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### Vagueness through definitions

Michael Freund

ISHA-IHPST,

Université de Paris IV, 28 rue Serpente, 75006 PARIS

Sharpness and vagueness

Most generally,

membership is not an

all-or-not-matter:

you have intermediatestates

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-wing

to-be-a-WMD

vague concepts

sharp concepts

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

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-religious

(qualitatively) vague concepts

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

Consider the concept

to-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 looked

for to account for categorial membership...

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

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

the conceptc applies less tox than to y

...

...

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 ?

1-Elementary definable concepts

2- Compositionality

3-Dynamically definable concepts

4-Conceptual dictionaries

1) A solution for elementary definable concepts

Elementary definableconcepts 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

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 agent’s knowledge:

dis known for every concept d of (c)

2) The elements of (c) are sufficient to acquire full knowledge of c:

cis fully determined by the d, d(c)

How is this construction operated ?

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)

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 birdb.

Counting the votes leads to

m birdt, m birdb, f birdb andf birdt

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 cyiff

- for all d (c)such thaty

+ transitive closure

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

Concept extension through membership orders

Definition: The object xfalls 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)

2-Compositionality of membership orders

Simple concepts can be linked together

to-be-a-french-doctor

to-be-rich-and-famous

by conjonction: c’&c

to-be-a-green-apple

to-be-a-flying-bird

by détermination: c’* c

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 c’*c y iff x c y andeither x

Example

x = a bat, y = an ostrich

c = to-be-a-bird, c’ = to-fly:

one hasx 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

Distance and membership function

c(x) = maximal length of a chain

x

xn Ext c

x x1 x2 x3 ...xn-1

c c’*c

c = 1- c/Nc, where Nc= supx c(x)

c (x) = 1 iff x Ext c

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

Example:

The set (m) of key-features

to-be-a-tall*tree,

to-be-northern

to-have-five points,

to-provide-syrup

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-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)

maple

The Gestalt Gmis represented

by the vertices and the edges

in italics, the ‘auxiliary’ features

is

tree

has

is

has

has

tall

leaves

resin

growing-country

have

provides

is

syrup

northern

fivepoints

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).

MAPLE

is

tree

is

has

has

has

resin

Again, these concepts may be given

different salience levels.

growing country

tall

leaves

provides

is

have

northern

fivepoints

syrup

We therefore associate with the concept to-be-a-maple

and its structured definition

the membership order induced by the

ordered set

(m) = {t*tr, n*gc, f*l, s*r}

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 closureof the relation:

for all d (c) such thaty

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)...

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 = d

such thatci (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.

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 < 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 cy iff Px(c) Py(c)

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...

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