Parts and Wholes

1. Parts and Wholes Patrick Lambrix Linköpings universitet

2. Outline • Mereology • Linguistics

3. Mereology • Ground mereology: parthood is a reflexive partial order x is a part of x. If x is a part of y and y is a part of x then x=y. If x is a part of y and y is a part of z then x is a part of z.

4. Mereology • x is proper part of y: x is part of y and y is not part of x • x overlaps y: there is a part of x that is also a part of y • x and y are disjoint: x and y do not overlap

5. Mereology • Binary Product: x . y: individual that is part of both x and y and any common part of x and y is a part of the product. • Binary Sum: x + y: individual that overlaps something iff it overlaps at least x or y. • Difference: x – y: largest individual contained in x that has no part in common with y.

6. Mereology • Generalized Product: Π x ’Fx’ : product of all objects satisfying F • Generalized Sum: σx ’Fx’ : sum of all objects satisfying F

7. Mereology • Universe U: the sum of all objects • Complement: the complement of x is U-x.

8. example u = x + y v = x + z w = y + z x y z x is proper part of u x is proper part of v u overlaps v u . v = x x + w = u + v u – w = x x and y disjoint y and z disjoint x and z disjoint

9. Classical Extensional Mereology Axioms: 0: any axiom set sufficient for first-order predicate calculus with identity 1: if x is proper part of y then y is not a proper part of x 2: proper part is transitive 3: if x is a proper part of y then there is a proper part z of y that is disjoint from x (WSP) 4: if there are objects satisfying F then there is an x such that for all y it holds that x overlaps y iff there is a z satisfying F that overlaps y.

10. Classical Extensional Mereology Ground mereology + supplementation principle + existence of sums strong supplementation: if x is not a part of y then there is a part z of x that is disjoint from y.

11. Classical Extensional Mereology Theorems • x is not a proper part of x. • x = y iff x is part of y and y is part of x • Transitivity of part-of • If x is proper part of y and y is part of z, then x is proper part of z

12. Classical Extensional Mereology Theorems • x = y iff for all z: z is part of x iff z is part of y • x = y iff for all z: x is part of z iff y is part of z

13. Classical Extensional Mereology Atoms: x is an atom if it has no proper parts Possible axioms: • Every individual has an atom as part. • Every individual has proper parts. • hybrid of 1and 2. Classical extensional mereology is neutral.

14. Extensional Mereology - criticisms • There are senses of part-of that are not transitive. • There is no guarantee of the existence of sum-individuals. • The criterion saying that individuals having all parts in common are identical, is generally false.

15. Extensional Mereology - criticisms • There are senses of part-of that are not transitive. The basic broad part-of is transitive. The non-transitivity comes from narrowing or specifying the part-of relation with things such as function.

16. Extensional Mereology - criticisms • There is no guarantee of the existence of sum-individuals. Existence of sums does not make theory inconsistent. Problem of applying principle to real world. May work well for portions and regions.

17. Extensional Mereology - criticisms • The criterion saying that individuals having all parts in common are identical, is generally false. Example: Tibbles <> Tib + Tail Tibbles and Tib + Tail share all parts

18. ’minimal’ requirements (Simons 1987) • Asymmetry of proper part • Transitivity of proper part • Weak supplementation principle

19. Approaches from linguistics • Winston, Chaffin, Hermann • Iris, Litowitz, Evens • Gerstl, Pribbenow

20. Winston – Chaffin - Hermann • Based on properties • Functional • Homeomerous • Separable

21. Winston – Chaffin - Hermann • Integral object – component • Collection – member • Mass – portion • Stuff – object • Activity – feature • Area – place

22. Integral object – component • Functional • Not homeomerous • Separable • Cup – handle • Car - wheel

23. Collection – member • Not functional • Not homeomerous • Separable • Forest – tree • Deck - card

24. Mass – portion • Not Functional • Homeomerous • Separable • Pie – slice of pie • Salt – grain of salt

25. Stuff - object • Not functional • Not homeomerous • Not separable • Steel - bike

26. Feature - activity • Functional • Not homeomerous • Not separable • Paying - shopping

27. Area - place • Not functional • Homeomerous • Not separable • Sweden – Linköping • Desert - oasis

28. Relation to other semantic relations Semantic relations inclusion possession attribution class meronymic spatial The six part-whole relations

29. Iris – Litowitz - Evens • Functional component • Segmented whole • Collections and members • Sets and subsets

30. Functional component • The part contributes to the whole not just as a structural unit but as essential to the purposeful activity of the whole. • Bike – wheel • Body - organ

31. Segmented whole • This part-whole relation implies the removability of the part or the divisibility of the whole. • Implies existence-dependence. Whole must exist before the part. • Pie – piece of pie • Sand – handful of sand

32. Collections and members • Relationship of member to collection or element to set. • Gaggle of geese – goose • Flock of sheep - sheep

33. Sets and sub-sets • The set A is a subset of the set B iff every member of A is a member of B.

34. Gerstl - Pribbenow • Induced by compositional structure - masses - collections - complexes • Independent of compositional structure - segment - portions

35. Masses - quantities • Homogeneous • Rice – a certain amount of rice w p

36. r r r p p p Collections - elements • Uniform • Rice – grain of rice w

37. Complexes - components • Heterogeneous • Car – engine w r1 r3 r2 p1 p2 p3

38. Segments • Results from the application of an external scheme. • Segmenting an entity in exterior, boundary interior parts • Beginning, middle, end

39. Portions • Portions are constructed by using a property dimension to select parts out of the whole. • Red parts of the painting • Annoying parts of a TV-show

40. Transitivity • Mereology: Part-of is transitive. xPy and yPz  xPz Sub-relations may not be. (xPy and F(x,y)) and (yPz and F(y,z)) does not necessarily lead to xPz and F(x,z)

41. Transitivity • Winston, Chaffin, Hermann: Transitivity holds within a single sense of part-of. No transitivity when mixing different senses. Part-of relations may be maintained when combining with other inclusion relations.

42. Transitivity • Iris, Litowitz, Evens: Transitivity holds for set – subset and segmented whole. No transitivity for functional parts and collection – element.

43. Literature • Simons, P., Parts - a study in ontology. Clarendon Press Oxford, 1987.

44. Literature • Winston, M., Chaffin, R., Herrmann, D., A taxonomy of part-whole relations. Cognitive Science, 11:417-444, 1987. • Iris, M., Litowitz, B., Evens, M., Problems of the part-whole relation. Relational Models of the Lexicon, Edited by Evens, M., Cambridge University Press, 1988.

45. Literature • Gerstl, P., Pribbenow, S., Midwinters, end games and body parts: a classification of part-whole relations. International Journal of Human-Computer Studies, 43:865-889, 1995.

46. Exercises • Prove theorems in mereology. • Are given models accepted in mereology? • Discuss the linguistic models. Describe the different categories in one model with respect to the categories in other models.

47. Exercise: Are these models valid? ………………………. …

48. Exercise: Is this a valid model?

49. Exercise: Are these models valid?

50. Exercises • Create a valid model in mereology with 15 elements. • Create a valid model in mereology with 12 elements.