition / ο
τω
.
(ii’) λwλt [
0
Presidentof
wt
0
USA] → ι
τω
(the individual office – role PUSA)
[λwλt [
0
Presidentof
wt
0
USA]]
wt
→ ι (the occupant of the office PUSA at w,t)
[
0
= [λwλt [
0
Presidentof
wt
0
USA]]
wt
0
G.W.Bush] → ο
λwλt [
0
= [λwλt [
0
Presidentof
wt
0
USA]]
wt
0
G.W.Bush] → ο
τω
.
(iii’) λwλt [
0
= [λwλt [
0
Presidentof
wt
0
USA]]
wt
0
G.W.Bush]
(ιι) ι
(ο ιι) ι ι
ο
Abstracting over t: (οτ)
Abstracting over w: ((οτ)ω), i.e., ο
τω
.
When being the President of USA, G.W. Bush is identical with the individual that holds the
office PUSA. If, however, John Kerry wanted to become the President of USA, he certainly
did not want to become G.W. Bush. He simply wanted to hold the office PUSA, i.e., he is
related not to the individual, but to the individual office. Now the paradoxical argument
mentioned in Chapter 2 is easily solved away:
“John Kerry wanted to become the President of USA”
λwλt [
0
Want
wt
0
J.Kerry [λwλt [
0
Become
wt
0
J.Kerry λwλt [
0
Presidentof
wt
0
USA]]]],
where Want / (ο ι ο
τω
)
τω
, Become / (ο ι ι
τω
)
τω
.
The construction λwλt [
0
Presidentof
wt
0
USA], i.e., a concept of the president of USA, is in
the de dicto supposition [6], ‘talking about’ the office itself, not its occupant in w,t.
Knowing is a relationinintension of an agent a to the meaning of the embedded
clause, i.e., to the construction of the respective proposition [8]. Thus when the President of
USA knows that John Kerry wanted to become the President of USA, he is related to the
construction of the proposition, and the former concept of the president of USA is used de re,
the latter de dicto.
The whole argument is analysed as follows:
λwλt [
0
= [λwλt [
0
Presidentof
wt
0
USA]]
wt
0
G.W.Bush]
λwλt [
0
Know
wt
[λwλt [
0
Presidentof
wt
0
USA]]
wt
0
[λwλt [
0
Want
wt
0
J.Kerry λwλt [
0
Become
wt
0
J.Kerry λwλt [
0
Presidentof
wt
0
USA]]]]]
12
Now the former concept of the President of USA is ‘free’ for substitution: we can substitute
0
G.W.Bush for [λwλt [
0
Presidentof
wt
0
USA]]
wt
, thus deducing that G.W. Bush knows that
John Kerry wanted to become the President of USA, but not that he wanted to become
G.W.Bush:
λwλt [
0
Know
wt
0
G.W.Bush
0
[λwλt [
0
Want
wt
0
J.Kerry [λwλt
0
Become
wt
0
J.Kerry λwλt [
0
Presidentof
wt
0
USA]]]]]
The undesirable substitution of
0
G.W.Bush for the latter occurrence of the construction
λwλt [
0
Presidentof
wt
0
USA] is blocked.
4. Theory of Concepts
Category of concepts has been almost neglected in the modern logic (perhaps only Bolzano,
Frege and Church studied the notion of a concept). A new impulse to examining concepts
came from computer science. Finnish logician Rauli Kauppi [15] axiomatised the classical
conception of concept as the entity determined by its extent and content. This theory is based
on the (primitive) relation of the intensional containment. GanterWille [9] theory defines
formal concept as the couple (extent, content) and makes use of the classical law of inversion
between extent and content, which holds for the conjunctive composition of attributes. Due to
this law a partial ordering can be defined on the set of formal concepts, which establishes a
concept lattice. Actually, ontologies viewed as classifications are based on this framework.
All these classical theories make use of the FOL apparatus classifying relations between
concepts that are not ontologically defined here.
Our conception defines concept as a closed construction [5, 18, 19], an algorithmically
structured procedure. An analogical approach can be also found in [20]. We do not consider
only general concepts of properties, but also a concept of a proposition, of an office, of a
number, etc. Simply, any closed construction (even an improper one) is a concept.
Comparison of the procedural theory of concepts with the classical settheoretical one can be
found in [7].
When building ontologies, we aim at conceptual analysis of entities talked about in the
given domain. In TIL, the analysis consists in formalising the meaning of an expression, i.e.,
in finding the construction of the denoted entity. If a sentence has a complete meaning, the
construction is a complete instruction of how to evaluate the truthconditions in any state of
affairs w,t. The meaning is a closed construction of the proposition of the form: λwλt C,
where C does not contain any free variables except w,t, and constructs a truth value. However,
not all the sentences of natural language denote propositions. Sometimes we are not able to
evaluate the truth conditions without knowing the (linguistic or situation of utterance) context.
In such a case the respective construction is open, contains free variables. For instance, the
sentence “He is happy” does not denote a proposition. Its analysis λwλt [
0
Happy
wt
x] contains
a free variable x. Only after x is evaluated by context supplying the respective individual, we
obtain a proposition. Thus the sentence does not express a concept of the proposition. On the
other hand, sentences with a complete meaning are complete instructions of arriving at the
proposition. They express concepts of a proposition.
Constructions (meanings) are assigned to expressions by linguistic convention. Closed
constructions are concepts assigned to expression with a complete meaning. Natural language
is not, however, perfect. In a vernacular we often confuse concepts with expressions. We say
that the concept of a computer had been invented, or that the concept of a whale changed, and
so on. As abstract entities, concepts cannot change or being invented. They can only be
discovered. We should rather say that the new expressions had been invented to express the
respective concept, or that the expression changed its meaning. Moreover, there are
13
homonyms (expressions with more concepts assigned), and synonyms (more expressions
expressing the same concept). Anyway, understanding an expression we know the respective
concept, we know what to do, which does not, however, mean that we know the result of the
procedure.
We have to make the notion of a concept still more precise before defining the
concept. Constructions are hyperintensionally individuated procedures, which is a fine
grained explication of meaning, but from the conceptual point of view it is rather too fine
grained. Some constructions are almost identical, not distinguishable in a natural language,
though not strictly identical. We define a relation of quasiidentity on the collection of all
constructions, and say that quasiidentical are constructions that are either αequivalent or η
equivalent. For instance, constructions λx[x >
0
0], λy[y >
0
0], λz[z >
0
0], etc., are α
equivalent. They define the class of positive numbers in a conceptually indistinguishable way.
Similarly, conceptually indistinguishable constructions are ηequivalent ones, like
0
+,
λxy [
0
+ x y], where the latter is an ηexpansion of the former. Each equivalent class of
constructions can be ordered, and we say that the first one is the construction in the canonical
normalised form. Concept is then defined as a canonical closed construction, the other quasi
identical constructions point at the concept.
Definition 4 (open / closed construction)
Let C be a construction. A variable x is οbound in C, if it is a subconstruction of a
construction C’ that is mentioned by trivialisation. A variable y is λbound in C, if it is a
subconstruction of a closure construction of the form λy and y is not οbound. A variable is
free in C, if it is neither οbound nor λbound. A construction without free variables is closed.
Examples. Construction [
0
+ x
0
1] is open: variable x is free here. Construction λx[
0
+ x
0
1] is
closed: variable x is λbound here. Construction
0
[λx[
0
+ x
0
1]] is closed: variable x is οbound
here.
Definition 5 (concept)
Concept is a closed construction in the canonical form.
Concept C
1
is contained in the concept C
2
, if C
1
is used as a constituent of C
2
.
Content of a concept C is a set of concepts contained in C.
Extent of a concept C is the entity constructed by C.
Extent of an empirical concept C
E
in a state of the world w,t is the value of the intension I
constructed by C in w,t (C
wt
).
Examples.
The concept of the greatest prime is strictly empty, it does not have any extent:
[
0
Sing λx ( [
0
Prime x] ∧ ∀y [[
0
Prime y] ⊃ [x ≥ y]] )].
(The Sing function ⎯ ‘the only x such that’ ⎯ returns the only member of a singleon, on
other sets (empty or sets of more than one member) it does not return anything.)
• The content of this concept is:
{
0
Sing,
0
Prime,
0
∀,
0
∧, λx ( [
0
Prime x] ∧ ∀y [[
0
Prime y] ⊃ [x ≥ y]] ),
0
⊃,
0
≥, and the
concept itself}.
The empirical concept of the President of the USA identifies the individual office:
λwλt [
0
Presidentof
wt
0
USA] → ι
τω
.
• Its content is the set {λwλt[
0
Presidentof
wt
0
USA],
0
Presidentof,
0
USA}.
• Its extent is the office.
• Its current extent is G.W. Bush. The concept used to be empirically empty before the
year 1789.
14
To make the Web search more effective, important conceptual properties and relations
should be followed. Among them, perhaps the most important is the relation called in the
Description logic subsuming (or in [15] intensional containment, known also as subconcept
superconcept). Conceptual relations are analytical. In other words, understanding, e.g., the
sentences like A cat is a feline, Whales are mammals, No bachelor is married, we do not have
to investigate the state of the world (or search in the Web) to evaluate them in any state of the
world as being true. This relation can be defined extensionally, i.e., in terms of the extents of
concepts. For instance, the property of being a cat has as its requisite the property of being a
feline: necessarily ⎯ in each state of affairs the population of cats is a subset of the
population of felines. Or, necessarily, the concept of the President of USA subsumes the
concept of the highest representative of USA. We have seen that except of DL, languages
based on FOL cannot specify this important relation. Subsuming is in TIL defined as follows:
Let C
1
, C
2
be empirical concepts. Then C
1
subsumes C
2
, denoted C
1
≥ C
2
,
iff in all the states
of affairs w, t the extent of C
1
is contained in the extent of C
2
. Formally:
Definition 6 (subsuming)
[
0
Subsume
0
C
1
0
C
2
] = ∀w∀t ∀x [[C
1wt
x] ⊃ [C
2wt
x]],
where (Subsume (ο ∗
n
∗
n
), C
1
→ (οα)
τω
, C
2
→ (οα)
τω
, x → α).
Thus for instance, since the concept of bachelor subsumes the concept of being never
married ex definitione, once we obtain a piece of information that Mr. X is a bachelor we will
not search any more for the information whether X is, or has ever been, married.
To follow such necessary relations between concepts, each important term of a domain
should be provided with an ontological definition of the entity denoted by the term. The
ontological definition is a complex concept composing as constituents primitive concepts of a
conceptual system [19], the given ontology. For instance, the concept woman can be defined
using primitive concepts person and female. The concept mother can be further defined
using the former, existential quantifier and the concept child:
0
woman = λwλt λx [[[
0
sexof
wt
x] =
0
female] ∧ [
0
person
wt
x]],
0
mother = λwλt λx [[
0
woman
wt
x] ∧ ∃y [[
0
childof
wt
x] y]]
where: x → ι, y → ι, woman / (οι)
τω
, female / (οι)
τω
, sexof / ((οι)
τω
ι)
τω
, person / (οι)
τω
,
childof / ((οι) ι)
τω
.
From the above definitions it follows that necessarily each woman is a female person, each
mother is a woman having children, etc.
Other analytical relations between concepts that can be defined using Web ontological
languages based on the Description logic are equivalence and incompatibility (disjointness in
DL):
Definition 7 (equivalence, incompatibility): Concepts C
1
, C
2
are equivalent, if they have
exactly the same extent (construct the same entity). Concepts C
1
, C
2
are incompatible, if in no
state of affairs w, t the extent of C
1
is a part of the extent of C
2
, and vice versa.
Note that concepts C
1
, C
2
are equivalent iff C
1
≥ C
2
and C
2
≤ C
1
, but not if their contents are
identical.
Examples:
Concepts bachelor and a married man are incompatible.
Concepts of the proposition that It is not necessary that if the President of USA is a
republican then he attacks Iraq and of the proposition that It is possible that the President of
USA is a republican and he does not attack Iraq are equivalent.
Some procedures can even fail, not producing any output. They are empty concepts.
There are several degrees of emptiness. From the strict emptiness, when the respective
concept does not identify anything (like the greatest prime) to emptiness, when the respective
15
procedure identifies an empty set. Empirical concepts always identify a nontrivial intension
of a type α
τω
. They cannot be (strictly) empty, they can be rather empirically empty when the
respective identified intension does not have any value or the value is an empty set in a
current state of the world w, t. A concept empirically empty in the actual state of affairs is,
e.g., the King of France. Not to search for extensions of empty concepts, classes of empty
concepts should be defined:
Strictly empty concept does not have an extension (the construction fails).
Empty concept has an empty extension.
Empirical concept C
E
is strictly empty in w,t, if it does not have an extent in w,t (the
intensional descent of I, i.e., C
wt
is improper.
Empirical concept C
E
is empty in w,t, if its extent in w,t is an empty set (the intensional
descent of I, i.e., C
wt
, constructs an empty set).
5. TIL Knowledge Representation in a MultiAgent World
A rational agent in a multiagent world is able to reason about the world (what holds true and
what does not), about its own cognitive state, and about that of other agents. Theory
formalizing reasoning of autonomous intelligent agents has thus to be able to ‘talk about’ and
quantify over the objects of propositional attitudes – structured meanings (constructions) of
the embedded clauses, iterate attitudes of distinct agents and express selfreferential
statements, like agent a knows that b knows that he believes that P. Last but not least, the
theory has to respect different inferential abilities of particular agents.
The agents have to communicate in a (pseudo) natural language, in order to
understand each other, and to provide relevant information to ‘whomever’, whenever and
whereever needed.
Obviously, any classical settheoretical theory is not able to meet these goals.
There are three kinds of knowing:
• implicit (of an agent who is a logical/mathematical genius, which leads to an ‘explosion’
of knowledge and the paradox of omniscience)
• explicit (which deprives an agent of any inferential capabilities)
• inferable (of a realistic agent with some inferential capabilities, who, however, is not
logically omniscient).
Modeltheoretic intensional logics (with Kripke or Montague semantics) conceive
possibleworld propositions (intensions) as objects of knowing. These approaches are apt for
modelling implicit knowledge, but cannot handle explicit or inferable knowledge in an
adequate way. Since equivalent formulas are indistinguishable, the problem of logical
omniscience cannot be avoided. Either an agent is bound to know all the logical consequences
of his/her/its known assumptions, or at least its equivalents, which is the tightest restriction to
the problem of omniscience obtainable by the settheoretical approach. Thus, for instance, if
an agent a knows that the number of inhabitants in Prague is equal to 1048576 people, he
should also assent to the statement that the number of inhabitants in Prague equals to 16
5
(the
hexnumber 100000). Well, you may say that the agent knows it only “implicitly”. But when
being ordered to behave according to the latter statement, for instance to organise an
emergency service for 100000 hexnumber of people, he must be aware of the fact so that to
be able to be active. Otherwise the system becomes chaotic and inconsistent.
Syntactic approaches adopting formulas as objects of knowing are the other extreme.
Though they are finegrained enough to be suitable for modelling explicit knowledge, they are
prone to inconsistencies when disquoting formulas, stemming from the need to model self
referential statements and the necessity to mention formulas within the theory. Moreover, an
agent a is deprived of any inferential abilities; it is just assigned the set of formulas⎯ its
16
explicit knowledge. Thus the agent a becomes an “agent idiot”, just a passive object. There
are some technically sophisticated ways of partly overcoming the above problems. But there
is a major philosophical objection to a syntactic approach: when knowing a statement S the
agent is not related to a piece of formal syntax, a formula, but to the meaning of S.
TIL approach to knowledge representation is technically as finegrained as the
syntactic approach, with two major distinctions:
• When knowing that S, an agent is related to the meaning of S, i.e., to TIL construction⎯
the hyperintensionally individuated mode of the presentation of the respective
proposition.
• We do not restrict the set of formulas the agent is said to know, instead we compute the
inferable knowledge relative to the inference rule(s) the agent is able to use.
Thus knowing is an object of type (ο ι *
n
)
τω
, a relationinintension of an individual to the
respective construction. The analysis of the above statement comes as follows:
λwλt [
0
Know
wt
0
a
0
[λwλt [
0
Card λx [
0
Inh
wt
x
0
Prague] =
0
1048576
10
]]].
The agent a is related to the construction
[λwλt [
0
Card λx [
0
Inh
wt
x
0
Prague] =
0
1048576
10
]],
which is mentioned here. Although the construction
[λwλt [
0
Card λx [
0
Inh
wt
x
0
Prague] =
0
100000
16
]]
identifies the same truthconditions, it is a distinct procedure, and the agent a does not have to
be able to evaluate it in a given state of affairs w,t, provided a does not master the rules of
transition from decimal to hexadecimal number system.
In what follows we just outline the TIL theory of computing agents’ inferable
knowledge relativized to particular inferential abilities, the sets of inference rules the agents
master.
Having an agent a equipped with a finite set of constructions (concepts of
propositions) K
exp
(a)
wt
(a’s current knowledge) and some intelligence (the set of inference
rules a masters), we compute the epistemic closure of K
exp
(a)
wt
. An exterior agent b
attempting to draw valid inferences about the interior agent a’s inferential knowledge needs a
closure principle to validate his inferences. To specify such a principle, we introduce the
functions Inf(R) / ((ο∗
n
) (ο∗
n
)) associating an input set C of constructions with the set of
constructions derivable from C using a set of rules R.
Let c (c → ∗
n
) now stand for an individual inferable piece of knowledge, d (d → (ο∗
n
))
for a stock of knowledge, and let R / (ο(∗
n
(ο∗
n
))) be a set of derivation rules, r → (∗
n
(ο∗
n
)) a
particular element of R. The following constructional schema specifies the function Inf(R):
λd λc [[d c] ∨ [∃r [
0
R r] ∧ (d —
r
c)]]
where (d —
r
c)
denotes derivation in accordance with r, i.e., the composition [[r d] = c]. The
schema can be read as follows: from any set d of constructions (λd) a construction c is
inferable (λc), if c belongs to d ([d c]), or c is derivable from d using a rule r.
For instance, let R contain the rule of disjunctive syllogism, the substitution rule, the
βreduction rule and the rule
20
C – C. Then Inf(R) is defined as follows:
0
Inf(R) =
λd λc [[d c] ∨ [ ∃c’ [d c’] ∧ [d
c,c’
[λwλt [¬(
2
c’)
wt
∨
(
2
c)
wt
]]]]].
There are technical complications. First, the stock of knowledge constructed by d is
usually a set of empirical concepts, i.e., constructions of intensions, propositions of types ο
τω
.
Second, since we are talking about the very objects of a’s epistemic attitudes, we need to
mention the constructions by trivialising them (which corresponds to calling a subprocedure
with formal parameters c, c’). To release the variables c, c’ bound by trivialisation, we have to
use the special substitution functions Sub, which realize a substitution of the actual values for
the formal parameters: if applied to the constructions C
1
, C
2
, C
3
, Sub / (∗
n
∗
n
∗
n
∗
n
) returns the
17
construction C such that C is the result of substituting C
1
for C
2
in C
3
. The doubleexecuting
variables ranging over propositional constructions
2
c,
2
c’ returns the respective propositions,
the intensional descent of which constructs a truthvalue. Finally, βreductions and the rule
transforming
20
C into C (
20
C – C) are to be performed. The upper index c,c’ is a notational
abbreviation of these devices.
Thus,
c,c’
[λwλt [¬(
2
c’)
wt
∨
(
2
c)
wt
]]
is to be unpacked as
[
0
Sub [
0
Tr c]
0
c [
0
Sub [
0
Tr c’]
0
c’
0
[λwλt [¬(
2
c’)
wt
∨ (
2
c)
wt
]]]].
Example Let a’s knowledge base contain the two facts (i) that Charles is bald and (ii) that
Charles is not bald or he is a king. Then, if a currently masters the rules R, he is able to
deduce that Charles is a king:
d →
v
{... [λwλt [
0
Bald
wt
0
Charles]], …,
[λwλt ¬[
0
Bald
wt
0
Charles] ∨ [
0
King
wt
0
Charles]], ...}
c’ →
v
[λwλt [
0
Bald
wt
0
Charles]]
c →
v
[λwλt [
0
King
wt
0
Charles]]
[
0
Sub [
0
Tr c]
0
c [
0
Sub [
0
Tr c’]
0
c’
0
[λwλt [¬(
2
c’)
wt
∨ (
2
c)
wt
]]]] →
v
[λwλt [¬
20
[λwλt [
0
Bald
wt
0
Charles]]
wt
∨
20
[λwλt [
0
King
wt
0
Charles]]
wt
]] =
(
20
C – C)
[λwλt [¬[λwλt [
0
Bald
wt
0
Charles]]
wt
∨ [λwλt [
0
King
wt
0
Charles]]
wt
]] =
(βreduction)
[λwλt [¬[
0
Bald
wt
0
Charles] ∨ [
0
King
wt
0
Charles]]].
Above we introduced the notion of a rule as a function of type (∗
n
(ο∗
n
)). We assume
that the rules R assigned to a are valid rules of inference and that the function Inf(R) meets the
following conditions for any agent a.
• Inf(R) is subclassical: if ϕ is derived from a stock of knowledge Γ, then ϕ is entailed by
Γ, i.e., if C
n
is the function assigning Γ with the set of its logical consequences, then
[
0
Inf(R)
Γ] ⊆ [C
n
Γ].
• Inf(R) is reflexive: Γ ⊆ [
0
Inf(R)
Γ].
(“a does not forget what a already knows.”)
• If Inf(R) is subclassical and reflexive, then it is monotonic:
if Γ ⊆ Γ’ then [
0
Inf(R)
Γ] ⊆ [
0
Inf(R)
wt
Γ’].
• Inf(R) is not idempotent: [
0
Inf(R) [
0
Inf(R)
A]] is not a subset of [
0
Inf(R)
A].
At this point we are able to recursively define the inferable knowledge of an agent a
mastering the rules R in a state w, t using the fixedpoint technique. The knowledge of an
agent a in the state w, t, whether explicit, inferable or implicit, is a set of propositional
constructions (concepts of propositions). The drawing of valid inferences about a’s inferable
knowledge is, for any w,t, executed stepwise. (for the sake of simplicity we now omit
trivialisations when no confusion can arise.) At step 0 we take a’s explicit knowledge as the
base of the induction K
0
(a)
wt
= K
exp
(a)
wt
. Step 1 consists in applying the function Inf(R) to this
knowledge, thus obtaining a new set of derived constructions K
1
(a)
wt
= [Inf(R) K
exp
(a)
wt
]. The
new set is a superset of the initial knowledge. But it is not necessarily equal to a’s inferable
knowledge yet: there may be more inferences to be drawn. Step 2 consists in applying Inf(R)
to the result of step 1 to obtain a new set: K
2
(a)
wt
= [Inf(R) K
1
(a)
wt
]. By iteration, an increasing
sequence of sets of constructions K
1
(a)
wt
⊆ K
2
(a)
wt
⊆ K
3
(a)
wt
… is obtained, such that each set
18
K
n+1
(a)
wt
depends only on the preceding set K
n
(a)
wt
. But at which step will the iteration stop?
There are two possibilities. Either there is a step m such that no more constructions can be
inferred: K
m+1
(a)
wt
= K
m
(a)
wt
, and K
m
(a)
wt
is the supremum of Inf(R). Or else there is no such
finite m, the sequence increasing ad infinitum for want of a maximum element. Still, even in
the latter case there exists a least upper bound of the sequence:
wt
k
kwt
aKaK )()(
1
U
∞
=
∞
=
This potentially infinite set is welldefined: it is the result of a potentially infinite
number of finite computational steps, and K
∞
(a)
wt
= [Inf(R) K
∞
(a)
wt
] holds. If the initial set of
explicit knowledge is a finite set of constructions, K
∞
(a)
wt
is countable.
In any case, the Inf(R) function is increasing and has a supremum. According to
Tarski’s fixedpoint theorem, there is a least fixed point of Inf(R) containing K
exp
(a)
wt
, and
since no more inferences can be drawn, this fixedpoint set is the whole inferable knowledge
of a in w, t.
Definition 8 (
inferable knowledge)
•
K
0
(a)
wt
= K
exp
(a)
wt
•
K
n+1
(a)
wt
= [ Inf(R) K
n
(a)
wt
]
•
nothing other …
The whole set of constructions validly inferable by a
⎯
a’s inferable knowledge
⎯
is
the fixed point of Inf(R):
K
inf
(a)
wt
= [ Inf(R) K
inf
(a)
wt
]
and it is the least fixed point of Inf(R) containing a’s explicit knowledge:
K
inf
(a)
wt
= µ λx [ Inf(R)
[ x ∪ K
exp
(a)
wt
]].
6. Conclusion
By way of conclusion, given an agent a furnished with a stock of recursively enumerable
explicit knowledge and a flawless command of only some rules R of inference, there is an
upper limit to the new knowledge it would be logically possible for the agent to derive from
the agent’s old knowledge: it is the closure, a’s inferable knowledge. If another agent b
masters the same set of rules R but its initial stock of knowledge is distinct, he may arrive at
another closure, even in case all the concepts of his initial knowledge are equivalent to a’s
concepts. This is as it should be: a and b would not understand each other providing the set R
does not contain a rule enabling them to “realise” the equivalence.
Without the assumption that every rule the agent uses is valid, we would have to
consider nonmonotonic reasoning as well. However, as long as we are modelling knowledge,
which we regard as factive and incapable of giving rise to inconsistent information bases, all
the reasoning must be monotonic. On the other hand, if we wish to model belief we must
allow the agent to use invalid rules of nonmonotonic reasoning potentially giving rise to
inconsistencies.
To pursue the research on knowledge management in a multiagent world, there is still
a lot to be done. First, we intend to use the method of conceptual lattices to partly order
classes of agents furnished with equivalent concepts and mastering the same set of rules. The
method of conceptual lattices should then facilitate involving dynamic aspects of the system
in a plausible way.
19
Apparently, we solved away the problem of logical omniscience. However, an agent is
resource bounded in many other aspects, in particular restricted time for calculation and
storage capacity. Thus the following subjects are a matter of further research:
•
Involving complexity problems (time and space limitations)
•
Dynamic aspects of the system (the assignment of the rules R to an agent is world / time
dependent, as well as its initial conceptual knowledge)
•
Doxastic logic of Beliefs
(managing hypotheses)
•
Belief revision and updating the base b (i.e., transition from a state w,t to w’,t’)
•
Nonmonotonic reasoning
•
Full axiomatisation of a ‘multiagent’ logic
•
Modelling uncertainty, vagueness
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––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
This research has been supported by the program "Information Society" of the Czech Academy of Sciences,
project No. 1ET101940420 "Logic and Artificial Intelligence for multiagent systems"
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