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The Theory of Granular Partitions:

A New Paradigm for Ontology

Barry Smith

Department of Philosophy

University at Buffalo

http://ontology.buffalo.edu

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The Theory of Granular Partitions:

A New Paradigm for Ontology

Barry Smith

Institute for Formal Ontology and Medical Information Science

University of Leipzig

http://ifomis.de

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Gesellschaft für Klassifikation

Classification

Classifying Studying

= Producing ClassificationsClassifications

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An ontology

• is a canonical representation of the types of entities in a given domain and of the types of relations between these entities:

• holy grail of a single benchmark ontology, which would make all databases intertranslatable

• an ontological Esperanto

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A Simple Partition

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A partition can be more or less refined

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Coarse-grained Partition

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Fine-Grained Partition

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Ontologies

Partitions are, roughly, what AI and database people call ontologies

but in which granularity is taken seriously

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• An organism is a totality of molecules

• An organism is a totality of cells

• An organism is a single unitary substance

• ... all of these express distinct granular partitions

An organism is a totality of atoms

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Ontological Zooming

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Universe/Periodic Table

animal

bird

canary

ostrich

fishfolk biology

partition of DNA space

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Universe/Periodic Table

animal

bird

canary

ostrich

fish

both are transparent partitions of one and the same reality

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Perspectivalism

Perspectivalism

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

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all express partitions which are transparent,

• at different levels of granularity,

• to the same reality beyond

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Ontology

• like cartography

• must work with maps at different scales and with maps picking out different dimensions of invariants

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If ontological realism is right• then there are very many map-like

• partitions, at different scales,

• which are all transparent to the reality beyond

• the mistake arises when one supposes

• that only one of these partitions is veridical

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There are not only map-like partitions of reality into material (spatial) chunks

• but also distinct partitions of reality into universals (genera, categories, kinds, types)

• mutually compatible ways of providing inventories of universals

• (among proteins, among cells, among organisms …)

• and distinct ways of partitioning the temporal dimension of processes

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Varieties of granular partitions

• Partonomies: inventories of the parts of individual entities

• Maps: partonomies of space

• Taxonomies: inventories of the universals covering a given domain of reality

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One example of ‘folk’ partition

• WordNet[1] • developed at the University of Princeton • defines concepts as clusters of terms called synsets. • Wordnet consists of some 100,000 synsets organized

hierarchically via:• A concept represented by the synset {x, x, …} is

said to be a hyponym of the concept represented by the synset {y, y,…} if native speakers of English accept sentences constructed from such frames as « An x is a kind of y ».

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A Formal Theory of Granular Partitions

Thomas Bittner and Barry Smith

http://ontology.buffalo.edu/smith/articles/partitions.pdf

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Partition

Definition:

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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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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Periodic Table

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Partitions have different granularity

Maps have different scales

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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: in fact they are the same table but pictured in maps of different scales

the job of the theory of granular partitions is to do justice to this identity in (granular) difference

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Some partitions are completely arbitrary

but 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)

planets, tennis balls

bona fide objects

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globe

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There are also Mixed Partitions

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Cerebral Cortex

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and also Reciprocal Partitions

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California Land Cover

Reciprocal partitions

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A Formal Theory of Granular Partitions

Thomas Bittner and Barry Smith

http://ontology.buffalo.edu/smith/articles/partitions.pdf

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Towards a Theory of Intentionality / Reference / Cognitive Directedness

GRANULAR PARTITIONS: THE SECOND DIMENSION

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Intentional directedness

… is effected via partitions

we reach out to objects because partitions are transparent

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Intentionality

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Intentional directedness always has a certain granularity

when I see an apple my partition does not recognize the molecules in the apple

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corrected

content, meaningrepresentations

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Intentional directedness

… is effected via partitions

we reach out to objects because partitions are transparent

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A Theory of Maps

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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 Theory of Websites

Barry SmithDepartment of PhilosophyUniversity at Buffalo

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A theory of language

• of assertive utterances (der Satz)

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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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The theory of partitions

is a theory of foregrounding,

of setting into relief

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You use the name ‘Mont Blanc’ to refer to a certain mountain

your utterance serves 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

Many parcels of reality are equally deserving of the name ‘Mont Blanc’

– Think of its foothills and glaciers

think of all the rabbits crawling over its surface

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Many but almost one

There are always outlying particles, questionable parts of things, not definitely included and not definitely not included.

Implies when referring to tokens and also when referring to types (outliers, Bordeaux 1997)

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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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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

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

in spite of being vague

have determinate truth-values

because the partitions they impose upon reality do not care about small (molecule-sized) differences

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Theory of vagueness

• How can -based • classifications be transparent,• if the world is shaped like this: •

•

?

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Species Genera as Tree

canary

animal

bird fish

ostrich

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Species-Genera as Map/Partition

animal

bird

canary

ostrich

fish

canary

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Tree and Map/Partition

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Ontological Zooming

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Universe/Periodic Table

animal

bird

canary

ostrich

fishfolk biology

partition of DNA space

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Granular Partitions, Vagueness and Approximation

Thomas Bittner and Barry Smith

http://ontology.buffalo.edu/smith/articles/svug.pdf

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THE END

THE END