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


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


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


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


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...


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


......

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


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 ?


1-Elementary definable concepts2- Compositionality

3-Dynamically definable concepts4-Conceptual dictionaries


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


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 ?


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)


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}


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


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


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


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


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


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


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


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


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:


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.


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}


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


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)...


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.


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)


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...

Vagueness through definitions

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