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 America Projection

no counties no county boundaries Part of the surface of the Earth photographed from space Projection establishes fiat boundaries Cell structure Map = Representation of cell structure County boundaries in reality P

Crisp and vague projection … Montana … crisp Himalayas Everest vague P1P1 PnPn 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 structureTargets in reality Hydrogen Lithium Projection

Projection of cells (2) … Wyoming Idaho Montana … Cell structure North America Projection

Multiple ways of projecting County partition Highway partition Big city partition

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 Humans Apes Dogs Mammals

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 projection L(o,z)  P(z,o) –There is no misprojection P(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 Star Venus Location is functional: If an object is located in two cells then these cells are identical, i.e., L(o,z 1 ) and L(o,z 2 )  z 1 = z 2 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,o 1 ) and P(z,o 2 )  o 1 = o 2

Preserve mereological structure Helium Noble gases Neon Potential of preserving mereological structure

Partitions should not distort mereological structure Humans Apes Dogs Mammals distortion If a cell is a proper subcell of another cell then the object targeted by 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 structureProjection preserves mereological structure

Projective completeness Empty cells Every cell has an object located within it:

Exhaustiveness Humans Apes Dogs Mammals Everything of kind  in the domain of the partition A is recognized by some cell in A HumansApes Cats Mammals

Example partitions:

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 Not a tree + double cell-labels at different levels of hierarchy Distorts mereological Structure Location 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) true The glass does not contain (drinkable amounts of) beer (super) false The glass contains tiny amounts of beer, microbes, mold, …

Partitions and context J = (‘There is no beer in the glass’, Partition) Glass Beer Glass Beer probe Cell ‘Beer’ does project Cell ‘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

In what follows we are interested in partitions relating to bona fide objects – to objects which were there before 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 Supervaluationism A 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 Gluts Consider: Rabbits are part of Mont Blanc in 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.

1855

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 a 1... a n of precisified referents, and (T2) F(a i ) 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(a i ) 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 Switzerland No. 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 Wojty » a is more intelligent than the Pope.

Reaching the Starting Gate In 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