Partitions

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

Projection of cells

…

Wyoming

Idaho

Montana

…

Cell structure North AmericaProjection

• 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

Projective relation to reality

Projection of cells (1)

Cell structure Targets in reality

Hydrogen

Lithium

Projection

Projection of cells (2)

…

Wyoming

Idaho

Montana

…

Cell structure North AmericaProjection

Multiple ways of projecting

CountypartitionHighwaypartition

Big citypartition

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

Theory A

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

Theory B

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

Selectivity

Granularity

Recognizing a whole without recognizing all of its parts

Classification of granular partitions

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

Foreground/Background

The theory of partitions is a theory of foregrounding,

of setting into relief

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’

Mont Blanc from Lake Annecy

Mount Everest Elevation Map

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.

Fiat Vagueness

Vagueness in the Fiat Realm

Vagueness in the Fiat Realm

In what follows we are interested in partitions relating to bona fide objects

– to objects which were therebefore we came along

Mont Blanc from Chatel

Mont Blanc (Tricot)

many precisifications

Mont Blanc (Tricot)

many precisifications

Mont Blanc (Tricot)

many precisifications

Mont Blanc (Tricot)

many precisifications

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

(philosophers

do not live in normal contexts)

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.

18551855

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.

Lake Constance

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).

No Gaps

‘Bald’, ‘cat’, ‘mountain’, ‘island,’ ‘lake’, are all vague

But corresponding (normal) judgments nonetheless have determinate truth-values.

There are (on one way of precisifying normal) no gaps