Combinatory Logic, Categorization and...

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. descles@paris4.sorbonne.fr

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

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

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’.

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

What is a combinator ? (3)

X u1 u2 ….. un

X u1u2…un

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 ]

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’.

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

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.

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.

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.

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 ]

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”

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

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 »»

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.

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, …

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

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)

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 …

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

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

1) All objects of Expansioninherit all concepts of Essence;

2) All typical objects inherit all concepts of Intension.

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

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

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

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)

Expans(f)

In Expans(f) :

‘y’ is an inferior of ‘x’

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) {=

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

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 ‘∆∆∆∆’.

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)

h = N1(g)

∆∆∆∆

δδδδ(g)

ττττ

Int (f)

Expans(f)

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)

with g j = N1(g i)

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).

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)

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.

ττττ(f)

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

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).

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’)

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.

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 »

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 »

No specified /Any object

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

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.

« 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]

(Π2 f) (N1g)

(Σ2f )(N1g)

g(Π∗Π∗Π∗Π∗f)

g(ΣΣΣΣ*f)

(N1g)(ΠΠΠΠ*f)

(N1g)(ΣΣΣΣ*f)

f(ΣΣΣΣ*)

contrary

disjunction

ττττ(f)

δδδδ(g1) (ττττ(f))

Int (f)

Expans(f)

δδδδ(g2) (ττττ(f)) u

Typical object

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

does not belongto Expans(f)

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…

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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