1. Relations Relations are entities obtaining between entities, their relata. In defining a relation, two parameters matter: -the number of relations, the arity of the relation -the type of the relata, expressed as an ordered list, it is called the signature of the relation. In addition, a number of axioms will encode the 'meaning' of the relation.

2. BFO SNAP ComponentSPAN Component Substances Dependents Space Regions Processuals Processes Events Space-Time Regions

3. Signatures in BFO Take the binary case, we find these main signatures: SNAP-SNAP -(snapi, snapi) -(snapi, snapj), inverse SPAN-SPAN idem SNAP-SPAN, inverse

4. Relations have families There are relations which may have several forms and variants, more specialized or similar versions. Forms and variants may require axiomatic specification. E.g., a relation of signature (snapi, snapj) may require i i. Forms may differ in signature. E.g., genidentity has two forms (snapi, snapj) or (span, span)

5. Relations in BFO Three main types of relations: -intra-ontological ones which obtain between relata of the same ontology (same SNAP, same SPAN) -trans-ontological ones which obtain between relata of different ontologies (different SNAP, different SPAN) -meta-ontological (or trans-ontological cross component) ones which obtain between relata of different ontologies of different types (SNAP-SPAN, SPAN-SNAP) Basic relations have forms of different type.

6. Ontological Dependence -specific/generic Holds when an entity necessitates the existence of another. Basic formal with different forms: -(SNAP, SNAP): property-substance The token redness of the sand necessitates the sand. -(SPAN, SNAP): process-participant The erosion of the rock necessitates the rock.

7. Part-to-Whole Basic relation exclusively intra-ontological: either SNAP- SNAP or SPAN-SPAN. SNAP-SNAP: only if SNAPs have the same index and the same granularity level. Needs additional trans-ontological relations: -temporary part accross SNAPs -cross granular parthood accross SNAPs SNAP-SNAP: -temporal part (and slice) -sub-process.

8. Genidentity SPAN-SPAN: There might be a form of geindentity among processes, either within a single ontology or accross a multiplicity. The such-has-to-have-come-forth-from relation. SNAP-SNAP: trans-ontological relation. Cut a chunk of matter in two, the sum of the remaining pieces is genidentical to the chunk before cutting

9. Causal Relations SNAP-SPAN: Causal Agency A substance produces or modifies causally a process. SPAN-SPAN: Processual causation A processual causes another processual to occur. SPAN-SNAP: Causal repercution A process results in the modification of a substance (it is mediated by processual causation). SNAP-SNAP: Causal Origin A substance is the causal origin of another. It is mediated by a chain of other types of causal relations.

10. SNAP-SPAN Relations Called earlier meta-ontological or trans- ontological cross component. Relate a SNAP entity to a SPAN entity or vice versa.

11. SNAP-SPAN Relations (2) SNAP-entities participate in Processes. In particular, they have histories. Dependent entities such as roles, plans, and status have executions.

12. SNAP-SPAN Relations (3) Time plays an important role has a medium. SNAP-entites exist at a time, SPAN-entites are located at a time. They co-incide at such times.

13. Signatures in BFO SNAP ComponentSPAN Component Substances Dependents Space Regions Processuals Processes Events Space-Time Regions

14. Signatures in BFO (2) SNAP ComponentSPAN Component Substances Dependents Space Regions Processuals Processes Events Space-Time Regions

15. Signatures in BFO (3) (SNAP, SPAN) (Substance, Processual) (Dependent, Processual) (Space Region, Processual) (Processual, Substance) (Processual, Dependent) (Processual, Space Regions) (SPAN, SNAP) (Substance, ST Region) (Dependent, ST Region) (Space Region, ST Region) (ST Region, Substance) (ST Region, Dependent) (ST Region, Space Regions)

16. Relations in BFO A number of relations following the lay out of possible signature with a varying degree of generality (some forms of others). Familly ressemblances, relations standing as inverse, more or less directly, some functional relations maybe. A few axes of variations which can be traced in the hierarchy: -activity / passivity -benefactor / malefactor -direct / indirect

17. SNAP-SPAN Participation Perpetration Initiation Perpetuation Termination Influence FacilitationHindering Mediating ObtainingTheater FromTo Genus Species

18. Participation (SNAP, SPAN) (Substance, Processual) A substance participates in a processual. A racer participates in a race. A voter participates in a poll.

19. Perpetration A substance perpetrates an action: direct and agentive participation in a processual. The referee fires the pistoll. The captain gives the order. (SNAP, SPAN) (Substance, Processual)

20. Initiation (SNAP, SPAN) (Substance, Processual) A substance initiates a processual. A referee starts the race. An attorney initiates the procedure of revision of the process of his or her client.

21. Perpetuation A substance preserve into being a process. The singer sings the song. A (heated) filament perpetuates the emission of light. (SNAP, SPAN) (Substance, Processual)

22. Termination (SNAP, SPAN) (Substance, Processual) A substance terminates a process. The operator switches off the projector in the theater, indicating thereby the end of the projection of the movie. The judge terminates the imprisonment of the pardonned convict.

23. Mediating (SNAP, SPAN) (Substance, Processual) A substance plays an indirect role in the unfolding of a process. The catalyst provides chemicals for the reaction. The diplomat organizes peace talks.

24. Influence (SNAP, SPAN) (Substance, Processual) A substance, although not participating, or a dependent entity (in fine dependent on a substance) has an influence of the course of action of a process or the occurrence of an event. The steepness of the slope bear effects on the movement of the troops. The generals influence the course of action.

25. Faciliation (SNAP, SPAN) (Substance, Processual) A substance as an indirect or secondary role of participation in the process (it can be a participate of a part of taht process). A driver give a ride to the diplomats, the pilote fly them to the rendez-vous. The peacekeepers facilitate the peace process.

26. Hindering (SNAP, SPAN) (Substance, Processual) A substance has an indirect negative effect on the unfolding of a process (by participating in other processes). Resistance hinders troops progression. Strikers prevents the airplane from departing.

27. Obtaining (SNAP, SPAN) (Substance, Processual) The most general relation between a dependent entity and a processual. The processual is an obtaining, realization or execution of the dependent. The power to legislate is realized by the chamber in passing a law. The benefactor role of antibiotics is executed during its journey in the body.

28. Theater (SNAP, SPAN) (Location, Processual) A location is the theater of the occurrence of processuals. Leipzig has several fair during the year. The Plain of Abraham have seen a great battle. Roughly: inverse of the taking-place relation.

29. Point of departure / arrival (SNAP, SPAN) (Substance, Processual) A place is the point of departure or of arrival of a process. Paris is the place where started the race. Roubaix is the place where finished the race. Roughly: inverses of starts-in / ends-at taking-place.

30. SPAN-SNAP Involvement Creation Sustain in existence Destruction Continuation Degradation Execution Taking place Starts in Ends at Genus Species Apparition Vanishing Demarcation Blurring Inhering Degradation

31. Involvement The most general relation between a processual and a substance. The former includes the latter in its unfolding. Some forms are as inverses of forms of participation. Races involve the racers. Wars involve civilians. (SPAN, SNAP) (Processual, Substance)

32. Creation (SPAN, SNAP) (Processual, Substance) A process brings into being a substance (substantial creation). The car accident creates a trauma. The work of the potter creates the vase (not the same relation as that between the potter and the vase).

33. Sustain in existence A process sustains in being a substance. The circulation of the blood sustains the body and its parts. Levying sustains the army. (SPAN, SNAP) (Processual, Substance)

34. Deterioration (SPAN, SNAP) (Processual, Substance) A process affects a substance with negative effects. Drinking coffee contributes to the deterioration of theeth. The flow of water erodes the rock.

35. Destruction A process brings out of being a substance. (Substantial destruction) The car accident destroys the car. The falling of the vase on the floor breaks it. (SPAN, SNAP) (Processual, Substance)

36. Demarcation (SPAN, SNAP) (Processual, Boundary) A process creates (fiat or bona fide) boundaries of substances. The signiature of the treaty settles borders between countries. The tracing of the area of operation by the surgeon defines a boundary, the incision performed by the surgeon yet another one.

37. Blurring (SPAN, SNAP) (Processual, Boundary) A process makes boundaries vague. The military statu quo creates no men's land. The successful transplant creates fuzzy boundaries between original tissues and grafted ones.

38. Inhering/material occurence? A process is projected on matter through this relation. It captures the relevant dependent entities inhering in substances. A warming process is a raise in temperature. Receiving an award mobilizes several statuses that of the awardee, of the awarder, etc. (SPAN, SNAP) (Processual, Dependent)

39. Execution/realization Some processuals are executions or realizations of dependent entities such as roles, status, powers, etc. The power of following the prescription is realized by the willing patient. Giving a lecture is a realization of the professor's status and an execution of its role. (SPAN, SNAP) (Processual, Dependent)

40. Apparition A process brings into being a dependent entity. (Qualitative Creation) The car accident re-shape the car. The backing of the clay gives the vase ist rigidity, its color. (SPAN, SNAP) (Processual, Dependent)

41. Continuation in existence A process continues the existence of a dependent entities. The projection of the slide continues the existence of the colored spot. The firing of the fireworks continues the coloration of the sky (supports change). (SPAN, SNAP) (Processual, Dependent)

42. Degradation (SPAN, SNAP) (Processual, Dependent) A process affects a substance's quality or status (a dependent entity) by lowering its degree. The opening of the window diminishes the temperature level in the room. The polishing of the vase diminishes its rugosity.

43. Vanishing (SPAN, SNAP) (Processual, Dependent) A process brings out of being a dependent entity. (Qualitative destruction) The car accident destroys the car's shape. The burning of the vase destroys its color.

44. Taking place (SPAN, SNAP) (Processual, Location) A process occurs in a given place or in an area. The Revolution took place in Paris. The wind blows beyond the 24th parallel.

45. Starts in / ends at (SPAN, SNAP) (Processual, Location) A process begins / ends at a location. The race started in Paris. The race ended in Roubaix.

46. Functionality of relations Some relations are functional, i.e., many-to-one. Taking-place is functional when understood as exact- location. Participating as the subject of a single doer processual (e.g., birth or death) is functional.

47. Functionality of relations (2) Functionality of the most general form of a relation seems rare. The question is which are they? It is possible to define a functional form of a relation. Take a many-to-many relation (e.g., participation), associates with each elements of its domain (i.e., participants in a processual) the sum of its image (I.e., processes) under the original relation.

48. Projection We may define projections as functions associating each element of the domain of a relation R with the sum of its images under R. The projection is said to be along R.

49. Projection (2) For instance, a process projects on the time of its duration, on the space-time it occupies (intra SPAN). It projects as well on: -the space it occupies at a given time (along taking-place) -the sum of its participants at a time (along mobilization) -the sum of dependent entities (along obtaining)

50. Projection (3) process material regionspatial region participants spatio-temporal region snap t1 span

51. Temporal Co-Incidence x y SNAP-t At time t, colony of bacteria in the throat. time SPAN Migration of a colony of bacteria from the mouth to the small intestine. B

52. Temporal Projection The co-incidence in time of relata required by many SNAP-SPAN relations motivates a form of incidence in time or temporal projection for each type of entity. SNAP entities: -SNAP-SPAN: (SNAP, Time) which amounts to existence at a time (projection along this relation). SPAN entities: -SPAN-SPAN: (Processual, Time) which is a form of location in time (projection along this relation).

53. Temporal Slices as the ‘other‘ leg of the bridge between SNAP and SPAN x y x, y participate in B time B

54. Temporal Slices as the ‘other‘ leg of the bridge between SNAP and SPAN time x y Time t At time t, a temporal slice of B co-incide temporally with SNAP-t.

55. Co-temporality It is a requisit for most SNAP-SPAN relations (participation, obtaining, etc.) Exception: complex relations or forms which are derived from elementary SNAP-SPAN relations and one or many trans-ontological SNAP relations or intra- ontological SPAN (crossing time). For instance, a relation between a person and the remains of a relative conceived defining with genidentity a direct relation to the relative (such as remebrance or love).

56. SNAP-SPAN Participation Perpetration Initiation Perpetuation Termination Influence FacilitationHindering Mediating ObtainingTheater FromTo Genus Species

57. SPAN-SNAP Involvement Creation Sustain in existence Destruction Continuation Degradation Execution Taking place Starts in Ends at Genus Species Apparition Vanishing Demarcation Blurring Inhering Degradation