Concepts, Language and Ontologies 

(from the logical point of view) 

 

Marie Duží   

VŠB-Technical University of OstravaCzech Republic

    

Motto: Es gibt eine und nur eine vollständige Analyse des Satzes.Wittgenstein, Tractatus, 3.25

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

Terminology – ontology: What are we talking about? (Current state: a mess, chaos !!)

        What kind of entity is a concept?

CONCEPT = universal ??CONCEPT = expression ??

CONCEPT = <Int, Ext>Int: Intension (intent, content) of a CONCEPT Ext: Extension (extent) of a CONCEPT

(Circular ”definition”)

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 conceptions:

Concept = an axiomatic theory

Content = the set of axioms, Extent = the set of models

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!

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

Functions, procedures, sets, CONCEPTS

(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 – ().

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 !

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 /

• 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 ...

Sources of mess (Confusing):

Expression (”icon of” an abstract entity) – written recipewith

Mode of presentation (structured procedure, concept ) – n

abstract way of cooking

withThe 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

Sources of mess (confusing): ‘The president of CR’ (Empirical) expression 

wt [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)

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

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 0[x : 00]] True, Improper / (1) – used

[x : 00] / 1 – mentioned

‘Charles knows that dividing x by 0 is improper’

wt [0Knowwt 0Charles 0[0Improper 0[x : 00]] ]

construction [0Improper 0[x : 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

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 (/ ). !!!

Example:

w t [ 0TennisPlayerwt [w t [0Presidentwt 0CR]]wt ]

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 /

w t [ 0TennisPlayerwt [w t [0Presidentwt 0CR]]wt ]

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, t0TennisPlayer A set of individuals (who play tennis) / () w t [0Presidentwt

0CR] not defined till Feb. 28thVaclav 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)

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

Examples:

The concept of a bachelor:wt x [ [ 0Marriedwt x] [0Manwt x] ] ()

contains 0Married, 0Man, wt 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

wt [ 0Woodenwt 0Horse ] Wooden / (() ())0Horse IC [wt [0Woodenwt 0Horse ] ]

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”.

 

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)

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. Linguistic 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 …

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.  

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, …

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:

wt x ([R1wt x] … [Rn

wt x]).

 Ganter-Wille: conjunctive conception -- a special (frequent) case

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)

• 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

Conclusion: possible applications

Our knowledge concern conceptsCorrect 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

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