Part 2 Tools of Ontology: Mereology, Topology, Dependence

Ontological Dependence

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

Ontological Dependence Substances are that which can exist on their own Moments require a support from substances in order to exist

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 A moment of visual extension is mutually dependent on a moment of color 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 OSD(x, y) := SD(x, y)   MSD(x, y) My headache is one-sidedly specifically dependent on me.

Substances and Moments Substances are the bearers or carriers of moments, … moments are said to ‘inhere’ in their substances

Ontological Dependence Substances are such that, while remaining numerically one and the same, they can admit contrary moments at different times … I am sometimes hungry, sometimes not

Substances can also gain and lose parts … as an organism may gain and lose molecules

Types of relations between parts 1. Dependence relations 2. Side-by-sideness relations 3. Fusion relations

Dependence cannot exist without a thinker a thought moment substance

Theory of vagueness Side-by-sideness found among both substances and moments

Fusion Topology

Topology, like mereology, applies both in the realm of substances and in the realm of moments

Mereotopology = topology on a mereological basis

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

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 moment, either

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

interior substance 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.

Moments, 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 moments depend on one substance (as a headache depends upon a head)

Relational moments John Mary kiss stand in relations of one-sided dependence to a plurality of substances simultaneously

Examples of relational moments kisses, thumps, conversations, dances, legal systems Such relational moments join their carriers together into collectives of greater or lesser duration

Mereology ‘Entity’ = absolutely general ontological term of art embracing at least: all substances, moments, and all the wholes and parts thereof, including boundaries

Totality of entities = domain of our variables but this domain falls into: 1. the 4-D domain of processes 2. the successive time-stamped 3- D domains of: substances plus sections through processes

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 AM2x ≤ y  y ≤ x  x = y AM3x ≤ 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

The sum axiom holds unrestrictedly, once the range of variables has been fixed either to the 4-D ontology or to one or other of the time- stamped 3-D ontologies

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 In the 4-D ontology: electricity, Christianity, your body’s metabolism In the 3-D ontology: the Beatles, the Gewandhausorchester, 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)) difference –x :=  z (  O(z, x)) complement

What is a Substance? Bundle theories: a substance is a whole made up of moments as parts. What holds the moments 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 AC1x ≤ c(x) expansiveness AC2 c(c(x)) ≤ c(x) idempotence AC3 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) boundary interior x

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