1 VT

2 Barry Smith Department of Philosophy (Buffalo) Institute for Formal Ontology and Medical Information Science (Leipzig) ontology.buffalo.edu ifomis.de The Theory of Granular Partitions

3 A Simple Partition

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6 A partition can be more or less refined

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9 Partition A partition is the drawing of a (typically complex) fiat boundary over a certain domain

10 GrGr

11 A partition is transparent It leaves the world exactly as it is

12 Artist’s Grid

13 Label/Address System A partition typically comes with labels and/or an address system

14 Montana

15 Cerebral Cortex

16 Mouse Chromosome Five

17 A partition can comprehend the whole of reality

18 Universe

19 It can do this in different ways

20 Periodic Table

21 Perspectivalism Different partitions may represent cuts through the same reality which are skew to each other

22 Universe/Periodic Table

23 Partitions can sometimes create objects fiat objects = objects determined by partitions

24 Kansas

25 = objects which exist independently of our partitions (objects with bona fide boundaries) bona fide objects

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27 California Land Cover

28 Artist’s Grid

29 a partition is transparent (veridical) = its fiat boundaries correspond at least to fiat boundaries on the side of the objects in its domain if we are lucky they correspond to bona fide boundaries (JOINTS OF REALITY)

30 Tibble’s Tail fiat boundary

31 Partitions are artefacts of our cognition = of our referring, perceiving, classifying, mapping activity

32 and they always have a certain granularity when I see an apple my partition does not recognize the molecules in the apple

33 Alberti’s Grid

34 Sets belong not to the realm of objects but to the realm of partitions Sets are not objects in reality, but mathematical tools for talking about reality

35 Idealism the road to idealism propositions, sets, noemata,...

36 Goodman: Many worlds Me: Many partitions

37 we have all been looking in the wrong direction

38 Dürer Reverse

39 Intentionality

40 Intentionality

41 corrected content, meaning representations

42 The mistaken view

43 The correct view set-like structures belong here

44 Alberti’s Grid

45 Not propositional attitudes but object attitudes the attitudes mediated by partitions (thus relatively coarse-grained)

46 Defining  Sets are (at best) special cases of partitions Cells are to partitions as singletons are to sets

47 Objects and cells objects are located in cells as guests are located in hotel rooms: L A (x, z) the analogue of the relation between an element and its singleton

48 an object x is recognized by a partition A: x  A :=  z (L A (x, z)) there is some cell in A in which x is located

49 Set as List Partition A set is a list partition (a set is, roughly, a partition minus labels and address system) The elements exist within the set without order or location —they can be permuted at will and the set remains identical

50 Partitions better than sets Partitions are better than sets

51 David Lewis on Sets Set theory rests on one central relation: the relation between element and singleton. Sets are mereological fusions of their singletons (Lewis, Parts of Classes, 1991)

52 Cantor’s Hell... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)

53 Cantor’s Hell... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)

54 Mystery Lewis:... since all classes are fusions of singletons, and nothing over and above the singletons they’re made of, our utter ignorance about the nature of the singletons amounts to utter ignorance about the nature of classes generally.

55 An object can be located in a cell within a partition in any number of ways: – object x exemplifies kind K – object x possesses property P – object x falls under concept C – object x is in location L

56 The theory of partitions is a theory of foregrounding, of setting into relief

57 You use the name ‘Mont Blanc’ to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

58 You use the name ‘Mont Blanc’ to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

59 You use the name ‘Mont Blanc’ to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

60 You use the name ‘Mont Blanc’ to refer to a certain mountain You see Mont Blanc from a distance In either case your attentions serve to foreground a certain portion of reality Setting into Relief

61 Foreground/Background

62 The Problem of the Many There is no single answer to the question as to what it is to which the term ‘Mont Blanc’ refers. Many parcels of reality are equally deserving of the name ‘Mont Blanc’ – Think of its foothills and glaciers, and the fragments of moistened rock gradually peeling away from its exterior; think of all the rabbits crawling over its surface

63 Mont Blanc from Lake Annecy

64 The world itself is not vague Rather, many of the terms we use to refer to objects in reality are such that, when we use these terms, we stand to the corresponding parcels of reality in a relation that is one-to-many rather than one-to-one. Something similar applies also when we perceive objects in reality.

65 Many but almost one David Lewis: There are always outlying particles, questionable parts of things, not definitely included and not definitely not included.

66 Granularity Cognitive acts of Setting into Relief: the Source of Partitions Partititions: the Source of Granularity Granularity: the Source of Vagueness

67 Objects and cells x  A :=  z (L A (x, z) there is some cell in A and x is located in that cell Recall: object x is recognized by partition A

68 John

69 Tracing Over Granularity: if x is recognized by a partition A, and y is part of x, it does not follow that y is recognized by A. When you think of John on the baseball field, then the cells in John’s arm and the fly next to his ear belong to the portion of the world that does not fall under the beam of your referential searchlight. They are traced over.

70 (Recall Husserl’s theory of Abschattungen) (Ship of Theseus: different partitions of the same unterliegende sachliche Tatbestandsmaterial)

71 John

72 Granularity the source of vagueness... your partition does not recognize parts beneath a certain size. This is why your partition is compatible with a range of possible views as to the ultimate constituents of the objects included in its foreground domain

73 Granularity the source of vagueness It is the coarse-grainedness of our partitions which allows us to ignore questions as to the lower-level constituents of the objects foregrounded by our uses of singular terms. This in its turn is what allows such objects to be specified vaguely Our attentions are focused on those matters which lie above whatever is the pertinent granularity threshold.

74 Mont Blanc from Chatel

75 Mont Blanc (Tricot)

76 Bill Clinton is one person – these are both supertrue Mont Blanc is one mountain

77 they are true h no matter which of the many aggregates of matter you assign as precisified referent

78 Bill Clinton is one person – both are true on the appropriate level of granularity (our normal, common-sense ontology is in perfect order as it stands) Mont Blanc is one mountain

79 Standard Supervaluationism A sentence is supertrue if and only if it is true under all precisifications. A sentence is superfalse if and only if it is false under all precisifications. A sentence which is true under some ways of precisifying and false under others is said to fall down a supervaluational truth-value gap. Its truth-value is indeterminate.

80 Are those rabbits part of Mont Blanc?

81 Example of Gaps On Standard Supervaluationism Rabbits are part of Mont Blanc falls down a supertruth-value gap

82 Different Contexts In a perceptual context it is supertrue that these rabbits are part of Mont Blanc In a normal context of explicit assertion it is superfalse that these rabbits are part of Mont Blanc In a real estate context in a hunting community it is supertrue that these rabbits are part of that mountain

83 So, even with Rabbits are part of Mont Blanc, there are no gaps. Are there any naturally occurring contexts with gaps?

84 Supervaluationism Contextualized We pay attention in different ways and to different things in different contexts So: the range of available precisified referents and the degree and the type of vagueness by which referring terms are affected will be dependent on context.

85 Supervaluationism Contextualized The range of admissible precisifications depends on context The evaluations of supervaluationism should be applied not to sentences taken in the abstract but to judgments taken in their concrete real-world contexts

86 No gaps The everyday judgments made in everyday contexts do not fall down supervaluational truth- value gaps because the sentences which might serve as vehicles for such judgments are in normal contexts not judgeable

87 Gaps and Gluts Consider: Rabbits are part of Mont Blanc is in a normal context unjudgeable Compare: Sakhalin Island is both Japanese and not Japanese

88 Problem cases An artist is commissioned to paint a picture of Jesus Christ and uses himself as a model. Consider the judgment: ‘This is an image of Jesus Christ’

89 No gaps Just as sentences with truth-value gaps are unjudgeable, so also are sentences with truth-value gluts. (Solution, here, to the liar paradox. Pragmatic approach to problematic cases (e.g. liar paradox) ontologically clarified by contextualized supervaluationism

90 Normal contexts including normal institutional contexts have immune systems which protect them against problematic utterances such utterances are not taken seriously as expressing judgments

91 Judgments exist only as occurring episodes within natural contexts... thus they are partly determined by the immune systems which natural contexts standardly possess

92 Judgments and Evolution Most naturally occurring contexts possess immune systems because those which did not would have been eliminated in the struggle for survival. But the semantics hereby implied has nothing to do with pragmatic eliminations of objective truth normally favored by proponents of evoluationary epistemology

93 Contextualized Supervaluationism A judgment p is supertrue if and only if: (T1) it successfully imposes in its context C a partition of reality assigning to its constituent singular terms corresponding families of precisified aggregates, and (T2) the corresponding families of aggregates are such that p is true however we select individuals from the many candidate precisifications.

94 Supertruth and superfalsehood are not symmetrical: A judgment p is superfalse if and only if either: (F0) it fails to impose in its context C a partition of reality in which families of aggregates corresponding to its constituent singular referring terms are recognized,

95 Falsehood or both: (F1) the judgment successfully imposes in its context C a partition of reality assigning to its constituent singular terms corresponding families of precisified aggregates, and (F2) the corresponding families of aggregates are such that p is false however we select individuals from the many candidate precisifications.

96 Pragmatic presupposition failure: In case (F0), p fails to reach the starting gate for purposes of supervaluation Consider: „Karol Wojtyła is more intelligent than the present Pope“

97 Lake Constance No international treaty establishes where the borders of Switzerland, Germany, and Austria in or around Lake Constance lie. Switzerland takes the view that the border runs through the middle of the Lake. Austria and Germany take the view that all three countries have shared sovereignty over the whole Lake.

98 Lake Constance If you buy a ticket to cross the lake by ferry in a Swiss railway station your ticket will take you only as far as the Swiss border (= only as far as the middle of the lake)

99 but for all normal contexts concerning fishing rights, taxation, shipping, death at sea, etc., there are treaties regulating how decisions are to be made (with built in immune-systems guarding against problematic utterances)

100 Lake Constance an ontological black hole in the middle of Europe

101 Lake Constance (D, CH, A) Switzerland Austria Germany

102 That Water is in Switzerland You point to a certain kilometer-wide volume of water in the center of the Lake, and you assert: [Q] That water is in Switzerland. Does [Q] assert a truth on some precisifications and a falsehood on others?

103 No By criterion (F0) above, [Q] is simply (super)false. Whoever uses [Q] to make a judgment in the context of currently operative international law is making the same sort of radical mistake as is someone who judges that Karol Wojtyła is more intelligent than the present Pope.

104 Reaching the Starting Gate In both cases reality is not such as to sustain a partition of the needed sort. The relevant judgment does not even reach the starting gate as concerns our ability to evaluate its truth and falsehood via assignments of specific portions of reality to its constituent singular terms.

105 Partitions do not care Our ordinary judgments, including our ordinary scientific judgments, have determinate truth-values because the partitions they impose upon reality do not care about the small (molecule-sized) differences between different precisified referents.

106 Again: Enduring types of (social, legal, administrative, planning) contexts have immune systems to prevent the appearance of the sort of problematic vagueness that is marked by gaps and gluts

107 No Gaps ‘Bald’, ‘cat’, ‘dead’, ‘mountain’ are all vague But corresponding (normal) judgments nonetheless have determinate truth- values. There are (on one way of precisifying ‘normal’ in the above) no truth-value gaps

108 p hilosophical contexts are not normal

109 DOWN WITH PHILOSOPHY !

110 An ontology is a canonical representation of the types of entities in a given domain and of the types of relations between these entities: holy grail of a single benchmark ontology, which would make all databases intertranslatable an ontological Esperanto

111 Ontological Zooming

112 Universe/Periodic Table animal bird canary ostrich fish folk biology partition of DNA space

113 Universe/Periodic Table animal bird canary ostrich fish both are transparent partitions of one and the same reality

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115 Ontology like cartography must work with maps at different scales and with maps picking out different dimensions of invariants

116 If ontological realism is right then there are very many map-like partitions, at different scales, which are all transparent to the reality beyond the mistake arises when one supposes that only one of these partitions is veridical

117 There are not only map-like partitions of reality into material (spatial) chunks but also distinct partitions of reality into universals (genera, categories, kinds, types) mutually compatible ways of providing inventories of universals (among proteins, among cells, among organisms …) and distinct ways of partitioning the temporal dimension of processes

118 Varieties of granular partitions Partonomies: inventories of the parts of individual entities Maps: partonomies of space Taxonomies: inventories of the universals covering a given domain of reality

119 One example of ‘folk’ partition WordNet[1][1] developed at the University of Princeton defines concepts as clusters of terms called synsets. Wordnet consists of some 100,000 synsets organized hierarchically via: A concept represented by the synset {x, x, …} is said to be a hyponym of the concept represented by the synset {y, y,…} if native speakers of English accept sentences constructed from such frames as « An x is a kind of y ».

120 A Formal Theory of Granular Partitions Thomas Bittner and Barry Smith

121 The Parable of the Two Tables from Arthur Eddington, The Nature of the Physical World (1928) Table No. 1 = the ordinary solid table made of wood Table No. 2 = the scientific table

122 The Parable of the Two Tables ‘My scientific table is mostly emptiness. Sparsely scattered in that emptiness are numerous electric charges rushing about with great speed; but their combined bulk amounts to less than a billionth of the bulk of the table itself.’

123 Eddington: Only the scientific table exists.

124 The Parable of the Two Tables Both of the tables exist – in the same place: in fact they are the same table but pictured in maps of different scales the job of the theory of granular partitions is to do justice to this identity in (granular) difference

125 Towards a Theory of Intentionality / Reference / Cognitive Directedness GRANULAR PARTITIONS: THE SECOND DIMENSION

126 Intentional directedness … is effected via partitions we reach out to objects because partitions are transparent

127 Applications Theory of selectivity of cognition (including natural language cognition) Theory of granularity (medical data, genetic data) Theory of transformations between partitions of the same reality (SNOMED, UMLS …)

128 THE END