combinatory logic, categorization and...
TRANSCRIPT
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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
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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
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• 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.
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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
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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).
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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, …
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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
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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.
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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
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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 » ;
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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?
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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’.
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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
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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.
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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)
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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
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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
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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)’ .
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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 ‘∆∆∆∆’ .
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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).
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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.
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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).
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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’.
Jean-Pierre Desclés, Berne oct. 2004 66
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
Jean-Pierre Desclés, Berne oct. 2004 68
« 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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Jean-Pierre Desclés, Berne oct. 2004 69
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
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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 »
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« 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
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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)).
Jean-Pierre Desclés, Berne oct. 2004 76
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]
Jean-Pierre Desclés, Berne oct. 2004 78
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
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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…
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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.
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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.