1. 1 A Unified Theory of Vagueness and Granularity Barry Smith http://ontology.buffalo.edu

2. 2 A Unified Theory of Vagueness and Granularity and Reference and Truth and Everything and Intentionality

3. 3 A Simple Partition

4. 4

5. 5

6. 6

7. 7

8. 8 Topological Operations on Partitions Gluing Restricting Extending ….

9. 9

10. 10

11. 11

12. 12 A partition can be more or less refined

13. 13

14. 14

15. 15 Partition A partition is the drawing of a (typically complex) fiat boundary over a certain domain

16. 16 GrGr

17. 17 Partitions are artefacts of our cognition = of our referring, perceiving, classifying, mapping activity

18. 18 A partition is transparent It leaves the world exactly as it is

19. 19 Artist’s Grid

20. 20 Label/Address System A partition typically comes with labels and/or an address system

21. 21 Mouse Chromosome Five

22. 22 A partition can comprehend the whole of reality

23. 23 Universe

24. 24 It can do this in different ways

25. 25 The Spinoza Partition

26. 26 Periodic Table

27. 27 Perspectivalism Different partitions may represent cuts through the same reality which are skew to each other

28. 28 (You can cut the cheese in different ways)

29. 29 Universe/Periodic Table

30. 30 Partitions have different granularity Maps have different scales Partitions are, roughly, what AI people call ‘ontologies’ (but in which granularity is taken seriously)

31. 31 Partitions can have empty cells

32. 32 01234… Partition of people in this room according to: number of years spent in jail

33. 33 01234… Partition of people in this room according to: number of days spent in jail

34. 34 Partitions vs. sets Two partitions can be distinguished purely in virtue of their labeling (Partitions are cognitive artefacts)

35. 35 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

36. 36 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.’

37. 37 Eddington: Only the scientific table exists.

38. 38 The Parable of the Two Tables Both of the tables exist – in the same place: they are pictured in maps of different scales (Eddington: only very, very, very large- scale maps are true of reality …) (Reductionism: only very, very, very large-scale maps are true of reality …)

39. 39 The Parable of the Two Intentionalities

40. 40 The Parable of the Two Intentionalities Cognition No. 1 = ordinary cognition, a relation between a mental act and an object (reference, semantics belongs here) Cognition No. 2 = scientific cognition, a matter of reflected light rays entering the brain through the retina

41. 41 Cognition No. 2 Scientific cognition is mostly emptiness. Sparsely scattered in that emptiness are numerous rays of light rushing into the retina with great speed …

42. 42 How to make folk psychology consistent with neuroscience? by means of a good theory of granularity

43. 43 The Parable of Two Intentionalities Both of the two cognitions exist – in the same place: they are pictured in maps of different scale

44. 44 Partitions can sometimes create objects fiat objects = objects determined by partitions

45. 45 Tibble’s Tail fiat boundary

46. 46 Canada Quebec Canada

47. 47 Some partitions create (or reflect) whole systems of fiat objects

48. 48 The Counties of Kansas

49. 49 The Counties of Kansas

50. 50 The Counties of Kansas

51. 51 The Counties of Kansas

52. 52 Some partitions are completely arbitrary but true of reality nonetheless partitions are windows true = transparent to reality

53. 53 The DER-DIE-DAS partition DER (masculine) moon lake atom DIE (feminine) sea sun earth DAS (neuter) girl fire dangerous thing

54. 54 = objects which exist independently of our partitions (objects with bona fide boundaries) bona fide objects

55. 55 globe

56. 56... rookbishoppawnknight... JohnPaulGeorgeRingo... updowncharmstrange...

57. 57 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 spatial location L – object x is in measurement-band B contrast the meagre resources of set theory 

58. 58 These are different ways in which cells can be projected onto reality

59. 59 Grids of Reality (Mercator 1569)

60. 60 Maps do not lie

61. 61 The railway tracks on the Circle Line are not in fact yellow:

62. 62 Every projection system is correct the point is merely to use it properly intelligence of the projective technique vs. stupidity of the interpreter

63. 63 Realism transparency: the grid of a partition helps us to see the world aright

64. 64 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)

65. 65 The Empty Mask (Magritte) mama mouse milk Mount Washington

66. 66 Cerebral Cortex

67. 67 Artist’s Grid

68. 68 Intentional directedness … is effected via partitions we reach out to objects because partitions are transparent

69. 69 and they always have a certain granularity when I see an apple my partition does not recognize the molecules in the apple

70. 70 Alberti’s Grid

71. 71 Towards a Theory of Intentionality / Reference / Cognitive Directedness

72. 72 we have all been looking in the wrong direction

73. 73 Dürer Reverse

74. 74 This is a mistake propositions, sets, noemata, meanings, models, concepts, senses,... content does not belong in the target position

75. 75 Intentionality this is the correct view

76. 76 corrected content, meaning representations our normal intentional directedness is a directedness towards the real world

77. 77 Intentionality and is normally many-rayed

78. 78 An example of a pseudo-problem in the history of philosophy: How can we ever transcend the realm of meanings / contents / ideas / sensations / noemata and reach out to the realm of objects in themselves ?

79. 79 we reach out to objects because partitions are transparent Intentional directedness … is effected via partitions

80. 80 12 34 Counting requires partitions

81. 81 Frege: “Numbers belong to the realm of concepts” Reinach: Numbers belong to the realm of Sachverhalte Smith: Numbers belong to the realm of partitions

82. 82 Measurement belongs to the realm of partitions... -20  -10 -10  0 0  10 10  20... massively increased... normal increased chronic...

83. 83 Sets belong to the realm of partitions Sets are not objects in reality, but mathematical tools for talking about reality

84. 84 Another mistake:

85. 85 The correct view set-like structures belong here

86. 86 Defining  Sets are (at best) special cases of partitions

87. 87 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

88. 88 Cells are to partitions as singletons are to sets

89. 89 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

90. 90 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)

91. 91 Cantor’s Hell... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)

92. 92 Cantor’s Hell... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)

93. 93 Partitions better than sets Partitions are as we can see better than sets

94. 94 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.

95. 95 The ‚mystery‘ of set theory arises from supposing that sets are objects This is the root, also, of Frege’s problem in the Grundgesetze This is the root of the catastrophic high- rise projects of post-Cantorian set theory

96. 96 Demolition

97. 97 Cantor’s Hell arises because set theory confuses the fiat boundaries generated by our partitions with the bona fide boundaries possessed by objects themselves

98. 98 Vagueness

99. 99 The theory of partitions is a theory of foregrounding, of setting into relief

100. 100 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

101. 101 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

102. 102 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

103. 103 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

104. 104 Foreground/Background but there is a problem

105. 105 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

106. 106 Many but almost one David Lewis: There are always outlying particles, questionable parts of things, not definitely included and not definitely not included.

107. 107 Granularity Cognitive acts of Setting into Relief: the Source of Partitions

108. 108 Granularity Cognitive acts of Setting into Relief: the Source of Partitions Partititions: the Source of Granularity

109. 109 Granularity Cognitive acts of Setting into Relief: the Source of Partitions Partititions: the Source of Granularity Granularity: the Source of Vagueness

110. 110 Treatment of Vagueness The relation of location between an object and a cell can be generalized to include partial location, rough location, temporary location, supervaluational location …

111. 111 John

112. 112 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. traced over. …they are traced over.

113. 113 John

114. 114 Granularity the source of vagueness... your partition does not recognize parts beneath a certain size. Hence: your partition is compatible with a range of possible views as to the ultimate constituents of the objects included in its foreground domain

115. 115 Granularity the source of vagueness 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 It is the coarse-grainedness of our partitions The coarse-grainedness of our partitions

116. 116 Our attentions are in every case focused upon whatever it is which lies above the pertinent granularity threshold

117. 117 John

118. 118 Partitions do not care 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. Our ordinary judgments,

119. 119 The Problem of the Many is Solved: Because our judgments are marked by granularity they still have determinate truth-values even in spite of the fact that our reference is always vague because there are always questionable parts of things

120. 120 Reference is a cognitive phenomenon whereby we become related to reality via transparent granular partitions

121. 121 Optical Hooks

122. 122 An (Irregular) Partition

123. 123 A Portion of Reality

124. 124 Cartographic Hooks

125. 125 A Map

126. 126

127. 127 A Sentence Blanche is shaking hands with Mary

128. 128 A Portion of Reality

129. 129 Semantic Hooks Blanche is shaking hands with Mary

130. 130 A Sentence Blanche is shaking hands with Mary

131. 131 Die Projektion 3.12... der Satz ist das Satzzeichen in seiner projektiven Beziehung zur Welt. 3.13 Zum Satz gehört alles, was zur Projektion gehört; aber nicht das Projizierte.

132. 132 Satz und Sachverhalt arb language world names simple objects

133. 133 Satz und Sachverhalt arb language world cells (in coarse-grained partitions) simple and complex objects

134. 134 Satz und Sachverhalt arb language world projection

135. 135 Satz und Sachverhalt arb Semantic Projection „ John kisses Mary “ John this kiss Mary

136. 136 Truth is easy

137. 137 Falsehood: A Realist Theory Falsehood is not: successful conformity with some non-existing state of affairs... it is the failure of an attempted conformity, resting on either 1. failure of projection, or 2. failure of coordination

138. 138 Satz und Sachverhalt arb Projection Failure „John kisses Mary“ John Mary nothing here

139. 139 Nothing really nothing

140. 140 Satz und Sachverhalt arb Projection Failure „John kisses Mary“ John Mary

141. 141 Coordination Failure arb „John kisses Mary“ Mary this kiss John Coordination Failure

142. 142 Realist Semantics We begin with a theory of propositions as articulated pictures of reality The theory of truth comes along as a free lunch We then show how to deal with the two kinds of failure which constitute falsehood

143. 143 THE END