Combinatory Logic, Categorization and Typicalitylalic.paris-sorbonne.fr/PUBLICATIONS/2004/presentation... · 2012-02-16 · Combinatory Logic, Categorization and Typicality Jean-Pierre

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Jean-Pierre Desclés, Berne oct. 2004 1

Combinatory Logic, Combinatory Logic, Categorization and TypicalityCategorization and Typicality

Jean-Pierre DesclésParis-Sorbonne University

LaLICC « Languages, Logic, Informatics, Cognition and Communication », CNRS / Paris-Sorbonne

Jean-Pierre. [email protected]

Swiss Society for Logic and Philosophy of Science, Berne, 14-15 october 2004


Summary

1. Combinatory Logic

2. Differences between Combinatory Logic and λλλλ-calculus

3. Categorization : a naive approach

4. Categorization : a new approach

5. Typical object and specification operator

6. Typical and atypical instances ; inheritance property

7. « Star » quantifiers vs fregean quantifiers

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1. Combinatory Logic1. Combinatory Logic


• with different compositions of operators ;

• where a composition is expressed by an abstract operator, called a Combinator;

• without using bound variables ;

• defined insidethe applicative language, without interpreting in specific domains.

COMBINATORY LOGIC = a logic of operators

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Combinatory expressions (e.c.)

The result of the application is presented by a simple concatenation of operator ‘X’ and operand ‘Y’, hence :

XY = def <X,Y>

We suppose left association: XYZ = (XY)Z ≠≠≠≠ X(YZ)

Rules:

(i) Basic expressions are e.c. ;

(ii) If ‘X’ and ‘Y’ are e.c. then <X,Y> is a e.c.


Church’s Functional Types

Rules: (i) The basic types are functional types ;

(ii) If ‘ αααα’ and ‘ ββββ’ are functional types then ‘Fαβαβαβαβ’ is a functional type.

Rule of application : @ < [Fαβαβαβαβ : X] , [αααα : Y] > =>ββββ [ββββ : Z]

When an operator ‘X’, with the type ‘Fαβαβαβαβ’, is aplying to an operand ‘Y’

with the type ‘αααα’, then the type of the type of the result ‘Z’ is ‘ββββ’.

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Application , Abstraction

[Fαβαβαβαβ : X], [αααα : Y] [ ββββ : XY], [αααα : Y]

--------------------------- -------------------------

[ββββ : XY] [F αβαβαβαβ : X]

Application Abstraction

Analogy with proposition calculus:

Modus Ponens ( ⊃⊃⊃⊃ - elimination ) ( ⊃⊃⊃⊃ - introduction )

αααα αααα hyp.

⊃⊃⊃⊃ αβαβαβαβ ββββ

-------- --------------

ββββ ⊃⊃⊃⊃ αβαβαβαβ


What is a combinator ? (1)

A combinator is an abstract operator which produces a new complex operatorfrom given operators.

Examples of elementary combinators :

IX ⇒⇒⇒⇒ββββ X identity

BXYZ ⇒⇒⇒⇒ββββ X(YZ) functional composition

WXY ⇒⇒⇒⇒ββββ XYY diagonalization

KXY ⇒⇒⇒⇒ββββ X cancellation

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What is a combinator ? (3)

X u1 u2 ….. un

X u1u2…un

a1 ap

Complex operator Successive operands

@@

@

@

. . .

« Equivalent » λλλλ-expressions

Every combinator can be expressed by a λλλλ -expression :

I =def λλλλf [ f ]

K =def λλλλf . λλλλx [ f ]

S =def λλλλg . λλλλf . λλλλx [ gx(fx) ]

C =def λλλλf . λλλλx . λλλλy [ fyx ]

B =def λλλλg. λλλλf . λλλλx [ g(fx) ]

W =def λλλλf . λλλλx [ fxx ]

C* = def λλλλx. λλλλf [ fx ]

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Properties of combinators• A combinator can be expressed by a λλλλ-expression ;

• A combinator is self-applicative ;

• There are basic combinators ;

• All combinators are defined from basic combinators ;

• Two basic combinators are sufficient, for instance : Sand K ;

• There is an « algebra » of combinators, generated from basic combinators ;

• For every combinator, there is a type schema (polymorphism).


Negation of a conceptLet ‘N 0’ the operator of proposition negation. From ‘N 0’ , we define the negation operator ‘N1’ of a concept :

1. N0(fa) hyp.2. BN0 fa B int. 3. [ N1 =def BN0 ] def. of N1

4. (N1f) a rempl.

The types of ‘N0’ = ‘FHH’; The type of ‘N1’ = ‘FFJHFJH’.

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Twice = def WB

Define the operator « twice » : twice f x = f(fx)

1. f(f x)2. Bff x B-intr.3. WBfx W-intr.4. [ twice = WB ] def.5. twice fx rempl.


2. Differences between 2. Differences between Curry’s Combinatory Logic and Curry’s Combinatory Logic and

Church’s Church’s λλλλλλλλ--CalculusCalculus

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No « intensional » equivalence

Combinatory Logic is an applicative language

• without bound variables, hence its more synthetic power ;

• wit an extensional equivalence with λλλλ-calculus ;

• but non « intensional » equivalence.


Example : Sxyz = xz(yz)

but, by an abstracting process (in Combinatory Logic) in introducing the combinators Sand K :

[x] Sxyz = S(SS(Ky))(Kz)[x] xz(yz) = S(SI(Kz))(K (yz))

hence : S(SS(Ky))(Kz) ≠≠≠≠ S(SI(Kz))(K (yz))

However, for all U : ([x] Sxyz)U = ([x] xz(yz))U

So, we get extensional equality but not an intensional equality.

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3. CATEGORIZATION :« naive » approach


Concept / Objects (in Frege’s tradition)

We start with concept in the sense of Frege.

A concept ‘f’ is a function from a domain D into true values :

f : D -> { T, ⊥⊥⊥⊥ }

In Frege’s work, individual entities are objects

but also classes of entities (extensions), truth values, courses-of-values … are objects.

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Logical types

We consider only :

J = type of individual entities ;

H = type of true values

FJH = type of concepts (unary predicates)

FJFJH = type of relations (or binary predicates)

FHFHH = type of conjunctive operators

FJJ = type of specification operators

FFJHH = type of fregean quantifiers


Concept and instances• A concept ‘f’ is an operator with the type ‘FJH’ ;

• An instance ‘x’ of the concept ‘f’ is an object, with type ‘J’, such that : f(x) = T.

• To every concept ‘f’ with the type FJH are associated its Extension and its Intension.

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Intension / Extension :a naive approach

There is a duality between intension and extension

=> Intension can be reduced to Extension

Int(f) ⊇⊇⊇⊇ Int(g) �� f -> g �� Ext(f) ⊆⊆⊆⊆ Ext(g)�� (∀∀∀∀x) [ (f(x) = T) => (g(x) = T) ]

Extensional equality : Ext(f) = Ext(g) => f = g


Inheritance in Semantic NetworkInheritance in Semantic Network

[ x -> f ] <=> [ Int(x) ⊇⊇⊇⊇ Int(f) ] <=> [ x ∈∈∈∈ Ext(f) ] <=> [ f ∈∈∈∈ Int(x) ]

if ‘x’ belongs to the extension of ‘f’, and if‘g’ is in the intension of ‘f’, then ‘x’ inherits ‘g’ and belongs to the extension of ‘g’, that is:

[Inher ] [ x ∈∈∈∈ Ext(f) ] & [ g ∈∈∈∈ Int(f)] => [ x ∈∈∈∈ Ext(g) ][ x ∈∈∈∈ Ext(f) ] & [ g ∈∈∈∈ Int(f)] => [ g ∈∈∈∈ Int(g) ]

Transitivity of inheritance :[ f(x) = true ] & [ g ∈∈∈∈ Int(f)] => [ g(x) = true ]

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be-man

be-mortal-beinghave-two-legs

Socrates


Inheritance Principle Inheritance Principle in Semantic Network (in AI)in Semantic Network (in AI)

Socrates -> “be-a-man” -> “be-a-mortal-being” in a semantic Network

Socrates ∈∈∈∈ Ext (“be-a-man”) ⊆⊆⊆⊆ Ext (“be-a-mortal-being”) Int (Socrates) ⊇⊇⊇⊇ Int (“be-a-man”) ⊇⊇⊇⊇ Int (“be-a-mortal-being”)

[ Socrates ∈∈∈∈ Ext(“be-a-man”) ]<=>[ “be-a-man” ∈∈∈∈ Int(Socrates) ][ Socrates ∈∈∈∈Ext(“be-a-mortal-being”) ]<=>[“be-a-mortal-being” ∈∈∈∈Int(Socrates) ]

It is clear that Socrates inherits all properties that are in the intension of the extension it belongs :

Socrates -> “be-a-man” -> “be-a-mortal-being” ------------------------------------------------------------------∴∴∴∴ Socrates -> “be-a-mortal-being”

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Problems with the naive Problems with the naive approach approach

of categorisationof categorisation


Indetermination in Natural Languages

A referential object is not at all always fully specified.

Natural Languages express no specification of reference by means of articles, quantifiers, relative clauses …:

a dog,

a whitedog,

a dog which belongs to Tintin

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A problem of InheritanceA problem of Inheritance‘Good’ Deduction: ‘Bad’ Deduction:

(1) All men have two feet (4) A man has two feet(2) Aristotle is a man (5) John is a man

------------------------------ (6) John has only one foot(3) (3) (3) (3) ∴∴∴∴ Aristotle has two feet ----------------------------

(7) * John has two feet

If we accept this general knowledge:(8) the property “to have two feet”

which is “incompatible” with :(9) the property “to have only one foot”

then arises the following contradiction:(9) John has only one footand John has two feet.

Jean-Pierre Desclés, Berne oct. 2004 28John

to-be-a-man

have two feet

have only one foot

Int (be-a-man)

Int (John) contradiction

John cannot inherit the property «John cannot inherit the property « havehave--twotwo--feetfeet »»

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Port Royal’s Logic (Arnauld and Nicole)

The « compréhension » of a general term is the set of attributes which it implies, or, the set of attributes which could not removed without destruction of idea.

The extenion (« étendue ») [here : « Expansion »] of a term is the set of things to which it is applicable, or what older logicians called inferiors. It is the set of its inferiors.

=> The confusion of their expositioin seems to be due to their use of the word « inferiors » which is itself metaphorical and unclear.


Is Frege an extensional logician ?

« One may perhaps get the impression from these explanations that the conflict between extensional and intensional logicians I am taking the side of latter. In fact I do hold that the concept is logically prior to its extension, and I regard as futile the attempt to base the extension of a concept as a class not on the concept but on individual things. »

« Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra of Logik, p. 455

From introduction of Montgomery Furth to The Basic Laws of Arithmetic, p. xl.

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f

a1, a2, ai, aj, an, …..

concept

Ext(f)

In Frege’s approach and « classic » set theory : every object in Ext(f) is fully specified.

f(ai) = Tfor i = 1,2, …n, …


f

a1, a2, ai, aj, an, …..

concept

Ext(f)

In this new approach : every ai in Ext(f) is also fully specified but exist no fully specified objects in Expansion.

Expans(f) ττττ(f)

Int (f)

typical object

x = no specified object

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4. CATEGORIZATION :4. CATEGORIZATION :a new approacha new approach


Notion of expansionInstances are specific or no specific.

• Following Port Royal’s Logic, we introduce Expansion of a concept (in French : « Etendue »)• Expansion contains all instances, specific or no specific :

Expans(f) = { x ; f(x) = T }

• Expansion generalizes extension to no specified instances; • Extension contains all specified instances• Extension is a part of expansion : Ext(f) ⊆⊆⊆⊆ Expans(f)

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Intension / Essence

The essenceof a concept is the class of all concepts such that

all objects which fall under the concept inherit necessarly these concepts.

=> Essence is a part of the intension

A concept in the intension is not necessarly inherited by an object at which is applied this concept, with the value « true ».

Characterizing and defining a concept is always a discussion about intension and essence of this concept.


Specification and Typicality

⇒All instances of a concept are not homogeneous :

• there are typical and atypical instances ;

• there are specified and no specified instances,;

• instances are more or less specified …

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More or less specified instances

A dog is less specified than this dog

A whitedog is more specified than a dog

=> «a white dog » is an inferior of « a dog »

We get a sequence of more specified instances :

a dog -> a whitedog

-> a whitedog which belongs to Tintin

-> this dog = Milou


Typical and atypical instances

In a category, all instances are not homogeneous :

• some instances are « good representations » of the concept ; as an object : these objects are prototypes of the concept ;

• others instances may be atypical, they cannot be « good representations », as objects, of the concept ;

• typical instances inherit all conceptsof intension

• atypical instances does not inherit all conceptsof intension

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Prototypes : Examples

• « Adam » is a prototype of « to be an human » ;

• « Eve » is a prototype of « to be a woman » ;

• « Doctor Fautus » is the prototype of the concept

« to be a very old scientist who is falling in love

with a young lady » ;


Expansion /Extension Intension / Essence

An object of Expansion is not necessarly fully specified. Only, the objects of Extensionare fully specified.

All objects of Expansiondo not inherit all concepts of Intension

but :

1) All objects of Expansioninherit all concepts of Essence;

2) All typical objects inherit all concepts of Intension.

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Problems

=> How to define and to handle

• specified instancesand no specified instances ?

• typical and a typical instances ?

=> How capture the relations « more typical than » and « more specified than » ?

⇒ How to reformulate Extensionand Intensionwith this new approach of categorization ?

⇒ How to relate Extensionto the notion of Expansion?


5. Typical object and 5. Typical object and specification operatorspecification operator

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Typical object : ττττ(f)

To every concept ‘f’ with the type ‘FJH’, we associate :

an object ττττ(f), which is « the best representation » as no specified object, of the concept ‘f’, :

ττττ(f) is the typical objectsuch that :

ττττ(f) is a the less specified object among instances of ‘f’;

ττττ(f) inherits all concepts contained in the intension of ‘f’ ;

ττττ(f) generates all typical (specified or not) instances of ‘f’.


Typical Object

The typical Object ττττ(f) of the concept ‘f’ is such that

∀∀∀∀g ∈∈∈∈ Int(f) :

1) It inherits all concepts ‘g’ which belong to Int(f) : g(ττττf) = T

2) It is a fixpoint : δδδδ(g)(ττττ(f)) = ττττ(f)

3) It generates all typical instances of Expans(f) by means of specifications associated to other concepts

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Specification operator : δδδδ(g)

Let ‘g’ a concept with the type FJH.

To ‘g’ is associated a function ‘d(g)’, with the type FJJ : ‘δδδδ(g)’ builds a more specified object ‘y’ from an object ‘x’

• If ‘x’ is an object, then the object ‘y’ is specified by the concept ‘g’:

y = δδδδ(g)(x) ;

• The object ‘y’ inherits the concept ‘g’ :

g(y) = g( δδδδ(g)(x) ) = T


Path of successive specifications

The object ‘y’ is specified, by means of a path ‘∆∆∆∆’ of successive determinations, from ‘x’ :

y = ∆∆∆∆(x) = ( δδδδ(gn) 0 …0 δδδδ(g2) 0 δδδδ(g1) ) (x)

The concepts and associated specifications δδδδ(gi) (i=1, 2, …,n) are the components of the path ‘∆∆∆∆’.

The successive specifications builds the object ‘y’ from ‘x’ and successive assertions :

g1(y) = g2 (y) = …= gn (y) = T

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The instance ‘y’ is more specified than the instance ‘x’

1) x and y are instances of the expansion :

x ∈∈∈∈ Expans(f) and y ∈∈∈∈ Expans(f)

2) Exist concepts g1, g2, …, gn such that :

y = (δδδδ(gn) 0 …0 δδδδ(g2) 0 δδδδ(g1)) (x)

with some conditions on specifications.


x1 = δδδδ(g1)(ττττ(f))

x2 = δδδδ(g2)(δδδδ(g1)(ττττ(f))

y = δδδδ(gn) (…. (δδδδ(g2)(δδδδ(g1)(ττττ(f)) …)

δδδδ(g1)

δδδδ(g2)

δδδδ(gn)

.

.

.

x

Expans(f)

In Expans(f) :

‘y’ is an inferior of ‘x’

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Fully specified or no specified instances of a concept ‘f’

‘x’ is fully specified iff the specification of ‘x’ is maximal : the object ‘x’ can be designated by a deictic operator : « this x »

=> ‘x’ belongs to the Extension : x ∈∈∈∈ Ext(f)

‘x’ is not (fully) specified when it cannot be designated by a deictic operator => ‘x ∉∉∉∉ Ext(f)

but a part Ext(x) of ‘Ext(x)’ may be associated to the object ‘x’ ∈∈∈∈ Expans(f)


x = a no specified instance of ‘f’

a1 a2 a3 … … an }

Fully specified instances of ‘f’

Ext(f) ⊇⊇⊇⊇ Ext(x) {=

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To every concept ‘f’, with type FJH, are associated :

(i) the object ‘ττττ(f)’, called « typical object », with the type ‘J’;

(ii) the specification operator ‘δδδδ(f)’, with the type ‘FJJ’.

• ‘ττττ’ is a constructive operator of a representative object of concept ; its type is : FFJHJ ;

• ‘δδδδ’ is a constructive operator of specification ; its type is : FFJHFJJ.

Constructive operators ττττ and δδδδ


The operator ‘ττττ’ is a fixpoint for ‘ Sδδδδ’

1. (δδδδ(f))(ττττ(f))

2. Sδδδδ ττττ f intr. Combinator S

3. [δδδδ(f)(ττττ(f)) = ττττ(f) ] pointfix property

4. [Sδδδδ ττττ (f) = ττττ (f ) ]

5. [Sδδδδ ττττ = ττττ ] by abstraction

Combinatory relation between ττττ and δδδδ

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6. Conflicts by specifications6. Conflicts by specifications


ττττ(f)

∆∆∆∆’

x = ∆∆∆∆(ττττ(f))

δδδδ(g)

y = δδδδ(g)(x)

A concept ‘g’ can conflict with a concept ofInt(f) - Ess(f)or with other specifications, in the path ‘∆∆∆∆’.

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Conflict with Intension

Let a concept ‘f’ with its Intension Int(f).

Let ‘g’ a concept such that ‘y = (δδδδg)(x)’ is an instance of ‘f’ (with ‘x’ ∈∈∈∈ Expans(f) and ‘x’ inherits all properties of Int (f)) .

If exists a concept ‘h’ of Int(f) – Ess(f) such that :

h = N1(g)

then ‘g’ conflicts with Int (f) .

In this case :[ h(y) = (N1g)(y) = N0(g(y)) = T ] ∧∧∧∧ [ g(y) = T ]

=> a contradiction about the object ‘y’ specified by ‘δδδδ(g)’ .


ττττ(f)

f

h = N1(g)

∆∆∆∆

δδδδ(g)

ττττ

Int (f)

Expans(f)

x

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Conflict in a path of specifications

Let a path ‘∆∆∆∆’ :

y = ∆∆∆∆(x) = (δδδδ(gn) 0 …0 δδδδ(gj) 0 … 0 δδδδ(gi) 0… δδδδ(g1)) (x)

The concept gi conflicts with the concept gj

when gj is the negation of gi (gj = N1(gi)) or the inverse (gj = N1(gi)) :

there is a contradiction in the components of the path ‘∆∆∆∆’ .


x1 = δδδδ(g1)(ττττ(f))

x2 = δδδδ(g2)(δδδδ(g1)(ττττ(f))

y = δδδδ(gn) (…. (δδδδ(g2)(δδδδ(g1)(ττττ(f)) …)

δδδδ(g1)

δδδδ(g2)

δδδδ(g i)

.

ττττ(f)

Expans(f)

δδδδ(g j)

xi

xj

with g j = N1(g i)

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Conflict with Essence

Let ‘f’ a concept with Ess(f) ⊆⊆⊆⊆ Int(f).

If a concept ‘g’ conflicts with a concept of Ess(f),

then exists ‘h’ in Ess(f) such that :

h = N1(g) ,

If ‘u = ( δδδδg)(x)’ is an instance of ‘f’,

then a contradiction arizes :

[ g(u) = T ] ∧∧∧∧ [h(u) = (N1(g))(x) = N0(gu) = T]

=> ‘u’ does not belong to Expans(f).


f

ττττ(f)

x u = δ δ δ δ(g)(x) δ δ δ δ(g)

ττττ

Ess(f)

h = N1(g)

Int (f)

Expans(f)

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Structured Class of Concepts and Objects

Let < FF , ->, ττττ, δ, δ, δ, δ, OO > where :

•• FF is a class of individual concepts structured by a preorder ‘->’ between concepts ;•• OO is a class of objects such that the concepts of FFcan be applied to;• ττττ is an operator which relates a concept to its associatestypical object ;• δδδδ is an operator which gives a specification to the objects.


f

ττττ(f)

x

y = δ δ δ δ(g1)(x)

δ δ δ δ(g1)

y = δ δ δ δ(g2)(x)

y = δ δ δ δ(g3)(x) δ δ δ δ(g2) δ δ δ δ(g3)

ττττ

Typical instances

g1∉∉∉∉Int(f)g2∉∉∉∉Int(f) ∧∧∧∧(∃∃∃∃ h2∈∈∈∈Int(f) -Ess(f); h2= Ng2g3∉∉∉∉Int(f) ∧∧∧∧(∃∃∃∃ h3∈∈∈∈ Ess(f) ⊆⊆⊆⊆ Int(f) ; h3 = Ng3

All instances

Ess(f)

h3 = Ng3

Int(f)h2 = Ng2

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Typical / atypical instances of a concept

Any instance of ‘f’ belongs to Expans(f) and it inherits all concepts of Ess(f).

• Any typical instanceof ‘f’ inherits every concept of Int(f).

• Any atypical instanceof ‘f’ does not inherit every concept of Int( f).


Typical / atypical instances of a concept (2)

Let a object ‘y’ specified from an instance ‘x’ of ‘f’ :

y= (δδδδg)(x))

• If ‘g’ does not conflict with any concept of Int(f), then ‘y’ belongs to Expans(f) and is a typical instanceof ‘f’ ;

• If ‘g’ conflicts with some concept of Int(f) – Ess(f), then ‘y’ belongs to Expans(f) but it is an atypical instanceof ‘f’ ;

• If ‘g’ conflicts with some concept of Ess(f), then ‘y’ does not belong toExpans(f) : ‘y’ is out of the category genrated by ττττ(f).

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A typical / atypical instance of an atypical instance

Let ‘x’ an atypical instance of a concept ‘f’.

Let y = ∆∆∆∆(x) an instance of ‘f’ (=> ‘y’ belongs to Expans(f) )

The object ‘y’ is a typical instance of ‘x’when every concept in the path ‘∆∆∆∆’ does not conflict with the other concepts in the path «∆∆∆∆’ » from ‘ ττττ(f)’ to ‘x’.

The object ‘y’ is an atypical instance of ‘x’when there is a concept ‘g’ in the path ‘∆∆∆∆’ which conflicts with a concept « g’ » in the path «∆∆∆∆’ » from ‘ ττττ(f)’ to ‘x’.


ττττ(f)

∆∆∆∆’

x = ∆∆∆∆’(ττττ(f))

∆∆∆∆

y = ∆∆∆∆ (x)

Let x an atypical instance of f

1) If ‘g’, in the path ‘ ∆∆∆∆’, conflicts with a concept « g’ » inin the path «∆∆∆∆’ »,then ‘y’ is an atypical instance of ‘x’.

2) The instance ‘y’ can be a typical instance of the instance ‘x’, but ‘x’ is an atypical instance of ‘f’.

δδδδ(g)

δδδδ(g’)

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7. «7. « StarStar » quantifiers» quantifiers


« Classical » quantifiers versus « star » quantifiers

• A « classical » quantifier is an operator whose the operand is a predicate and the result is a proposition or a predicate

• A « star » quantifier is a specification operator which apply to a term.

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Illative quantifiers « classic » An illative quantifier is a version of fregean quantifiers (or classical quantifiers) without using bound variables

Classical quantifiers Illative quantifiers Logicalwith bound without bound Types variables variables∀∀∀∀x [ f(x) ] ΠΠΠΠ1 f FFJHH∃∃∃∃x [ f(x) ] ΣΣΣΣ1 f

∀∀∀∀x [ f(x) => g(x) ] ΠΠΠΠ2 fg FFJHFJHH∃∃∃∃x [ f(x) ∧∧∧∧ g(x) ] ΣΣΣΣ2 fg


Rules for illative quantifiers

FFJHH : ΣΣΣΣ1111 FJH : f FFJHH : ΠΠΠΠ1 FJH : f

----------------------------- ---------------------------

H : ΣΣΣΣ1 f H : ΠΠΠΠ1 f

« Something is f » « Anything is f »

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Illative quantifiers ΣΣΣΣ2 and ΠΠΠΠ2

FFJHFFJHH : ΣΣΣΣ2222 FJH : f FFJHFFJHH : ΠΠΠΠ2222 FJH : f

------------------------------------ -------------------------------------

FFJHH : ΣΣΣΣ2f FJH : g FFJHH : ΠΠΠΠ2f FJH : g

-------------------------------------------- ---------------------------------------------

H : ΣΣΣΣ2fg H : ΠΠΠΠ2fg

« Some f is g » « Any f is g »


« Star » Quantifiers ΣΣΣΣ* and ΠΠΠΠ*

A « star » quantifier is an operator which builds up a no specified object from an object :

FJJ : ΣΣΣΣ* J : a FJJ : ΠΠΠΠ * J : a

----------------------- -----------------------

FJH : g J : ΣΣΣΣ*a FJH : g J : ΠΠΠΠ*a

--------------------------------------------- ---------------------------------------------

H : g (ΣΣΣΣ*f) H : g ( ΠΠΠΠ*f)

« Some f is g » « Any f is g »

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No specified /Any object

a

ΣΣΣΣ*a ΠΠΠΠ*a

No specified Object, abstract from a

Any objectabstract from a

object


ττττ(f)

ΣΣΣΣ *(ττττ (f))

{a1 a2 … … an}Ext(f)

ΣΣΣΣ*

Typical instances of f

ΣΣΣΣ*(ττττ (f)) is an no specifiedobject such that :f (ΣΣΣΣ*(ττττ (f))) = T

a1, a2, …, an are completely determinate Objects, such that

f(a1) = f(a2) = …= f(an) = T

Abstractionby no specification

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ττττ(f)

ΠΠΠΠ*(ττττ (f))

{a1 a2 … … an}Ext((ΠΠΠΠ*(ττττ (f)))

ΠΠΠΠ*

Typical instances of f

ΠΠΠΠ*(ττττ (f)) is an object whatever such thatf (ΠΠΠΠ*(ττττ (f))) = T

a1, a2, …, an are completely determinate objects, substituable to the no determinate object ΠΠΠΠ*(ττττ (f)).


Rules for « star » quantifiers

g(ΠΠΠΠ*(ττττ(f))) g(x)

------------- [e-ΠΠΠΠ*] -------------- [i- ΣΣΣΣ*]

g(x) g(ΣΣΣΣ*(ττττ(f)))

‘x’ is any typical instance ‘x’ is a no specified instanceof ‘f’ of ‘f’

ΠΠΠΠ*(ττττ(f)) is whatever ; ΣΣΣΣ*(ττττ(f)) is no specified

It is an object. It is an object.

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« Classical » Universal QuantifierΠΠΠΠ2

reduces to the « Star » Quantifier ΠΠΠΠ*

[ ΠΠΠΠ2 = BC* ΠΠΠΠ* ] (law) ΠΠΠΠ2 is defined in terms of the quantifier ΠΠΠΠ*

(ΠΠΠΠ2f)g =>ββββ g(ΠΠΠΠ*f)

1. (ΠΠΠΠ2f)g hyp.

2. [ ΠΠΠΠ2 = BC* ΠΠΠΠ* ] def. of ΠΠΠΠ2

3. BC* ΠΠΠΠ* fg rempl.

4. C* (ΠΠΠΠ* f) g [B-e]

5. g(ΠΠΠΠ* f) [C*-e]


The classical existential Quantifier ΣΣΣΣ2 reduces to the existential Star Quantifier ΣΣΣΣ*

[ ΣΣΣΣ2 = BC* ΣΣΣΣ* ] (law) Reduction of ΣΣΣΣ2 to ΣΣΣΣ*

(ΣΣΣΣ2f)g =>ββββ g(ΣΣΣΣ*f)

1. (ΣΣΣΣ2f)g hyp.

2. [ ΣΣΣΣ2 = BC* ΣΣΣΣ* ] def. of ΣΣΣΣ2

3. BC* ΣΣΣΣ* fg rempl.

4. C*(ΣΣΣΣ* f) g [B-e]

5. g(ΣΣΣΣ* f) [C*-e]

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Π2fg

Σ2fg

(Π2 f) (N1g)

(Σ2f )(N1g)

Σ1f

g(Π∗Π∗Π∗Π∗f)

g(ΣΣΣΣ*f)

(N1g)(ΠΠΠΠ*f)

(N1g)(ΣΣΣΣ*f)

f(ΣΣΣΣ*)

contrary

disjunction


ττττ(f)

f

δδδδ(g1) (ττττ(f))

Int (f)

Expans(f)

z

δδδδ(g2) (ττττ(f)) u

Typical object

Ext (f)Extττττ (f)Typical fully specified instances

does not belongto Expans(f)

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Power of Combinatory Logic

A very flexible and sound language for expressing :

• Complex concepts from given operators ;

• Intrinsic properties of operators ;

• Relations between operators (with isotypicality principle) ;

• Without using bound variables : no telescopage of bound variables, no side effects…


Using Combinatory Logic• Logic : Study of paradoxes, recursive functions, quantification, semiotic analysis of variables; new developments for alternative logics;

• Computer Sciences: Study of the semantics of programming languages; Applicative style of programming : ML, CAML, HASKELL …

• Linguistics : Formal expression of relations between grammatical and lexical operators; Cognitive and Applicative Grammar (CAG); relations (analysis and synthesis) between levels of representations;

• Cognitive Sciences and AI: Representations of knowledges; representation of meaning for lexical predicates (verbs, prepositions…);

• Analysis of philosophical concepts: Combinatory analysis of the Unum Argumentumof Anselme of Cantorbery’s Proslogion…

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DESCLES, Jean-Pierre, “De la notion d’opération à celle d’opérateur ou à la recherche deformalismes intrinsèques”,Mathématiques et sciences humaines, Paris, 1981, pp. 5-32.

DESCLES, Jean-Pierre, « Approximation et typicalité », L’a-peu-près, Aspects anciens etmodernes de l’approximation, Editions de l’Ecole des Hautes Etudes en Sciences Sociales,Paris, 1988, pp. 183-195.

DESCLES, Jean-Pierre,Langages applicatifs, langues naturelles et cognition, Paris, Hermès,1990.

DESCLES, Jean-Pierre, « La double négation dans l'Unum Argumentum analysé à l'aide dela logique combinatoire"Travaux du Centre de Recherches Semiologiques, n°59, pp. 33-74,Université de Neuchâtel, septembre, 1991.

DESCLES, Jean-Pierre, « La logique combinatoire typée est-elle un « bon » formalismed’analyse des langues naturelles et des représentations cognitives ? » in LENTIN, 1997, pp.179-223.

DESCLES, Jean-Pierre, « Logique combinatoire, types, preuves et langage naturel »,inTravaux de logique, Introduction aux logiques non classiques, Centre de Recherchessémiologiques, Université de Neuchâtel, 1997, pp. 91-160.

DESCLES, Jean-Pierre, « Categorization : A Logical Approach of a Cognitive Problem”,Journal of Cognitive Science, Vol. 3, n° 2, 2002, pp. 85-137.

DESCLES, Jean-Pierre, “Analyse non frégéenne de la quantification”, in Pierre Jorday(éditeur) Quantification dans la logique moderne, L’Harmattan, Paris, pp. 264-312.


DESCLES, Jean-Pierre, « Combinatory Logic, Language, and Cognitive Representations », in Paul Weingartner (editor) Alternative Logics. Do Sciences Need Them ?, Springer, 2003, pp. 115-148.

DESCLES, Jean-Pierre, et Zlatka GUENTCHEVA, « Quantification Without Bound Variables », in Böttner, Thümmel (editors), Variable-free Semantics, Secolo Verlag, Rolandsmauer, 13-14, Osnabrück, 2000, pp. 210-233.

FREUND Michael, Jean-Pierre DESCLES, Anca PASCU, Jérôme CARDOT, « Typicality, Contextual Inferneces and Object Determination Logic », soumis à publication, 2004, 26 pages.

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Combinatory Logic, Categorization and Typicalitylalic.paris-sorbonne.fr/PUBLICATIONS/2004/presentation... · 2012-02-16 · Combinatory Logic, Categorization and Typicality Jean-Pierre