1 Barry Smith Department of Philosophy (Buffalo) Institute for Formal Ontology and Medical Information Science (Leipzig) ontology.buffalo.edu ifomis.de Granularity and Knowledge Representation

2 A Simple Partition

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5 A partition can be more or less refined

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

9 GrGr

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

11 Artist’s Grid

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

13 Montana

14 Cerebral Cortex

15 Mouse Chromosome Five

16 A partition can comprehend the whole of reality

17 Universe

18 It can do this in different ways

19 Periodic Table

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

21 Universe/Periodic Table

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

23 Kansas

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

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26 California Land Cover

27 Artist’s Grid

28 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)

29 Tibble’s Tail fiat boundary

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

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

32 Alberti’s Grid

33 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

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

35 Goodman: Many worlds Me: Many partitions

36 we have all been looking in the wrong direction

37 Dürer Reverse

38 Intentionality

39 Intentionality

40 corrected content, meaning representations

41 The mistaken view

42 The correct view set-like structures belong here

43 Alberti’s Grid

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

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

46 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

47 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

48 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

49 Partitions better than sets Partitions are better than sets

50 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)

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

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

53 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.

54 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

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

56 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

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 Foreground/Background

61 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

62 Mont Blanc from Lake Annecy

63 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.

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

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

66 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

67 John

68 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.

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

70 John

71 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

72 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.

73 Mont Blanc from Chatel

74 Mont Blanc (Tricot)

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

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

77 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

78 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.

79 Are those rabbits part of Mont Blanc?

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

81 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

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

83 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.

84 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

85 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

86 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

87 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’

88 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

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

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

91 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

92 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.

93 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,

94 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.

95 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“

96 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.

97 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)

98 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)

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

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

101 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?

102 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.

103 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.

104 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.

105 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

106 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

107 p hilosophical contexts are not normal

108 DOWN WITH PHILOSOPHY !

109 THE END

110 The Counties of England: An Irregular Partition

111 CartographicHooks

112 A Map

113 Optical Hooks

114 Semantic Hooks Blanche is shaking hands with Mary

115 Second Order Vagueness Partitions, too, can have vague boundaries (this is part of what allows us to share partitions) (part of what allows us to have truthmakers in common for our separate judgments)

116 What happens when we use several contexts at once? This is after all a normal thing to happen (need theory of amalgamation of partitions)