1 A Unified Theory of Vagueness and Granularity Barry Smith

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

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 Mouse Chromosome Five

15 A partition can comprehend the whole of reality

16 Universe

17 It can do this in different ways

18 The Spinoza Partition

19 Periodic Table

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

21 (You can cut the cheese in different ways)

22 Universe/Periodic Table

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

24 Partitions can have empty cells

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

26 Partition of people in this room according to: number of days spent in jail

27 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

28 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.’

29 Eddington: Only the scientific table exists.

30 The Parable of the Two Tables Both of the tables exist – in the same place: they are pictured in maps of different scale

31 The Parable of 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

32 The Parable of Two Intentionalities How to make folk psychology consistent with neuroscience?

33 The Parable of Two Intentionalities Scientific cognition is mostly emptiness. Sparsely scattered in that emptiness are numerous rays of light rushing into the retina with great speed … Both of the two cognitions exist – in the same place: they are pictured in maps of different scale

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

35 Tibble’s Tail fiat boundary

36 Canada Quebec Canada

37 Some partitions are completely arbitrary but true (transparent) nonetheless

38 Kansas

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

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

41 globe

42 California Land Cover Reciprocal partitions

43... rookbishoppawnknight... JohnPaulGeorgeRingo... updowncharmstrange...

44 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 

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

46 Grids of Reality (Mercator 1569)

47 Every projection system is correct the point is merely to use it properly intelligence of the projective technique vs. stupidity of the interpreter (maps do not lie)

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

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

50 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)

51 The Empty Mask (Magritte) mama mouse milk Mount Washington

52 Cerebral Cortex

53 Artist’s Grid

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

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

56 Alberti’s Grid

57 Towards a Theory of Intentionality / Reference / Cognitive Directedness

58 we have all been looking in the wrong direction

59 Dürer Reverse

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

61 Intentionality this is the correct view

62 corrected content, meaning representations

63 Intentionality

64 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 ?

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

Counting requires partitions

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

68 Measurement belongs to the realm of partitions   0 0   massively increased... normal increased chronic...

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

70 Another mistake:

71 The correct view set-like structures belong here

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

73 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

74 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

75 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)

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

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

78 Partitions better than sets Partitions are as we can see better than sets

79 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.

80 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

81 Demolition

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

83 Cantor’s Hell arises because set theory confuses the fiat boundaries generated by our partitions (e.g. by our setting certain phenomena into relief in terms of the ‘real numbers’) with the bona fide boundaries possessed by objects themselves

84 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

85 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

86 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

87 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

88 Foreground/Background but there is a problem

89 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

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

91 Granularity Cognitive acts of Setting into Relief: the Source of Partitions

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

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

94 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 …

95 John

96 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.

97 John

98 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

99 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.

100 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.

101 A Theory of Reference as a theory of intentionality mediated via transparent partitions

102 Optical Hooks

103 An (Irregular) Partition

104 A Portion of Reality

105 Cartographic Hooks

106 A Map

107 A Sentence Blanche is shaking hands with Mary

108 A Portion of Reality

109 Semantic Hooks Blanche is shaking hands with Mary

110 A Sentence Blanche is shaking hands with Mary

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

112 Satz und Sachverhalt arb language world names simple objects

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

114 Satz und Sachverhalt arb language world projection

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

116 Truth is easy

117 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

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

119 Nothing really nothing

120 Satz und Sachverhalt arb Projection Failure „John kisses Mary“ John Mary

121 Coordination Failure arb „John kisses Mary“ Mary this kiss John Coordination Failure

122 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

123 THE END