Swiss Society for Logic and Philosophy of Science, Berne, 14-15 october 2004
Combinatory Logic, Categorization and Typicality Jean-Pierre Desclés Paris-Sorbonne University LaLICC « Languages, Logic, Informatics, Cognition and Communication », CNRS / Paris-Sorbonne Jean-Pierre.
[email protected] Jean-Pierre Desclés, Berne oct. 2004
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Summary 1. Combinatory Logic 2. Differences between Combinatory Logic and λ-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 Jean-Pierre Desclés, Berne oct. 2004
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1. Combinatory Logic
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COMBINATORY LOGIC = a logic of operators • with different compositions of operators ; • where a composition is expressed by an abstract operator, called a Combinator ; • without using bound variables ; • defined inside the applicative language, without interpreting in specific domains.
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Combinatory expressions (e.c.) Rules : (i) Basic expressions are e.c. ; (ii) If ‘X’ and ‘Y’ are e.c. then is a e.c.
The result of the application is presented by a simple concatenation of operator ‘X’ and operand ‘Y’, hence : XY = def We suppose left association : XYZ = (XY)Z ≠ X(YZ) Jean-Pierre Desclés, Berne oct. 2004
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Church’s Functional Types Rules : (i) The basic types are functional types ; αβ’ (ii) If ‘α α’ and ‘β β ’ are functional types then ‘Fαβ αβ is a functional type. Rule of application : @ < [Fαβ αβ : X] , [α α : Y] > =>β [β β : Z] When an operator ‘X’, with the type ‘Fαβ αβ’, αβ is aplying to an operand ‘Y’ with the type ‘α α’, then the type of the type of the result ‘Z’ is ‘β β ’. Jean-Pierre Desclés, Berne oct. 2004
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Application , Abstraction Application [Fαβ αβ : X],
Abstraction
[α α : Y]
[β β : XY],
[α α : Y]
---------------------------
-------------------------
[β β : XY]
[Fαβ αβ : X]
Analogy with proposition calculus : Modus Ponens ( ⊃ - elimination )
( ⊃ - introduction )
α
α
⊃ αβ
β
--------
--------------
β
⊃ αβ
hyp.
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What is a combinator ? (1) A combinator is an abstract operator which produces a new complex operator from given operators. Examples of elementary combinators :
IX
⇒β X
identity
BXYZ ⇒β
X(YZ)
functional composition
WXY ⇒β
XYY
diagonalization
KXY ⇒β
X
cancellation Jean-Pierre Desclés, Berne oct. 2004
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What is a combinator ? (3) @ @
.
.
.
X u1u2…un @ @
X
u1
u2 …..
un
a1
Complex operator
ap
Successive operands
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« Equivalent » λ-expressions Every combinator can be expressed by a λ -expression :
I
=def
λf [ f ]
K
=def
λf . λx [ f ]
S
=def
λg . λf . λx [ gx(fx) ]
C
=def
λf . λx . λy [ fyx ]
B
=def
λg. λf . λx [ g(fx) ]
W
=def
λf . λx [ fxx ]
C*
=def
λx. λf [ fx ]
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Properties of combinators • A combinator can be expressed by a λ-expression ; • A combinator is self-applicative ; • There are basic combinators ; • All combinators are defined from basic combinators ; • Two basic combinators are sufficient, for instance : S and K ; • There is an « algebra » of combinators, generated from basic combinators ; • For every combinator, there is a type schema (polymorphism). Jean-Pierre Desclés, Berne oct. 2004
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Negation of a concept Let ‘N0’ the operator of proposition negation. From ‘N0’ , we define the negation operator ‘N1’ of a concept : 1. N0(fa) 2. BN0 fa 3. [ N1 =def BN0 ] 4. (N1f) a
hyp. B int. def. of N1 rempl.
The types of ‘N0’ = ‘FHH’; The type of ‘N1’ = ‘FFJHFJH’. Jean-Pierre Desclés, Berne oct. 2004
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Twice = def WB
Define the operator « twice » : twice f x = f(fx) 1. 2. 3. 4. 5.
f(f x) Bff x WBfx [ twice = WB ] twice fx Jean-Pierre Desclés, Berne oct. 2004
B-intr. W-intr. def. rempl. 13
2. Differences between Curry’s Combinatory Logic and Church’s λ-Calculus
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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 λ-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 S and K : [x] Sxyz [x] xz(yz) hence :
= =
S(SS(Ky))(Kz) S(SI(Kz))(K(yz))
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, coursesof-values … are objects. Jean-Pierre Desclés, Berne oct. 2004
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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 Jean-Pierre Desclés, Berne oct. 2004
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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 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 ] Jean-Pierre Desclés, Berne oct. 2004
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be-mortal-being
have-two-legs
be-man
Socrates
Inheritance Principle 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” Jean-Pierre Desclés, Berne oct. 2004
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Problems with the naive approach of categorisation Jean-Pierre Desclés, Berne oct. 2004
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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 white dog, a dog which belongs to Tintin
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A problem of Inheritance ‘Good’ Deduction:
‘Bad’ Deduction:
(1) All men have two feet (2) Aristotle is a man -----------------------------(3) ∴ Aristotle has two feet
(4) (5) (6)
A man has two feet John is a man John has only one foot ---------------------------* John has two feet
(7)
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 foot and John has two feet. Jean-Pierre Desclés, Berne oct. 2004
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John cannot inherit the property « have have--two two--feet »
Int(be-a-man) have two feet
Int(John)
to-be-a-man
contradiction
have only one foot John
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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. Jean-Pierre Desclés, Berne oct. 2004
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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. Jean-Pierre Desclés, Berne oct. 2004
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In Frege’s approach and « classic » set theory : every object in Ext(f) is fully specified. concept
f
Ext(f)
a1, a2,
ai,
f(ai) = T for i = 1,2, …n, …
aj,
an, …..
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In this new approach : every ai in Ext(f) is also fully specified but exist no fully specified objects in Expansion.
Int(f) concept
f Expans(f)
τ(f)
typical object
Ext(f)
x = no specified object
a1, a2,
ai,
aj, Jean-Pierre Desclés, Berne oct. 2004
an, ….. 32
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4. CATEGORIZATION : a new approach
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Notion of expansion Instances 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) Jean-Pierre Desclés, Berne oct. 2004
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Intension / Essence The essence of 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. Jean-Pierre Desclés, Berne oct. 2004
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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 white dog 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 white dog -> a white dog which belongs to Tintin -> this dog = Milou Jean-Pierre Desclés, Berne oct. 2004
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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 concepts of intension • atypical instances does not inherit all concepts of 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 Extension are fully specified.
All objects of Expansion do not inherit all concepts of Intension but : 1) All objects of Expansion inherit all concepts of Essence; 2) All typical objects inherit all concepts of Intension. Jean-Pierre Desclés, Berne oct. 2004
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Problems => How to define and to handle • specified instances and no specified instances ? • typical and a typical instances ? => How capture the relations « more typical than » and « more specified than » ? ⇒ How to reformulate Extension and Intension with this new approach of categorization ? ⇒ How to relate Extension to the notion of Expansion ? Jean-Pierre Desclés, Berne oct. 2004
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5. Typical object and specification operator
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Typical object : τ(f) To every concept ‘f’ with the type ‘FJH’, we associate : an object τ(f), which is « the best representation » as no specified object, of the concept ‘f’, : τ(f) is the typical object such that : τ(f) is a the less specified object among instances of ‘f’; τ(f) inherits all concepts contained in the intension of ‘f’ ; τ(f) generates all typical (specified or not) instances of ‘f’.
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Typical Object The typical Object τ(f) of the concept ‘f’ is such that ∀g ∈ Int(f) : 1) It inherits all concepts ‘g’ which belong to Int(f) : g(ττf) = T 2) It is a fixpoint : δ(g)(ττ(f)) = τ(f) 3) It generates all typical instances of Expans(f) by means of specifications associated to other concepts
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Specification operator : δ(g) Let ‘g’ a concept with the type FJH. To ‘g’ is associated a function ‘d(g)’, with the type FJJ : ‘δ δ(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 = δ(g)(x) ; • The object ‘y’ inherits the concept ‘g’ : g(y) = g( δ(g)(x) ) = T Jean-Pierre Desclés, Berne oct. 2004
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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) = ( δ(gn) 0 …0 δ(g2) 0 δ(g1) ) (x) The concepts and associated specifications δ(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 Jean-Pierre Desclés, Berne oct. 2004
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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 = (δ δ(gn) 0 …0 δ(g2) 0 δ(g1)) (x) with some conditions on specifications. Jean-Pierre Desclés, Berne oct. 2004
x
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In Expans(f) :
δ(g1) x1 = δ(g1)(ττ(f))
‘y’ is an inferior of ‘x’
δ(g2)
Expans(f)
x2 = δ(g2)(δ δ(g1)(ττ(f))
. . . δ(gn)
y = δ(gn) (…. (δ δ(g2)(δ δ(g1)(ττ(f)) …)
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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) Jean-Pierre Desclés, Berne oct. 2004
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x = a no specified instance of ‘f’
Ext(f) ⊇ Ext(x) = { a1
a2
a3
…
…
an }
Fully specified instances of ‘f’ Jean-Pierre Desclés, Berne oct. 2004
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Constructive operators τ and δ To every concept ‘f’, with type FJH, are associated : (i) the object ‘ττ(f)’, called « typical object », with the type ‘J’; (ii) the specification operator ‘δ δ(f)’, with the type ‘FJJ’. • ‘ττ’ is a constructive operator of a representative object of concept ; its type is : FFJHJ ; • ‘δ δ’ is a constructive operator of specification ; its type is : FFJHFJJ. Jean-Pierre Desclés, Berne oct. 2004
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Combinatory relation between τ and δ The operator ‘ττ’ is a fixpoint for ‘Sδ’ 1.
(δ δ(f))(ττ(f))
2.
Sδτf
intr. Combinator S
3.
[δ δ(f)(ττ(f)) = τ(f) ]
pointfix property
4.
[S δ τ (f) = τ (f ) ]
5.
[S δ τ = τ ] Jean-Pierre Desclés, Berne oct. 2004
by abstraction 52
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6. Conflicts by specifications
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τ(f) ∆’ x = ∆(ττ(f)) δ(g)
A concept ‘g’ can conflict with a concept of Int(f) - Ess(f) or with other specifications, in the path ‘∆ ∆’.
y = δ(g)(x)
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Conflict with Intension Let a concept ‘f’ with its Intension Int(f). Let ‘g’ a concept such that ‘y = (δ δg)(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 ‘δ δ(g)’ . Jean-Pierre Desclés, Berne oct. 2004
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h = N1(g)
Int(f) f τ τ(f) ∆
Expans(f)
δ(g) x Jean-Pierre Desclés, Berne oct. 2004
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Conflict in a path of specifications Let a path ‘∆ ∆’ : y = ∆(x) = (δ δ(gn) 0 …0 δ(gj) 0 … 0 δ(gi) 0… δ(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 ‘∆ ∆’ . Jean-Pierre Desclés, Berne oct. 2004
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τ(f) δ(g1) x1 = δ(g1)(ττ(f)) δ(g2)
Expans(f)
x2 = δ(g2)(δ δ(g1)(ττ(f))
. δ(g i)
xi δ(g j)
with g j = N1(g i)
xj y = δ(gn) (…. (δ δ(g2)(δ δ(g1)(ττ(f)) …) Jean-Pierre Desclés, Berne oct. 2004
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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 = (δ δg)(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). Jean-Pierre Desclés, Berne oct. 2004
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Int(f) Ess(f)
h = N1(g) f τ τ(f)
Expans(f) x
δ(g) δ
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u = δ(g)(x) δ
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Structured Class of Concepts and Objects Let < F , ->, τ, δ, O > where : • F is a class of individual concepts structured by a preorder ‘->’ between concepts ; • O is a class of objects such that the concepts of F can be applied to; • τ is an operator which relates a concept to its associates typical object ; • δ is an operator which gives a specification to the objects. Jean-Pierre Desclés, Berne oct. 2004
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Int(f) h2 = Ng2 Ess(f) h3 = Ng3
g1∉ ∉Int(f) g2∉ ∉Int(f) ∧(∃ ∃ h2∈ ∈Int(f) -Ess(f); h2= Ng2 g3∉ ∉Int(f) ∧(∃ ∃ h3∈ ∈ Ess(f) ⊆ Int(f) ; h3 = Ng3
δ(g δ 2)
f τ τ(f)
x δ(g δ 1)
δ(g δ 3)
y = δ(g δ 3)(x)
y = δ(g δ 1)(x) y = δ( δ g2)(x) Typical instances
All instances
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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 instance of ‘f’ inherits every concept of Int(f). • Any atypical instance of ‘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= (δ δg)(x)) • If ‘g’ does not conflict with any concept of Int(f), then ‘y’ belongs to Expans(f) and is a typical instance of ‘f’ ; • If ‘g’ conflicts with some concept of Int(f) – Ess(f), then ‘y’ belongs to Expans(f) but it is an atypical instance of ‘f’ ; • If ‘g’ conflicts with some concept of Ess(f), then ‘y’ does not belong to Expans(f) : ‘y’ is out of the category genrated by τ(f). Jean-Pierre Desclés, Berne oct. 2004
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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 ‘ττ(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 ‘ττ(f)’ to ‘x’. Jean-Pierre Desclés, Berne oct. 2004
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τ(f)
∆’
δ(g’)
x = ∆’(ττ(f))
∆
δ(g)
Let x an atypical instance of f 1) If ‘g’, in the path ‘∆ ∆’, conflicts with a concept « g’ » in in 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’.
y = ∆ (x) Jean-Pierre Desclés, Berne oct. 2004
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7. « Star » quantifiers
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« 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 with bound variables ∀x [ f(x) ] ∃x [ f(x) ]
Illative quantifiers without bound variables Π1 f Σ1 f
Logical Types
∀x [ f(x) => g(x) ] ∃x [ f(x) ∧ g(x) ]
Π2 fg Σ2 fg
FFJHFJHH
FFJHH
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Rules for illative quantifiers FFJHH : Σ1
FJH : f
FFJHH : Π1 FJH : f
-----------------------------
---------------------------
H : Σ1 f
H : Π1 f
« Something is f »
« Anything is f »
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Illative quantifiers Σ2 and Π2 FFJHFFJHH : Σ2
FFJHFFJHH : Π2
FJH : f
FJH : f
------------------------------------
-------------------------------------
FFJHH : Σ2f
FFJHH : Π2f
FJH : g
FJH : g
--------------------------------------------
---------------------------------------------
H : Σ2fg
H : Π2fg
« Some f is g »
« Any f is g »
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« Star » Quantifiers Σ* and Π* A « star » quantifier is an operator which builds up a no specified object from an object : FJJ : Σ*
FJJ : Π * J : a
J:a
----------------------FJH : g
J : Σ*a
----------------------FJH : g
J : Π*a
---------------------------------------------
---------------------------------------------
H : g (Σ Σ*f)
H : g (Π Π*f)
« Some f is g »
« Any f is g »
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No specified /Any object Any object abstract from a
No specified Object, abstract from a
Π*a
Σ*a
a object Jean-Pierre Desclés, Berne oct. 2004
τ(f) Σ*
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Σ*(ττ (f)) is an no specified object such that : f (Σ Σ*(ττ (f))) = T
Σ *(ττ (f)) Abstraction by no specification {a1
a2
… … Ext(f)
an}
Typical instances of f a1, a2, …, an are completely determinate Objects, such that f(a1) = f(a2) = …= f(an) = T Jean-Pierre Desclés, Berne oct. 2004
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τ(f)
Π*(ττ (f)) is an object whatever such that f (Π Π*(ττ (f))) = T
Π* Π*(ττ (f)) {a1
a2
… … Ext((Π Π*(ττ (f))) Typical instances of f
an}
a1, a2, …, an are completely determinate objects, substituable to the no determinate object Π*(ττ (f)). Jean-Pierre Desclés, Berne oct. 2004
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Rules for « star » quantifiers g(Π Π*(ττ(f)))
g(x)
------------- [e-Π Π*]
-------------- [i-Σ Σ*]
g(x)
g(Σ Σ*(ττ(f)))
‘x’ is any typical instance of ‘f’
‘x’ is a no specified instance of ‘f’
Π*(ττ(f)) is whatever ;
Σ*(ττ(f)) is no specified
It is an object.
It is an object.
Jean-Pierre Desclés, Berne oct. 2004
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« Classical » Universal Quantifier Π2 reduces to the « Star » Quantifier Π* [ Π2 = BC* Π* ] (law) Π2 is defined in terms of the quantifier Π* (Π Π2f)g =>β β g(Π Π*f) 1.
(Π Π2f)g
hyp.
2.
[ Π2 = BC* Π* ]
def. of Π2
3.
BC* Π* fg
rempl.
4.
C* (Π Π* f) g
[B-e]
5.
g(Π Π* f)
[C*-e]
Jean-Pierre Desclés, Berne oct. 2004
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The classical existential Quantifier Σ2 reduces to the existential Star Quantifier Σ* [ Σ2 = BC* Σ* ] (law) Reduction of Σ2 to Σ* (Σ Σ2f)g =>β β g(Σ Σ*f) 1.
(Σ Σ2f)g
hyp.
2.
[ Σ2 = BC* Σ* ]
def. of Σ2
3.
BC* Σ* fg
rempl.
4.
C*(Σ Σ* f) g
[B-e]
5.
g(Σ Σ* f)
[C*-e]
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Π2fg
contrary
(Π2 f) (N1g)
Σ1f disjunction (Σ2f )(N1g)
Σ2fg
(N1g)(Π Π*f)
g(Π∗ Π∗f) Π∗ f(Σ Σ*)
(N1g)(Σ Σ*f)
g(Σ Σ*f) Jean-Pierre Desclés, Berne oct. 2004
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Int(f)
f Expans(f)
τ(f) δ(g1) (ττ(f))
z
Typical object does not belong to Expans(f)
δ(g2) (ττ(f))
Extτ (f) Typical fully specified instances Jean-Pierre Desclés, Berne oct. 2004
u
Ext (f) 80
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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… Jean-Pierre Desclés, Berne oct. 2004
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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 Argumentum of Anselme of Cantorbery’s Proslogion… Jean-Pierre Desclés, Berne oct. 2004
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DESCLES, Jean-Pierre, “De la notion d’opération à celle d’opérateur ou à la recherche de formalismes intrinsèques”, Mathématiques et sciences humaines, Paris, 1981, pp. 5-32. DESCLES, Jean-Pierre, « Approximation et typicalité », L’a-peu-près, Aspects anciens et modernes 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 de la 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 » formalisme d’analyse des langues naturelles et des représentations cognitives ? » in LENTIN, 1997, pp. 179-223. DESCLES, Jean-Pierre, « Logique combinatoire, types, preuves et langage naturel », in Travaux de logique, Introduction aux logiques non classiques, Centre de Recherches sémiologiques, 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. Jean-Pierre Desclés, Berne oct. 2004
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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. Jean-Pierre Desclés, Berne oct. 2004
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