The language of set theory is a FOLWUT language

The language of set theory is a FOLWUT language
VT The language of set theory is a FOLWUT language

Against Fantology: The Case of Properties
Barry Smith Ontologie Eins Null

Armstrong’s spreadsheet ontology

F G H I J K L M N O P Q R S T U V a b c d e f g h i j k

F G H I J K L M N O P Q R S T U V a x b c d e f g h i j k

F G H I J K L M N O P Q R S T U V a x b c d e f g h i j k and so on …

Fantology The doctrine, usually tacit, according to which ‘Fa’ (or ‘Rab’) is the key to ontological structure The syntax of first-order predicate logic is a mirror of reality (Fantology a special case of linguistic Kantianism: the structure of language is they key to the structure of [knowable] reality)

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

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

Formal Ontology and Symbolic Logic
Great advances of Frege, Russell, Wittgenstein, Peano (in logic, and in philosophy of mathematics) Leibnizian idea of a universal characteristic …symbols are a good thing

But Russell, Wittgenstein, Armstrong …
Entails is a logical relation Part-whole is an ontological relation First mistake of fantology: All form is logical form

First-order logic F(a) R(a,b) F(a) v R(a,b)
Either a F’s or a stands in R to b

Standard semantics F stands for a property a stands for an individual
properties belong to Platonic realm of forms or properties are sets of individuals for which F(a) is true

Armstrong Departs from fantology with his Aristotelian doctrine of universals as immanent to particulars

Fantology The forms F(a) and R(a,b) are the basic clue to ontology
(Confusion of logical form and ontological form) Armstrong recognizes that semantics and epistemology have no implications for ontology

... but He is nonetheless a prisoner of fantological syntax

Fantology infects computer science, too
here I will concentrate on the role of fantology within analytical metaphysics

Fantology Works very well in mathematics
Platonist theories of properties here are very attractive

Fantology Fa All generality belongs to the predicate
‘a’ is a mere name Contrast this with the way scientists use names: The electron has a negative charge DNA-Binding Requirements of the Yeast Protein Rap1p as selected In Silico from Ribosomal Protein Gene Promoter Sequences

For extreme fantologists ‘a’ leaves no room for ontological complexity
Hence: reality is made of atoms Hence: all probability is combinatoric All true ontology is the ontology of ultimate universal furniture – the ontology of a future, perfected physics Fantology cannot do justice to the existence of different levels of granularity of reality Thus fantology is conducive to reductionism in philosophy

Fantology Tends to make you believe in some future state of ‚total science‘ when the values of ‚F‘ and ‚a‘, all of them, will be revealed to the elect (A science as a totality of propositions closed under logical consequence)

Armstrong The ontology of the basic furniture of reality – must be expressible in F(a), R(a,b) form Armstrong’s original definition of truthmaking: the state of affairs that p makes q true := p & necessarily(p  q) p ranges over truths of basic science q ranges over all truths

For the fantologist “(F(a)”, “R(a,b)” … is the language for ontology
This language reflects the structure of reality The fantologist sees reality as being made up of atoms plus abstract (1- and n-place) ‘properties’ or ‘attributes’ Truthmaker theory should set us free from this mistake

Fantology Fa The particular corresponds to a bare name
noumenal view of particulars (distinction between thin and thick particulars) aversion to idea of substances as spatially extended and spatially located

Fantology Fa noumenal view of particulars
Cf. Wittgenstein’s Tractatus (doctrine of simples)

Fantalogy Fa Quine’s distinction between ontology and ideology
physical objects do not instantiate universals; they are just occupied regions of spacetime predicates are just ideology (no singular terms for universals)

Fantology All form is logical form All necessity is logical necessity
Cf. Wittgenstein‘s doctrine of the independence of states of affairs

Fantology Fa To understand properties is to understand predication
(effectively in terms of functional application à la Frege)

Contrast Aristotle Predication in the category of substance:
John is a man, Henry is an ox Predication in the category of accident: John is hungry, Henry is asleep

For Fantology no predication in the category of substance (or the two types of predication are confused) Armstrong: property universals are all we need no need for kind universals (Armstrong’s four-dimensionalism implies that there are no substances)

There is only one form of states of affairs
if there was predication in the category of substance, this would mess up Armstrong’s doctrine of states of affairs

Armstrong’s own view State of affairs = Substance + universals
Substances are the locus of particularity Universals explain invariance/similarity (Both particulars and universals are abstractions from states of affairs)

Fantology Fa yields the form of the basic ingredients of reality
Thick particulars: a + F + G + H + … Thin particulars: a (‘irreducible particularity’)

For Armstrong tropes are a bad theory for understanding states of affairs (for understanding how the basic ingredients of reality are joined together) For Armstrong tropes are congealed states of affairs (Propositions of the form ‘Fa’ are the key to basic reality)

Fantology Fa: This should be the form of laws of nature (things + universal powers) (not, for instance, differential equations) Therefore, again, a noumenal view of science Armstrong not able to name even one example of a really existing univeral or of a really existing particular Compare again Wittgenstein

Fantology leads not only to Armstrong’s atoms + properties view of the basic ingredients of reality but also to trope bundle views (where the a is deleted, and the F, G, H… are seen as having particularity) Compare: Leibniz’s monadology (each monad is a bundle of concepts)

Fantology Fa No clear way to deal with time
Armstrong talks not of processes but of particulars gaining or losing properties But gaining and losing are themselves processes

Booleanism if F stands for a property and G stands for a property then
F&G stands for a property FvG stands for a property not-F stands for a property FG stands for a property and so on

Strong Booleanism There is a complete lattice of properties:
self-identity FvG F G F&G non-self-identity

Strong Booleanism There is a complete lattice of properties:
self-identity FvG not-F F G not-G F&G non-self-identity

Booleanism responsible, among other things, for Russell’s paradox
Armstrong free from Booleanism With his sparse theory of properties

That Lewis and Armstrong
arrived at their sparse view of properties against the solid wall of fantological Booleanist orthodoxy is a miracle of modern intellectual history analogous to a 5 stone weakling climbing up to breathe the free air at the top of Mount Everest with 1000 ton weights attached to his feet

leading them back, on this point,
to where Aristotelians were from the very beginning

Fantology (given its roots in mathematics)
has no satisfactory way of dealing with time hence leads to banishment of time from the ontology (as in Armstrong’s four-dimensionalism)

A better view In order to do justice to time we need to recognize both properties and processes better: do not use the word ‘property’ at all, talk rather of SPQR entities (both universals and instances) and of process-universals and process-instances

SPQR states, powers, qualities, roles,
functions, dispositions, plans, shapes … Plus Processes = the expressions, realizations of all of these things in time

The (Aristotelian) Ontological Sextet
Substances SPQR entities Processes Universals Substance-universals SQPR-universals, Process-types Particulars Individual Substances SPQR-instances (Tropes…) Process-tokens

Armstrong´s view: Properties Substances SPQR entities Processes
Universals Properties Particulars

The trope view: Tropes, bundles Substances SPQR entities Processes
Universals Particulars Tropes, bundles

The set-theoretical view:
Substances SPQR entities Processes Universals Sets Particulars Elements

Armstrong Property universals are all we need
No need to distinguish kind universals No need to distinguish predications in the category of substance from predications in the category of accident

Substances SPQR entities Processes Universals Substance-universals SQPR-universals, Process-types Particulars Individual Substances SPQR-instances (Tropes…) Process-tokens

Gene Ontology Cellular Component Ontology: subcellular structures, locations, and macromolecular complexes; examples: nucleus, telomere Substances Molecular Function Ontology: tasks performed by individual gene products; transcription factor, DNA helicase SPQR Biological Process Ontology: broad biological goals accomplished by ordered assemblies of molecular functions; mitosis, purine metabolism Processes

Fantology implies a poor treatment of relations
R(a,b) in terms of adicity What is the adicity of your headache (A relation between your consciousness and various processes taking place in an around your brain) ?

Fantology implies a neglect of environments
John kisses Mary always in some environment (= roughly, in some spatial region: a room, a car …) Spatial regions are, like substances, three-dimensional endurants

Fantology leads you to talk nonsense about family resemblances

Fantology emphasizes the linguistic over the perceptual/physiognomic
(the digitalized over the analogue)

The limitations of fantology
lead one into the temptations of possible world metaphysics, and other similar fantasies

Fantology leads one to talk nonsense about possible worlds
Definition: A possible world W is a pair (L,D) consisting of a set of first-order propositions L and a set of ground-level assertions D. … Informally, the set L is called the laws of W, and the set D is called the database of W. Other informal terms might be used: L may be called the set of axioms or database constraints for W. (John Sowa)

Substances SPQR entities Processes Universals Substance-universals SQPR-universals, Process-types Particulars Individual Substances (including environments) SPQR-instances (Tropes…) Process-tokens

The set-theoretical view:
Substances SPQR entities Processes Universals Sets Particulars Elements

Fantology is a form of linguistic Kantianism Semantic Fantology
is a form of set-theoretical Kantianism The [knowable] world = the [set-theoretic] model of a formal theory

Arguments against Set Theory
Lesniewski’s Argument: Even set theorists do not understand their own creations; thus they do not know how one important family of sets (the set of real numbers, for example) relates in size to other sets (the set of natural numbers, for example). Still no generally accepted correct axiomatization of set theory, Questions re Axiom of Choice, etc.

Set theory is Booleanism unremediated
Booleanism without any remediating features whatsoever

There are skew partitions (true) of the same reality
for example reflecting different granularities of analysis. If we identify entities in the world with sets, we cannot do justice to the identity of one and the same object as partitioned on different levels. Mereology, in contrast, can allow the simultaneous truth of: An organism is a totality of cells. An organism is a totality of molecules. France is the totality of its 7 regions. France is the totality of its 116 provinces.

Dominance of set-theoretic ontology as an account of classes
means that there is no analytic-philosophical treatment of multi-variate statistical classification in spite of the fact that this is the major approach to classification in all sorts of natural and social sciences

The application of set theory to a subject-matter
presupposes the isolation of some basic level of Urelemente, which make possible the simulation of the structures appearing on higher levels by means of sets of successively higher types.

But there is no such basic level of Urelemente in many spheres to which we might wish to direct ontological analysis, and in many spheres there is no unidirectional (upward) growth of complexity generated by simple combination.

Set theory reduces all complexity to combination or unification
Set theory is a general theory of the structures which arise when objects are conceived as being united together ad libitum on successively higher levels, each object serving as member or element of objects on the next higher level.

Set theory is of course of considerable mathematical interest
It is however an open question whether there is any theoretical interest attached to the possibility of such ad libitum unification from the perspective of ontology. For the concrete varieties of complexity which in fact confront us are subject always in their construction to quite subtle sorts of constraints, constraints which vary from context to context.

Set theory allows unrestricted (Boolean) combinations
therefore gives as far more objects than we need {all red things, the number 6}

Sets are abstract entities
Sets are timeless (they do not change) Thus a philosopher who countenances them in his ground-floor ontology has already renounced the advantages of a theory which is committed only to changing realia. He is thereby left with the problem of connecting up the abstracta he countenances with the real entities with which they are in different ways associated.

Against Set Theory as a Vehicle for Semantics
There are some who would argue that we can understand a theory (for example in logic) only when we have given a set-theoretic semantics for that theory. (This is rather like saying that we can understand French only when we have translated it into English.) And how, on this basis, can we understand the language of set theory itself?

Truth for empirical sentences
has classically been understood in terms of a correspondence relation (i.e. of some sort of isomorphism) between a judgment or assertion on the one hand and a certain portion of reality on the other. But reality evidently does not come ready-parcelled into judgment-shaped portions Hence practitioners of logical semantics have treated not of truth as such (understood as truth to an autonomous reality), but of truth in a model, where the model is a specially constructed set-theoretic reality-surrogate.

Other problems If sets don't change, then a set-theoretical ontology cannot do justice the causal-historical continuous order Since sets divide the world into elements (points) this implies a certain unfaithfulness to boundary phenomena/continua Can’t do justice to gradations/prototypes

Mereology can deal more adequately with real-world collections
Consider the collection of trees that is a certain forst. What is its cardinality? Are two trees that share a common root system one or two?

Mereology can deal more adequately with fields
(e.g. in quantum field theory) since it does not presuppose the isolation of urelemente/atoms at the bottom of a structural hierarchy

The standard set-theoretic account of the continuum
initiated by Cantor and Dedekind and contained in all standard textbooks of the theory of sets, will be inadequate for at least the following reasons:

The experienced continuum
does not sustain the sorts of cardinal number constructions imposed by the Dedekindian approach. The experienced continuum is not isomorphic to any real-number structure; standard mathematical oppositions, such as that between a dense and a continuous series, here find no application.

Set theory can yield at best a model
of the experienced continuum and similar structures, not a theory of these structures themselves (for the latter are after all not sets).

The experienced continuum
is in every case a concrete, changing phenomenon, a phenomenon existing in time, a whole which can gain and lose parts.

The application of set theory
to a subject-domain presupposes the isolation of some basic level of Urelemente in such a way as to make possible a simulation of all structures appearing on higher levels by means of sets of successively higher types.

Theory of the continuum
Set theory: out of unextended building blocks an extended whole can somehow be constructed. But the experienced continuum is not organized out of particles or atoms, rather, the wholes, including the medium of space, come before the parts which these wholes might contain and which might be distinguished on various levels within them.

Set theory leads to paradoxes
In mereology, paradoxes do not arise, since every collection is part of itself, and there cannot be a collection that is not a part of itself

The alternative to fantology
must take the spatiality and materiality and modular complexity of substances seriously Mereology plus Granularity plus theory of spatial extension ‘a’ refers to something that is complex

Mereology allows a nicer treatment of both plurals and mass nouns than set theory (but mereology, too, has problems dealing with time, and with granularity)

Mereology is much simpler than set theory
Whereas set theory has two distinct operators: element-of and subset-of, mereology has only one basic operator: part-of

Mereology makes no distinction between an individual and a singleton set
nor between different ways of building up collections by level of nesting: {a,b,c} is identical to {a, {{{b}}, {c}}}. Nelson Goodman: "No distinction of individuals without distinction of content."

How are the 6 categories of entity related together?
Via FORMAL RELATIONS such as instantiation, part-whole, expression (between a function and a process) …

A better syntax variables x, y, z … range over
universals and particulars in all 6 categories predicates stand only for FORMAL relations such as instantiates, part-of, connected-to, is-a-boundary-of, is-a-niche-for, etc. FORMAL relations are not extra ingredients of being (compare jigsaw puzzle pieces and the relations between them)

This suggests a new syntax:
=(x,y) Part(x,y) Inst(x,y) Dep(x,y) Isa(x,y) John is wise: Inst(John, wisdom) John is a man: Isa(John, man)

Compare the syntax of set theory
(x,y) one (formal) predicate

Compare the syntax of set theory
Basic Formal Ontology (BFO) has the syntax of set theory but with a PLURALITY of formal relations Note that logic gives us no clue as to what these are (they must include: location in space, location at a time …)

Compare the syntax of description logics
isa(x,y) one formal predicate

Compare the syntax of description logics
isa means: is an instance of is a sub-class of (is subsumed by) sometimes also: may be a is a part of (all symbolized by means of a single relational constant)

Compare Davidson’s treatment of events
New syntax: =(x,y) Part(x,y) Inst(x,y) Dep(x,y) Compare Davidson’s treatment of events Does(John,e)

Fantology All form is logical form
To understand how the world hangs together you need to understand ... and, or, not, all, some ... No. You need formal-ontological relations like partial identity, spatial location, temporal location, instantiation, ...

But what ARE the formal relations?
Clue: Anatomy vs. Physiology SNAP vs. SPAN Synchronic vs. diachronic ontology

Different ontological perspectives
SNAP vs SPAN Universals vs. Particulars Different levels of granularity: molecular, cellular, organism ...

A hypothesis (first rough version)
Formal relations are those relations which are not captured by either SNAP or SPAN because they traverse the SNAP-SPAN divide they glue SNAP and SPAN entities together

Formal relations are those relations which are not captured by either an inventory of universals or an inventory of particulars because they traverse the universal-particular divide they glue universals and particulars together ---- above all instantiation

Formal relations are those relations which are not captured by an ontology on any single level of granularity because they traverse the granular divide they glue together entities on different levels of granularity --- above all parthood

This generates a first list of formal relations,
e.g. dependence, but we find some of these relations also within SNAP or within SNAP

The idea (modified version)
Formal relations are the relations that hold SNAP and SPAN entities/ontologies together and analogous relations … they come for free, they do not add anything to being = they are links between categories

Example: Ontological Dependence
(SPAN, SNAP): process->substance The erosion of the rock necessitates the existence of the rock (SNAP, SNAP): SPQR->substance The token redness of the sand necessitates the existence of the sand

Generating a typology Two main types of formal relations:
inter-ontological („transcendental“): obtain between entities of different ontologies intra-ontological: obtain between entities of the same ontology (intra-SNAP, intra-SPAN)

Three parameters: - the arity of the relation
- the types of the relata, expressed as an ordered list, called the signature of the relation - the formal nature of the relation

Principal Signatures In the binary case: SNAP-SNAP
- (SNAPi, SNAPi), i = i - (SNAPi, SNAPi), i < j, i > j SPAN-SPAN SNAP-SPAN SPAN-SNAP

Transtemporal relations
Examples: Genidentity (transtemporal generalization of identity/part-whole) Successive causality

Genidentity The such-as-to-have-come-forth-from relation.
Signature: SNAPi-SNAPj Cut a chunk of matter in two, the sum of the remaining pieces is genidentical to the chunk before cutting Also SPAN-SPAN? Is there a form of genidentity among processes?

Successive Causality SPAN-SPAN: Process causation
One process causes another process SPAN-SNAP: Causal repercussion A process results in the modification of a substance (always mediated by process causation) SNAP-SPAN: Agent causation A substance produces causally a process SNAP-SNAP: Causal origin One substance is the causal origin of another (mediated by other types of causal relations)

do not apply on all levels of granularity Successive Causality
SPAN-SPAN: Process causation SPAN-SNAP: Causal repercussion SNAP-SPAN: Agent causation SNAP-SNAP: Causal origin do not apply on all levels of granularity

Our main target: Temporally extended relations
Participation (holds between a substance and a process such as an action or a life or history) Realization (holds between SPQR… entities and their SPAN expressions)

Substance->Process
PARTICIPATION (a species of dependence)

Participation (SNAP-SPAN)
A substance (SNAP) participates in a process (SPAN) A runner participates in a race An organ participates in a sickness

Axes of variation direct/mediated benefactor/malefactor
activity/passivity (agentive) direct/mediated benefactor/malefactor (conducive to existence) [MEDICINE]

SNAP-SPAN Participation Perpetration (+agentive) Influence Patiency
Initiation Termination Facilitation Perpetuation Hindrance Mediation

Substances Mesoscopic reality is divided at its natural joints
into substances: animals, bones, rocks, potatoes, brains

The Ontology of Substances
Substances form natural kinds (universals, species + genera)

Processes Processes merge into one another
Process kinds merge into one another … few clean joints either between instances or between types

Processes t i m e

Nouns and verbs Substances and processes Continuants and occurrents
Endurants and perdurants In preparing an inventory of reality we keep track of these two different categories of entities in two different ways

Substances and processes
t i m e process demand different sorts of inventories

Substances demand 3-D partonomies
space

Processes demand 4D-partonomies
t i m e

Processes a whistling, a blushing, a speech
a run, the warming of this stone

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

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

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?

The Four-Dimensionalist Ontology (SPAN)
t i m e

The Time-Stamped Ontology (SNAP)
here time exists outside the ontology, as an index or time-stamp

SNAP and SPAN Substances+SPQR, and processes
Continuants, and occurrents Endurants, and perdurants In preparing an inventory of reality we keep track of these two different categories of entities in two different ways

Fourdimensionalism denies this
– time is just another dimension, analogous to the three spatial dimensions – only processes exist – substances are analyzed away as worms/fibers within the four-dimensional process plenum – there is no change

There are no substances
Bill Clinton does not exist Rather: there exists within the four-dimensional plenum a continuous succession of processes which are similar in Billclintonizing way

is right in everything it says
Fourdimensionalism is right in everything it says But incomplete

It needs to be supplemented
Cf. Quantum mechanics: particle vs. wave ontologies

Two Orthogonal, Complementary Perspectives
SNAP and SPAN

SNAP and SPAN the tumor and its growth the surgeon and the operation
the virus and its spread the temperature and its rise the disease and its course the therapy and its application

Axiom Part-relations never traverse the SNAP-SPAN divide
No process is ever part of a substance and vice versa No quality is ever part of a process and vice versa Process and function belong to two different orders of being

SNAP and SPAN SNAP entities - have continuous existence in time
- preserve their identity through change - exist in toto if they exist at all SPAN entities - have temporal parts - unfold themselves phase by phase - exist only in their phases/stages

SNAP vs. SPAN SNAP: a SNAPshot ontology of endurants existing at a time SPAN: a four-dimensionalist ontology of processes

Your life is 4-dimensional
You are a substance Your life is a process You are 3-dimensional Your life is 4-dimensional

Three kinds of SNAP entities
Substances SPQR… entities Spatial regions, contexts, niches, environments

one-place SPQR… entities
tropes, individual properties (‘abstract particulars’) a blush my knowledge of French the whiteness of this cheese the warmth of this stone

relational SPQR… entities
John Mary love stand in relations of one-sided dependence to a plurality of substances simultaneously

SNAP entities provide the principles of individuation/segmentation for SPAN entities No change without some THING or QUALITY which changes identity-based change

Processes, too, are dependent on substances
One-place vs. relational processes One-place processes: getting warmer getting hungrier

Examples of relational processes
kissings, thumps, conversations, dances, Such relational processes join their carriers together into collectives of greater or lesser duration

SNAP and SPAN ontologies are partial only
Each is a window on that dimension of reality which is visible through the given ontology (Realist perspectivalism)

SNAP: Entities existing in toto at a time

SPAN: Entities extended in time

Relations between SNAP and SPAN
SNAP-entities participate in processes they have lives, histories

SPQR… entities and their SPAN realizations
the expression of a function the exercise of a role the execution of a plan the realization of a disposition

SNAP function role plan disposition therapy disease

expression exercise execution realization application course SPAN

Special role of formal relations
Only they are represented by predicates in the first-order logic representation of our ontology But what are formal relations?

Different ontological perspectives
SNAP vs SPAN Universals vs. Particulars Different levels of granularity: molecular, cellular, organism ...

Formal relations are those relations which are not captured by either SNAP or SPAN because they traverse the SNAP-SPAN divide they glue SNAP and SPAN entities together