Prezentace aplikace PowerPoint

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Concepts, Language and Ontologies   (from the
logical point of view)     Marie
Duí       VB-Technical University of
Ostrava Czech Republic         Motto Es gibt
eine und nur eine vollständige Analyse des
Satzes. Wittgenstein, Tractatus, 3.25
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Content

• Terminology - Ontology
• Traditional theories of concept
• What kind of entity is a concept ?
• What is the content and extent of a concept ?
• Does the Law of inverse proportion always hold ?
• Transparent intensional logic (Pavel Tichý)
• Theory of concepts (Pavel Materna)
• Concepts and language
• Ontological vs. linguistic definition
• Conceptual lattices
• Conclusion An outline of applications

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Terminology ontology What are we talking
about? (Current state a mess, chaos
!!)         What kind of entity is a
concept? CONCEPT universal ?? CONCEPT
expression ?? CONCEPT ltInt, Extgt Int
Intension (intent, content) of a CONCEPT Ext
Extension (extent) of a CONCEPT
(Circular definition)
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What kind of entity is the content and extent of
a concept?

• Content subexpressions ??
• Content Intension possible world semantics
  ??
• Content (Intent) Kauppi a pre-concept not
  defined
• Content Ganter-Wille database-like
  attributes The way of combining them only
  conjunctive
• Extent objects falling under the concept ?
• objects satisfying attributes of the
  content
• More sophisticated conceptionsConcept an
  axiomatic theory
• Content the set of axioms, Extent the set of
  models

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Traditional conception. Concept is something
that consists of an intent and extent Worrisome
questions a) What is that something? b) What
exactly the extent and intent (content) is? c)
How shall we handle modal and temporal
variability of the extent? d) Does the law of
inverse proportion between the intent and the
extent always hold? Bolzano The way of
composing contained constituents is important!
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Our approach Transparent Intensional Logic
(TIL) Pavel Tichý Platonism and realism
(nominalists are hostile) Platonic
heaven (beyond space and
time) Actualised, discovered potential
named abstract objects (in any language
natural, formal, demonstrative,
...)   Expression ? sense (meaning) concept ?
denotation   Back to old-fashioned classics
(Bolzano, Frege, Russell, Church, Gödel, )
Functions, procedures, sets, CONCEPTS
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(Infinite) Hierarchy of entities (of our
ontology) 1st order Unstructured entities
(from the algorithmic point of view,
though having parts, members, ) a) basic
entities (non-functional) members of basic
types ? True, False ? individuals
(universal universe of discourse) ? time
points (real numbers) ? possible worlds
(consistent maximum sets of thinkable facts) b)
(partial) functions (mappings) (?1,,?n) ? ?
denoted (? ?1?n). (?-)sets are
mapped by characteristic functions (??).
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Intensions vs. extensions (still members of 1st
order)

• ?-intension member of a type ((??)?
• denoted ???
• ?-extension not a function from ?
• Examples of intensions
• student / (??)?? - property of individuals
• the president of CR / ??? - individual office
• Charles is a student / ??? proposition
• age of / (??)?? attribute (empirical function)
• Not to confuse with Intension (intent, content),
• Extension (extent) of a concept !

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Structured procedures

• 2nd order Constructions of 1st order entities,
  all of them belong to type ?1
• Variables x, y, z ... ? any type (not only
  individuals!)
• Trivialisation 0X ? basic object X, function X
• Closure ? x1 ... xn C ? Function / (?
  ?1...?n) ?1 ?n ?
• Composition C X1 Xn ? Value of the
  function (? ?1...?n) ?1
  ?n ?
• Example
• ?x 0 x 01, x, 01, 05 / ?1 ( / belong
  to)
• x ? ?, ?x 0 x 01 ? (? ?) (? construct)
• ?x 0 x 01 05 ? 6 / ?

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• 3rd order Constructions of 1st and 2nd order
  entities, all of them belong to type ?2
• Examples
• 0?x 0 x 01 / ?2, constructs ?x 0 x 01
  / ?1
• Adding 1 is an arithmetic procedure
• Ar / (? ?1) class of arithmetic 1st order
  constructions
• 0Ar 0?x 0 x 01 / ?2, constructs True
• And so on ...

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Sources of mess (Confusing) Expression (icon
of an abstract entity) written
recipe with Mode of presentation (structured
procedure, concept ) ?n abstract way of
cooking with The product of the procedure
(mostly 1st order, unstructured) with
(property of) meals _____________________________
_________________________ Process of executing
the procedure cooking in space and time
with (case the product being a
function) The value of the above (at an
argument) particular dumplings
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Sources of mess (confusing)

• The president of CR (Empirical)
  expression
•  
• ?w?t 0Presidentwt 0CR meaning concept
•  
• office / ??? intension (
  denotation)
• (but extent of the concept)
• Nobody (Havel till Feb.) Value of the
  intension (in w,t)
• result of empirical information retrieval (e.g.
  web search)

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Using vs. Mentioning (entities of our
ontology)  1st order        basic
entities only mentioned 03, 0Charles       
functional entities a) used to obtain its
value (by composition) ?x x 01 05 ?
6 0Even 05 ? False talking
about the value de re b) mentioned (talking
about the whole function de dicto)
Adding 1 is a bijective mapping 0Bij ?x
x 01 ? True Bij / (? (??)) But in both
cases construction ?x x 01 is used
(either de dicto or de re) to construct the
function
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2nd order        Constructions
(concepts) a)    used to construct (identify) a
(1st order) entity 0Bij ?x x 01
Construction ?x x 01 is used de
dicto, function adding is mentioned
?x x 01 05 Construction ?x x 01
is used de re, function adding is
used b)  mentioned (talking about concept
construction) Dividing x by 0 is improper
(does not yield any result) 0Improper
0x 00 ? True, Improper / (??1) used
x 00 / ?1 mentioned
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Charles knows that dividing x by 0 is
improper ?w?t 0Knowwt 0Charles 00Improper 0x
00 construction 0Improper 0x 00
mentioned Our knowledge, deductive (inference)
abilities concern primarily concepts, i.e.,
constructions, i.e., procedures not only their
outcomes - truth-values, intensions,
propositions, Modes of presentation, ways of
presenting are important Do we know the Number
? ? the ratio of the circumference of a circle
to its diameter
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Non-traditional Theory of Concepts (Materna).
Did we answer the fundamental ontological
question What is a concept? Concept is a
closed construction (roughly up to
renaming bound variables, ) What is the
content (intent) and extent of the concept?   A
concept C1 is (intensionally) contained in a
concept C2, iff C1 is a sub-construction of
C2, denoted C1 ?IC C2. Content (intension) of a
concept C is the set of concepts that are
contained in C. Extent (extension) of a concept
C is the object E, which is constructed by
C. An empirical concept is such a concept CE,
the extent of which is an ?-intension (/ ???).
!!!
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• Example
• ?w ?t 0TennisPlayerwt ?w ?t 0Presidentwt
  0CRwt
• Content Extent
• ----------------------------------------------
  ----------------
• 0TennisPlayer Ind. property / (??)??
• ?w ?t 0Preswt 0CR Ind. Office / ???
• 0President emp. function / (? ?)??
• 0CR individuum / ? (for the sake of
  simplicity)
• The whole concept proposition / ???

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?w ?t 0TennisPlayerwt ?w ?t 0Presidentwt
0CRwt Vaclav II. The
extent of an empirical concept CE in a world/time
w,t the value of its extent Int in w,t
Intwt Out of the scope of an a priory LOGIC
! Empirical investigation   Content Extent in
w, t 0TennisPlayer A set of individuals (who
play tennis) / (??) ?w ?t 0Presidentwt
0CR not defined till Feb. 28th Vaclav Klaus
now / ? 0President function / (? ?)
0CR Individuum (for the sake of
simplicity) The whole concept Truth-value
True / ? A simple concept of a (1st order)
object X is 0X. (Primitive concept with respect
to a Conceptual System)
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Relation of intensional containment (?IC) is the
relation of partial ordering on the set of
concepts (reflexive, anti-symmetric and
transitive) Can a (semantic) conceptual lattice
(following the law of inverse proportion) be
built up using ?IC ? NO. Just an enumeration of
contained concepts does not suffice. We have to
specify the way in which the contained concepts
are composed together to form a structured
complex and apply correct logical inference
rules on the whole concept. Set-theoretical
approach does not suffice It cannot render the
structural (procedural) character of
concepts. Analogy We deal with the difference
between a (structured) algorithm and its
(flat) output
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Examples The concept of a bachelor ?w?t ?x
? 0Marriedwt x ? 0Manwt x ? (??)??
contains 0Married, 0Man, ?w?t ?x ?
0Marriedwt x, student of the university of
Prague vs. student of the university of Prague
or Brno Man who understands all European
languages vs. Man who understands all living
European languages (Bolzano) cities and
districts of the Czech republic vs. cities and
districts in Moravia Wooden horse vs horse
! Adjectives either modify a property, or
create a new property ?w?t 0Woodenwt 0Horse
Wooden / ((??) (??)??)?? 0Horse ?IC ?w?t
0Woodenwt 0Horse
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Concepts and Language. Assignment expression ?
concept (meaning) is given by a linguistic
convention, it is an empirical relation. Thus
the answer to another question Do concepts
change? is NO just the above assignment of
concepts to expressions can change, meaning of
an expression changes, we even invent new
expressions to name some newly discovered
concepts, and some old expressions cease to be
used. Hence a (living) language develops, and
moreover, each domain of interest uses actually
its own jargon, we are building particular
ontologies.  
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Ontological vs. linguistic definition

• Each complex nonempty concept C is
• An ontological definition of its extent O,
• concept C defines the object O constructed by C.
• Example Ontological definition of (the class of)
  prime numbers / (??) is
• ?x ( 0Nat x ? 0Card ?y (0Nat y ? 0Div x
  y) 02 )
• Ontological definition does not define an
  expression but an object (intension / extension)

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By a definition we usually understand the
following schema Expression E1 (definiendum) df
expression E2 (definiens). From the logical
point of view this is a linguistic definition.
Thus simple expressions often do not express
primitive simple concepts (trivialisation of a
denoted object), but complex concepts.   Linguisti
c definition assigns to E1 as its meaning the
ontological definition of the object denoted by
E2. Examples Cat df Domestic carnivorous
animal, a feline, Prime ?x ( 0Nat x ?
0Card ?y (0Nat y ? 0Div x y) 02 )
Primes df natural numbers that have exactly
two factors. Number ? df the ratio of the
circumference of a circle to its
diameter Accountant is a man who masters
financial operations
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Conceptual lattices Requisites and typical
properties Reqpr P Q ?wt ?x Qwt x ? Pwt
x (P is a requisite of Q) Reqof P U
?wt 0Ewt U ? ?x Uwt x ? Pwt x (P
is a requisite of U, E is the property (of an
office) of existence) TPpr P Q G ?wt ?x
?Gwt x ? Qwt x ? Pwt x (P is
typical for Q, unless G) TPof P U G ?wt
0Ewt U ? ?x ?Gwt x ? Uwt x ? Pwt x
(P is typical for U, unless G) Artificial
intelligence the condition G -- the guard
of a rule.  
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A typical property of a bird is flying, unless it
is a penguin or an ostrich. A typical property of
a swan is being white, unless it has been born in
Australia or New Zealand.   Being a ruler of
France is a requisite of the King of
France. Being a carnivorous animal is a requisite
of a cat.   It follows from the concept of a
cat that my Mikes is a carnivorous animal,
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Semantic partial ordering on the set of
(equivalent) concepts   Let C1 and C2 be
empirical concepts such that C1 constructs a
requisite R of the extent I constructed by C2
Then C1 is weaker than or equivalent to C2,
denoted C1 ? C2.   Claim Let properties EC1,
EC2 be extents of concepts C1, C2, respectively,
such that C1 ? C2. Then necessarily,
i.e., in all world/times w,t, EC2wt ? EC1wt
The law of inverse proportion.   A special
case (finite number of requisites) a concept C
can construct I by means of conjuncting Ri
?w?t ?x (R1wt x ? ? Rnwt
x).  Ganter-Wille conjunctive conception -- a
special (frequent) case
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• Our Theory provides
• an explication of classical approaches
• an essential extension of classical theories
  Ganter-Wille, Kauppi, intuitionistic
• TIL essential extension overcomes the following
• shortcomings
• (all of that under one hat)

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• Extensional systems do not distinguish analytical
  vs empirical? using a modal, temporal or
  intensional logic (S5, Ty2, Montague, TIL, )
• 1st order predicate logic - not mentioning
  (functions, relations, concepts) ? using
  higher-order logic of which order ? ? type
  system
• Denotational approach not disting. synonymous
  vs. equivalent ? procedural declarative
  semantics (structured meanings)
• Formalistic approach not handling fine-grained
  distinction between a formal scheme of a set of
  constructions vs. the construction itself ?
  transparent approach (formal, but
  non-formalistic)
• Classical systems of predicate logics do not
  handle partial functions and empty concepts ?
  TIL partiality being propagated up

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Conclusion possible applications

• Our knowledge concern concepts
• Correct fine-grained logical (i.e. conceptual)
  analysis is a necessary condition of knowledge
  acquisition, inferring (implicit) knowledge and
  performing correct semantic information retrieval
• Problem Practical applicability of the method in
  the web environment comprising huge amount of
  heterogeneous documents.
• ARG Methods of reducing the dimension of the
  problem.
• Poset of pairs ??Documents, ?Expressions?
  (Galois definition) ordered by the relation
  of occurring in
• ? Lattice of areas of interests together with
  their vocabularies

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• The next step to be done is a linguistic one
• It consists in a disambiguation of the
  vocabulary, creation of the so-called
  intelligent thesaurus a semantic dictionary
  in which each important term is provided with the
  ontological definition of the denoted object
  concept (logical construction) expressed by the
  expression is assigned to it.
• Using inference rules of the given system ?
  requisites and typical properties
• ? Semantic conceptual lattice