Concepts Language and Ontologies from the logical point

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 Concepts, Language and Ontologies (from the logical point of view) Marie Duží VŠB-Technical

Concepts, Language and Ontologies (from the logical point of view) Marie Duží VŠB-Technical University of Ostrava Czech 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

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 !!)

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 =

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:

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

Our approach: Transparent Intensional Logic (TIL) Pavel Tichý Platonism and realism (nominalists are hostile) Functions, procedures, Platonic „heaven“ sets, CONCEPT (beyond space and time) S 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, …)

(Infinite) Hierarchy of entities (of our ontology): 1 st order: Unstructured entities (from the

(Infinite) Hierarchy of entities (of our ontology): 1 st 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 1 st order) • -intension: member of a

Intensions vs. extensions (still members of 1 st 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 • 2 nd order: Constructions of 1 st order entities, all of

Structured procedures • 2 nd order: Constructions of 1 st order entities, all of them belong to type 1 – Variables: x, y, z. . . any type (not only individuals!) – Trivialisation: 0 X basic object X, function X – Closure: [ x 1. . . xn C] Function / ( 1. . . n) 1 n – Composition: [C X 1 … Xn] Value of the function ( 1. . . n) 1 n Example: x [0+ x 01], x, 01, 05 / 1 x , x [0+ x 01] ( ) [ x [0+ x 01] 05 ] 6 / ( ‘/’ = belong to) (‘ ‘ = construct)

 • 3 rd order: Constructions of 1 st and 2 nd order entities,

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

Sources of mess (Confusing): Expression (”icon of” an abstract entity) – written recipe with

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 1 st 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’ w t [0 Presidentwt 0 CR]

Sources of mess (confusing): ‘The president of CR’ w t [0 Presidentwt 0 CR] office / (Empirical) expression meaning = concept 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) 1 st order: · basic entities: only

Using vs. Mentioning (entities of our ontology) 1 st order: · basic entities: only mentioned – 03, 0 Charles · functional entities: a) used to obtain its value (by composition) [ x [x + 01] 05] 6 [0 Even 05] False ”talking about” the value – de re b) mentioned (”talking about” the whole function – de dicto) ‘Adding 1 is a bijective mapping’ [0 Bij [ 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

2 nd order: Constructions (concepts) a) used to construct (identify) a (1 st order)

2 nd order: Constructions (concepts) a) used to construct (identify) a (1 st order) entity [0 Bij [ 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)’: [0 Improper 0[x : 00]] True, Improper / ( 1) – used [x : 00] / 1 – mentioned

‘Charles knows that dividing x by 0 is improper’ w t [0 Knowwt 0

‘Charles knows that dividing x by 0 is improper’ w t [0 Knowwt 0 Charles 0[0 Improper 0[x : 00]] ] construction [0 Improper 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

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 C 1 is (intensionally) contained in a concept C 2, iff C 1 is a sub-construction of C 2, denoted C 1 IC C 2. 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 [ 0 Tennis. Playerwt [ w t [0 Presidentwt 0 CR]]wt

Example: w t [ 0 Tennis. Playerwt [ w t [0 Presidentwt 0 CR]]wt ] Content Extent ------------------------------- 0 Tennis. Player Ind. property / ( ) 0 Pres 0 CR] w t [ Ind. Office / wt 0 President emp. function / ( ) 0 CR individuum / (for the sake of simplicity) The whole concept proposition /

 w t [ 0 Tennis. Playerwt [ w t [0 Presidentwt 0 CR]]wt

w t [ 0 Tennis. Playerwt [ w t [0 Presidentwt 0 CR]]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 0 Tennis. Player 0 0 w t [ Presidentwt CR] 0 President 0 CR The whole concept Extent in w, t A set of individuals (who play tennis) / ( ) not defined till Feb. 28 th Vaclav Klaus now / function / ( ) Individuum (for the sake of simplicity) Truth-value True / A simple concept of a (1 st order) object X is 0 X. (Primitive concept with respect to a Conceptual System)

Relation of intensional containment ( IC) is the relation of partial ordering on the

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: w t x [ [ 0 Marriedwt x]

Examples: The concept of a bachelor: w t x [ [ 0 Marriedwt x] [0 Manwt x] ] ( ) contains 0 Married, 0 Man, w t x [ 0 Marriedwt 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 [ 0 Woodenwt 0 Horse ] 0 Horse [ w t [0 Wooden 0 Horse ] ] IC wt Wooden / (( ) )

Concepts and Language. Assignment ‘expression concept (=meaning)’ is given by a linguistic convention, it

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

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 ( [0 Nat x] [0 Card y ([0 Nat y] [0 Div 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 E 1 (definiendum) =df

By a ‘definition’ we usually understand the following schema: Expression E 1 (definiendum) =df expression E 2 (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 E 1 as its meaning the ontological definition of the object denoted by E 2. Examples: Cat =df Domestic carnivorous animal, a feline, … Prime: x ( [0 Nat x] [0 Card y ([0 Nat y] [0 Div 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]

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 [[0 Ewt 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 [[0 Ewt 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

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 C 1 and C

Semantic partial ordering on the set of (equivalent) concepts Let C 1 and C 2 be empirical concepts such that C 1 constructs a requisite R of the extent I constructed by C 2: Then C 1 is weaker than or equivalent to C 2, denoted C 1 C 2. Claim: Let properties EC 1, EC 2 be extents of concepts C 1, C 2, respectively, such that C 1 C 2. Then necessarily, i. e. , in all world/times w, t, EC 2 wt EC 1 wt The law of inverse proportion. A special case: (finite number of requisites) a concept C can construct I by means of ”conjuncting” Ri: Ganter-Wille: w t x ([R 1 wt x] … [Rnwt x]). conjunctive conception -- a special (frequent) case

Our Theory provides: • an explication of classical approaches • an essential extension of

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

• Extensional systems do not distinguish analytical vs empirical using a modal, temporal or intensional logic (S 5, Ty 2, Montague, TIL, …) • 1 st 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 concepts Correct fine-grained logical (i. e. conceptual) analysis

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

The next step to be done is a linguistic one: It consists in a

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