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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
Problems with the naive Problems with the naive approach approach
of categorisationof categorisation
Jean-Pierre Desclés, Berne oct. 2004 26
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.
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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, …
Jean-Pierre Desclés, Berne oct. 2004 32
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?
Jean-Pierre Desclés, Berne oct. 2004 42
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’ :
•• 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.
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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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
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 82
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.
Jean-Pierre Desclés, Berne oct. 2004 84
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.