Mereotopologies:
Fiat and Bona Fide Boundaries
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
substance
interior
boundary
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
… perhaps also holes
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
substances compete for space:
(unlike shadows and holes)
no two substances can occupy the same
spatial region at the same time.
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
They can admit contrary moments at
different times.
Collectives,
like substances,
may gain and lose parts or members
may undergo other sorts of changes
through time.
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
Collectives/heaps
are the duals of undetached parts
Both involve fiat boundaries
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
substance
undetached part
collective of
substances
special sorts of undetached parts
ulcers
tumors
lesions
…
Fiat boundaries
physical (bona
fide) boundary
fiat boundary
Holes, too, involve fiat
boundaries
A hole in the ground
Solid physical boundaries at the floor
and walls
but with a lid that is not made of matter:
hole
Holes involve two kinds of
boundaries
bona fide boundaries which exist
independently of our demarcating acts
fiat boundaries which exist only because
we put them there
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
Mountain
bona fide upper boundaries
with a fiat base:
where does the mountain start ?
... a mountain is not a substance
nose
...and it’s not an accident, either
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
Mountain
bona fide upper boundaries
with a fiat base:
Architects Plan for a House
fiat upper boundaries
with a bona fide base:
where does the mountain start ?
... a mountain is not a substance
nose
...and it’s not a process, either
One-place qualities and processes
depend on one substance
(as a headache depends upon a head)
Relational qualities and processes
kiss
John
Mary
stand in relations of one-sided
dependence to a plurality of
substances simultaneously
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
Mereology
‘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
‘x is part of y’ in symbols: ‘x ≤ y’
We define overlap as the
sharing of common parts:
O(x, y) := z(z ≤ x  z ≤ y)
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.
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
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
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
electricity, Christianity, your body’s
metabolism
the Beatles, the population of Erie County,
the species cat
Other Boolean Relations
x  y := z(z ≤ x  z ≤ y) binary sum
x  y := z(z ≤ x  z ≤ y)
product
Other Boolean Relations
x – y := z (z ≤ x  O(z, y))
–x := z (O(z, x))
difference
complement
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
Topology
How can we transform a sheet of rubber in
ways which do not involve cutting or
tearing?
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’
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
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
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
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 Properties
Topology and Boundaries
Open set: (0, 1)
Closed set: [0, 1]
Open object:
Closed object:
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
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 Closure
AC1
AC2
AC3
x ≤ c(x)
expansiveness
c(c(x)) ≤ c(x)
idempotence
c(x  y) = c(x)  c(y)
additivity
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.
Boundary
b(x) := c(x)  c(–x)
The boundary of an entity is also the
boundary of the complement of the entity
Interior
i(x) := x – b(x)
x
interior
boundary
An entity and its complement
-x
x
The entity alone
x
The complement alone
-x
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
Topological transformations =
transformations which take open objects to
open objects
e.g. moving, shrinking
x
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
a hole requires a host
A closed object need not be
connected
…. nor must it be free of holes
…. or slits
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
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 = yz  w(w ≤ (c(y)c(z))))
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)))
Problems
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
Strong connectedness
Scn(x) := Cn*(i(x))
An object is strongly connected if its
interior is connected*
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
Substances are located in spatial regions
More needed
Some substances have a causal integrity
without being completely disconnected
from other substances:
heart
lung
Siamese twin
Time
Substances can preserve their numerical
identity over time
Full treatment needs an account of:
spatial location
transtemporal identity
causal integrity, matter
internal organization –
modular order of bodily systems
1. Basic Formal Ontology
mereotopology
dependence
granularity/partition theory
SNAP/SPAN
action/participation
plans/functions/executions
systems/modularity
causality/powers/dispositions
environments/niches
normativity
Medical Being
mereotopology: anatomy with holes, layers
dependence
granularity/partition theory: molecules, genes, cells …
SNAP/SPAN: anatomy, physiology …
action/participation: doctor, patient, drug …
plans/functions/executions: therapy, application of
therapy …
causality/powers/dispositions: prevention
environment: environmental influences on disease
normativity: health, disease, ‘normal’ liver
Chapters
Main Body Systems (contd.)
nervous system
respiratory system
immune system
How do these systems relate together?
(a medico-ontological analogue of the mindbody problem)
WHAT IS A SYSTEM? Fiat object?
The ontologist’s job
is not to mimic or replace or usurp science
not to discover statistical or functional laws
it is to establish the categories involved in given
domains of reality and the relations between
them
via:
taxonomies
and:
partonomies
and by addressing NORMATIVE ISSUES such
as: what holds in the standard case
Rules for Good Ontology
These are rules of thumb:
They represent ideals to be
approximated to in practice
(and often come with trade-offs)
Naturalness
A good ontology should include in its basic
category scheme only those categories
which are instantiated by entities in reality
(it should reflect nature at its joints)
A good first test:
the categories in question should be
reflected in Technically Extended
English
= English as extended by the various
technical vocabularies of medical and
scientific disciplines
Basic categories
are reflected by morphologically simple terms:
dog
pain
foot
blood
hunger
hot
red
diabetes
No theoretical artifacts
A good ontology should not include in its
basic category scheme
artifacts of logical, mathematical or
philosophical theories (such as: transfinite
cardinals, instantaneous rabbit-slices, nonexistent golden mountains, functions
across possible worlds, and the like).
Problem cases:
Fictional entities?
Absences?
Holes?
A good category scheme
a
should not be a mish-m sh of natural and
philosophical taxa
(keep views separate:
basic views, domain-specific views,
theoretical-artefactual views)
Two sorts of semantics
1. intended interpretation (the world)
2. control for consistency …
Tools are just tools
If specific logical or mathematical or
conceptual tools are needed, for example
for semantic purposes,
then these should be clearly recognized as
tools and thus not be seen as having
consequences for basic ontology.
(Possible worlds …)
Perspectivalism
Perspectivalism
Different partitions may represent
cuts through the same reality which
are skew to each other
Ontology
like cartography
must work with maps at different scales and
with maps picking out different dimensions
of invariants
Varieties of granular partitions
Partonomies: inventories of the parts of
individual entities
Maps: partonomies of space
Taxonomies: inventories of the universals
covering a given domain of reality
Cheese-paring principle
A good ontology should have the
resources to do justice to the fact that the
world can be sliced in many ways
Example of cheese-paring
substance
action
(relational
process)
agent
(substance
plus role)
substance
patient
(substance
plus role)
linked by mutual dependence
Double-Counting
in realm of substances
person
*ear, nose, throat, arm
*family, clinical trial population
fiat parts and aggregates should be
explicitly marked as involving doublecounting
Double-Counting
in realm of processes
process
*beginning, end, first phase
*series of clinical trials, World Cup
fiat parts and aggregates on the same
level of granularity should be explicitly
marked as involving double-counting
Rule: No Crossing Categories
If C is a core category then an instance of C
is always an instance of C whichever view
of C we take
If C is a core category then an instance of C
is always an instance of C whichever
granularity we take
If C is a core category then all parts and
aggregates of instances of C are also
instances of C
Rule: Respect Granularity
spatial region
substance
quality
parts of spatial regions are always spatial regions
Respect Granularity
spatial region
substance
quality
parts of substances are always substances
Relations crossing the SNAP/SPAN
border are not part-relations
substance John
John’s life
physiological
processes
Relations between entities at
different granularities
are part relations
Hence we have two sorts of part-relations:
1. within a granularity
2. between granularities
Where granularities start and stop is
determined:
by the formation of scientific disciplines
by fiat?
How to treat cross-categorial
structures?
which ontology do they belong to?
How to treat higher-order attributions
Universals have instances
Universal A depends for its instantiation on
the instantiation of universal B
Roughly: these are meta-assertions
(that they have special truthmakers of their
own is an illusion of language)
Universals have instances
is not an extra assertion
rather it is something which shows itself via
the syntax of a good ontological language
(cf. Wittgenstein’s Tractatus)
Rules for good syntax in
formalizing ontology
entities of the same category should be
represented by means of symbols of the
same type
some symbols will not represent entities at
all (V, , =, , etc.)
Problems arise for partial ontologies
only if they come along with the claim to
be complete
(reductionists are nearly always correct in
what they hold to exist -but incorrect when they hold that nothing
else exists)
Even reductionists
are right as far as they go
(even their peculiar maps of reality,
as consisting of processes,
or of spacetime worms,
are transparent to reality)
The only problem with such maps is that
they are not complete
Rule: Representations
A representation is never identical with the
object which it is a representation of
Rule: Fallibilism
Ontologists are seeking principles that are
true of reality,
but this does not mean that they have
special powers for discovering the truth.
Ontology is, like physics or chemistry, part
of a piecemeal, on-going process of
exploration, hypothesis-formation, testing
and revision.
Fallibilism
Ontological claims advanced as true today
may well be rejected tomorrow in light of
further discoveries or of new and better
arguments
Ontology is like a small window on reality
which, in fits and starts, gets bigger and
more refined as we proceed
Rule: Adequatism
A good ontology should be adequatist:
its taxonomies and partonomies should
comprehend the entities in reality at all levels
of aggregation,
from the microphysical to the cosmological,
and including also the middle world (the
mesocosmos) of human-scale entities in
between.
Adequatists: Aristotle, Ingarden, Chisholm; Johansson,
Smith
Nothing in life is certain
except
death
and taxes
Fictionalism is always wrong
Either an entity exists, or it does not exist
Either an entity type exists, or it does not exist
Rule: Quine is wrong
There is no entity without identity
We have no identity criteria for
people
taxes
plans
diseases
Quine’s slogan
-- “no entity without an identity criterion” -represents a confusion of ontology and
epistemology
Compare: “no truth without a truth criterion”
Rules Governing Taxonomies
Every (coherent, tested) ontology for
a given domain at a given level of
granularity
should be representable as a tree in
the mathematical sense
Problem cases: shapes, colors ?
Natural scientific classifications are
principled
Principled classifications satisfy the
no-diamonds rule:
A
E
F
B
C
G
D
H
Good
Bad
Counterexample in the realm of
artifacts ?
urban structures
buildings
car parks
multi-story car-parks
Eliminating counter-examples
urban structures
buildings
parking areas
multi-story car-parks
“Ontoclean”
Rule: No ‘others’
All category labels should be positive
No category labels like:
entities which do not fall under the other
categories
Tree structure
Higher nodes within the tree represent
more general universals, lower nodes
represent less general universals.
Branches connecting nodes represent the
relations of inclusion of a lower category in
a higher:
man is included in mammal
mammal is included in animal
and so on.
An Ontology (Taxonomy) should be
Principled
Suppose that in counting off the cars passing
beneath you on the highway, your checklist
includes one box labeled red cars and another
box labeled Chevrolets.
The resultant inventory will be unprincipled;
you will almost certainly be guilty of counting
some cars twice.
Unprincipled = the two modes of classification
belong to two distinct classifications made for
two distinct purposes
An Ontology (Taxonomy) should be
Principled
Principled = Constructed for a single purpose
Principled = Generative (recursive?)
Principled = Double-counting clearly marked
Principled = SNAP-SPAN opposition reflected (so
mereological determinateness is guaranteed)
Principled = Clear rules when a new category must be
admitted
What else?
CYC is not principled
Well-formedness rule
Each tree is unified
in the sense that it has a single top-most
or maximal node, representing the
maximum category
comprehending all the categories
represented by the nodes lower down the
tree
Why trees?
A taxonomy (ontology) with two maximal
nodes would be in need of completion by
some extra, higher-level node
representing the union of these two
maxima.
Otherwise it would not be one taxonomy at
all, but rather two separate taxonomies
(e.g. SNAP and SPAN)
‘Entity’
= label for the highest-level category of
ontology.
Everything which exists is an entity
Alternative top-level terms favored by
different ontologists: ‘thing,’ ‘object,’ ‘item,’
‘element,’ ‘existent.’
Use of ‘entity’ is dangerous (see Frege)
Rule: Seek to establish a basis in
minimal nodes (leaves)
Leaves of the tree represent the lowest
categories (infima species)
= categories in which no sub-categories are
included.
‘Has a basis in minimal nodes’ = the
categories at the lowest level of the tree
exhaust the maximum category
Rule: Aim for Exhaustiveness
The chemical classification of the noble
gases is exhausted by:
Helium, Neon, Argon, Krypton, Xenon and
Radon.
…normally very hard to achieve
For a taxonomy with a basis in
minimal nodes
every intermediate node in the tree is
identifiable as a combination of minimal
nodes.
More well-formedness principles
There should be a finite number of steps
between the maximal category and each
minimal category.
There should be the same number of steps
between the topmost node of the tree and
all its lowest-level nodes.
Well-Formedness
The taxonomy as a whole is thereby divided
into homogeneous levels,
each level represents a jointly exhaustive
and pairwise disjoint partition of the
corresponding domain of categories on the
side of objects in the world.
Relations can also hold across
granularities
Microbial processes in the human body
sustain the human body in existence
Neurophysiological processes in the brain
cause and provide the substratum for
cognitive processes
Trees of universals
(species-genus hierarchies)
capture the way the world is (realism)
– they depict the invariant
structures/patterns/regularities in reality
BUT: species-genus hierarchies
may capture the way the world should
be
– by depicting the
structures/patterns/regularities in the
realm of standards, ideal cases,
recipes
(a hierarchy of medical therapies)
Extended Aristotelian Realism
The general terms of technically extended
English (or many of them),
including terms like ‘Coca Cola’,
correspond to universals (species and
genera, invariant patterns) in reality