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Formal Ontology. Schedule. Sep. 4: Introduction: Mereology , Dependence and Geospatial Ontology Reading: Basic Tools of Formal Ontology Ontological Tools for Geographic Representation. Schedule.

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Formal Ontology

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Formal ontology

Formal Ontology



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

  • Reading:

  • Basic Tools of Formal Ontology

  • Ontological Tools for Geographic Representation



  • Sep. 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



  • Oct. 2: Granularity

  • Reading: 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]



  • 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



  • Nov. 9 Day-long Saturday Workshop

  • 9am 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



  • Institute for Formal Ontology and Medical Information Science

  • Some background

The manchester school

The Manchester School

  • Kevin Mulligan

  • Peter Simons

  • Barry Smith

  • in Manchester 1973-76

  • working on the ontology of Edmund Husserl

Edmund husserl

Edmund Husserl

Logical investigations 1900 01

Logical Investigations¸1900/01

  • the theory of part and whole

  • the theory of dependence

  • the theory of boundary, continuity and contact

Formal ontology1

Formal Ontology

  • (term coined by Husserl)

  • the theory of those ontological structures

  • (such as part-whole, universal-particular)

  • which apply to all domains whatsoever

Formal ontology vs formal logic

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

Formal ontology vs formal logic1

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

Formal ontology and symbolic logic

Formal Ontology and Symbolic Logic

  • Great advances of Frege, Russell, Wittgenstein

  • Leibnizian idea of a universal characteristic

  • …symbols are a good thing



  • don’t confuse Logical with Ontological Form

  • Russell

  • Part-whole is not a logical relation

Formal ontology

  • for Frege, Russell, Lesniewski,

  • Wittgenstein, Quine

  • Logic is a ‘Zoology of Facts’

  • Formal theories are theories of reality

  • with one intended interpretation

  • = the world


after starting off on the right road

Logic took a wrong turn

Logic took a wrong turn

Formal ontology

Logic took a wrong turn

Formal ontology

  • Tarski, Carnap, Putnam, Sowa, Gruber:

  • Forget reality!

  • Lose yourself in ‘models’!

Ifomis ontology

IFOMIS Ontology

  • is an ontology of reality

  • Standard Information Systems Ontologies

  • are ontologies of mere 'models'

Standard information systems ontologies

Standard Information Systems Ontologies:

  • programming real ontology into computers is hard

  • therefore: we will simplify ontology

  • and not care about reality at all

Painting the emperor s palace is hard

Painting the Emperor´s Palace ishard



  • we will not try to paint the Palace at all

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

Ifomis strategy

IFOMIS Strategy

  • get real ontology right first

  • and then investigate ways in which this real ontology can be translated into computer-useable form later


A language to map these

a language to map these

  • Formal ontological structures in reality

A directly depicting language



a directly depicting language

  • ‘John’ ‘( ) is red’


Wittgenstein s tractatus

are pictures of

Wittgenstein’s Tractatus

  • Propositions

  • States of affairs

Parts and moments

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

Formal ontology

A directly depicting language1

A directly depicting language

  • may contain an analogue of conjunction

  • p and q

  • _______

  • p

But it can contain no negation

but it can contain no negation

  • p

  • _______

  • p

And also no disjunction

and also no disjunction

  • p or q

  • ______

  • p

The idea of a directly depicting language

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

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 ...)

A network of domain ontologies

A Network of Domain Ontologies

  • Material (Regional) Ontologies

Basic Formal Ontology




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

  • terminology standardization

  • taxonomy standardization

  • supports reasoning about reality

Formal ontology


  • 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)

A network of domain ontologies1

A Network of Domain Ontologies

A network of domain ontologies2

A Network of Domain Ontologies

A network of domain ontologies3

A Network of Domain Ontologies

A network of domain ontologies4

A Network of Domain Ontologies

A network of domain ontologies5

A Network of Domain Ontologies

Extended formal ontology bfo extended by mind

Extended formal ontology(BFO Extended by Mind)

Bfo extended by mind

BFO Extended by Mind


Bfo extended by mind1

BFO Extended by Mind











is complicated

What is the best language to describe this complexity

What is the best language to describe this complexity?

Anglocentric realism

Anglocentric Realism

  • We have a huge amount of knowledge of reality,

  • at many different levels of granularity,

  • from microphysics to cosmology

Anglocentric realism1

Anglocentric Realism

  • TEE = Technically Extended English

  • = English extended by the technical vocabularies of

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

Anglocentric realism2

Anglocentric Realism

  • Our knowledge of reality as expressed in Technically Extended English

  • is increasing by the hour



  • … there are problems with TEE as a formal representation language

  • (cf. Tarski)

Nouns and verbs

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

Natural language

t i m e


Natural language

  • glues them together indiscriminately


Snapshot vs video

t i m e


Snapshot vs. Video




  • Mesoscopic reality is

  • divided at its natural joints

  • into substances:

  • animals, bones, rocks, potatoes

The ontology of substances

The Ontology of Substances

  • Substances form natural kinds

  • (universals, species + genera)



  • Processes merge into one another

  • Process kinds merge into one another

  • … few clean joints either between instances or between types


t i m e


Substances and processes

t i m e


Substances and processes

demand different sorts of inventories

Substances demand 3 d partonomies


Substances demand 3-D partonomies

Processes demand 4d partonomies

t i m e

Processes demand 4D-partonomies



  • a whistling, a blushing, a speech

  • a run, the warming of this stone

Processes may have temporal parts

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

Substances do not have temporal parts

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

Substances have spatial parts

Substances have spatial parts

Formal ontology

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

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

Formal ontology

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

  • within a directly depicting language?

You are a substance

You are a substance

  • Your life is a process

  • You are 3-dimensional

  • Your life is 4-dimensional

Substances and processes form two distinct orders of being

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?

Main problem

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

Main problem1

Main problem

  • There is a polymorphous ontological promiscuity of the English sentence,

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

Two alternative basic ontologies

Two alternative basic ontologies

  • SNAP and SPAN

  • SNAP = substances plus qualities

  • SPAN = processes

These represent two views

These represent two views

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

The four dimensionalist ontology

t i m e

The Four-Dimensionalist Ontology

Boundaries are mostly fiat

t i m e

boundaries are mostly fiat

everything is flux

Mereology works without restriction everywhere here

clinical trial

t i m e

mereology works without restriction everywhere here

The time stamped ontology

The Time-Stamped Ontology




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

Mereology works without restriction in every instantaneous 3 d section through reality

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

Three views partitions of the same reality

Three views/partitions of the same reality

All contain huge amounts of knowledge of this reality

all contain huge amounts of knowledge of this reality

against Kant

Ontological zooming

Ontological Zooming

  • The dimension of granularity

Part 2

Part 2

Tools of Ontology:

Mereology, Topology, Dependence

Ontological dependence

Ontological Dependence


+ qualities


Ontological dependence1

Ontological Dependence

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

Ontological dependence2

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

Specific dependence

Specific Dependence

  • O := overlap

  • x := x is necessarily such that

  • E! := existence

  • SD(x, y) := O(x, y) x(E!x  E!y)

Mutual specific dependence

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)

One sided specific dependence

One-Sided Specific Dependence

  • OSD(x, y) := SD(x, y) MSD(x, y)

  • My headache is one-sidedly specifically dependent on me.

Substances qualities processes

Substances, Qualities, Processes

  • Substances are the bearers or carriers of qualities and processes,

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

Ontological dependence3

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



  • can also gain and lose parts

  • … as an organism may gain and lose molecules

Types of relations between parts

Types of relations between parts

  • 1. Dependence relations

  • 2. Side-by-sideness relations

  • 3. Fusion relations




a thinking

cannot exist without a thinker


Theory of vagueness

Theory of vagueness


found among substances

and among qualities and processes




Topology like mereology

Topology, like mereology,

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



  • = topology on a mereological basis

Substances undetached parts and heaps

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

Formal ontology


undetached part

collective of


Special sorts of undetached parts

special sorts of undetached parts

  • ulcers

  • tumors

  • lesions

Fiat boundaries

physical (bona fide) boundary

fiat boundary

Fiat boundaries



  • of bona fide boundaries:

  • an animal’s skin, the surface of the planet

  • of fiat boundaries:

  • the boundaries of postal districts and census tracts



  • bona fide upper boundaries

  • with a fiat base:

Architects plan for a house

Architects Plan for a House

  • fiat upper boundaries

  • with a bona fide base:

Where does the mountain start

where does the mountain start ?

... a mountain is not a substance

Formal ontology


...and it’s not a quality, either

A substance has a complete physical boundary

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

Formal ontology




A substance takes up space

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

A substance has spatial parts

A substance has spatial parts

  • … perhaps also holes

Each substance is such as to have divisible bulk

Each substance is such as to have divisible bulk:

  • it can in principle be divided into separate spatially extended substances

By virtue of their divisible bulk

By virtue of theirdivisible bulk

  • substances compete for space:

  • (unlike shadows and holes)

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

Substances vs collectives

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.

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

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

  • They can admit contrary qualities at different times.



  • like substances,

  • may gain and lose parts or members

  • may undergo other sorts of changes through time.

Qualities and processes too may form collectives

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

One place qualities and processes

One-place qualities and processes

  • depend on one substance

  • (as a headache depends upon a head)

Formal ontology




  • Relational qualities and processes

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

Examples of relational qualities and processes

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



  • ‘Entity’ = absolutely general ontological term of art

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

Primitive notion of part

Primitive notion of part

  • ‘x is part of y’ in symbols: ‘x ≤ y’

We define overlap as the sharing of common parts

We define overlap as the sharing of common parts:

  • O(x, y) := z(z ≤ x  z ≤ y)

Axioms for basic mereology

Axioms for basic mereology

  • AM1 x ≤ x

  • AM2x ≤ y  y ≤ x  x = y

  • AM3x ≤ y  y ≤ z  x ≤ z

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



  • 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

Formal ontology


  • 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

Definition of sum

Definition of Sum

  • x(x) := yz(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

Examples of sums

Examples of sums

  • electricity, Christianity, your body’s metabolism

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

Other boolean relations

Other Boolean Relations

  • x  y := z(z ≤ x  z ≤ y) binary sum

  • x  y := z(z ≤ x  z ≤ y)product

Other boolean relations1

Other Boolean Relations

  • x – y := z (z ≤ x  O(z, y)) difference

  • –x := z (O(z, x)) complement

What is a substance

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



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



  • 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’



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

Examples of topological spatial properties

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

Examples of topological spatial properties1

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.

Topological properties

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 …

Topological properties1

Topological Properties

Topology and boundaries

Topology and Boundaries

  • Open set: (0, 1)

  • Closed set: [0, 1]

  • Open object:

  • Closed object:



  • = 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

Axioms for closure

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

Axioms for closure1

Axioms for Closure

  • AC1x ≤ c(x) expansiveness

  • AC2 c(c(x)) ≤ c(x) idempotence

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

Axioms for closure2

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.



  • b(x) := c(x)  c(–x)

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






  • i(x) := x – b(x)

An entity and its complement

An entity and its complement

  • -x


The entity alone

The entity alone


The complement alone

The complement alone

  • -x

Closed and open objects

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

Definining topology

Definining Topology

  • Topological transformations = transformations which take open objects to open objects

  • e.g. moving, shrinking


Closed objects

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

Contrast holes

Contrast holes

  • a hole requires a host

A closed object need not be connected

A closed object need not be connected

Nor must it be free of holes

…. nor must it be free of holes

Or slits

…. or slits



  • 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



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

  • Cn(x) :=

  • yz(x = yz w(w ≤ (c(y)c(z))))



  • 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)))





  • 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

Strong connectedness

Strong connectedness

  • Scn(x) := Cn*(i(x))

  • An object is strongly connected if its interior is connected*

Definition of substance

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)

More needed

More needed

  • Substances are located in spatial regions

More needed1

More needed

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

  • heart

  • lung

  • Siamese twin

Formal ontology


  • Substances can preserve their numerical identity over time

  • Full treatment needs an account of:

  • spatial location

  • transtemporal identity

  • causal integrity, matter

  • internal organization

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