1 Formal Ontology 2 Schedule Sep. 4: Introduction: Mereology, Dependence and Geospatial Ontology Reading: Basic Tools of Formal Ontology Ontological.
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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
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