Vagueness through definitions
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Transcript of Vagueness through definitions
Vagueness through definitions
Michael FreundISHA-IHPST,
Université de Paris IV, 28 rue Serpente, 75006 PARIS
Cejkovice, 09/ 2009 2
It is only in the simplest cases that a concept separate objects in to
distinct classes without any bridge between them
to-be-a-dogto-be-a-toothbrush
to-be-an-integerto-be-gold
to-be-from-Mozartto-be-a-verb
sharp concepts
Most generally, membership is not an
all-or-not-matter: you have intermediate states
to-be-a-heapto-be-tall
to-be-a-lieto-be-left-wingto-be-a-WMD
vague concepts
Sharpness and vagueness
Cejkovice, 09/ 2009 3
are both vague concepts... However theirvagueness have a different flavour
Vagueness, though, is not a uniform notion
to-be-a-sand-heap to-be-a-lie
Vagueness may be qualified as quantitative in the first caseand as qualitative in the second one.
Fuzzy concepts are vague concepts for which associated membership can be measured
through a fuzzy function
Cejkovice, 09/ 2009 4
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-lieto-be-clever
to-be-a-causeto-be-religious
(qualitatively) vague concepts
Cejkovice, 09/ 2009 5
The treatment of vagueness clearlydepends 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 simplestcases, may be not suitable to other kinds of vague concepts
Cejkovice, 09/ 2009 6
Consider the conceptto-be-weapon-of-mass-destruction
and the following object
Up to which degree does this gun deserve to be called a WMD ?
Membership functions should not be systematically lookedfor to account for categorial membership...
Cejkovice, 09/ 2009 7
A universal criterion in the treatmentof 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
Cejkovice, 09/ 2009 8
......
Any concept c induces a comparison orderamong 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 orderx c y:
x falls at most as much as y under the concept c x <c y:
the concept c applies less to x than to y
Cejkovice, 09/ 2009 9
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 categorizationtheory 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 dothe results conform with experimental studies ?
Cejkovice, 09/ 2009 10
1-Elementary definable concepts2- Compositionality
3-Dynamically definable concepts4-Conceptual dictionaries
Cejkovice, 09/ 2009 11
to-be-a-bird
to-be-a-vertebrateto-have-feathersto-have-a-beakto-have-wings
to-be-a-tent to-be-a-houseto-be-made-of-cloth
to-be-gold
to-be-a-metalto-be-yellow
to-be-precious
Elementary definable concepts are introduced with the help ofsimpler or already known elementary concepts
1) A solution for elementary definable concepts
A bat has less birdhood than a robin, and more birhood than a mouse
Cejkovice, 09/ 2009 12
With any elementary definable concept is therefore associatedan auxiliary set of defining features
c (c)
to-be-a-bird {to-have feathers, to-have-a-beak, to-have-wings}
1) The elements of (c) are part of the agent’s 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 ?
Cejkovice, 09/ 2009 13
A simple solution is to use skeptical choiceand 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 setx 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)
Cejkovice, 09/ 2009 14
Example:
vertebrate oviparous warm-blood beak wings
mouse x x
tortoise x x x
bat x x x
flie x x
Using skeptical procedure leads to m bird b.
Counting the votes leads tom bird t, m bird b, f bird b and f bird t
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}
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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 birdsthat 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 <d x, there exists d’ (c), d’
more salient than d, such that x <d’ y
+ transitive closure
Cejkovice, 09/ 2009 16
vertebrate
oviparous warm-blooded beak wings
mouse x x
tortoise x x x
bat x x x
flie x x
Then we have m bird b, f b m and m bird t, andneither b birdt, nor t bird b.
vertebrate
Suppose the salience order on (bird) is given by
beak wings
oviparous
warm-blooded
Cejkovice, 09/ 2009 17
Definition: The object x falls under the concept c if x is c-maximal.
An object x falls under a definable concept iff it falls under each of its defining feature
Ext(c) = d (c), Ext(d)
The extension Ext(c) of c (the category associated with c) is the set of c-maximal objects of the universe
Concept extension through membership orders
Cejkovice, 09/ 2009 18
2-Compositionality of membership orders
Simple concepts can be linked together
by conjonction: c’&c to-be-a-french-doctorto-be-rich-and-famous
by détermination: c’* cto-be-a-green-appleto-be-a-flying-bird
Cejkovice, 09/ 2009 19
By compositionality, the membership orderassociated 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 c’*c y iff x c y and either x <c y, or x c’ y
Cejkovice, 09/ 2009 20
x = a bat, y = an ostrich c = to-be-a-bird, c’ = to-fly:
one has x c’*c 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 (c’*c)
c c c’*c c
Example
Cejkovice, 09/ 2009 21
Distance and membership function
c(x) = maximal length of a chainx <c x1 <c x2 <c ... <c xn with xn Ext c
xn Ext c x x1 x2 x3 ...xn-1
c = 1- c/Nc, where Nc= supx c(x)
c (x) = 1 iff x Ext c
c c’*c
Cejkovice, 09/ 2009 22
3-Dynamically definable concepts
Elementary definable concepts constitute a very restricted family of concepts.
Definitions do not consist in a simple sequence ofdefining 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
Cejkovice, 09/ 2009 23
maple: tall tree growing in northern countries whose leaves have five points, and whose resin is used to
produce a syrup.
The set (m) of key-featuresto-be-a-tall*tree,
to-be-northern to-have-five points,
to-provide-syrup
maple
tree
growing-country
northern
leaves
fivepoints
resin
syrup
has
is
hashas
have provides is
The Gestalt Gm is representedby the vertices and the edges
in italics, the ‘auxiliary’ features
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-tall*tree...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)
tall
is
Example:
Cejkovice, 09/ 2009 24
tree
resingrowing country
northern fivepoints
leaves
syrup
has
have providesis
has has
MAPLE
is
The maplehood of an item x may be evaluated by evaluating membership relative to the composed concepts
t*tr =(to-be-a-tall)*(to-be-a-tree),n*gc=(to-be-northern) *(to-have-a-growing-country),
f*l =(to-have-five points) *(to-have-leaves),s*r =(to-produce-syrup)*(to-have-resin).
tall
is
Again, these concepts may be given different salience levels.
Cejkovice, 09/ 2009 25
We therefore associate with the concept to-be-a-maple and its structured definition
the membership order induced by theordered set
(m) = {t*tr, n*gc, f*l, s*r}
Cejkovice, 09/ 2009 26
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 <d x, there exists d’ (c), d’ more salient than d,
such that x <d’ y
Cejkovice, 09/ 2009 27
4- Conceptual dictionaries
The ‘target’ membership order c is computed fromthe orders d, d (ci), which are supposed to be known
from the agent
What if the defining features of the definable concept c are themselves definable ?
(c) ={c1, c2, ..., cn}
In particular, c (ci)...
Cejkovice, 09/ 2009 28
A conceptual dictionary is a pair (C , ) where: C set of concepts,
: C ---->0(C), such that
there is no infinite sequence c1, c2, ...cn,...with ci (ci-1).
Set ‘c < d’ if there exists a sequence c0 = c, c1, c2, ..., cn = dsuch that ci (ci+1)(c is ‘simpler’ than d)
Then < 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.
Cejkovice, 09/ 2009 29
P = set of minimal elements (the primitive concepts of the dictionary)P(c) = set of primitive elements p such that p < c
Pz(c) = set of elements of P(c) that apply to the object z
Ext c = Ext p, p P(c)
Membership and membership orders associated with conceptual dictionaries
If no salience order is set on (c),
x c y iff Px(c) Py(c)
Cejkovice, 09/ 2009 30
This construction takes care of a large family of concepts...However...
To-kill = ? to cause death
Conclusion
- Not all concepts are definable
The extensional properties of a concept are not sufficientto acquire full knowledge of this concept...