1. Parts and Classes in Biomedical Ontology Barry Smith http://ontologist.com

2. GO:0003673: cell fate commitment Definition: The commitment of cells to specific cell fates and their capacity to differentiate into particular kinds of cells.

3. GO: asymmetric protein localization involved in cell fate commitment

4. The intended meaning of part-of as explained in the GO Usage Guide is: “part of means can be a part of, not is always a part of: the parent need not always encompass the child. For example, in the component ontology, replication fork is a part of the nucleoplasm; however, it is only a part of the nucleoplasm at particular times during the cell cycle”

5. So, GO ‘part of’ means: can be a part of, not is always a part of

6. But what about: GO: a flagellum is part-of cells here ‘part of’ means: some kinds of cells always have flagella as parts

7. And what about:

8. GO: Cellular Component Ontology is part-of Gene Ontology GO: Biological Process Ontology is part-of Gene Ontology GO: Molecular Process Ontology is part-of Gene Ontology here ‘part of’ means: one vocabulary is included in another vocabulary

9. GO’s three meanings of part-of 1. A time-dependent mereological inclusion relation between instances A sometimes_part_of B = def  t  x  y (inst(x, A, t) & inst(y, B, t) & part(x, y, t)). 2. Some (types of) Bs have As as parts: A part_of GO B = def  C (C is_a B & A part_of C) 3. Inclusion relations between vocabularies

10. GO’s use of ‘part of’ illustrates the following problems One term being used to represent a plurality of different relations One lexically simple term being used to represent lexically complex concept A term with an established use (inside and outside biomedical ontology) being used with a new non-standard use WHY SHOULD WE CARE?

11. Because we want to use GO to support reasoning

12. GO’s Usage Guide lists four ‘logical relationships’ between ‘is a’ and ‘part of’: (1) (A part_of B & C is_a B)  A part_of C (2) is_a is transitive (3) part_of is transitive (4) NOT: (A is_a B & C part_of A)  C part_of B

13. Of these four logical relationships, only (2) is_a is transitive is valid, and even this law is mis-expressed by GO as: if A is an instance of B and B is an instance of C then A is an instance of C so that GO confuses classes with instances

14. (3) part_of GO is transitive fails because of plastid part_of GO cytoplasm cytoplasm part_of GO cell (sensu Animalia) But not: plastid part_of GO cell (sensu Animalia).

15. GO built by biologists who deliberately did not want to take account of any of the results of non- biologists working in fields such as ‘ontology’ But still: GO belongs to the world of KR The ‘K’ of KR is characteristically a very odd fragment of what (e.g. scientists) would recognize as ‘knowledge’

16. The world of KR is world of classes exclusively (e.g. WordNet) Dictionary makers live in a world of classes exclusively Terminologists live in a world of classes exclusively Description logic lives in a world of classes (almost) exclusively

17. GO’s confusion about part-of 1. A time-dependent mereological inclusion relation between instances A sometimes_part_of B = def  t  x  y (inst(x, A, t) & inst(y, B, t) & part(x, y, t)). 2. Some (types of) Bs have As as parts: A part_of GO B = def  C (C is_a B & A part_of C) 3. Inclusion relations between vocabularies illustrate the need to take not just classes but also instances into account

18. Entities

19. universals (classes, types, roles …) particulars (individuals, tokens, instances …) Axiom: Nothing is both a universal and a particular

20. Two Kinds of Elite Entities classes, within the realm of universals instances within the realm of particulars

21. Entities classes

22. Entities classes* *natural, biological

23. Entities classes of objects different axioms for classes of functions, processes, etc.

24. Entities classes instances

25. Classes are natural kinds Instances are natural exemplars of natural kinds (problem of non-standard instances must be dealt with also)

26. Entities classes instances penumbra of borderline cases

27. Entities classes instances junk example of junk: beachball desk

28. Primitive opposition between universals and particulars variables A, B, … range over universals variables x, y, … range over particulars

29. Primitive relations: inst and part inst(Jane, human being) part(Jane’s heart, Jane’s body) A class is anything that is instantiated An instance as anything (any individual) that instantiates some class

30. Entities human Jane inst

31. Entities human Jane’s heart part Jane

32. Axioms for part Axioms governing part (= ‘proper part’) (1) it is irreflexive (2) it is asymmetric (3) it is transitive (+ usual mereological axioms) part is the usual mereological relation among individuals

33. Definitions class(A) = def  x inst(x, A) instance(x) = def  A inst(x, A) Theorem: Nothing can be both an instance and a class

34. Axiom of Extensionality Classes which share identical instances are identical (need to take care of the factor of time)

35. Entities classes x, y, … differentiae (roles, qualities…)

36. Differentiae Aristotelian Definitions An A is a B which exemplifies C C is a differentia No differentia is a class exemp(individual, differentia) exemp(Jane, rationality) objects exemplify roles

37. role

38. A is_a B genus(A) species(A) classes instances

39. A is_a B = def  x (inst(x, A)  inst(x, B)) genus(A)= def  B (B is_a A & B  A) species(A)= def  B (A is_a B & B  A) classes instances

40. nearest species nearestspecies(A, B)= def A is_a B &  C ((A is_a C & C is_a B)  (C = A or C = B) B A

41. Definitions highest genus lowest species

42. lowest species and highest genus lowestspecies(A)= def species(A) & not-genus(A) highestgenus(A)= def genus(A) & not-species(A) Theorem: class(A)  genus(A) or lowestspecies(A)

43. Axioms Every class has at least one instance Distinct lowest species never share instances SINGLE INHERITANCE: (nearestspecies(A, B) & nearestspecies (A, C))  B = C

44. Axioms governing inst genus(A) & inst(x, A)   B nearestspecies(B, A) & inst(x, B) EVERY GENUS HAS AN INSTANTIATED SPECIES nearestspecies(A, B)  A’s instances are properly included in B’s instances EACH SPECIES HAS A SMALLER CLASS OF INSTANCES THAN ITS GENUS

45. Axioms nearestspecies(B, A)   C (nearestspecies(C, A) & B  C) EVERY GENUS HAS AT LEAST TWO CHILDREN nearestspecies(B, A) & nearestspecies(C, A) & B  C)  not-  x (inst(x, B) & inst(x, C)) SPECIES OF A COMMON GENUS NEVER SHARE INSTANCES

46. Theorems (genus(A) & inst(x, A))   B (lowestspecies(B) & B is_a A & inst(x, B)) EVERY INSTANCE IS ALSO AN INSTANCE OF SOME LOWEST SPECIES (genus(A) & lowestspecies(B) &  x(inst(x, A) & inst(x, B))  B is_a A) IF AN INSTANCE OF A LOWEST SPECIES IS AN INSTANCE OF A GENUS THEN THE LOWEST SPECIES IS A CHILD OF THE GENUS

47. Theorems A is_a B & A is_a C  (B = C or B is_a C or C is_a B) CLASSES WHICH SHARE A CHILD IN COMMON ARE EITHER IDENTICAL OR ONE IS SUBORDINATED TO THE OTHER

48. Theorems (genus(A) & genus(B) &  x(inst(x, A) & inst(x, B)))   C(C is_a A & C is_a B) IF TWO GENERA HAVE A COMMON INSTANCE THEN THEY HAVE A COMMON CHILD

49. Theorems class(A) & class(B)  (A = B or A is_a B or B is_a A or not-  x(inst(x, A) & inst(x, B))) DISTINCT CLASSES EITHER STAND IN A PARENT-CHILD RELATIONSHIP OR THEY HAVE NO INSTANCES IN COMMON

50. The axioms and theorems above are non-trivial Almost all of them can be found in Aristotle Taken over by the Foundational Model of Anatomy Forgotten because of dominance of set theory and its dark progeny (KR, description logic, model-theoretic semantics, ‘conceptual modeling’, etc.)

51. Definition of is_a A is_a B = def A and B are classes &  x (inst(x, A)  inst(x, B))

52. Part_of as a relation between classes is more problematic testis part_of human being ? heart part_of human being ?

53. WordNet can’t deal with optional body parts, like warts, freckles, etc. Nor with temporary optional body parts like pony-tail or five-o'clock shadow.

54. Part_for and Has_part from Smith and Rosse, “The Role of Foundational Relations in the Alignment of Biomedical Ontologies” A part_for B =def  x ( inst(x, A)   y ( inst(y, B) & part(x, y) ) ) B has_part A =def  y ( inst(y, B)   x ( inst(x, A) & part(x, y) ) ) human testis part_for human being, But not: human being has_part human testis. human being has_part heart, But not: heart part_for human being.

55. Part_of A part_of B =def A part_for B & B has_part A This defines an Egli-Milner order It guarantees that As exist only as parts of Bs and that Bs are structurally organized in such a way that As must appear in them as parts. part_of NOT best understood as a relation between classes!

56. Foundational Model of Anatomy distinguishes canonical anatomy – deals with classes and with instances (generically) plus instantiated anatomy (deals with individual cases) plus various variant anatomies to deal with standard sorts of deviant instances

57. Relations in Foundational Model of Anatomy clinical part of constitutional part of forms general part of member of necessary part necessary whole part of possible part regional part of related part segmental composition of segmental contribution to systemic part of

58. Recall: problems with GO’s use of ‘part of’ One term used to represent a plurality of different relations A term with an established use adopted with a new non-standard use Lexically simple term is used to represent lexically complex concepts  LEADS TO CIRCULARITY: if ‘part of’ means ‘can be part of’ then ‘can be part of’ means ‘can be can be part of’ …

59. Circular definitions endemic in biomedical terminology systems (GO, Snomed, etc.) Circular definitions can be cheaply produced in large numbers to impress funding agencies

60. UMLS Semantic Type Semantic Type: Idea or Concept Definition: An abstract concept, such as a social, religious or philosophical concept. problem: circularity (many other problems: Florence is an idea or concept)

61. UMLS-SN Semantic Relation Semantic Relation: conceptual_part_of Definition: Conceptually a portion, division, or component of some larger whole. Inverse: has_conceptual_part definition is semantically incoherent

62. UMLS-SN Semantic Relation Semantic Relation: associated_with Definition: has a significant or salient relationship to. Inverse: associated_with confuses entity with our cognition of the entity

63. UMLS-SN Semantic Relation Semantic Relation: isa Definition: The basic hierarchical link in the Network. If one item "isa" another item then the first item is more specific in meaning than the second item. Inverse: inverse_isa confuses class with concept/meaning is not a definition

64. UMLS-SN Semantic Relation Semantic Relation: part_of Definition: Composes, with one or more other physical units, some larger whole. This includes component of, division of, portion of, fragment of, section of, and layer of. Inverse: has_part bad inverse

65. UMLS-SN Semantic Relation Semantic Relation: location_of Definition: The position, site, or region of an entity or the site of a process. Inverse: has_location danger of confusing instance/class levels

66. UMLS-SN Semantic Relation Semantic Relation: occurs_in Definition: Takes place in or happens under given conditions, circumstances, or time periods, or in a given location or population. This includes appears in, transpires, comes about, is present in, and exists in. Inverse: has_occurrence same problem confusion of objects and processes (exists in)

67. UMLS-SN Semantic Relation Semantic Relation: prevents TUI: T148 Definition: Stops, hinders or eliminates an action or condition. Inverse: prevented_by bad inverse: contraception prevents pregnancy pregnancy prevented by contraception

68. UMLS-SN Semantic Relation Semantic Relation: process_of Definition: Action, function, or state of. Inverse: has_process avoids circularity by introducing confusion

69. UMLS-SN Semantic Relation Semantic Relation: produces Definition: Brings forth, generates or creates. This includes yields, secretes, emits, biosynthesizes, generates, releases, discharges, and creates. Inverse: produced_by bad inverse: artificial insemmination produces pregnancy pregnancy produced by artificial insemmination

70. The UMLS Semantic Network is ‘an upper-level ontology … in which all concepts are given a consistent and semantically coherent representation’. Alexa McCray, “An upper level ontology for the biomedical domain”. Comp Functional Genomics 2003; 4: 80-84.

71. Conclusion Work on biomedical ontologies and terminologies has focused almost exclusively on classes (often confusingly referred to as ‘concepts’). The class-orientation of KR goes with the assumption that all that need be said about classes can be said without appeal to formal features of instantiation of the sorts described above. KR-facts (e.g. about ‘conceptual parts’) are pulled out of the air in an unprincipled way. This leads to an impoverished regime of definitions in which the use of identical terms (like ‘part’) masks underlying incompatibilities.

72. THE END

74. Conclusion 1/2 Matters have not been helped by the fact that description logic has been oriented primarily around reasoning with classes. Certainly if we are to produce information systems with the requisite computational properties, then this entails recourse to a logical framework like that of description logic. At the same time we must ensure that the data that serves as input to such systems is organized formally in a way that sustains rather than hinders successful alignment with other systems. There are two complementary tasks: REFERENCE ONTOLOGY and APPLICATION ONTOLOGY

75. Classes vs. Sums Classes are distinguished by granularity: they divide up the corresponding domain into whole units or members, whose interior parts and structure are traced over. The class of human beings is instantiated only by human beings as single, whole units. A mereological sum is not granular in this sense.

76. Classes vs. Sets Both classes and sets are marked by granularity – but sets are timeless Each class or set is laid across reality like a grid consisting (1) of a number of slots or pigeonholes each (2) occupied by some member. But a set is determined by its members. This means that it is (1) associated with a specific number of slots, each of which (2) must be occupied by some specific member. A set is thus specified in a double sense. A class survives the turnover in its instances, and so it is specified in neither of these senses, since both (1) the number of associated slots and (2) the individuals occupying these slots may vary with time. A class is not determined by its instances as a state is not determined by its citizens.

77. Classes vs. Sets A set with n members has in every case exactly 2n subsets The subclasses of a class are limited in number (which classes are subsumed by a larger class is a matter for empirical science to determine)

78. Classes vs. sets A set is an abstract structure, existing outside time and space. The set of human beings existing at t is (timelessly) a different entity from the set of human beings existing at t because of births and deaths. A class can survive changes in the stock of its instances because classes exist in time. (An organism can similarly survive changes in the stock of cells or molecules by which it is constituted.) D1*A is_a B =def  t  x ( inst(x, A, t)  inst(x, B, t) ), D1* will take care of false positives such as adult is_a child

79. We can prove: is_a is reflexive and antisymmetric Axiom: part_of is irreflexive We can prove that part_of is asymmetric We can prove that both is_a and part_of are transitive

80. GO’s curators accordingly now consider removing the corresponding assertion from its Usage Guide. As concerns (3), consider: plastid part_ofGO cytoplasm cytoplasm part_ofGO cell (sensu Animalia) But not:plastid part_ofGO cell (sensu Animalia). While ‘cell (sensu Animalia)’ is not a term in GO, it does conform to GO’s rules for term formation, and this suggests reason for some uncertainty also as to the validity of (3). GO justifies its rejection of (4) with the following example: meiotic chromosome is_a chromosome synaptonemal complex part_ofGO meiotic chromosome But not necessarily: synaptonemal complex part_ofGO chromosome.

81. On the reading of GO’s ‘part of’ as meaning ‘can be part of’, however, it seems that synaptonemal complex is ‘part of’ chromosome. And if the reading of GO’s ‘part of’given in D11 is correct, then (4) can indeed be proved as a matter of logic. We suggest that it is only by appeal to formal definitions that these and related uncertainties (detailed in [[19]]) can be resolved. Formal definitions would help to ensure also that when the terms of controlled vocabularies like GO are mapped into the UMLS Metathesaurus then this is done in ways that support the drawing of reliable inferences concerning relations between these terms and existing terms in the Metathesaurus.[19]

82. The growth of bioinformatics has led to an increasing number of evolving ontologies which must be correlated with the existing terminology systems developed for clinical medicine. A critical requirement for such correlations is the alignment of the fundamental ontological relations used in such systems, above all the relations of class subsumption (is_a) and partonomic inclusion (part_of). To achieve this end, however, existing clinical and evolving bioinformatics terminologies need to call upon formalisms whose significance was not evident at the time these resources were originally conceived.

83. Both is_a and part_of are ubiquitous in bioinformatics ontologies and terminologies. Yet their treatment is inconsistent and problematic, and in some cases the two relations are not clearly distinguished at all. SNOMED-RT, for example, has: both testes is_a testis. UMLS has: plant leaves is_a plant. In this communication we argue that a coherent treatment of is_a and part_of must be based on explicit formal definitions which take into account not only the classes involved as terms of these relations but also the instances of these classes. We base our arguments on the lessons we learned during the evolution of the Digital Anatomist Foundational Model of Anatomy (FMA, for short [[1]]) in which we have refined the treatment of these relations over time and distinguished between classes and instances in terms of canonical and instantiated anatomy. [[2]][1][2]

84. Our objectives are to define canonical and instantiated anatomy before giving formal definitions of is_a and part_of in terms of a theory of instantiation. We then discuss in this light issues of universal relevance to ontologies, such as classes vs. wholes and sets, granularity, idealization, and the role of time and change. After illustrating problematic usage of is_a and part_of we draw conclusions for ontology alignment, pointing to the need for supplementing Description Logic-based reasoning implementations with rigorous manual auditing of underlying data-sources based on formal analyses in terms of instance-level relations and on clear and intuitive principles of curation. Canonical and Instantiated Anatomy

85. Canonical anatomy is a field of anatomy (science) that comprises the synthesis of generalizations based on anatomical observations that describe idealized anatomy (structure). These generalizations have been implicitly sanctioned by their usage in anatomical discourse. Instantiated anatomy is a field of anatomy (science) that comprises anatomical data pertaining to individual instances of organisms and their parts. Instantiated anatomy is needed to support the application of biomedical knowledge in clinical care and in fields such as image analysis. The corresponding instance-data is not incorporated into the FMA, which deals with idealizations at a higher level of abstraction. In introducing the relation between canonical and instantiated anatomy, however, the FMA provides the key to an adequate formal treatment of is_a and part_of, for the latter can be defined and formally interrelated only when the relation of instantiation between instances and classes is taken into account.

86. Formal Theory of Is_a and Part_of We use the term entity as a universal ontological term of art embracing objects, processes, functions, structures, times and places, and we distinguish among entities in general two special sub-totalities, called instances and classes, respectively. Instances are individuals (particulars, tokens) of special sorts. Thus each is a simply located entity, bound to a specific (normally topologically connected) location in space and time. [[3]] Classes (also called universals, kinds, types) are multiply located; they exist in their respective instances.[3]

87. To formalize these notions we use standard first-order logic with variables x, y, x1, etc. ranging over instances, and A, B, A1, etc. ranging over classes. Our system rests on two primitive relations of inst and part. Inst is the relation of instantiation between instances and classes, illustrated by: Jane is an instance of human being. Part is the relation of parthood among instances, illustrated by: Jane’s heart is part of Jane’s body. We define a class as anything that is instantiated; an instance as anything (any individual) that instantiates some class. The principal axioms governing inst are: (1) that it holds in every case between an instance and a class, in that order; and (2) that nothing can be both an instance and a class. The axioms governing part (also called ‘proper part’) can be specified as follows [[4]]. It is (1) irreflexive (no entity is part of itself), (2) asymmetric (if part(x, y) then not-part(y, x)), and (3) transitive (if part(x, y) and part(y, z), then part(x, z)). In addition, it satisfies: (4) a principle governing the formation of sums of parts (for example of binary sums x+y), and (5) a remainder axiom, to the effect that if part(x, y) then there is some part z of y which does not share parts in common with x.[4] We use the standard quantifiers of first-order logic: , abbreviating for some value of, and , abbreviating for all values of. The device of quantification allows us to take account of instantiation in generic fashion, i.e. without the need to take specific instances into account. The full formalism requires general axioms specifying the properties of classes as natural kinds (rather than arbitrary collections) [[5]], together with more specific axioms dealing with the different sorts of classes (of objects, functions, processes, pathways, sites, etc.) in the different domains of biomedical ontology. It also requires an axiom of extensionality, to the effect that classes which share identical instances are themselves identical.[5]

88. We can now define is_a, the relation of class subsumption: D1A is_a B =def  x ( inst(x, A)  inst(x, B) ) where ‘  ’ abbreviates: if... then.... To say that A is_a B is to say that every instance of A is an instance of B. To define part_of is more tricky. We start by defining: D2A part_for B =def  x ( inst(x, A)   y ( inst(y, B) & part(x, y) ) ). D2 provides information primarily about As; it tells us that As do not exist except as instance-level parts of Bs. Conversely: D3B has_part A =def  y ( inst(y, B)   x ( inst(x, A) & part(x, y) ) ) provides information primarily about Bs; it tells us that Bs do not exist except with As as instance-level parts.

89. Because there are female as well as male human beings, we can state: human testis part_for human being, but we cannot state: human being has_part human testis. Because non-human vertebrates also have hearts, we can state human being has_part heart, but not: heart part_for human being. We now define the relation part_of by combining D2 and D3: D4A part_of B =def A part_for B & B has_part A Thus A part_of B if and only if: (i) for any instance x of A there is some instance y of B which is such that x stands to y in the instance-level part relation, and vice versa: (ii) for any instance y of B there is some instance x of A which is such that x stands to y in this same relation. This yields a strong structural mereological tie between the classes A and B (defining a so-called Egli- Milner order [[6]]). It guarantees that As exist only as parts of Bs and that Bs are structurally organized in such a way that As must appear in them as parts. That partonomies like those associated with the FMA are structured by the full part_of relation is ensured by the fact that here all terms for body parts are assumed to have an implicit prefix designating the type of organism involved.[6]

90. Sometimes we need to capture mereological relations involving specific numbers of instances. Thus in a case like human being has_part brain, we need to express that each instance of human being has exactly one instance of brain as part: inst(x, human)   y  z((inst(z, brain) & part(z, x))  z = y) with generalizations to represent a human being’s canonical organization as having two lungs, ten fingers, and so on.

91. Both is_a and part_of are standardly treated as relations between classes. The formal structure of D4 makes it clear, however, that the latter does not signify that classes stand in some special class-level mereological inclusion relation. Rather, it expresses more fundamental part-relations – captured in D2 and D3 – between the underlying instances. A distinction analogous to that between D2, D3 and D4 is indispensable to the formal definition of many other foundational relations of biomedical ontologies – including 53 of the 54 relations contained in the UMLS Semantic Network (UMLS-SN, Version 2003AB) [[7]]. In particular, reference to instances is a necessary first step in the rigorous implementation, in systems like the FMA, of mereotopological relations such as spatial occupation and spatial adjacency, as also of concepts such as junction, boundary, cluster, and the like. [1,[8],[9]][7][8][9]

92. We can then prove that is_a is reflexive (for every class A we have: A is_a A), and antisymmetric (if A is_a B and B is_a A, then A and B are identical). We need to add as axiom that part_of is irreflexive (that no class is part_of itself). From this we can prove that part_of is also asymmetric (if A part_of B then not-B part_of A). We can prove also that both is_a and part_of are transitive: thus if A is_a B and B is_a C, then A is_a C, and if A part_of B and B part_of C then A part_of C. Classes vs. Wholes: Granularity and Idealization has been allowed to mask underlying incompatibilities. Matters have not been helped by the fact that description logic, the prevalent framework for terminology-based reasoning systems, has with some recent exceptions (e.g. [[20]]) been oriented primarily around reasoning with classes.[20] Certainly if we are to produce information systems with the requisite computational properties, then this entails recourse to a logical framework like that of description logic. At the same time, however, we must ensure that the data that serves as input to such systems is organized formally in a way that sustains rather than hinders successful alignment with other systems. The way forward is to recognize, as does the FMA, that these are two distinct tasks, both of which are equally important to the construction of biomedical ontologies and terminologies.

93. A rigorous system of formal definitions to support biomedical ontology alignment must clarify also the relations between the concept of class, mereological whole and set. Here, too, the reference to instances is indispensable. For classes are distinguished by the fact that they capture their instances in a way which involves the factor of granularity, which means: in such a way as to divide up the corresponding domain into whole units or members, whose interior parts and structure are traced over. [[10]] A mereological sum is not granular in this sense. The mereological sum of human beings comprehends also all instance-level parts (including organs, cells, molecules, and so on). The class of human beings, in contrast, is instantiated only by human beings as single, whole units.[10]

94. The instances (units, members) in a class are marked out by the fact that, in the Aristotelian terms used by the FMA, they share a common essence. [[11],[12]] Which classes exist in a given domain is a matter for empirical research. Hence a good first clue to the existence of a class is provided by the fact that there exists a corresponding term that has either been sanctioned by (in our case anatomical) science or can be inferred from terms so sanctioned by the need to fill gaps in the taxonomy or partonomy (for example terms for higher-level classes and for not previously named classes instantiated by macroscopic parts of the body) [[13]]. In anatomy and related disciplines a supplementary clue may be provided through the association of given classes with the structural genes whose coordinated expression gives rise to the corresponding instances.[11][12][13] Each class-definition in the FMA specifies the essence shared by the corresponding instances via the specification of (i) a genus, which is some wider class to which the given class belongs, together with (ii) the differentiae which mark out its instances within this wider class.

95. Biological classes are marked always by an opposition between standard or prototypical instances and a surrounding penumbra of non-standard instances (not all instances of the class human being are marked by the presence of amputation stumps or pituitary tumors). To do justice to these matters FMA introduces the factor of idealization, which means (in first approximation) that the classes of the FMA’s Anatomy Taxonomy AT include only those instances to which canonical anatomy applies. This means that we need to revise definitions D1–D4 by restricting the range of variables x, y,... to the realm of individuals which satisfy the generalizations of canonical anatomy, so that the same abstraction of anatomy (structure) will be represented in all the instances of any given AT-class. This device of specifying different ranges of variables gives us the means also to represent the generalizations belonging to the different branches of canonical anatomy, for example to canonical anatomy for male vs. female human beings, for human beings at various developmental stages, and for organisms in other species. It can allow us also to represent the generalizations governing the anatomical variants yielded by the presence of, for example, coronary arteries or bronchopulmonary segments, which deviate from canonical anatomical patterns of organization.

96. Classes vs. Sets: Granularity and Time Sets in the mathematical sense, too, are marked by the factor of granularity, which means that each set comprehends its members as single, whole units. A class or set is laid across reality like a grid consisting (1) of a number of slots or pigeonholes each (2) occupied by some member. (This informal talk of grids and slots is formalized in [[14]] in terms of the theory of granular partitions.) Classes are distinguished from sets, however, by the fact that a set is determined by its members. This means that it is (1) associated with a specific number of slots, each of which (2) must be occupied by some specific member. A set is thus specified in a double sense. A class, in contrast, survives the turnover in its instances, and so it is specified in neither of these senses, since both (1) the number of associated slots and (2) the individuals occupying these slots may vary with time.[14] Sets are distinguished from classes also in this: a set with n members has in every case exactly 2n subsets, constituted by all the combinations of these members. The subclasses of a class, on the other hand, are limited in number, and which classes are subsumed by a larger class is a matter for empirical science to determine. Leaves (lowest nodes) in the taxonomy are (changing) collections of instances. As we move up the taxonomy we encounter in succession collections of such collections of instances, collections of collections of such collections, etc., organized in a nested hierarchy reaching up to the maximal class or ‘root’. We can visualize the classes at different levels as being analogous to geopolitical entities (towns, counties, states) as represented on a map. Instances correspond in this analogy to the corresponding populations: a class is not determined by its instances as a state is not determined by its citizens.

97. Classes are distinguished from sets also by their relation to time. A set is an abstract structure, existing outside time and space, and this is so even when its members are parts of concrete reality. Since each set is determined by its members, the set of human beings existing at t is (timelessly) a different entity from the set of human beings existing at t because of births and deaths. Matters are different with regard to classes. The class human being can survive the change in the stock of its instances which occurs when John and Jane die, because classes exist in time. John and Jane themselves can similarly survive changes in the stock of cells or molecules by which they are constituted. To do justice to the fact that classes in the biological domain endure even when their extensions change, a full definition of the is_a relation must involve a temporally indexed reading of inst (with variables t, t, etc., ranging over times): D1*A is_a B =def  t  x ( inst(x, A, t)  inst(x, B, t) ), so that A is_a B means: at all times t, if x is an instance of A at t then x is an instance of B at t. D1* will also take care of false positives such as adult is_a child, which an untensed reading of D1 would otherwise allow. In general, all statements of inst and part relations involving objects in biomedical ontologies, like all the data of instantiated anatomy, are indexed by times.

98. Taxonomy and Partonomy A taxonomy such as AT is formally speaking a tree in the mathematical sense. It satisfies axioms to the effect that (1) it has a root or unique maximal genus (here: anatomical entity) and (2) all other classes are connected to this root via finite chains of is_a relations satisfying a principle of single inheritance. A partonomy, in contrast, is a partial order in the mathematical sense, with top (here: organism – the class instantiated by mereologically maximal entities), to which all other classes are connected via chains of part_of relations. We can then define the concepts of root and leaf of a taxonomy and top and bottom of a partonomy as follows. D5root(A) =def  B (B is_a A) D6leaf(A) =def  B (B is_a A  A = B) D7top(A) =def  B (A = B or B part_of A) & not-  B (A part_of B) D8bottom(A) =def not-  B (B part_of A).

99. We can then postulate axioms to the effect that every class includes some leaf as subclass, and that every instance of every class instantiates some leaf:  A  B ( leaf(B) & B is_a A )  A  x ( inst(x, A)   B (leaf(B) & inst(x, B) ) ) The taxonomical union AÈB of classes A and B is defined as the minimal class satisfying the condition that it contains both A and B as subclasses. Such a class always exists, since A and B are in any case subclasses of the root. The taxonomic union of femur and liver, for example, is organ. The partonomic union of two classes A+B is the class, if it exists, whose instances are sums x+y of instances of classes A and B respectively. While every pair of classes has a taxonomic union, only some classes have a partonomic union, since entities of the form x+y are instances of classes only in some highly restricted cases, for example: left lung = upper-lobe-of- left-lung + lower-lobe-of-left-lung. Such examples characteristically involve the phenomenon of fiat boundaries. [[15],[16]][15][16]

100. As concerns taxonomic intersection, a class is never immediately subordinated to more than one higher class within a tree. This means that if two classes overlap in sharing some common sub-class, then this is because one is a subclass of the other. A  B, the taxonomic intersection of A and B, if it exists, is then simply the smaller of these two classes. We can add further an axiom to the effect that, if two classes are such as to overlap in sharing some common instances, then this, too, is because one is a subclass of the other:  x (inst(x, A) Ù inst(x, B))  A is_a B or B is_a A. Classes can overlap partonomically, on the other hand, in such a way that there is a class which stands in the part_of relation to both, though neither stands in this relation to the other: D9A1 partonomic_overlap A2 = def  A (A part_of A1 & A part_of A2). For example: pelvis and vertebral column overlap in the sacrum and coccyx. Most classes in the biomedical domain do not overlap partonomically in this sense, yet it is this difference in behavior between taxonomic and partonomic overlap which captures the essential difference between the tree structure of taxonomies and the partial order structure of partonomies.

101. Conclusion Practitioners in the biomedical sciences move easily between the realm of classes and the realm of instances existing in time and space. For historical reasons, however, work on biomedical ontologies and terminologies – which grew out of work on medical dictionaries and nomenclatures – has focused almost exclusively on classes (or ‘concepts’) atemporally conceived. This class- orientation is common in knowledge representation, and its predominance has led to the entrenchment of an assumption according to which all that need be said about classes can be said without appeal to formal features of instantiation of the sorts described above. This, however, has fostered an impoverished regime ofof definitions in which the use of identical terms in different systems has been allowed to mask underlying incompatibilities. Matters have not been helped by the fact that des­crip­tion logic, the pre­valent frame­work for ter­mi­no­logy-based reas­on­ing sys­tems, has with some recent exceptions (e.g. [[i]]) been oriented primarily around reasoning with classes.[i] Certainly if we are to produce information systems with the requisite computational properties, then this entails recourse to a logical frame­work like that of description logic. At the same time, however, we must ensure that the data that serves as input to such sys­tems is organized formally in a way that sus­tains rather than hinders successful alignment with other systems. The way forward is to recognize, as does the FMA, that these are two distinct tasks, both of which are equally important to the construction of biomedical ontologies and terminologies.

102. The problem of ontology alignment GO SCOP SWISS-PROT SNOMED MeSH FMA … all remain at the level of TERMINOLOGY (two reasons: legacy of dictionaries + DL) What we need is a REFERENCE ONTOLOGY = a formal theory of the foundational relations which hold TERMINOLOGY ONTOLOGIES and APPLICATION ONTOLOGIES together

103. Analogous distinctions required for nearly all foundational relations of ontologies and semantic networks: A causes B A is associated with B A is located in B etc. Reference to instances is necessary in defining mereotopological relations such as spatial occupation and spatial adjacency

104. Instances are elite individuals Which classes (and thus which instances) exist in a given domain is a matter for empirical research. Cf. Lewis/Armstrong “sparse theory of universals”

105. Dextension(A) = {x | inst(x, A)} D9differentia(A) = def  B  C nearestspecies(B, C) & A  B & A  C & extension(C) = extension(B)  extension(C)

106. The genus together with the differentia of a species constitutes the essence of the species. differentia (A)  not-class(A)

107. Mathematical Structure Each class hierarchy constitute a supremum-semilattices with respect to is_a.

108. Axioms (Berg) A1 lowestspecies(A)   x inst(x, A) A2 lowestspecies(A) & lowestspecies(B) & A  B  (not-  x inst(x, A) & inst(x, B)) A3 nearestspecies(A, B) & nearestspecies (A, C)  B = C A4 genus(A) & inst(x, A)   B nearestspecies(B, A) & inst(x, B) A5nearestspecies(A, B)  the extension of A is a subset of the extension of B

109. Axioms (Berg) genus(A) & inst(x, A)   B nearestspecies(B, A) & inst(x, B) EVERY GENUS HAS AN INSTANTIATED SPECIES nearestspecies(A, B)  the extension of A is a subset of the extension of B EACH SPECIES HAS A SMALLER CLASS OF INSTANCES THAN ITS GENUS

110. Axioms (Berg) nearestspecies(B, A)   C (nearestspecies(C, A) & B  C EVERY GENUS HAS AT LEAST TWO CHILDREN nearestspecies(B, A) & nearestspecies(C, A) & B  C)  not-  x (inst(x, B) & inst(x, C)) SPECIES OF A COMMON A8There is no infinite sequence such that nearestspecies(Ai, Ai+1) for all i  1 A9There is no infinite sequence such that nearestspecies(Ai+1, Ai) for all i  1

111. Theorems (Berg) T1nearestspecies(A, B)  the extension of A is a proper subset of the extension of B T2  A  x inst(x, A) T3nearestspecies(A, B)  not-  C (nearestspecies(A, C) & nearestspecies(C, B)) T4lowestspecies(A1) & lowestspecies(A2) & nearestspecies(A1, B)  not-  C(nearestspecies (B, C) & nearestspecies (C, A2)

112. Theorems (Berg) T5(genus(A) & inst(x, A))   B (lowestspecies(B) & B is_a A & inst(x, B)) T6(genus(A) & lowestspecies(B) &  x (inst(x, A) & inst(x, B))  B is_a A T7A is_a B & A is_a C  (B = C or B is_a C or C is_a B T8(genus(A) & genus(B) &  x(inst(x, A) & inst(x, B)))   C(C is_a A & C is_a B) T9class(A) & class(B)  (A = B or A is_a B or B is_a A or not-  x(inst(x, A) & inst(x, B)))

113. WordNet NOT: wheel PART OF car WordNet represents part-of quite sparingly It normally gives trivial holonymic relations which are just true by definition). wheel PART OF wheeled vehicle steering wheel PART OF steering system

114. WordNet With has_part relations it is more generous: car, auto, automobile, machine, motorcar -- HAS PART: air bag HAS PART: glove compartment etc.

115. Circular definitions and associated problems in general endemic in biomedical terminology systemsConfusion of use and mention Confusion of concepts and objects Confusion of concepts and classes Confusion of terms and objects Confusion knowledge with what is known Confusion of object-level with machine-level Simple stupidity … all of which lead to poor coding

116. UMLS-SN

117. UMLS-SN Semantic Relations Semantic Relation: affects TUI: T151 Definition: Produces a direct effect on. Implied here is the altering or influencing of an existing condition, state, situation, or entity. This includes has a role in, alters, influences, predisposes, catalyzes, stimulates, regulates, depresses, impedes, enhances, contributes to, leads to, and modifies. Inverse: affected_by

118. UMLS-SN Semantic Relations Semantic Relation: carries_out TUI: T141 Definition: Executes a function or performs a procedure or activity. This includes transacts, operates on, handles, and executes. Inverse: carried_out_by

119. UMLS-SN Semantic Relations Semantic Relation: causes TUI: T147 Definition: Brings about a condition or an effect. Implied here is that an agent, such as for example, a pharmacologic substance or an organism, has brought about the effect. This includes induces, effects, evokes, and etiology. Inverse: caused_by

120. UMLS-SN Semantic Relations Semantic Relation: consists_of TUI: T172 Definition: Is structurally made up of in whole or in part of some material or matter. This includes composed of, made of, and formed of. Inverse: constitutes

121. UMLS-SN Semantic Relations Semantic Relation: contains TUI: T134 Definition: Holds or is the receptacle for fluids or other substances. This includes is filled with, holds, and is occupied by. Inverse: contained_in

122. UMLS-SN Semantic Relations Semantic Relation: derivative_of TUI: T178 Definition: In chemistry, a substance structurally related to another or that can be made from the other substance. This is used only for structural relationships. This does not include functional relationships such as metabolite of, by product of, nor analog of. Inverse: has_derivative

123. UMLS-SN Semantic Relations Semantic Relation: developmental_form_of TUI: T179 Definition: An earlier stage in the individual maturation of. Inverse: has_developmental_form

124. UMLS-SN Semantic Relations Semantic Relation: evaluation_of TUI: T161 Definition: Judgment of the value or degree of some attribute or process. Inverse: has_evaluation

125. UMLS-SN Semantic Relations Semantic Relation: exhibits TUI: T145 Definition: Shows or demonstrates. Inverse: exhibited_by

126. UMLS-SN Semantic Relations Semantic Relation: functionally_related_to TUI: T139 Definition: Related by the carrying out of some function or activity. Inverse: functionally_related_to

127. UMLS-SN Semantic Relations Semantic Relation: indicates TUI: T156 Definition: Gives evidence for the presence at some time of an entity or process. Inverse: indicated_by

128. UMLS-SN Semantic Relations Semantic Relation: ingredient_of TUI: T202 Definition: Is a component of, as in a constituent of a preparation. Inverse: has_ingredient

129. UMLS-SN Semantic Relations Semantic Relation: issue_in TUI: T165 Definition: Is an issue in or a point of discussion, study, debate, or dispute. Inverse: has_issue

130. UMLS-SN Semantic Relations Semantic Relation: manifestation_of TUI: T150 Definition: That part of a phenomenon which is directly observable or concretely or visibly expressed, or which gives evidence to the underlying process. This includes expression of, display of, and exhibition of. Inverse: has_manifestation

131. UMLS-SN Semantic Relations Semantic Relation: property_of TUI: T159 Definition: Characteristic of, or quality of. Inverse: has_property

132. UMLS-SN Semantic Relations Semantic Relation: result_of TUI: T157 Definition: The condition, product, or state occurring as a consequence, effect, or conclusion of an activity or process. This includes product of, effect of, sequel of, outcome of, culmination of, and completion of. Inverse: has_result

133. UMLS-SN Semantic Relations Semantic Relation: surrounds TUI: T176 Definition: Establishes the boundaries for, or defines the limits of another physical structure. This includes limits, bounds, confines, encloses, and circumscribes. Inverse: surrounded_by

134. UMLS-SN Semantic Relations Semantic Relation: traverses TUI: T177 Definition: Crosses or extends across another physical structure or area. This includes crosses over and crosses through. Inverse: traversed_by

135. UMLS-SN Semantic Relations Semantic Relation: performs TUI: T188 Definition: Executes, accomplishes, or achieves an activity. Inverse: performed_by

136. UMLS-SN Semantic Relations Semantic Relation: physically_related_to TUI: T132 Definition: Related by virtue of some physical attribute or characteristic. Inverse: physically_related_to

137. UMLS-SN Semantic Relations Semantic Relation: conceptually_related_to Definition: Related by some abstract concept, thought, or idea. Inverse: conceptually_related_to

138. Prototypicality Biological classes are marked always by an opposition between standard or prototypical instances and a surrounding penumbra of non- standard instances How solve this problem: restrict range of instance variables x, y, to standard instances? Recognize degrees of instancehood? (Impose topology/theory of vagueness on classes?)