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Logic of Determination of Objects (LDO) :
How to articulate “extension”with “intension”
and “objects”with “concepts”
Jean-Pierre Descl´es and Anca Pascu
Abstract. From a logical viewpoint, object is never defined, even by a negative definition. This paper is a theoretical contribution about object using a
new constructivist logical approach called Logic of Determination of Objects
(LDO) founded on a basic operation, called determination. This new logic
takes into account cognitive problems such as the inheritance of properties
by non typical occurrences or by indeterminate atypical objects in opposition to prototypes that are typical completely determinate objects. We show
how extensional classes, intensions, more and less determined objects, more
or less typical representatives of a concept and prototypes are defined and
organized, using a determination operation that constructs a class of indeterminate objects from an object representation of a concept called typical
object.
Mathematics Subject Classification (2000). Primary 03B40; Secondary 03B65.
Keywords. Typical/atypical instance, extensional class, determination, indeterminate object, more or less determinate object.
1. Introduction
The aim of the Logic of Determination of Objects (LDO) [6], [8], [9] is to provide
conceptual and logical answers to the following questions and issues: What is a
typical instance? What is an atypical instance? What is a prototypical instance?
How does one define a typicality relation between instances without using a degree of membership? Articulate intension with extension? Define “family resemblances”? Manage exceptions? Manage prototypical properties? Manage multiple
inheritances? To study these problems, we adopt the general framework of operators and operands with types in which it becomes possible to define not only
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Jean-Pierre Descl´es and Anca Pascu
operators of predication (predicates), but also operators of determination of objects, and operators for building indeterminate objects from concepts, as well as
explicitly articulating, for given concepts, the “structured intension of concepts”
with the “structured extension of concepts” without taking into consideration the
fact that intension is defined only by duality with the extension. The LDO is a
non-classical logic of construction of objects. It contains a theory of typicality with
an extended system of quantification.1
“Classical” First Order Predicate Logic (FOPL) is characterized by the following features:
1. It is a formalism with linked variables and deduction calculus entailing different complex operations (the operation of renaming variables);
2. The notions of typical and atypical instances are not represented;
3. All terms are interpreted by “fully determinate objects” (denotations of
proper names) and predicates by classes of fully determinate objects since
the notion of a “more or less determinate object” is unknown;
4. The operations of determination are not taken into account, while they are
used by natural languages (by adjectives, relatives, deictics, etc.);
5. FOPL is “extensional” , the equivalence between concepts being exactly defined by the equality of their extensions. In other words, the intension associated with a concept is defined by a complete duality with its extension. Thus
the notion of “intension” becomes useless.
LDO is constructed as an answer to the absence of elements mentioned above.
The LDO motivations are issued from the low power of FOPL to represent a
semantics for natural languages:
• The inadequacy between logical categories and language categories (adjectives, intransitive verbs as unary predicates);
• The absence of “determination” as a logical operation (a book; a red book; a
book which is on the table);
• The absence of typicality (The sentence French men are arrogant men does
not imply that All Frenchmen are arrogant).
2. Basic notions of LDO
LDO is a typed applicative system in the sense of Curry ([1]). It can be regarded
as a formal theory of concepts and objects.
LDO is a typed applicative system LDO = (F, O, T ) where: F is the set of
concepts; O is the set of objects; T is a type theory.
A concept is an operator, an object is always an operand. Types are associated
with concepts and objects.
1 This
system is not presented in this paper.
Logic of Determination of Objects (LDO)
3
2.1. Types theory of LDO
We adopt a theory of types (the functional types of Church) according to Curry
[1].
• Primitive types are: J individual entity type, H truth value (sentence) type;
• F: functional type constructor;
• Rules:
– Primitive types are types;
– If α and β are types, then Fαβ is a type;
– All types are obtained by one of the above rules.
In LDO:
- All objects are operands of type J; all propositions are of type H;
- All concepts are operators of type FJH.
An expression X of type α is specified by: X : α.
Definition 2.1. (Application of a concept to an object) We denote:
> if f is applied to x (“x falls under f ” )
(f x) =
⊥ otherwise ( “x does not fall under f ” )
(2.1)
The applicative scheme which expresses the application of a concept to an
object is:
f : FJH
x:J
(f x) : H
(2.2)
Remark 2.2. In LDO, N1 is the operator of negation defined as:
((N1 f )x) = > if and only if (f x) = ⊥
It has the classical logic property: (N1 (N1 g)) = g
In LDO, N0 is the negation of a sentence defined as:
(N0 (f x)) = > if and only if (f x) = ⊥
LDO is an applicative language of operators applied to operands of different
types (see [1], [4]); it is composed of:
1. Predicates defined on individual objects (concepts of type FJH) and the relators between individuals with respective types FJFJH, FJFJFJH, etc.);
2. The type of propositions is H;
3. Connectives between propositions are of the type FHFHH;
4. Fregean quantifiers: simple quantifiers with the type FFJHH; restricted quantifiers with the type FFJHFFJHH);
5. Operators of negation with the type FHH (classical negation or intuitionist
negation) defined only on propositions.
6. Objects of type J.
LDO is also an illative (inferential) language with inferential rules presented
in section 4.
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Jean-Pierre Descl´es and Anca Pascu
2.2. Basic operators of LDO
The constructor of the “typical object” : the operator τ . (see [2], [3]). This operator, denoted by τ and called the constructor of the typical object builds an object
totally indeterminate starting from a concept. Its type is FFJHJ; it canonically
associates to each concept f , an indeterminate object τ f , called “typical object”.
Its applicative scheme is :
τ : FFJHJ
f : FJH
(2.3)
τf : J
The object τ f , is the “best representative” object of the concept f ; it is
totally indeterminate, typical and abstractly represents the concept f in the form
of an “any typical object whatever”2 . The typical object τ f associated with f is
unique.
For example, if we take as concept f , the concept to-be-a-man then, the
typical object associated is a-man. For the concept f , to-be-a-computer, τ f is
a-computer:
τ : FFJHJ
to-be-a-man : FJH
(2.4)
a-man : J
The operator of determination: the operator δ. The operator δ, called the constructor of determination operators, builds a determination operator, starting from
a given concept. The operator δ canonically associates a determination operator of
the type FJJ to each concept f . The type of operator δ is FFJHFJJ. Its applicative
scheme is:
δ : FFJHFJJ
f : FJH
(2.5)
δf : FJJ
A determination operator δf is an operator which being applied to an object
x constructs another object y: y = ((δf ) x)3 . The object y is more determinate
than the object x, by means of the determination added by δf .
For example, if the concept f is to-be-red, then δf is red; if f is to-be-on-thetable, then δf is which-is-on-the-table.
The determination δf ,to-be-red applied to the object a-book gives the more
determinate object a-red-book:
red : FJJ
a-book : J
a-red-book : J
(2.6)
Definition 2.3. (The composition of determinations)
Let x be an object and δf , δg two determinations. In this case, we can write:
2 This
expression was choosen to encode the notion captured by the word “quelconque” in French.
We thank one of our reviewers who made important remarks regarding this expression.
3 We use the prefixed notation of a function, that is (f x) for f (x).
Logic of Determination of Objects (LDO)
5
((δg ◦ δf ) x) = (δg((δf ) x))
The composition of determinations is associative and supposed to be commutative.
Definition 2.4. (Chain of determination)
A chain of determination ∆ is a finite string of determinations which can be
composed of each other:
∆ = δg1 ◦ δg2 ◦ · · · ◦ δgn
Definition 2.5. ( More or less determinate object)
A more or less determinate object is an object recursively obtained starting
from the object τ f by:
1. τ f is a more or less determinate object;
2. If ∆ is a chain of determinations, then y = (∆ x) = ((δg1 ◦ δg2 ◦ . . . δgn )x) =
(δg1 (δg2 . . . (δgn x) . . . )) is a more or less determinate object;
3. Each more or less determinate object is obtained by the above rules.
Remark 2.6.
If x is a more or less determinate object obtained by a chain of determinations,
its associated concept will be denoted by x
ˆ.
2.3. Concepts and objects
The set of concepts F is provided with three relations:
• →
called comprehension :
The comprehension
f →g
modelises the intuitive notion that “the concept f directly comprises the
concept g” or “the concept g is directly comprised by the concept f ” . This
relation is: reflexive and antisymmetric. It is not transitive.
• ` called direct necessary comprehension
The direct necessary comprehension
f `g
modelises the fact that “the concept f directly contains in a necessary manner
the concept g or “the concept g is necessarly directly contained in the concept
f ” . The concept g is an “essential” component of f . This relation is: reflexive,
antisymmetric and transitive. The definition of the essence of a concept is
given in 2.4, definition 2.10.
• 7−→ called compatibility :
The compatibility f 7−→ g modelises the notion that the determination
δg can be applied to τ f or to each object x obtained by determination starting
from τ f . This relation is reflexive and antisymmetric.
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Jean-Pierre Descl´es and Anca Pascu
In this way, the set of concepts (F, →, `, 7−→) is structured by relations:
→, `, 7−→.
The set of objects, O, contains a subset, Odet , which is the set of “totally
determinate objects” .
Definition 2.7. (fully determinate object)
An object x is totally determinate if and only if for each determination δg
with g ∈ F :
(δg x) = x
In LDO, objects are of two kinds:
• more or less determinate object: x ∈ O;
• fully determinate object: x ∈ Odet .
Nevertheless, all of them are of type J.
The set O is structured by the binary relation ≥ defined as :
x ≥ y if and only if there is g ∈ F such that y = (δg x)
This relation is reflexive, antisymmetric and transitive.
2.4. Classes of concepts associated with a concept f
Definition 2.8. (Characteristic intension of a concept f )
The characteristic intension of a concept f is the set of concepts which caracterize f , in a sense a pack of properties of f .
Int-caract f = {g/f → g}
For example if f is to-be-a-man, then g can be to-have-two-legs; if f is to-bea-bird, then g can be to-fly.
Definition 2.9. (The intension of a concept f )
Int f = {g/f →? g}
where ? stands for the transitive closure of the relation.
Definition 2.10. (The essence of a concept f )
The essence of a concept f is the set of concepts necessarily comprised in f .
If we remove a concept g from the essence of f , we destroy the concept f ; it is not
the same. If a concept g is in the essence of a concept f , then the negation of g
cannot belong to this essence.
Ess f = {g/f ` g}
For example if f is to-be-a-man, then g can be be-derived-from-two-male-andfemale-human-cells. The concept g is essential for f . If f is to-be-a-bird, then g
can be to − lay − eggs. But the concept to-have-two-legs is not essential for the
concept to-be-a-man, since there are one-legged-men. Also the concept to-fly is not
essential for the concept to-be-a-bird, since there are birds which cannot fly.
Logic of Determination of Objects (LDO)
7
Remark 2.11. Ess f ⊂ Int f .
Definition 2.12. (Concepts compatible with a concept f )
Comp f = {g/f 7−→ g}
For exemple, the concept to-be-married is compatible with the concept tobe-a-man. One can determine the object τ f , a-man, by the determination to-bemarried. We obtain the object a-married-man more determinate than the object
a-man. Although we cannot determine a-man by the concept to-have-a-Laplacetransformation.
Remark 2.13. Ess f ⊂Int f ⊂ Comp f .
Classes of objects associated with a concept f and with the object τ f .
Definition 2.14. (Expansion (Etendue in French4 ))
The expansion of f , denoted by Exp(f ), is the set of all objects of O (more
or less determinate or totally determinate) to which f can be applied:
Exp (f ) = {x ∈ O /(f x) = >}
The expansion of τ f , denoted by Exp(τ f ), is the set of all objects of O (more
or less determinate or totally determinate) which can be constructed starting from
τf:
Exp (τ f ) = {x ∈ O /x = (∆ τ f )}
Definition 2.15. (Extension)
The extension of f , denoted by Ext(f ), is the set of all totally determinate
objects to which the concept f can be applied:
Ext (f ) = {x ∈ Odet /(f x) = >}
The extension of τ f , denoted by Ext(τ f ), is the set of all totally determinate
objects which can be constructed starting from τ f :
Ext (τ f ) = {x ∈ Odet /x = (∆ τ f )}
Remark 2.16. Ext (f ) ⊂ Exp f ; Ext (τ f ) ⊂ Exp (τ f )
In this paper we assume that:
Ext (f ) = Ext (τ f )
Exp (f ) = Exp (τ f )
A representation of objects of LDO is given in the figure ??. From τ f , a
more or less determinate object x is obtained by the chain of determination ∆. It
belongs to Exp(τ f ). We obtain the fully determinate object x0 by applying ∆1 to
x. The object x0 belongs to Ext(τ f ):
For example, if the object τ f is to-be-a-man, then x can be a-professor-atSorbonne and x0 can be Jean-Pierre Descl´
es.
4 The
Port Royal logic talks about “tendue”.
8
Jean-Pierre Descl´es and Anca Pascu
2.5. Theory of typicality in LDO
Definition 2.17. (Typical object of a concept f )
An object x, x ∈ O is called typical object of the concept f if and only if:
• For each chain of determination ∆ = (δg1 ◦ δg2 ◦ · · · ◦ δgi ◦ · · · ◦ δgn ) such that
x = (∆ τ f );
• For each i = 1, . . . , n, gi is such that:
– Either: if gi ∈ Int f , then N1 gi ∈
/ Int f 5 and N1 gi ∈
/ Int x
ˆ (a canary
versus a bird (see figure 2));
– Or: if gi ∈ Int f , N1 gi ∈ Int f , then gi ∈ Int-caract x
ˆ (Julie versus an
ostrich (see figure 2)).
Definition 2.18. (Atypical object of a concept f )
An object x, x ∈ O is called atypical object of the concept f if and only if:
• There is a chain of determination ∆ = (δg1 ◦ δg2 ◦ · · · ◦ δgi ◦ · · · ◦ δgn ) such
that x = (∆ τ f ) and
– Either: there is gi i = 1, . . . , n such that N1 gi ∈ Int f (an ostrich versus
a bird (see figure 2));
– Or: there is gi i = 1, . . . , n such that:
∗ If gi ∈ Int f and N1 gi ∈ Int f , then N1 gi ∈ Int-caract f and
gi ∈ Int-caract x
ˆ (Rita versus an ostrich (see figure 2));
– Or: there is gi i = 1, . . . , n such that:
∗ If gi ∈ Int f , then there is a y such that x = (∆1 y) and y =
(∆2 τ f ) and y is an atypical f (Rita versus a bird passing by an
ostrich (see figure 2)).
2.6. Other operators in LDO
The operator stating the typicality/atypicality of an object: T Y P /AT Y P .
• T Y P (f )(x) = > means that “x is a typical object of f of type FFJHFJH”;
for example T Y P (to-be-a-bird)(a-canary) = >;
• AT Y P (f )(x) = > means that “x is an atypical object of f of type FFJHFJH”;
for example AT Y P (to-be-a-bird)(an-ostrich) = >.
The applicative scheme for T Y P is:
T Y P :FFJHFJH
f :FJH
T Y P (f ):FJH
x:J
T Y P (f )(x):H
3. LDO axioms
The axioms of LDO are:
5N
1
is the functional negation, the negation of a concept.
Logic of Determination of Objects (LDO)
9
Aτ δ1: [(∀f )(f ∈ F)] [(∃! τ f )(τ f ∈ O) and (∃δf )(δf : O −→ O)];
For each concept f there is the object τ f and the determination operator
δf and they are unique.
Aτ δ2: [(∀f )(f ∈ F)] [(δf (τ f )) = (τ f )];
τ f is a fixed point of δf .
Aτ δ3: [(∀f, g), (f, g ∈ F)(∀x ∈ O)] [[ (f x) = > ∧ g ∈ Ess (f )] ⇒ ((δg)x) = x];
An object x falling under f is a fixed point for each determination
constructed by an essential concept g of f .
Aτ δ4: [(∀f )(f ∈ F] [δf ◦ δf = δf ];
Idempotency of determination composition.
Aτ δ5: [(∀f, g, h)(f, g, h ∈ F)] [(δh ◦ δg) ◦ δf = δh ◦ (δg ◦ δf )]; [(∀f, g)(f, g ∈ F)] [δf ◦
δg = δg ◦ δf ];
Associativity and commutativity of determination composition.
Aτ δ6: [((∀f )(f ∈ F)] [(f (τ f )) = > iff Extτ (τ f ) 6= ∅]
This axiom states the existence of occurrences of a concept.
4. LDO rules
The rules of LDO are divided into:
Rules modelizing FOPL connectives and rules specific to the LDO. The rules
specific to the LDO are dived into three categories: rules for typicality/atypicality,
rules for quantifiers and rules relating operators. In this paper we present the rules
for typicality/atypicality. Other rules have been presented in other publications.
Syntax of the rules :
• The general form of the rules is that of natural deduction:
–
X
(4.1)
Y
–
X1 , X2 , . . . , Xn
(4.2)
Y, Z, . . . , U
where X1 , X2 , . . . , Xn = X1 ∧ X2 ∧ · · · ∧ Xn and
Y, Z, . . . , U = Y ∧ Z ∧ · · · ∧ U (both antecedent and consequent are
conjunctions.)
• Rules are constructed with propositions ((f x)), with set operators (g ∈
Int(f )) and with binary operators : T Y P and AT Y P .
If X = {C1 , . . . , Cn /prop} where C1 , . . . , Cn are set conditions and prop is a
logical proposition, it means : there is prop under C1 , . . . , Cn .
From the semantical point of view, the rules express :
1. The translation of a definition ([K])
0
2. The inheritance mechanism ([H11], [H12], [H1∗ ], [H2], [H2∗ ], [H2 ?], [H3 ], [H3 ],
[H3∗ ]) ;
3. Determination properties ([D1], [D2],[C1]);
4. Negation ([N1]).
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Jean-Pierre Descl´es and Anca Pascu
Inheritance of an essential concept:
(f x), g ∈ Ess(f )
(g x)
[E]
Inheritance of a characteristic concept:
h ∈ Int-caract(x)
(h x)
[K]
Inheritance for typical objects:
T Y P (f )(x), g ∈ Int(f ), N1 g 6∈ Int(f )
(g x)
[H11]
T Y P (f )(x), g ∈ Int(f ), N1 g ∈ Int(f ), g ∈ Int-caract(f )
[H12]
(g x)
Rule H12 can be replaced by H120
T Y P (f )(x), {g, N1 g} ⊂ Int(f ), g ∈ Int-spec(f )
(g x)
[H120 ]
Categorisation of an object as being typical:
(f x), {g ∈ Int(f ), N1 g 6∈ Int(f )/(g x)}
T Y P (f )(x), ∃∆, x = (∆ (τ f ))
[H1∗ ]
Non-necessary inheritance for an atypical object :
AT Y P (f )(x), g ∈ Int(f )
?(g x)
[H2 ?]
The symbol ?(g x) stands for (g x) ∨ ((N1 g) x) (either (g x) or ((N1 g) x)) in
the following sense: without aditional information, we deduce (g x).
Categorisation of an object as being atypical
(f x), g ∈ Int(f ), N1 g ∈ Int(f ), ((N1 g)x)
AT Y P (f )(x), (N0 (g x))
[H2∗ ]
Categorisation of an object as being atypical (particular case of H2∗ )
Logic of Determination of Objects (LDO)
(f x), h ∈ Int-caract(x), h = (N1 g), g ∈ Int(f )
AT Y P (f )(x), N0 (g x), ∃∆, x = (δ(N1 g))(τ f )) ◦ ∆
11
[H2]
Inheritance of an atypical object
AT Y P (f )(x), (h x), N1 g 6∈ Int(h), h ∈ Int(g)
(g x)
[H3]
AT Y P (f )(x), N1 g 6∈ Int-caract(f ), g ∈ Int(f )
(g x)
[H30 ]
Compared to the rule H2 ?, the rule H30 is a case where it is possible to make
a conclusion about inheritance with supplementary information.
AT Y P (f )(x), ∃∆, y = (∆x)
AT Y P (f )(y)
[H3∗ ]
An object constructed from an atypical object of f , always remains an atypical object of f .
Determination
(f x), (g x)
(f ((δg) x)), (g ((δf ) x))
[D1]
(f x), (g x), T Y P (f )(x)
x = ((δg)(τ f ))
[D2]
Composition of determinations
z = ((δg) y), y = ((δh) x)
z = (((δg) ◦ (δh)) x)
[C1]
Negation
((N1 u)x)
(N0 (u x))
(N0 (N0 (u x)))
(u x))
(N0 (u x)
((N1 u)x)
(N1 (N1 u))
u
[N1- N0]
[NN0 - NN1]
12
Jean-Pierre Descl´es and Anca Pascu
5. Example
The following sentences are associated with the semantic network (the network of
concepts) of the figure 1 6 .
(a) Birds are flying animals.
(b) Ostriches are birds, but they do not fly.
(c) Canaries are birds.
(d) Kiki is a canary.
(e) Julie is an ostrich.
(f ) Rita is an ostrich but it flies.
6 The
vertices are concepts and the arrows stand for direct comprehension f → g.
Logic of Determination of Objects (LDO)
13
to-be-an-animal
s
6
to-be-a-bird
sto-be6-a-canary
s
to-be-Kiki
s
to-flys
*
>
6
to-be-an-ostrich s
not-to-fly
s
J
]
J
J
J
J
J
J
s
Js
to-be-Rita
to-be-Julie
Figure 1: A semantic network.
14
Jean-Pierre Descl´es and Anca Pascu
Remark 5.1. The network of objects can have more vertices than the network of
concepts with which it is associated: one can associate a concept x
ˆ to a more or
less determinate object x obtained by determinations from τ f ; this concept is not
in the concept network.
Let be the network of concepts (F, →) of figure 1. Associated caracteristic
intensions are:
Int-caract (to-be-an-animal) = ∅
Int-caract (to-fly) = ∅
Int-caract (not-to-fly) = ∅
Int-caract (to-be-a-bird) = {to-be-an-animal, to-fly}
Int-caract (to-be-ostrich) = {to-be-a-bird, not-to-fly}
Int-caract (to-be-a-canary) = {to-be-a-bird}
Int-caract (to-be-Kiki) = {to-be-a-canary}
Int-caract (to-be-Julie) = {to-be-an-ostrich}
Int-caract (to-be-Rita) = {to-be-an-ostrich, to-fly}
Corresponding intentions are obtained by the reflexive and transitive closure
of the relation →:
Int (to-be-an-animal) = {to-be-an-animal}
Int (to-be-a-bird) = {to-be-a-bird, to-be-an-animal, to-fly}
Int(to-fly) = {to-fly}
Int(not-to-fly) = {not-to-fly}
Int (to-be-a-canary) = {to-be-a-canary, to-be-a-bird, to-be-an-animal, to-fly}
Int (to-be-an-ostrich) = {(to-be-an-ostrich, to-be-an-animal, to-be-a-bird, tofly, not-to-fly}
Int (to-be-Kiki) = {to-be-Kiki, to-be-a-canary, to-be-a-bird, to-be-an-animal,
to-fly}
Int (to-be-Julie) = {to-be-Julie, to-be-an-ostrich, to-be-a-bird, to-be-an-animal,
to-fly, not-to-fly}
Int (to-be-Rita) = {to-be-Rita, to-be-an-ostrich, to-be-a-bird, to-be-an-animal,
to-fly, not-to-fly}
The network of objects obtained by determinations from the concepts of
figure 1 is represented in figure 2.
For each concept f of the network of figure 1, the set of compatible concepts
is:
Comp (to-be-an-animal) = {to-be-an-animal, to-fly, not-to-fly}
Comp (to-be-a-bird) = Int (to-be-a-bird) ∪ {not-to-fly}
Comp (to-fly) = {to-fly}
Comp (not-to-fly) = {not-to-fly}
Comp (to-be-a-canary) = Int (to-be-a-canary)
Comp (to-be-an-ostrich) = Int (to-be-an-ostrich)
Comp (to-be-Kiki) = Int (to-be-Kiki)
Comp (to-be-Julie) = Int (to-be-Julie)
Comp (to-be-Rita) = Int (to-be-Rita)
Logic of Determination of Objects (LDO)
15
τ to-be-an-animal
= an-animal
u
∆ ⊃ {(δ to-fly)}
?
?
≥
u
τ to-be-a-bird
= a-bird
∆
∆1 ◦ (δ not-to-fly)
2
?
?
=
=
≥
≥
u τ to-be-a-canary =
uτ to-be-an-ostrich =
JJ an-ostrich
a-canary
JJ
JJ
JJ
∆3
∆4
JJ ∆5 ◦ (δ to-fly)
JJ
?
?
^
J^
J
≥u
≥
≥u
u
Kiki
Julie
Rita
Figure 1: Network of objects obtained from concepts of the figure 1.
16
Jean-Pierre Descl´es and Anca Pascu
Determinations by concepts compatible with a concept f can be applied to
each object which falls under this concept:
• The chain ∆ contains all determinations which are applied to the indeterminate object “an-animal” in order to obtain the indeterminate object “a-bird”
including “δ to-fly”;
• The determination “δ not-to-fly” is the determination which can be applied
to the object “a-bird”, and then, by appling the chain ∆1 , one obtains the
object “an-ostrich” . In the same way, by applying the determination “δ tofly” to the object “an-ostrich” and then, by the chain ∆5 one obtains the
object “Rita” ;
• The chains ∆2 , ∆3 , ∆4 do not contain either “δ to-fly” , or “δ not-to-fly” .
Example. We show how to obtain by the inferences of LDO that “Julie, an ostrich
which does not inherit all the concepts of a typical bird, is atypical as bird and it
cannot fly” .
1
2
3
4
5
6
7
8
9
10
11
12
(to-be-an-ostrich Julie)
(to-be-a bird an-ostrich)
not-to-fly ∈ Int-caract (an-ostrich)
to-fly ∈ Int (to-be-a bird)
T Y P (to-be-an-ostrich)(Julie)
to-be-a-bird ∈ Int (to-be-an-ostrich)
(to-be-a-bird Julie)
(not-to-fly an-ostrich)
not-to-fly ∈ Int (to-be-an-ostrich)
(not-to-fly Julie)
((to-be-a-bird) (δ(not-to-fly )) Julie)
AT Y P (to-be-a-bird)(Julie)
So, Julie is a bird which cannot fly :
((to-be-a-bird) (which does not fly) Julie)
hypothesis
hypothesis
hypothesis
hypothesis 1, by default
definition of Int,2
H11,5,6
K,3
definition of Int-caract,8
H12,1,5,9
D1,7,10
H2 11
Example. Rita is an ostrich which flies, so it does not inherit all the concepts of
typical ostriches, since it is an atypical ostrich. It is still an atypical bird.
1
2
3
4
5
6
(to-be-an-ostrich Rita)
(to-fly Rita)
not-to-fly ∈ Int-caract ((to-be-an-ostrich)
hypothesis
hypothesis
hypothesis
(not-to-fly Rita)
K 2, 3
(to-fly Rita)
?
contradiction 4,5
∗
The contradiction is solved by means of the rule H2 which does not permit
the inheritance of the concept g, but permits the inheritance of N1 g for atypical
instances.
Logic of Determination of Objects (LDO)
17
1
2
3
4
(to-be-an-ostrich Rita)
(to-fly Rita)
not-to-fly ∈ Int-caract(to-be-an-ostrich)
to-fly = (N1 (N1 -to-fly))
hypothesis
hypothesis
hypothesis
5
6
7
8
not-to-fly ∈ Int-caract(to-be-an-ostrich)
(N0 (not-to-fly Rita))
AT Y P (to-be-an-ostrich)(Rita)
AT Y P (to-be-a-bird)(Rita)
consequence of 3
N0 en 2
H2∗ ,1,2,3
H3∗ ,1,2,4,5
6. Conclusion
LDO is an extension of FOPL inside the framework of the logic of operators. The
features of FOPL are:
1. All instances of f are typical instances: Exp (f ) = Ext (f ) = Expτ (f );
2. There is an isomorphism between the ordered set h Int(f ), →? i and the
subclasses of h Ext(f ) ⊆ i; the relation →? between concepts is “extensional”
following the duality principle:
Ext (g) ⊆ Ext (f ) ⇐⇒ f →? g ⇐⇒ Int (g) ⊇ Int (f )
3. The essence of a concept is identical to its intension: Ess(f ) = Int(f ) (we
may have, in some situations, Ess(f ) = {f });
4. The typical object (τ f ) is identical to the conceptual object of Hilbert (f )
[7]; this conceptual object belongs to the extension: τ f = f .
Comparing these features of FOPL with LDO, can be noted the main features
of LDO:
1. There are typical and atypical instances (objects) of f : Exp (f ) 6= Ext (f ) 6=
Expτ (f );
2. There is not a one-to-one correspondence between the ordered set
hInt (f ), →? i and the subclasses of hExt (f ), ⊆i; the relation →? between concepts is not “extensional”; the duality principle does not apply;
3. The essence of a concept is not identical to its intension: Ess (f ) ⊂ Int (f );
4. The typical object τ f is not identical to the conceptual object of Hilbert f ;
this typical object does not belong to the extension, hence the necessity to
work with the expansion and indeterminate objects;
5. A concept is an operator described by : hf, τ f, δf, Ess (f ), Int (f ), →? i
In LDO, we introduce quantifiers (typical quantifiers Π? and Σ? ) defined in the
general framework of Combinatory Logic [1]. Different rules for managing the
formal behaviour of these two quantifiers are introduced without without any
use of linked variables and without any epistemological difficulties (see [1], [5])
concerning variables.
18
Jean-Pierre Descl´es and Anca Pascu
References
[1] H. B. Curry, R. Feys, Combinatory Logic vol I. North Holland, Amsterdam, 1958.
[2] J. P. Descl´es, Implication entre concepts, la notion de typicalit. Travaux de linguistique
et de littrature, XXIV,1 Paris, (1986), 79–102.
[3] J. P. Descl´es, Thorie de la typicalit´e. L’A-peu-pr`es - Aspects anciens et modernes de
l’approximation. Editions de l’EHESS, Paris, (1987), 183-195.
[4] J. P. Descl´es, Langages applicatifs, langues naturelles et cognition. Herm`es, Paris,
1990.
[5] J. P. Descl´es, K. S. Cheong, Analyse critique de la notion de variable. Math´ematiques
et Sciences humaines 173 Paris, (2006), 43-102.
[6] J. P. Descl´es, A. Pascu, Logic of Determination of Objects - The Meaning of Variable
in Quantification. International Journal on Artificial Intelligence Tools 6, (2006), 10411052.
[7] D. Hilbert, P. Bernays, Grundlagen der Mathematik II. Springer-Verlag, 1939.
[8] A. Pascu, Logique de D´etermination d’Objets: concepts de base et math´ematisation en
vue d’une mod´elisation objet. Th`ese de doctorat, Univesit´e de Paris-Sorbonne, (2001).
[9] A. Pascu, Les objets dans la repr´esentation des connaissances. Application aux processus de cat´egorisation en informatique et sciences humaines. Habilitation `
a diriger
des recherches, Paris-Sorbonne, (2006).
Acknowledgment
Many thanks to our reviewers for their interesting remarks.
Jean-Pierre Descl´es
Universit de Paris Sorbonne LaLIC (Langues, Logiques, Informatique et Cognition)
28, rue Serpente
75006 Paris
France
e-mail: [email protected]
Anca Pascu
Universit de Bretagne Occidentale Brest LALIC
20, rue Duquesne
93837 29238 Brest Cedex 03
France
e-mail: [email protected]