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

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.

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 John Mary kiss 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 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

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)) difference –x :=  z (  O(z, x)) 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 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 – 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 mind- body 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, non- existent golden mountains, functions across possible worlds, and the like).

Problem cases: Fictional entities? Absences? Holes?

A good category scheme m s h-msh should not be a m i s h-m a 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 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) 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 double- counting

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 regionqualitysubstance parts of spatial regions are always spatial regions

Respect Granularity spatial regionqualitysubstance parts of substances are always substances

Relations crossing the SNAP/SPAN border are not part-relations John’s life substance John physiological processes sustaining in existence

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 G B C D H Good Bad

Counterexample in the realm of artifacts ? urban structures buildings car parks multi-story car-parks

Eliminating counter-examples urban structures buildingsparking 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