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Formal Ontology

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Schedule

Sep. 4: Introduction: Mereology , Dependence and Geospatial Ontology

Reading:

Basic Tools of Formal Ontology Ontological Tools for Geographic Representation

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ScheduleSep. 5: (Thursday) 4pm Metaphysics talk by David

Hershenov (Jointly with Philosophy Department Colloquium)

Sep. 11: Talk by Peter Forrest on Mereology and Time. (Jointly with Philosophy Department Colloquium)

Sep. 18: Truthmaking and the Semantics of Maps

Sep. 25: Vagueness

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ScheduleOct. 2: GranularityReading: A Theory of Granular Partitions

[Oct. 9 University Convocation: No meeting]

Oct. 16: Talk by Chuck Dement on: " The Ontology of Formal Ontology"

[Oct. 23 No meeting][Oct. 30 No meeting]

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Schedule

Nov. 6: 2pm “SNAP and SPAN”: Cognitive Science Colloquium Talk, 280 Park

Nov 6: 4pm Discussion of "SNAP and SPAN“

Nov. 8 (Friday): 4pm Talk by Berit Brogaard

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Schedule

Nov. 9 Day-long Saturday Workshop9am Achille Varzi: " From Ontology to Metaphysics"

10.45 am Berit Brogaard

12.30 Pizza Lunch

1pm Achille Varzi: "Ontology and Logical Form"

3-5pm Barry Smith

Nov. 13 Final Lecture

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IFOMIS

Institute for Formal Ontology and Medical Information Science

Some background

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The Manchester School

Kevin Mulligan

Peter Simons

Barry Smith

in Manchester 1973-76

working on the ontology of Edmund Husserl

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Edmund Husserl

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Logical Investigations¸1900/01

– the theory of part and whole– the theory of dependence– the theory of boundary, continuity and contact

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Formal Ontology

(term coined by Husserl)

the theory of those ontological structures

(such as part-whole, universal-particular)

which apply to all domains whatsoever

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Formal Ontology vs. Formal Logic

Formal ontology deals with the interconnections of things

with objects and properties, parts and wholes, relations and collectives

Formal logic deals with the interconnections of truths

with consistency and validity, or and not

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Formal Ontology vs. Formal Logic

Formal ontology deals with formal ontological structures

Formal logic deals with formal logical structures

‘formal’ = obtain in all material spheres of reality

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Formal Ontology and Symbolic Logic

Great advances of Frege, Russell, Wittgenstein

Leibnizian idea of a universal characteristic

…symbols are a good thing

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Warning

don’t confuse Logical with Ontological Form

Russell

Part-whole is not a logical relation

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for Frege, Russell, Lesniewski,

Wittgenstein, Quine

Logic is a ‘Zoology of Facts’

Formal theories are theories of reality

with one intended interpretation

= the world

tragicallyafter starting off on the right road

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Logic took a wrong turn

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Logic took a wrong turn

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Tarski, Carnap, Putnam, Sowa, Gruber:

Forget reality!

Lose yourself in ‘models’!

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IFOMIS Ontology

is an ontology of reality

Standard Information Systems Ontologies

are ontologies of mere 'models'

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Standard Information Systems Ontologies:

programming real ontology into computers is hard

therefore: we will simplify ontology

and not care about reality at all

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Painting the Emperor´s Palace is hard

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therefore

we will not try to paint the Palace at all

... we will be satisfied instead with a grainy snapshot of some other building

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IFOMIS Strategy

get real ontology right first

and then investigate ways in which this real ontology can be translated into computer-

useable form later

NOT ALLOW ISSUES OF COMPUTER-TRACTABILITY TO DETERMINE THE

CONTENT OF ONTOLOGY

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a language to map these

Formal ontological structures in reality

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a directly depicting language

‘John’ ‘( ) is red’

Object Property

Frege

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Wittgenstein’s Tractatus

Propositions

States of affairs

are pictures of

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Parts and Moments

in a directly depicting language

all well-formed parts of a true formula are also true

(The Oil-Painting Principle)

A new sort of mereological inference rule – the key to the idea of a directly depicting language

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A directly depicting language

may contain an analogue of conjunction

p and q_______

p

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but it can contain no negation

p_______

p

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and also no disjunction

p or q______

p

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The idea of a directly depicting language

suggests a new method

of constituent ontology:

to study a domain ontologically

is to establish the parts, qualities and processes of the domain

and the interrelations between them

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BFO and GOL

Basic Formal Ontology (BFO)

BFO as an ontological theory of reality designed as a real constraint on domain ontologies

(as opposed to conceptual modeling ...)

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A Network of Domain Ontologies

Material (Regional) Ontologies

Basic Formal Ontology

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Ontology

seeks an INVENTORY OF REALITY

Relevance of ontology for information systems, e.g.:

terminology standardization

taxonomy standardization

supports reasoning about reality

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Basic Formal Ontology

= a formal ontological theory, expressed in a directly depicting language, of all non-intentional parts of reality

(an ontology of the whole of reality but leaving aside minds and meanings)

BFO

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A Network of Domain Ontologies

BFO

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A Network of Domain Ontologies

GeO

BFO

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A Network of Domain Ontologies

MedO

GeO

BFO

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A Network of Domain Ontologies

CellO

MedO

GeO

BFO

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A Network of Domain Ontologies

GenO CellO

MedO

GeO

BFO

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Extended formal ontology(BFO Extended by Mind)

GenO CellO

MedO

GeO

BFOBFOBFO+Mind

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BFO Extended by Mind

GenO CellO

MedO

GeO

BFOBFOBFO+MindEcO

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BFO Extended by Mind

B(Gen)O B(Cell)O

B(Med)O

B(Chem)O

BFOBFOBFO+MindEcO

LexO

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Reality

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Reality

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Reality

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Reality

is complicated

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What is the best language to describe this complexity?

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Anglocentric Realism

We have a huge amount of knowledge of reality,

at many different levels of granularity,

from microphysics to cosmology

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Anglocentric Realism

TEE = Technically Extended English

= English extended by the technical vocabularies of

meteorology, chemistry, genetics, medicine, astronomy, engineering, etc.

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Anglocentric Realism

Our knowledge of reality as expressed in Technically Extended English

is increasing by the hour

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Unfortunately

… there are problems with TEE as a formal representation language

(cf. Tarski)

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Nouns and verbs

Substances and processes

Continuants and occurrents

In preparing an inventory of reality

we keep track of these two different categories of entities in two different ways

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Natural language

glues them together indiscriminately

substance

t i m

e

process

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Snapshot vs. Video

substance

t i m

e

process

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Substances

Mesoscopic reality is

divided at its natural joints

into substances:

animals, bones, rocks, potatoes

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The Ontology of Substances

Substances form natural kinds

(universals, species + genera)

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Processes

Processes merge into one another

Process kinds merge into one another

… few clean joints either between instances or between types

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Processes

t i m e

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Substances and processes

t i m

e

process

demand different sorts of inventories

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Substances demand 3-D partonomies

space

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Processes demand 4D-partonomies

t i m e

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Processes

a whistling, a blushing, a speech

a run, the warming of this stone

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Processes may have temporal parts

The first 5 minutes of my headache is a temporal part of my headache

The first game of the match is a temporal part of the whole match

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Substances do not have temporal parts

The first 5-minute phase of my existence is not a temporal part of me

It is a temporal part of that complex process which is my life

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Substances have spatial parts

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How do we glue these two different sorts of entities together mereologically?

How do we include them both in a single inventory of reality?

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How do we fit these two entities together within a single system of representations?

within a directly depicting language?

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You are a substance

Your life is a process

You are 3-dimensional

Your life is 4-dimensional

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Substances and processes form two distinct orders of being

Substances exist as a whole at every point in time at which they exist at all

Processes unfold through time, and are never present in full at any given instant during which they exist.

When do both exist to be inventoried together?

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Main problem

English swings back and forth between two distinct depictions of reality

… imposing both 3-D partitions (yielding substances) and 4-D partitions (yielding processes) at the same time

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Main problem

There is a polymorphous ontological promiscuity of the English sentence,

which is inherited also by the form ‘F(a)’

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Two alternative basic ontologies

SNAP and SPAN

SNAP = substances plus qualities

SPAN = processes

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These represent two views

of the same rich and messy reality, the reality captured promiscuously by TEE

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The Four-Dimensionalist Ontology

t i m e

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boundaries are mostly fiat

t i m e

everything is flux

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mereology works without restriction everywhere here

t i m e

clinical trial

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The Time-Stamped Ontology

t1

t3

t2

here time exists outside the ontology, as an index or time-stamp

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mereology works without restriction in every instantaneous 3-D section through

reality

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Three views/partitions of the same reality

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all contain huge amounts of knowledge of this reality

against Kant

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Ontological Zooming

The dimension of granularity

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Part 2

Tools of Ontology:

Mereology, Topology, Dependence

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Ontological Dependence

processes

+ qualitiessubstances

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Ontological Dependence

How to link together the domain of substances and the domain of processes?

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Ontological Dependence

Substances are that which can exist on their own

Processes require a support from substances in order to exist

This holds for qualities, too

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Specific Dependence

O := overlap

x := x is necessarily such that

E! := existence

SD(x, y) := O(x, y) x(E!x E!y)

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Mutual specific dependence

Each token of visual extension is mutually dependent on a token color quality

The north pole of a magnet is mutually dependent on the south pole

MSD(x, y) := SD(x, y) SD(y, x)

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One-Sided Specific Dependence

OSD(x, y) := SD(x, y) MSD(x, y)

My headache is one-sidedly specifically dependent on me.

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Substances, Qualities, Processes

Substances are the bearers or carriers of qualities and processes,

… the latter are said to ‘inhere’ in their substances

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Ontological Dependence

Substances are such that, while remaining numerically one and the same, they can admit contrary qualities at different times

… I am sometimes hungry, sometimes not

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Substances

can also gain and lose parts

… as an organism may gain and lose molecules

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Types of relations between parts

1. Dependence relations

2. Side-by-sideness relations

3. Fusion relations

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Dependence

cannot exist without a thinker

a thinking process

substance

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Theory of vaguenessSide-by-sideness

found among substances and among qualities and processes

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Fusion

Topology

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Topology, like mereology,

applies both in the realm of substances and in the realms of qualities and processes

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Mereotopology

= topology on a mereological basis

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Substances, Undetached Parts and Heaps

Substances are unities.

They enjoy a natural completeness

in contrast to their undetached parts (arms, legs)

and to heaps or aggregates

… these are topological distinctions

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substance

undetached part

collective of substances

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special sorts of undetached parts

ulcers

tumors

lesions

…

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Fiat boundaries

physical (bona fide) boundary

fiat boundary

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Examples

of bona fide boundaries:

an animal’s skin, the surface of the planet

of fiat boundaries:

the boundaries of postal districts and census tracts

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Mountain

bona fide upper boundaries with a fiat base:

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Architects Plan for a House

fiat upper boundaries with a bona fide base:

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where does the mountain start ?

... a mountain is not a substance

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nose

...and it’s not a quality, either

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A substance has a complete physical boundary

The latter is a special sort of part of a substance

… a boundary part

something like a maximally thin extremal slice

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interiorsubstance

boundary

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A substance takes up space.

A substance occupies a place or topoid (which enjoys an analogous completeness or rounded-offness)

A substance enjoys a place at a time

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A substance has spatial parts

… perhaps also holes

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Each substance is such as to have divisible bulk:

it can in principle be divided into separate spatially extended substances

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By virtue of their divisible bulk

substances compete for space:

(unlike shadows and holes)

no two substances can occupy the same spatial region at the same time.

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Substances vs. Collectives

Collectives = unified aggregates: families, jazz bands, empires

Collectives are real constituents of reality (contra sets)

but still they are not additional constituents, over and above the substances which are their parts.

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Collectives inherit some, but not all, of the ontological marks of

substances

They can admit contrary qualities at different times.

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Collectives,

like substances,

may gain and lose parts or members

may undergo other sorts of changes through time.

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Qualities and processes, too, may form collectives

a musical chord is a collective of individual tones

football matches, wars, plagues are collectives of actions involving human beings

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One-place qualities and processes

depend on one substance

(as a headache depends upon a head)

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Relational qualities and processes

John Mary

kiss

stand in relations of one-sided dependence to a plurality of substances simultaneously

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Examples of relational qualities and processes

kisses, thumps, conversations,

dances, legal systems

Such real relational entities

join their carriers together into collectives of greater or lesser duration

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Mereology

‘Entity’ = absolutely general ontological term of art

embracing at least: all substances, qualities, processes, and all the wholes and parts thereof, including boundaries

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Primitive notion of part

‘x is part of y’ in symbols: ‘x ≤ y’

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We define overlap as the sharing of common parts:

O(x, y) := z(z ≤ x z ≤ y)

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Axioms for basic mereology

AM1 x ≤ x

AM2 x ≤ y y ≤ x x = y

AM3 x ≤ y y ≤ z x ≤ z

Parthood is a reflexive, antisymmetric, and transitive relation, a partial ordering.

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Extensionality

AM4 z(z ≤ x O(z, y)) x ≤ y

If every part of x overlaps with y

then x is part of y

cf. status and bronze

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Sum

AM5 x(x)

y(z(O(y,z) x(x O(x,z))))

For every satisfied property or condition there exists an entity, the sum of all the -ers

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Definition of Sum

x(x) := yz(O(y,z) x(x O(x,z)))

The sum of all the -ers is that entity which overlaps with z if and only if there is some -er which overlaps with z

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Examples of sums

electricity, Christianity, your body’s metabolism

the Beatles, the population of Erie County, the species cat

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Other Boolean Relations

x y := z(z ≤ x z ≤ y) binary sum

x y := z(z ≤ x z ≤ y) product

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Other Boolean Relations

x – y := z (z ≤ x O(z, y)) difference

–x := z (O(z, x)) complement

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What is a Substance?

Bundle theories: a substance is a whole made up of tropes as parts.

What holds the tropes together?

... problem of unity

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Topology

How can we transform a sheet of rubber in ways which do not involve cutting or tearing?

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Topology

We can invert it, stretch or compress it, move it, bend it, twist it. Certain properties will be invariant under such transformations –

‘topological spatial properties’

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Topology

Such properties will fail to be invariant under transformations which involve cutting or tearing or gluing together of parts or the drilling of holes

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Examples of topological spatial properties

The property of being a (single, connected) body

The property of possessing holes (tunnels, internal cavities)

The property of being a heap

The property of being an undetached part of a body

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Examples of topological spatial properties

It is a topological spatial property of a pack of playing cards that it consists of this or that number of separate cards

It is a topological spatial property of my arm that it is connected to my body.

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Topological Properties

Analogous topological properties are manifested also in the temporal realm:

they are those properties of temporal structures which are invariant under transformations of

slowing down, speeding up, temporal translocation …

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Topological Properties

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Topology and Boundaries

Open set: (0, 1)

Closed set: [0, 1]

Open object:

Closed object:

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Closure

= an operation which when applied to an entity x yields a whole which comprehends both x and its boundaries

use notion of closure to understand structure of reality in an operation-free way

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Axioms for Closure

AC1: each entity is part of its closure

AC2: the closure of the closure adds nothing to the closure of an object

AC3: the closure of the sum of two objects is equal to the sum of their closures

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Axioms for Closure

AC1 x ≤ c(x) expansiveness

AC2 c(c(x)) ≤ c(x) idempotence

AC3 c(x y) = c(x) c(y) additivity

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Axioms for Closure

These axioms define in mereological terms a well-known kind of structure, that of a closure algebra, which is the algebraic equivalent of the simplest kind of topological space.

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Boundary

b(x) := c(x) c(–x)

The boundary of an entity is also the boundary of the complement of the entity

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Interior

i(x) := x – b(x)

boundary

interiorx

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An entity and its complement

-x

x

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The entity alone

x

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The complement alone

-x

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Closed and Open Objects

x is closed := x is identical with its closure

x is open := x is identical with its interior

The complement of a closed object is open

The complement of an open object is closed

Some objects are partly open and partly closed

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Definining Topology

Topological transformations = transformations which take open objects to open objects

e.g. moving, shrinking

x

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Closed Objects

A closed object is an independent constituent of reality:

It is an object which exists on its own, without the need for any other object which would serve as its host

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Contrast holes

a hole requires a host

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A closed object need not be connected

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…. nor must it be free of holes

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…. or slits

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Connectedness

Definition

An object is connected

if we can proceed from any part of the object to any other

and remain within the confines of the object itself

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Connectedness

A connected object is such that all ways of splitting the object into two parts yield parts whose closures overlap

Cn(x) :=

yz(x = yz w(w ≤ (c(y)c(z))))

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Connectedness*A connected* object is such that,

given any way of splitting the object into two parts x and y,

either x overlaps with the closure of y

or y overlaps with the closure of x

Cn*(x) := yz(x = y z (w(w ≤ x w ≤ c(y)) w(w ≤ y w ≤ c(x)))

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Problems

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Problem

A whole made up of two adjacent spheres which are momentarily in contact with each other will satisfy either condition of connectedness

Strong connectedness rules out cases such as this

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Strong connectedness

Scn(x) := Cn*(i(x))

An object is strongly connected if its interior is connected*

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Definition of Substance

A substance is a maximally strongly connected non-dependent entity:

S(x) := Scn(x) y(x ≤ y Scn(y) x = y) zSD(x, z)

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More needed

Substances are located in spatial regions

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More needed

Some substances have a causal integrity without being completely disconnected from other substances:

heart

lung

Siamese twin

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Time

Substances can preserve their numerical identity over time

Full treatment needs an account of:

spatial location

transtemporal identity

causal integrity, matter

internal organization