Vagueness through definitions

1. Vagueness through definitions Michael Freund ISHA-IHPST, Université de Paris IV, 28 rue Serpente, 75006 PARIS

2. 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

3. 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

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-lie to-be-clever to-be-a-cause to-be-religious (qualitatively) vague concepts

5. 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

6. 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...

7. 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

8. 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 <c y: the conceptc applies less tox than to y ...     ...

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 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 ?

10. 1-Elementary definable concepts 2- Compositionality 3-Dynamically definable concepts 4-Conceptual dictionaries

11. 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

12. 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 ?

13. 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)

14. 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

15. 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 <dx, there exists d’(c), d’ more salient than d, such that x <d’y + transitive closure

16. 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

17. 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)

18. 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

19. 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 <c y, orx  c’ y

20. 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

21. Distance and membership function c(x) = maximal length of a chain x <c x1 <c x2 <c ... <c xnwith xn  Ext c 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

22. 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

23. 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

24. 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

25. 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}

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 closureof the relation: for all d (c) such thaty <d x, there exists d’(c), d’ more salient than d, such that x <d’ y

27. 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)...

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 = 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.

29. 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)

30. 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...