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Theory of granular partitions
• There is a projective relation between cognitive subjects and reality
Major assumptions:
• Humans ‘see’ reality through a grid
• The ‘grid’ is usually not regular and raster shaped
• no counties • no county boundaries
Part of the surface of the Earth photographed from space
Projection establishes fiat boundaries
Cell structure
Map =Representationof cell structure
County boundaries in reality
P
Crisp and vague projection
…Montana
…
crisp
Himalayas
EverestvagueP1
Pn
Vague reference is always reference to fiat boundaries!
Theory of granular partitions• Major assumptions
– Projection is an active process:• it brings certain features of reality into the foreground of our
attention (and leaves others in the background)
• it brings fiat objects into existence
– This projective relation can reflect the mereological structure of reality
Theory of granular partitions (4)
• Core components (master conditions)– Cell structures (Theory A)
– Projective relation to reality (Theory B)
• Subcell relation• Minimal, maximal cell• Trees, Venn-diagrams
• Projection and location• Projection is a partial, functional, (sometimes)
mereology-preserving relation
Systems of cells
• Subcell relation– Reflexive, transitive, antisymmtric
• The cell structure of a granular partition– Has a unique maximal cell (top-most node, root)
• ‘Idaho’ in the county partition of Idaho• The periodic table as a whole
– Each cell is connected to the root by a finite chain
– Every pair of cells is either in subcell or disjointness relation
Cell structures and trees
Cell structures can be represented as trees and vice versa
Animal
Bird Fish
Canary
Ostrich
Shark
Salmon
Projection and location
H u m a ns A p es U n ico rns
M a m m a ls
Humans Apes
Dogs
Mammals
),Humans''( HumansP
lysuccessfulproject
NOT does Unicorn'' cell The
???),'Unicorn(' P
recognized
NOT is species The
???)L(Dogs,
Dog
)Humans'',(HumansL
recognized
is species The Human
Misprojection
…
Idaho
Montana
Wyoming
…
P(‘Montana’,Idaho) but NOT L(Idaho,’Montana’)
P(‘Idaho’,Montana) but NOT L(Montana,’Idaho’)
P(‘Wyoming’,Wyoming) AND L(Wyoming,’Wyoming’)
• A granular partition projects transparently onto reality if and only if
Transparency of projection (1)
– Location presupposes projectionL(o,z) P(z,o)
– There is no misprojectionP(z,o) L(o,z)
Transparency of projection (2)
• Still: there may be irregularities of correspondence
– There may be cells that do not project (e.g. ‘unicorn’)
– Multiple cells may target the same object
– There may be ‘forgotten’ objects (e.g. the species dog above)
Functionality constraints (1)
Morning Star
Evening StarVenus
Location is functional: If an object is located in two cells then these cells are identical, i.e., L(o,z1) and L(o,z2) z1 = z2
Two cells projecting onto the same object
Functionality constraints (2)
China
Republic of China (Formosa)
People’s Republic of China
The same name for two different things:
Projection is functional: If two objects are targeted by the same cell then they are identical, i.e., P(z,o1) and P(z,o2) o1 = o2
Preserve mereological structure
Helium
Noble gases
Neon
EmptyNeonHelium
gasesNobleNeon
gasesNobleHelium
EmptyNeHe
NGNe
NGHe
Potential of preserving mereological structure
Partitions should not distort mereological structure
M am m als A p es U n ico rn s
H u m an s
Humans Apes
Dogs
Mammals
HumansMammal
Humans''Mammal''
distortion
If a cell is a proper subcell of another cell then the object targetedby the first is a proper part of the object targeted by the second.
Features of granular partitions
• Selectivity– Only a few features are in the foreground of
attention
• Granularity– Recognizing a whole without recognizing all of
its parts
• Preserve mereological structure
Theory of granular partitions (4)
Classes of granular partitions according to
• Degree of preservation of mereological structure
• Degree of completeness of correspondence
• Degree of redundancy
Mereological monotony
…
Helium
Noble gases
Neon
…
Helium
Noble gases
Neon
Projection does not distort mereological structure
21212,21,1 o and )( and )( zzozoLzoL Projection preserves mereological structure
Projective completeness
Empty cells
function totala is Projection
scompletnes Projective
),(:),( zoLoAzZ
Every cell has an objectlocated within it:
Exhaustiveness
Humans Apes
Dogs
Mammals
Everything of kind in the domain of the partition A is recognized by some cell in A
),( and ),(:
and )(
zoLAzZz
Φ(o)ADo
Humans Apes Cats
Mammals
Example partitions:
… lake* narrow ocean pond
pool
sea
…
body of water
loch
tarn reservoir lake* millpond
tank
* = term appears twice
Properties of cadastral partitions
• Cell structure: stored in database
• Projection carves out land-parcels (geodetic projection)
• Properties– Transparent: projection and
location are total functions– Exhaustive (no no-mans lands)
– Mereologically monotone
Folk categorization of water bodies
… lake* narrow ocean pond
pool
sea
…
body of water
loch
tarn reservoir lake* millpond
tank
* = term appears twice
Not a tree +double cell-labelsat different levelsof hierarchy
Distorts mereologicalStructureLocation is not a function
Science
= the endeavour to construct partitions of reality which satisfy the conditions of
mereological monotony (tree structure)
exhaustiveness (every object recognized)
functionality (one object per cell)
…but no God’s eye partition
– every partition we create has some granularity
Vagueness
Sentence: ‘There is no beer in the glass.’
Drunkard:Hygiene inspector:
Judgments = Sentence + Context
(super) trueThe glass does not contain (drinkable amounts of) beer
(super) falseThe glass contains tinyamounts of beer, microbes, mold, …
Partitions and contextJ = (‘There is no beer in the glass’, Partition)
Glass
Beer
Glass
Beer
probe
Cell ‘Beer’ does projectCell ‘Beer’ does not project
J is true in this context J is false in this context
Setting into Relief
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
But what is it, that you foreground?
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’
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.
Beverly Hills
Moreover:
The way you partition (carve up) the world when you think of Mont Blanc cannot be under-stood along any simple geographical lines.
It is not as if one connected, compact (hole-free) portion of reality is set into relief in relation to its surroundings,
as Beverly Hills is set into relief within the wider surrounding territory of Los Angeles County.
Granularity the source of vagueness
This is because 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
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 always focused on those matters which lie above whatever is the pertinent granularity threshold.
In what follows we are interested in partitions relating to bona fide objects
– to objects which were therebefore we came along
Mont Blanc is one mountain
Bill Clinton is one person
– these are both supertrue
Mont Blanc is one mountain
many precisifications
. . . yet:
Standard Supertruth
they are true no matter which of the many aggregates of matter you assign as precisified referent
Standard SupervaluationismA sentence is supertrue if and only if it is true under all such precisifications. A sentence is superfalse if and only if it is true under all such 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.
Philosophers of vagueness
like to make a big song and dance about sentences which fall down supertruth-value gaps
-- such sentences are what is characteristic of vagueness
Example of Gaps
EXAMPLE
On Standard Supervaluationism
Rabbits are part of Mont Blanc
falls down a supertruth-value gap
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 might be supertrue that these rabbits are part of that mountain
Supervaluationism Contextualized
Supervaluations depend on contexts
We pay attention in different ways and to different things in different contexts
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.
Some sentences are unjudgeable
• The umbrella in your cocktail is part of your meal• The neutrino passing through your gullet is part of
your body.• President Chirac’s hat is part of France• John is exactly bald.• The Morning Star is not a star• The Morning Star does not have magic powers
and neither does the Evening Star
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
Gaps and GlutsConsider:
Rabbits are part of Mont Blancin a normal context inhabited by you or me
Compare:Sakhalin Island is both Japanese and not Japanese
Just as sentences with truth-value gaps are unjudgeable, so also are sentences with truth-value gluts.
Contextualized Supervaluationism
A judgment F(a) is supertrue if and only if:
• (T1) it successfully imposes in its context C a partition of reality assigning to ‘a’ a range a1 ... an of precisified referents, and
• (T2) F(ai) is true for each such precisified referent.
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 of the right sort
Falsehood
or both:
• (F1) it does impose such a partition
• (F2) and the corresponding families of aggregates are such that F(ai) is false for each such precisified referent
In case (F0), p fails even to reach the starting gate for purposes of supervaluation
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 takes the view that all three countries have shared sovereignty over the whole Lake.
• Germany takes the view that Germany takes no view on the matter.
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?
That Water is in SwitzerlandNo. 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 Wojtya is more intelligent than the Pope.
Reaching the Starting GateIn 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.
John is bald• This slurry is part of Mont Blanc
• Geraldine died before midnight
• John is bald
It is part of what we mean when we say that John is, as far as baldness is concerned, a borderline case that ‘John is bald’ is unjudgeable.
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).