Vagueness through definitions

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Vagueness through definitions
Michael Freund ISHA-IHPST, Université de Paris
IV, 28 rue Serpente, 75006 PARIS
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Sharpness and vagueness
Most generally, membership is not an
all-or-not-matter you have intermediate states
It is only in the simplest cases that a concept
separate objects in to distinct classes without
any bridge between them
to-be-a-dog to-be-a-toothbrush to-be-an-integer to
-be-gold to-be-from-Mozart to-be-a-verb
to-be-a-heap to-be-tall to-be-a-lie to-be-left-win
g to-be-a-WMD
sharp concepts
vague concepts
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Vagueness, though, is not a uniform notion
to-be-a-sand-heap
to-be-a-lie
are both vague concepts... However
their vagueness have a different flavour
Vagueness may be qualified as quantitative in the
first case and as qualitative in the second one.
Fuzzy concepts are vague concepts for which
associated membership can be measured through a
fuzzy function
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to-be-rich, to-be-tall, to-be-a-heap, to-be-hot
fuzzy concepts
For some other concepts, however, vagueness in
membership does not easily lead to a measurable
magnitude
to-be-a-lie to-be-clever to-be-a-cause to-be-relig
ious
(qualitatively) vague concepts
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The treatment of vagueness clearly depends of the
type of vagueness one has to deal with
Fuzzy concepts only represent a subfamily of
vague concepts
They received a adequate treatment through fuzzy
logics
The numerical treatment, applied in the
simplest cases, may be not suitable to other
kinds of vague concepts
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Consider the concept to-be-weapon-of-mass-destruct
ion
and the following object
Up to which degree does this gun deserve to be
called a WMD ?
Membership functions should not be systematically
looked for to account for categorial membership...
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A universal criterion in the treatment of
membership for vague concepts is comparison
We are unable to attribute a precise membership
degree to a sword or a gun as weapons of mass
destruction, but we nevertheless consider that
the concept of WMD applies more to a gun than to
a sword.
Similarly, it may be difficult to decide to what
point Jack or Peter are rich, but we may still
agree that Jack is richer than Peter
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Any concept c induces a comparison order among
the objects of the universe of discourse
Categorizing relatively to a concept amounts to
ordering the objects depending on the strength
with which the concept applies to them.
?c a partial weak order x ?c y x falls at most
as much as y under the concept c
x ltc y the concept c applies less to x than to
y
?
?
?
?
?...
...?
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The understanding of a concept requires the
knowledge of its associated membership order
- How can we determine this order ?
- Can we efficiently model the classical problems
of categorization theory in the framework of
membership orders ?
- In particular, what solutions do we propose to
the problem of compositionality ?
- Is our theory in adequacy with common sense,
and do the results conform with experimental
studies ?
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1-Elementary definable concepts 2-
Compositionality 3-Dynamically definable
concepts 4-Conceptual dictionaries
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1) A solution for elementary definable concepts
Elementary definable concepts are introduced with
the help of simpler or already known elementary
concepts
to-be-a-vertebrate to-have-feathers to-have-a-beak
to-have-wings
A bat has less birdhood than a robin, and more
birhood than a mouse
to-be-a-bird
to-be-a-metal to-be-yellow to-be-precious
to-be-gold
to-be-a-house to-be-made-of-cloth
to-be-a-tent
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With any elementary definable concept is
therefore associated an auxiliary set of
defining features
c
?(c)
to-be-a-bird
to-have feathers, to-have-a-beak, to-have-wings

• The elements of ?(c) are part of the agents
  knowledge
• ?d is known for every concept d of ?(c)

2) The elements of ?(c) are sufficient to acquire
full knowledge of c ?c is fully determined by
the ?d, d? ?(c)
How is this construction operated ?
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A simple solution is to use skeptical choice and
set ?c ? ?d, d? ?(c)
x ?bird y iff x ?vertebrate y, and x ?beak y,
and x ?feathers y, and x ?wings y.
An other solution is to simply count the votes,
and set x ?c y iff the number of voters choosing
y is not smaller than the number of voters
choosing x ( d x ?d y) ( d y ?d x)
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Example
Suppose that for an agent ?(to-be-bird)
to-be-vertebrate, to-be-oviparous,
to-be-warm-blooded, to-have-a-beak,
to-have-wings
Using skeptical procedure leads to m ?bird b.

Counting the votes leads to m ?bird t, m ?bird
b, f ?bird b and f ?bird t
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However, it is necessary to take into account the
relative salience of the features that are used
in the definition of c
For a child, to-have-wings (or to-fly) is a
feature of birds that is more salient than any
other one, so that a flie may appear as having
more birdhood than a tortoise...
Solution

• (c) being partially ordered by a salience order,
• set x ?cy iff
• for all d ? ?(c) such that y ltd x, there exists
  d? ?(c), d more salient than d, such that x
  ltd y

transitive closure
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Suppose the salience order on ?(bird) is given by
vertebrate
wings
beak
Then we have m ?bird b, f ?b m and m ?bird t,
and neither b ?birdt, nor t ?bird b.
oviparous
warm-blooded
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Concept extension through membership orders
Definition The object x falls under the concept
c if x is ?c-maximal.
The extension Ext(c) of c (the category
associated with c) is the set of ?c-maximal
objects of the universe
An object x falls under a definable concept iff
it falls under each of its defining feature
Ext(c) ?d ? ?(c), Ext(d)
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2-Compositionality of membership orders
Simple concepts can be linked together
to-be-a-french-doctor to-be-rich-and-famous
by conjonction cc
to-be-a-green-apple to-be-a-flying-bird
by détermination c c
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By compositionality, the membership
order associated with the composed concept
depends on the membership orders of its
constituents

• c c f(?c ?c)
• c c g(?c, ?c)

The first attempts of classical fuzzy logics to
account for compositionality through t-norms led
to disputable solutions...
cf Kamp-Partee, Prototype theory and
compositionality, Cognition (57) 1995
We associate with c c the lexicographic order
that gives priority to c x ? cc y iff x ?c y
and either x ltc y, or x ? c y
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Example
x a bat, y an ostrich c to-be-a-bird, c
to-fly one has x ? cc y
to-be-a-flying-bird applies better to an
ostrich than to a bat

• One has then full compositionality
• Ext (c c) Ext c? Ext c Ext (cc)
• ?c ? ?c ? ? cc ? ?c

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Distance and membership function
?c(x) maximal length of a chain x ltc x1 ltc x2
ltc ... ltc xn with xn ? Ext c
xn Ext c

x x1 x2 x3 ...xn-1
?c ? ? cc
?c 1- ?c/Nc, where Nc supx ?c(x)
?c (x) 1 iff x ? Ext c
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3-Dynamically definable concepts
Elementary definable concepts constitute a very
restricted family of concepts.
Definitions do not consist in a simple sequence
of defining features a whole apparatus is
underlying the definition , giving it its
specific dynamics
A description set of a concept therefore consists
of several key-concept together with a
well-defined Gestalt
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Example
The set ?(m) of key-features to-be-a-talltree,
to-be-northern to-have-five points, to-provide-s
yrup
maple tall tree growing in northern countries
whose leaves have five points, and whose resin is
used to produce a syrup.
Membership of an object x relatively to the
concept to-be-a-maple depends on its own
membership relatively to the concept
to-be-a-talltree... as well as on the membership
of auxiliary objects (the leaves of x, the
resin of x) relative to auxiliary concepts
(to-have-five-points, to-provide-a-syrup)
maple
The Gestalt Gm is represented by the vertices and
the edges in italics, the auxiliary features
is
tree
has
is
has
has
tall
growing-country
leaves
resin
have
provides
is
northern
syrup
fivepoints
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The maplehood of an item x may be evaluated by
evaluating membership relative to the composed
concepts ttr (to-be-a-tall)(to-be-a-tree), ng
c(to-be-northern) (to-have-a-growing-country), f
l (to-have-five points) (to-have-leaves), sr
(to-produce-syrup)(to-have-resin).
MAPLE
is
tree
has
has
has
is
resin
Again, these concepts may be given different
salience levels.
growing country
leaves
tall
provides
is
have
northern
fivepoints
syrup
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We therefore associate with the concept
to-be-a-maple and its structured definition the
membership order induced by the ordered set ?(m)
ttr, ngc, fl, sr
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This procedure takes care of the categorial
membership associated with any concept c whose
defining structure may be modelled by an ordered
set ?(c) of simple or compound concepts
We define x ?cy as the transitive closure of the
relation for all d ? ?(c) such that y ltd x,
there exists d? ?(c), d more salient than d,
such that x ltd y
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4- Conceptual dictionaries
What if the defining features of the definable
concept c are themselves definable ? ? (c) c1,
c2, ..., cn
The target membership order ?c is computed
from the orders ?d, d? ? (ci), which are supposed
to be known from the agent
In particular, c? ? (ci)...
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A conceptual dictionary is a pair (C , ?) where
C set of concepts, ? C ----gt?0(C), such
that there is no infinite sequence c1, c2,
...cn,...with ci ? ? (ci-1).
Set c lt? d if there exists a sequence c0 c,
c1, c2, ..., cn d such that ci ? ? (ci1) (c is
simpler than d)
Then lt? is a strict partial order with no
infinite descending chain its minimal elements
are the primitive concepts of the
dictionary, that is the concepts c such that ?
(c) ?
A defining chain of c descending chain of
maximal length
Every defining chain of c ends up with a
primitive concept.
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Membership and membership orders associated with
conceptual dictionaries
P set of minimal elements (the primitive
concepts of the dictionary) P(c) set of
primitive elements p such that p lt? c Pz(c)
set of elements of P(c) that apply to the object z
Ext c ? Ext p, p ? P(c)
If no salience order is set on ? (c), x ?c y
iff Px(c) ? Py(c)
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Conclusion
This construction takes care of a large family of
concepts... However...
To-kill ? to cause death
- Not all concepts are definable
The extensional properties of a concept are not
sufficient to acquire full knowledge of this
concept...