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A Unified Theory of Vagueness and Granularity
Barry Smithhttp://ontology.buffalo.edu
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A Simple Partition
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A partition can be more or less refined
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Partition
A partition is the drawing of a (typically complex) fiat boundary over a certain domain
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GrGr
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Partitions are artefacts of our cognition
= of our referring, perceiving, classifying, mapping activity
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A partition is transparent
It leaves the world exactly as it is
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Artist’s Grid
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Label/Address System
A partition typically comes with labels and/or an address system
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Mouse Chromosome Five
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A partition can comprehend the whole of reality
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Universe
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It can do this in different ways
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The Spinoza Partition
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Periodic Table
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Perspectivalism
PerspectivalismDifferent partitions may represent cuts through the same reality which are skew to each other
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(You can cut the cheese in different ways)
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Universe/Periodic Table
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Partitions have different granularity
Maps have different scales
Partitions are, roughly, what AI people call ‘ontologies’ (but in which granularity is taken seriously)
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Partitions can have empty cells
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0 1 2 3 4 …
Partition of people in this room according to: number of years spent in jail
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Partition of people in this room according to: number of days spent in jail
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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
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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.’
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Eddington:
Only the scientific table exists.
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The Parable of the Two Tables
Both of the tables exist – in the same place: they are pictured in maps of different scale
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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
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The Parable of Two Intentionalities
How to make folk psychology consistent with neuroscience?
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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
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Partitions can sometimes create objects
fiat objects = objects determined by partitions
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Tibble’s Tail
fiat boundary
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Canada
Que
bec
Canada
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Some partitions are completely arbitrary
but true (transparent) nonetheless
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Kansas
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The DER-DIE-DAS partition
DER
(masculine)
moon
lake
atom
DIE
(feminine)
sea
sun
earth
DAS
(neuter)
girl
firedangerous thing
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= objects which exist independently of our partitions
(objects with bona fide boundaries)
bona fide objects
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globe
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California Land Cover
Reciprocal partitions
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... rook bishop pawn knight ...
John Paul George Ringo
... up down charm strange ...
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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
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These are different ways in which cells can be projected onto reality
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Grids of Reality (Mercator 1569)
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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)
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The railway tracks on the Circle Line are not in fact yellow:
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Realismtransparency: the grid of a partition helps us to see the world aright
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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)
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The Empty Mask (Magritte)
mama
mouse
milk
Mount Washington
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Cerebral Cortex
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Artist’s Grid
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Intentional directedness
… is effected via partitions
we reach out to objects because partitions are transparent
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and they always have a certain granularity
when I see an apple my partition does not recognize the molecules in the apple
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Alberti’s Grid
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Towards a Theory of Intentionality / Reference / Cognitive Directedness
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we have all been looking in the wrong direction
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Dürer Reverse
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This is a mistake
propositions,sets, noemata,
meanings, models,
concepts, senses, ...
content does not belong in the target position
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Intentionality
this is the correct view
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corrected
content, meaningrepresentations
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Intentionality
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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 ?
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Intentional directedness
… is effected via partitions
we reach out to objects because partitions are transparent
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1 2 3 4
Counting requires partitions
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Frege: “Numbers belong to the realm of concepts”
Reinach: Numbers belong to the realm of Sachverhalte
Smith: Numbers belong to the realm of partitions
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Measurement belongs to the realm of partitions
... -20-10 -10 0 0 10 10 20 ...
massivelyincreased... normal increased chronic ...
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Sets belong to the realm of partitions
Sets are not objects in reality, but mathematical tools for talking about reality
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Another mistake:
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The correct view
set-like structures belong here
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Defining
Sets are (at best) special cases of partitions
Cells are to partitions as singletons are to sets
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Objects and cells
objects are located in cells as guests are located in hotel rooms:
LA(x, z)
the analogue of the relation between an element and its singleton
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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 withoutorder or location—they can be permuted at will and the set remains identical
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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)
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Cantor’s Hell
... the relation between an element and its singleton is
“enveloped in mystery”
(Lewis, Parts of Classes)
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Cantor’s Hell
... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)
... the relation between an element and its singleton is “enveloped in mystery” (Lewis, Parts of Classes)
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Partitions better than sets
Partitions are
as we can see
better than sets
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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.
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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
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Demolition
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The theory of partitions
is a theory of foregrounding,
of setting into relief
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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
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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
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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
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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
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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
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Foreground/Background
but there is a problem
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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
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Many but almost oneDavid Lewis:
There are always outlying particles, questionable parts of things, not definitely included and not definitely not included.
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Granularity
Cognitive acts of Setting into Relief: the Source of Partitions
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Granularity
Cognitive acts of Setting into Relief: the Source of Partitions
Partititions: the Source of Granularity
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Granularity
Cognitive acts of Setting into Relief: the Source of Partitions
Partititions: the Source of Granularity
Granularity: the Source of Vagueness
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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 …
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John
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Tracing OverGranularity: 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.
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John
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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
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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.
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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.
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A Theory of Reference
as a theory of intentionality mediated via transparent partitions
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Optical Hooks
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An (Irregular) Partition
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A Portion of Reality
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Cartographic Hooks
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A Map
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A Sentence
Blanche is shaking hands with Mary
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A Portion of Reality
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Semantic Hooks
Blanche is shaking hands with Mary
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A Sentence
Blanche is shaking hands with Mary
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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.
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Satz und Sachverhalt
a r b
language
world
names
simple objects
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Satz und Sachverhalt
a r b
language
world
cells(in coarse-grained partitions)
simpleand complex objects
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Satz und Sachverhalt
a r b
language
world
projection
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Satz und Sachverhalt
a r b
Semantic Projection
„ John kisses Mary “
John this kiss Mary
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Truth is easy
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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, or2. failure of coordination
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Satz und Sachverhalt
a r b
Projection Failure
„John kisses Mary“
John Mary
nothing here
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Nothing
really nothing
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Satz und Sachverhalt
a r b
Projection Failure
„John kisses Mary“
John Mary
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Coordination Failure
a r b„John kisses Mary“
Mary this kiss John
Coordination Failure
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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
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THE END
THE END
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