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Barry Smith

Department of Philosophy (Buffalo)

Institute for Formal Ontology and Medical Information Science (Leipzig)

ontology.buffalo.edu

ifomis.de

Granularity and Knowledge Representation

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A Simple Partition

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A partition can be more or less refined

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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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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Montana

Montana

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Cerebral Cortex

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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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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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Universe/Periodic Table

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Partitions can sometimes create objects

fiat objects = objects determined by partitions

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Kansas

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= objects which exist independently of our partitions

(objects with bona fide boundaries)

bona fide objects

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California Land Cover

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Artist’s Grid

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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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Tibble’s Tail

fiat boundary

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Partitions are artefacts of our cognition

= of our referring, perceiving, classifying, mapping activity

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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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Sets belong not to the realm of objects but to the realm of partitions

Sets are not objects in reality, but mathematical tools for talking about reality

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Idealism

the road to idealism

propositions,sets,

noemata, ...

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Goodman: Many worlds

Me: Many partitions

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we have all been looking in the wrong direction

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Dürer Reverse

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Intentionality

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Intentionality

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corrected

content, meaningrepresentations

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The mistaken view

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The correct view

set-like structures belong here

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Alberti’s Grid

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Not propositional attitudes

but object attitudes

the attitudes mediated by partitions

(thus relatively coarse-grained)

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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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an object x is recognized by a partition A:

x A := z (LA(x, z))

there is some cell in A in which x is located

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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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Partitions better than sets

Partitions are

better than sets

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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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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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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 location L

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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

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

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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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Mont Blanc from Lake Annecy

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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.

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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

• Partititions: the Source of Granularity

• Granularity: the Source of Vagueness

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Objects and cells

x A := z (LA(x, z)

there is some cell in A and x is located in that cell

Recall:object x is recognized by partition A

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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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(Recall Husserl’s theory of Abschattungen)

(Ship of Theseus: different partitions of the same unterliegende sachliche

Tatbestandsmaterial)

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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 vaguenessIt 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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Mont Blanc from Chatel

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Mont Blanc (Tricot)

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Bill Clinton is one person

– these are both supertrue

Mont Blanc is one mountain

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they are trueh

no matter which of the many aggregates of matter you assign as precisified referent

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Bill Clinton is one person

– both are true on the appropriate level of granularity(our normal, common-sense ontology is in perfect order as it stands)

Mont Blanc is one mountain

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Standard Supervaluationism

A sentence is supertrue if and only if it is true under all precisifications. A sentence is superfalse if and only if it is false under all 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.

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Are those rabbits part of Mont Blanc?

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Example of Gaps

On Standard Supervaluationism

Rabbits are part of Mont Blanc

falls down a supertruth-value gap

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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 is supertrue that these rabbits are part of that mountain

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So, even with

Rabbits are part of Mont Blanc,

there are no gaps.

Are there any naturally occurring contextswith gaps?

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Supervaluationism Contextualized

We pay attention in different ways and to different things in different contexts

So: 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.

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Supervaluationism Contextualized

The range of admissible precisifications depends on context

The evaluations of supervaluationism should be applied not to sentences taken in the abstract but to judgments taken in their concrete real-world contexts

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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

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Gaps and Gluts

Consider:Rabbits are part of Mont Blanc

is in a normal context unjudgeable

Compare:Sakhalin Island is both Japanese and

not Japanese

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Problem cases

An artist is commissioned to paint a picture of Jesus Christ and uses himself as a model. Consider the judgment:

‘This is an image of Jesus Christ’

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No gaps

Just as sentences with truth-value gaps are unjudgeable, so also are sentences with truth-value gluts. (Solution, here, to the liar paradox.

Pragmatic approach to problematic cases (e.g. liar paradox) ontologically clarified by contextualized supervaluationism

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Normal contexts

including normal institutional contexts have immune systems which protect them against problematic utterances

such utterances are not taken seriously as expressing judgments

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Judgments

exist only as occurring episodes within natural contexts

... thus they are partly determined by the immune systems which natural contexts standardly possess

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Judgments and Evolution

Most naturally occurring contexts possess immune systems because those which did not would have been eliminated in the struggle for survival.

But the semantics hereby implied has nothing to do with pragmatic eliminations of objective truth normally favored by proponents of evoluationary epistemology

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Contextualized Supervaluationism

A judgment p is supertrue if and only if:• (T1) it successfully imposes in its context

C a partition of reality assigning to its constituent singular terms corresponding families of precisified aggregates, and

• (T2) the corresponding families of aggregates are such that p is true however we select individuals from the many candidate precisifications.

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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 in which families of aggregates corresponding to its constituent singular referring terms are recognized,

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Falsehoodor both:

• (F1) the judgment successfully imposes in its context C a partition of reality assigning to its constituent singular terms corresponding families of precisified aggregates, and

• (F2) the corresponding families of aggregates are such that p is false however we select individuals from the many candidate precisifications.

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Pragmatic presupposition failure:

In case (F0), p fails to reach the starting gate for purposes of supervaluation

Consider: „Karol Wojtyła is more intelligent than the present Pope“

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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 and Germany take the view that all three countries have shared sovereignty over the whole Lake.

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Lake Constance

If you buy a ticket to cross the lake by ferry in a Swiss railway station your ticket will take you only as far as the Swiss border (= only as far as the middle of the lake)

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but for all normal contextsconcerning

fishing rights,

taxation,

shipping,

death at sea, etc., there are treaties regulating how decisions are to be made (with built in immune-systems guarding against problematic utterances)

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Lake Constance

an ontological black hole in the middle of Europe

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Lake Constance (D, CH, A)

SwitzerlandAustria

Germany

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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?

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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 Wojtyła is more intelligent than the present Pope.

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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.

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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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Again:

Enduring types of (social, legal, administrative, planning) contexts have immune systems to prevent the appearance of the sort of problematic vagueness that is marked by gaps and gluts

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No Gaps

‘Bald’, ‘cat’, ‘dead’, ‘mountain’ are all vague

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

There are (on one way of precisifying ‘normal’ in the above) no truth-value gaps

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philosophical contexts are not normal

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DOWN

WITH

PHILOSOPHY !

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THE END

THE END

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The Counties of England: An Irregular Partition

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CartographicHooks

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A Map

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Optical Hooks

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Semantic Hooks

Blanche is shaking hands with Mary

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Second Order Vagueness

Partitions, too, can have vague boundaries (this is part of what allows us to share partitions) (part of what allows us to have truthmakers in common for our separate judgments)

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What happens when we use several contexts at once?

This is after all a normal thing to happen

(need theory of amalgamation of partitions)