the ontology of holes. temple at corinth pipe the ontology of places
TRANSCRIPT
The Ontology of Holes
Temple at Corinth
Pipe
The Ontology of Places
Towards an Ontology of Places
Aristotle
Ontology of the place (topos) of an individual substance (ousia)
What is it for a substance to be (or to fit snugly) in a location or context?
Aristotle
Place
has size but not matter.
It has shape or form—exactly the shape or form of the thing that is located in it—but it lacks bulk.
Aristotle
We say that a thing is … in the air
because of the surface of the air which surrounds it;
for if all the air were its place, the place of a thing would not be equal to the thing —
which it is supposed to be. (211a24-28)
A place contains its body.
The body relates to its place in something like the way the liquid in an urn relates to the urn, or the hand relates to the glove.
Aristotle: A place
exactly surrounds the thing,
but the place does not depend specifically upon the thing, since the latter can be replaced by another thing, which is then said to be
in the same place.
When a thing is in a surrounding body of air or water
‘it is primarily in the inner surface of the surrounding body.’ The boundaries of the two—the outer surface of the thing and the inner surface of its surrounding body—exactly coincide (211a30-33): the place of a substance is the inner boundary of the immediately surrounding or containing body.
Places are holes
Places are holes
Places are holes
A hole in the ground
Solid physical boundaries at the floor and walls
but with a fiat lid:
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
niches, environments are holes
and some holes can move
Where are Places?Concrete Entity
[Exists in Space and Time]Concrete Entity
[Exists in Space and Time]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 4-D Ontology[Perdure. Unfold in Time]Entity in 4-D Ontology
[Perdure. Unfold in Time]
Processual EntityProcessual EntitySpatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3Spatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3
Spatial Regionof Dimension 0,1,2,3
Spatial Regionof Dimension 0,1,2,3 Dependent EntityDependent Entity
Independent EntityIndependent Entity
Quality (Your Redness, My Tallness)[Form Quality Regions/Scales]
Quality (Your Redness, My Tallness)[Form Quality Regions/Scales]
Role, Function, PowerHave realizations (called: Processes)
Role, Function, PowerHave realizations (called: Processes)
Substance[maximally connected causal unity]
Substance[maximally connected causal unity]
Boundary of Substance *Fiat or Bona Fide or MixedBoundary of Substance *
Fiat or Bona Fide or Mixed
Aggregate of Substances * (includes masses of stuff? liquids?)
Aggregate of Substances * (includes masses of stuff? liquids?)
Fiat Part of Substance * Nose, Ear, Mountain
Fiat Part of Substance * Nose, Ear, Mountain
Process [Has Unity]Clinical trial; exercise of role
Process [Has Unity]Clinical trial; exercise of role
Fiat Part of Process*Fiat Part of Process*
Aggregate of Processes*Aggregate of Processes*
Instantaneous Temporal Boundary of Process (= Ingarden’s 'Event’)*
Instantaneous Temporal Boundary of Process (= Ingarden’s 'Event’)*
Quasi-ProcessJohn’s Youth. John’s Life
Quasi-ProcessJohn’s Youth. John’s Life
Quasi-Quality Prices, Values, Obligations
Quasi-Quality Prices, Values, Obligations
Quasi-SubstanceChurch, College, Corporation
Quasi-SubstanceChurch, College, Corporation
Quasi-Role/Function/PowerThe Functions of the PresidentQuasi-Role/Function/Power
The Functions of the President
Where are Places (Holes)?Concrete Entity
[Exists in Space and Time]Concrete Entity
[Exists in Space and Time]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 4-D Ontology[Perdure. Unfold in Time]Entity in 4-D Ontology
[Perdure. Unfold in Time]
Processual EntityProcessual EntitySpatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3Spatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3
Spatial Regionof Dimension 0,1,2,3
Spatial Regionof Dimension 0,1,2,3 Dependent EntityDependent Entity
Independent EntityIndependent Entity
Quality (Your Redness, My Tallness)[Form Quality Regions/Scales]
Quality (Your Redness, My Tallness)[Form Quality Regions/Scales]
Role, Function, PowerHave realizations (called: Processes)
Role, Function, PowerHave realizations (called: Processes)
Substance[maximally connected causal unity]
Substance[maximally connected causal unity]
Boundary of Substance *Fiat or Bona Fide or MixedBoundary of Substance *
Fiat or Bona Fide or Mixed
Aggregate of Substances * (includes masses of stuff? liquids?)
Aggregate of Substances * (includes masses of stuff? liquids?)
Fiat Part of Substance * Nose, Ear, Mountain
Fiat Part of Substance * Nose, Ear, Mountain
Process [Has Unity]Clinical trial; exercise of role
Process [Has Unity]Clinical trial; exercise of role
Fiat Part of Process*Fiat Part of Process*
Aggregate of Processes*Aggregate of Processes*
Instantaneous Temporal Boundary of Process (= Ingarden’s 'Event’)*
Instantaneous Temporal Boundary of Process (= Ingarden’s 'Event’)*
Quasi-ProcessJohn’s Youth. John’s Life
Quasi-ProcessJohn’s Youth. John’s Life
Quasi-Quality Prices, Values, Obligations
Quasi-Quality Prices, Values, Obligations
Quasi-SubstanceChurch, College, Corporation
Quasi-SubstanceChurch, College, Corporation
Quasi-Role/Function/PowerThe Functions of the PresidentQuasi-Role/Function/Power
The Functions of the President
Where are Places?Concrete Entity
[Exists in Space and Time]Concrete Entity
[Exists in Space and Time]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 4-D Ontology[Perdure. Unfold in Time]Entity in 4-D Ontology
[Perdure. Unfold in Time]
Processual EntityProcessual EntitySpatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3Spatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3
Spatial Regionof Dimension
0,1,2,3
Spatial Regionof Dimension
0,1,2,3
Dependent EntityDependent Entity
Independent EntityIndependent Entity
Types of PlacesConcrete Entity
[Exists in Space and Time]Concrete Entity
[Exists in Space and Time]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 3-D Ontology[Endure. No Temporal Parts]
Entity in 4-D Ontology[Perdure. Unfold in Time]Entity in 4-D Ontology
[Perdure. Unfold in Time]
Processual EntityProcessual EntitySpatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3Spatio-Temporal Region
Dim = T, T+0, T+1, T+2, T+3
Generalized Spatial Regionof Dimension
0,1,2,3
Generalized Spatial Regionof Dimension
0,1,2,3
Dependent EntityDependent Entity
Independent EntityIndependent Entity
StationaryStationaryMobileMobile
Armchair Ontology
Armchair Ontology
artefacts and niches
Positive and negative parts
positivepart
negativepartor hole
(made of matter)
(not made of matter)
Formal Ontology
atomism vs. holism
set theory
mereology
Environments a Neglected Major Category
Substances
Qualities
Processes
Environments
Accidents
environmentplaceniche
habitatsettinghole
spatial regioninterior
Applications of these concepts
in biology, ecologyin anthropologyin lawin politicsin medicinein embryology
Ecological Niche Concepts
niche as particular place or subdivision of an environment that an organism or population occupies
vs.
niche as function of an organism or population within an ecological community
Elton
the ‘niche’ of an animal means its place in the biotic environment, its relations to food and enemies. [...] When an ecologist says ‘there goes a badger’ he should include in his thoughts some definite idea of the animal’s place in the community to which it belongs, just as if he had said ‘there goes the vicar’ (Elton 1927, pp. 63f.)
The Niche as Hypervolume
temperature
hum
idity
foli
age
den
sity
The Niche as Hypervolume
temperature
hum
idity
foli
age
den
sity
The Niche as Hypervolume
temperature
hum
idity
foli
age
den
sity
The Niche as Hypervolume
temperature
hum
idity
foli
age
den
sity
Hypervolume niche is a location in an attribute space
defined by a specific constellation of environmental variables such as degree of slope, exposure to sunlight, soil fertility, foliage density...
… John found his niche as a mid-level accounts manager in a small-town bank …
But every hypervolume niche must be realized in some specific spatial
location
Niche type must be tokenized in space
J. J. Gibson’s Ecological Psychology
The terrestrial environment is [best] described in terms of a medium, substances, and the surfaces that separate them. (Gibson 1979, p. 16)
Affordances
J. J. Gibson has provided a valuable account of the perceived world, which he presented as a prelude to his accounts of human visual perception
A key part of his account is the concept of affordances
Affordances
“The affordances of the environment are what it offers the animal, what it provides or furnishes, either for good or evil.”
James J. Gibson, The Ecological Approach to Visual Perception
Affordances
“The verb to afford is found in the dictionary, but affordance is not. I have made it up.”
“I mean by it something that refers both to the animal and the environment in a way that no existing term does.” (p. 127)
Gibson’s theory of surface layout
‘a sort of applied geometry that is appropriate for the study of perception and behavior’ (1979, p. 33)
ground, open environment, enclosure, detached object, attached object, hollow object, place, sheet, fissure, stick, fiber, dihedral, etc.
Gibson’s theory of surface layout
systems of barriers, doors, pathways to which the behavior of human beings is specifically attuned,
temperature gradients, patterns of movement of air or water molecules
positive and negative features of the environment
(Casati-Varzi theory of holes)
Monadology
Monadological Niche-Concepts
(niche as small world)
Uexküll
Cf. the ‘first principle’ of Jakob von Uexküll’s Umweltlehre:
all animals, from the simplest to the most complex, are fitted into their unique worlds with equal completeness.
A simple world corresponds to a simple animal, a well-articulated world to a complex one (p. 10).
Uexküll
Theoretical Biology (1928, p. 2): “All reality is subjective appearance -- this must serve as the fundamental insight of biology, too.”
Uexküll
I am afraid that if I publicly proclaim this perspective, that they will treat me à la Galileo,
and either lock me up in a madhouse or else ridicule me as an arch-reactionary.
Uexküll
However I must just once say my piece. Perhaps no one will understand me. Nevertheless, it remains a fact: ‘Epur non si move.’I do not move around the sun, but rather the sun rises and sets in my arch of the sky. It is always another sun, always a new space in which it moves.
Uexküll
the real world is much richer than the naïve person suspects
because there is spread out around every living thing its own world of appearance, …
which manifests so many variations that he could devote his whole life to the study of these worlds without there ever being an end in sight.
Uexküll
when once we have made a beginning in showing in regard to a few animals what environments surround them like solid but invisible glass houses, then we will soon be able to people the world around us with numberless other shimmering worlds, which will intensify the riches of our world a further thousandfold.
Uexküll
In this way biology offers to the naive man an unlimited enrichment of his world,
while the physicist makes of him a beggar. (Uexküll 1928, p. 62)
Gestalt Psychology
Psychological Environment vs. Geographical Environment
Koffka
The Ride Across Lake Constance
On a winter evening amidst a driving snowstorm a man on horseback arrived at an inn, happy to have reached shelter after hours of riding over the wind-swept plain on which the blanket of snow had covered all paths and landmarks.
The Ride Across Lake Constance
The landlord who came to the door viewed the stranger with surprise and asked him whence he came. The man pointed in the direction straight away from the inn, whereupon the landlord, in a tone of awe and wonder, said: ‘Do you know that you have just ridden across the Lake of Constance?’ At which the rider dropped stone dead at his feet.
The Ride Across Lake Constance
Koffka: In what environment, then, did the behaviour of the stranger take place? The Lake of Constance.
Certainly [... and it is] interesting for the geographer that this behaviour took place in this particular locality.
But not for the psychologist as the student of behaviour
The Ride Across Lake Constance
[the latter] will have to say: There is a second sense to the word environment according to which our horseman did not ride across the lake at all, but across an ordinary snow-swept plain.His behaviour was a riding-over-a-plain, but not a riding-over-a-lake. (Koffka 1935, pp. 27f.)
Psychologists’ Confusion
What we experience, according to Gestaltists such as Koffka, are not objects in physical reality (objects in the geographic environment). Rather, we experience, precisely, Gestalten, created objects, which differ from objects in physical reality inter alia because they arise through the application of special Gestalt ‘laws of organization’.
Same confusion in Scheler
Scheler’s theory of the milieu of practical life (influenced Heideggers writings on being-in-the-world)
Same confusion in Scheler
The ‘things’ which are relevant to our acting, what we always refer to when, for example, we trace certain deeds of human beings (or dispositions towards such deeds) to their ‘milieu’, have of course not the slightest to do either with Kant’s ‘thing in itself’ or with the objects conceived by science (through the supposition of which science ‘explains’ natural facts).
Same confusion in Scheler
The sun of the milieu of human beings is not the sun of astronomy. The meat that is stolen, bought, or what have you, is not a sum of cells and tissues with the chemicophysical processes which take place within them. The sun of the milieu is different at the North Pole, in moderate zones, and at the equator, and its beams are felt as different beams.
Same confusion in Scheler
] A ‘milieu-thing’ belongs to an ‘intermediate realm’ lying between our perceptual content and its objects on the one hand and those objectively thought objects on the other. (Scheler 1954, p. 159, Eng. trans. p. 140)
Husserl, Scheler, Heidegger ...
Niche as Lebenswelt
Roger Barker: Niche as Behavioral Setting
Niches are recurrent settings which serve as the environments for our everyday activities:
my swimming pool,
your table in the cafeteria,
the 5pm train to Long Island.
Each setting is associated with certain standing patterns of behavior.
Niches, for Barker,
are natural units in no way imposed by an investigator.
To laymen they are as objective as rivers and forests
— they are parts of the objective environment that are experienced as directly as rain and sandy beaches are experienced. (Barker 1968, p. 11)
Settings
Each setting has a boundary which separates an organized internal (foreground) pattern from a differing external (background) pattern.
Nesting
Many settings occur in assemblies:
A unit in the middle range of a nesting structure is simultaneously both circumjacent and interjacent,
both whole and part,
both entity and environment.
NestingAn organ—the liver, for example—is a whole in relation to its own component pattern of cells,
and is a part in relation to the circumjacent organism that it, with other organs, composes;
it forms the environment of its cells, and is, itself, environed by the organism. (Barker 1968, p. 154)
Ontological Clarification of Barker
Two notions of ‘spaceship’
John is in the spaceship
Marks of (bodily) substance
i. Rounded-offness
ii. Occupies space
iii. Complete boundary
iv. May have substantial parts (nesting)
v. May be included in larger substances
vi. Has a life (manifests contrary accidents at different times)
Marks of Niches
(i) A niche enjoys a certain natural completeness or rounded-offness,
being neither too small nor too large
—in contrast to the arbitrary undetached parts of environmental settings and to arbitrary heaps or aggregates of environmental settings.
(ii) A niche takes up space,
it occupies a physical-temporal locale,
and is such as to have spatial parts.
Within this physical-temporal locale is a privileged locus—a hole—
into which the tenant or occupant of the setting fits exactly.
(iii) A niche
has an outer boundary:
there are objects which fall clearly within it,
and other objects which fall clearly outside it.
(The boundary itself need not be crisp.)
(iv) An environmental setting
may have actual parts which are also environmental settings.
(v) An environmental setting
may similarly be a proper part of larger, circumcluding environmental settings.
(vi) Niche has a life
is now warm, now cold
now at peace, now at war ….
Marks of (bodily) substance
i. Rounded-offness
ii. Occupies space
iii. Complete boundary
iv. May have substantial parts (nesting)
v. May be included in larger substances
vi. Has a life; is now warm, now cold
Double Hole Structure
Medium (filling the environing hole)
Tenant (occupying the central hole)
Retainer (a boundary of some surrounding structure)
The Structure of Niches
media and retainers
the medium of the bear’s niche is a
circumscribed body of air
Two Types of Boundary
Fiat boundary Physical boundary
Four Basic Niche Types
1 2 3 4
1: a womb;2: a snail’s shell; 3: the niche of a pasturing cow; 4: the niche around a buzzard
Types of Niches
a pond, a nest, a cave, a hut, an air-conditioned apartment building
the history of evolution as a history of the development of niches
all vacant niches must have a retainerdependence of niche on tenant(s) the armchair nichetransforming niches of type 2 into niches of type 1
Four Basic Niche Types
1 2 3 4
1: a house;2: a snail’s shell; 3: the niche of a pasturing cow; 4: the earth’s atmosphere
stationary niches
1: your office when the door is closed; 2: a rabbit hole; 3: a seat at Yankee stadium; 4: the Klingon Empire
Niche as territory (home range)
nested hierarchy of stationery niches
Robert Ardrey, The Territorial Imperative
mobile niches
Four Basic Niche Types
1 2 3 4
1: a womb;2: a snail’s shell; 3: the niche of a pasturing cow; 4: the niche around a buzzard
Niche Construction
Lewontin: niches normally arise in symbiosis with the activities of organisms or groups of organisms;
they are not already there, like vacant rooms in a gigantic evolutionary hotel, awaiting organisms who would evolve into them.
“ecosystem engineering”
maintenance of niches (screwdrivers, paintings)
niches on different levels of the food chain
a. at the bottom of the hiearchy is the saprophytic chain, in which micro-organisms live on dead organic matter;
b. above this is the primary relation between animals and the plants they consume;
c. above this is the predator chain, in which animals of one sort eat smaller animals of another sort;
d. crosscutting all of these is the parasite chain, in which a smaller organism consumes part of a larger host organism.
Token Science
selection theory is concerned with phenomena at the level of populations; it is ‘concerned with what properties are selected for and against in a population. We do not describe single organisms and their physical constituents one by one.’ genotypes vs. genotokens
niche theory and set theory
Fiat Boundaries
fish and bird niches as volumes of space
demarcatory vs. behavioral fiat boundaries
trade-off between security/comfort and freedom of movement
Varieties of Controlled Airspace
Apertures, Mouths, and Sphincters
security vs. freedom of movement plantsbarnacles and snails fish and birdsskin or hide
Security vs. Freedom
the mouth of the bear, the threshold of your office
freedom of movement and fiat boundaries (of niches and of organisms)
the alimentary canal: hole or part ?
Double Hole Structure
Medium (filling the environing hole)
Tenant (occupying the central hole)
Retainer (a boundary of some surrounding structure)
The Medium for Life
a medium is a medium only relative to a given type of niche
a medium requires either a retainer (in the case of a vacant niche) or a tenant (in the case of an occupied niche)
when a tenant leaves its niche the gap left by the tenant is filled immediately by the surrounding mediumMichelangelo’s Davidexamples of media: air, smoke, water
Mixed Media
mixed media (including radioactive impurities, as well as as vitamins, amino acids, salts, and sugars)
Scrooge, crowds, plastic balls
every medium is maximal
what does the job of filling out the niche whose medium is made of air or water? Answer: bodies of vacuum
Vacuum
vacuum satisfies our operational test for being a medium.
vacuum = a volume of space which contains no particlesvacuum = fiat extensionbodies of vacuum are made of: nothing. vacuum is thus impenetrablevacuum contains no boundaries of the bona fide sort
application of niche concept
to formal theories of computer security (niches as ambients, firewalls as ambient boundaries)
to theories of situatedness, situation semantics, context-based semantics, information as context-dependent
in theories of planning (the meaning of life)
Beyond Biology
stacked retainers 1. underground nuclear shelters, 2. soccer
stadiums, 3. heliports, 4.bodies of controlled air space around airports. the niche of the astronaut is the interior of her spacesuitthe niche of the astronaut-plus-spacesuit is the interior of the spaceshipthe niche of the spaceship is the relevant region of spacetenants and niches are categorially disjoint
Lexical Semantics
the fruit is in the bowlthe bird is in the nestthe lion is in the cagethe pencil is in the cupthe fish is in the riverthe river is in the valleythe water is in the lakethe car is in the garagethe fetus is in the cavity in the uterine liningthe colony of whooping crane is in its breeding grounds
Lexical Semantics
‘She swam across the bay in which the submarine was buried and which supplied oysters for the local population.’
The Aristotle-Brentano Principle:
boundaries are ontologically parasitic on (i.e., cannot exist in isolation from) their hosts
The Touching Principle: two entities can be in contact (externally connected) only if one of them is not closed
The niche around the sleeping bear
There are relations of spatial overlap which do not imply corresponding relations of mereological overlap.
Niches are bounded not just spatially, and not just via physical material (the walls of the cave), but also via thresholds in quality-continua (for instance, temperature).
Hence:
distinct niches, may occupy the same spatial region.
Niches are exclusive environments: they cannot be shared by distinct entities (though distinct entities may have overlapping niches, both in the mereological and in the spatial sense of ‘overlap’).
bowling balls inside a closed velvet-lined case
twin fetuses inside a mother’s womb
Vagueness
A niche for an entity y may have proper parts that are not niches for y
What of the outer boundaries of niches?
Indoor vs. outdoor niches
Ecological subjects
A niche for a sum y+z is not ipso facto a niche for each of the summed parts. y+z = John’s head the head plus the rest of John’s body
Not every entity has its own niche. Those which do are natural units:
A substance (body, thing) is a maximally connected tenant, a tenant which is such that no larger connected tenant includes it as a proper part.
You are a substance, but your heart is only a connected tenant within your interior.
A group is a tenant including substances as proper parts.
Extending Mereology
mereology, formalized in terms of the single primitive relation: part of
mereotopology, obtained by adding extra primitive relation boundary for
theory of location, obtained by adding extra primitive relation located at
formal ecology, obtained by adding extra primitive relation niche for
Aim
To define structural properties such as:
open, closed, connected, compact, spatial coincidence, integrity, aggregate, boundary
Primitives
mereological predicate
P(x, y) (read: “x is part of y”), a
topological predicate
B(x, y) (“x is a boundary for y”),
locational predicate
L(x, y) (“x is located at y”).
For: exact location.
Some theorems:
T1 B(x, y) B(x, –y).
T2 B(x, y) B(y, z) B(x, z).
T3 P(x, y) B(y, z) B(x, z).
Defined Terms
D1 O(x, y)=df z (P(z, x) P(z, y))overlap
D2 xx =df xy (O(y, x) z (z O(z, y))) sum
D3 x+y =df z (P(z, x) P(z, y)) sum of x and y
D4 x–y =df z (P(z, x) O(z, y)) difference
D5 l(x) =df x(L(y, x)) location of x
Defined Terms
D6 b(x) =df z B(z, x) boundary of x
D7 i(x) =df x–b(x) interior of x
D8 c(x) =df x+b(x) closure of x
D9 Cl(x) =df x=c(x) closedness
D10 Rg(x) =df c(x) = c(i(x)) i(x) = i(c(x))regularity
Defined Terms
D11 C(x, y) =df O(x, y) O(c(x), y) O(x, c(y))
connection
D12 EC(x, y) =df C(x, y) O(x, y)
external connection
D13 IP(x, y) =df P(x, y) z(B(z, y) O(x, z))
interior parthood
D14 Cn(x) =df yz (x=y+z C(y, z))
self-connectedness
niche predicates
N(x, y), read: “x is a niche for y”.
N(x), read: “x is a niche”. This could be defined in terms of the binary predicate, but only if every niche has a tenant
‘N(x, y)’ and ‘N(x)’, where ‘’ ranges over organism-types.
medium and retainer
M(x, y)
“x is a medium for y”
R(x, y)
“x is a retainer for y”
free niche
D15 N*(x, y) =df N(x, y) zR(z, x)free niche for y
D16 N*(x) =df N(x) zR(z, x)
Every niche is either a free niche, in this sense, or else it has a retainer—
which will imply that it has a solid physical boundary for at least a portion of its exterior surface.
further definitions
D17 t(x) =df y N(x, y) tenant of x
D18 r(x) =df z R(z, x) retainer of x
D19 m(x) =df z M(z, x) medium of x
The Axioms
A1 N(x, y) O(l(x), l(y)) disjointness
A2 N(x, y) IP(l(y), l(x+y)) spatial containment
A3 N(x, y) C(x, y) connection of niche
A4 N(x, y) Cl(y) closure of tenant
A5 N(x, y) Cn(x) connectedness of niche
A6 N(x, y) Rg(y) regularity of tenant
A7 N(x, y) Rg(x) regularity of niche
A8 N(x, y) N(x, z) y = z functionality
Every occupied niche is a niche.
A9 yN(x, y) N(x)
There are no vacant fiat niches
A10 N*(x) yN(x, y)
Every fiat niche is a niche for something.
Media and retainers
A11 M(x, y) N(y)
A12 R(x, y) N(y)
Media are media of niches and retainers are retainers of niches.
Parts
A13 M(x, y) P(z, x) M(z, y)A14 R(x, y) P(z, x) R(z, y)
The parts of a medium for a given niche are themselves media for that niche and the parts of a retainer are themselves retainers.
A15 N(x) x = z(M(z, x) R(z, x))
Niches have no parts other than media and retainers.
Retainers and boundaries
A16 R(z, x) B(z, x)
Retainers are boundaries of niches (though not all boundaries of niches are retainers).
A17 N(x) zM(z, x)
Every niche has a medium (though a niche may lack a retainer).
A18 m(x) = m(y) x = y
No two niches have the same medium (though we leave it open whether two niches can have the same retainer).
Retainers and tenants
A19 N(x, y) R(z, x) C(z, y)
Retainers and tenants are not connected to each other, i.e., they do not share any physical parts or boundaries (for they are in every case separated by a medium.)
A20 M(z, x) R(w, x) EC(l(z), l(w))
The location of a retainer is externally connected (i.e., connected without overlap) to the location of the medium.
Axioms
A2' N(x, y) IP(l(y), l(m(x) + y))It is the medium of an occupied niche that surrounds
the tenant.
A3' N(x, y) C(m(x), y)It is the medium of an occupied niche that is connected
to the tenant. This actually follows from A3 in view of A19.
Axioms
A5' N(x) Cn(m(x))The medium of a niche is self-connected (though it
need not be compact, i.e., fill the entire environing hole: consider the bat flying in the bear’s cave).
A7' N(x) Rg(m(x))The medium of a niche is regular.
Theorems
T1 N(x) y(N(x, y) R(y, x))Every niche has either a tenant or a retainer. This is a
consequence of A10.
T2 M(x, y) z(N(y, z) R(z, y))Every medium requires either a tenant or a retainer.
This follows from T1 via A12.
Theorems
T3 M(z, x) P(z, x)
T4 R(z, x) P(z, x) Media and retainers are parts of niches. More
generally:
T5 M(x, y) P(z, x) P(z, y)
T6 R(x, y) P(z, x) P(z, y) All parts of a medium and all parts of a retainer are
parts of the relevant niche.
Theorems
T7 N(x, y) M(y, x)
T8 N(x, y) R(y, x)The tenant of a niche is neither a medium nor a retainer
thereof.
T9 M(z, x) R(w, x) EC(z, w)
The retainer of a niche is externally connected to the medium.
T10 R(z, x) B(z, m(x))
Retainers are boundaries of media.
Against multiplication of niches
T11 R(x, y) N(y – x)
A niche minus (part of) its retainer is not a niche.
This excludes the possibility that the difference between two niches might lie entirely in their retainers, which would result in an undue multiplication of niches with what are putatively the same boundaries.
Open Problems
X1 N(x, y) N(x', y') N(x + x', y + y') Mereological summing of niches is never additive.
cats whose niches come together to form a new, fused niche: the new niche is not just the mereological sum of the two separate niches;
for even assuming that the fiat boundaries of the two niches survive the fusion and continue to exist within the interior of the new niche, they are still not a part of it but are rather extrinsic to it.
Open Problems
X2 M(x, y) B(z, x) R(z, y) B(z, t(x))The boundaries of a medium are either retainers of the niche or boundaries of the tenant.
This would only be true if ‘B’ were understood as standing for physical boundaries, and only if one assumed that a medium has no holes except for the central holes occupied by the tenants.
(But consider again the bat in the bear’s cave, or a cage floating in the sea through which fish can swim.)
Open Problems
X3 M(x, y) B(z, x) EC(z, x)A medium never contains its own physical boundaries.
X4 B(b(m(x)), x)
Any boundary of the medium of a niche is a boundary of the niche itself.
This is false if we consider that the medium need not fill the environing hole completely. (The bat flying in the cave would not be part of the medium of the bear’s niche, yet the surface of the bat would not be part of the retainer either.)