1 From Aristotle to Analytic Metaphysics – From Frege to Tarski: A Critical Introduction to...

From Aristotle to Analytic Metaphysics – From Frege to Tarski:

A Critical Introduction to Ontology and First-Order Logic

Barry Smith

Aristotle

author of The Categories

Aristotle

From Species to Genera

canary

animal

Species Genera as Tree

canary

animal

bird fish

ostrich

Species-genusgenus trees can be represented also as map-like partitions

canary

animal

canarycanary

Species Genera as Tree

canary

animal

bird fish

ostrich

Species-Genera as Map/Partition

animal

canary

ostrich

canary

If Aristotelian realism is right,

then such partitions are transparent to the reality beyond

Tree and Map/Partition

Alberti’s Grid

c.1450

Coarse-grained Partition

Fine-Grained Partition

Scientific theories

comprehend in their underlying category systems veridical partitions of reality

often there are many veridical partitions of reality,

cross-cutting each other,

differing only in nuances)

What is a gene?

GDB: a gene is a DNA fragment that can be transcribed and translated into a protein

Genbank: a gene is a DNA region of biological interest with a name and that carries a genetic trait or phenotype

(from Schulze-Kremer)

GO does not tell us which of these is correct, or indeed whether either is correct, and it does not tell us how to integrate data from the corresponding sources

Question:

what other sorts of partitions have this feature of transparency?

the partitions of common sense (folk biology, folk physics, folk psychology ...)

Answer:

Aristotle

the ontologist of common-sense reality

Aristotle

The world we grasp in natural language

= the world as apprehended via that conceptualization we call common sense

= the normal environment (the niche) shared by children and adults in everyday perceiving and acting

The world of mothers, milk, and mice ...

The Empty Mask (Magritte)

Mount Washington

our common-sense partition of the world of common sense is transparent

(common sense, like science, is [mostly*] true)

mothers exist ...

* “mostly” because of the problem of vagueness

Problem of vagueness solved

by recognizing that our categories apply to reality in such a way as to respect an opposition

... between standard or focal or prototypical instances

... and non-standard or ‘fringe’ instances

ostrich

Natural categories have borderline cases

sparrow

... they have a kernel/penumbra structure

kernel of focal

instances

penumbra of borderline cases

animal

canary

ostrich

every cell in every common-sense partition is subject to this same kernel-penumbra structure:

What is common-sense reality?

the mesoscopic space of everyday human action and perception

– a space centered on objects organized into hierarchies of species and genera

... and subject to prototypicality

but more:

in addition to objects (substances),

which pertain to what a thing is at all times at which it exists:

cow man rock planet

the common-sense world contains also accidents

which pertain to how a thing is at some time at which it exists:

red hot suntanned spinning

An accident

= what holds of a substance per accidens

quid? substance quantum? quantity quale? qualityad quid? relationubi? placequando? timein quo situ? status/contextin quo habitu? habitusquid agit? actionquid patitur? passion

Nine Accidental Categories

= relations of inherence(one-sided existential dependence)

hunger

Substances are the bearers of accidents

Both substances and accidents

instantiate universals at higher and lower levels of generality

siamese

mammal

organism

substancespecies, genera

animal

instances

Common nouns

pekinese

mammal

organism

substance

animal

common nouns

proper names

siamese

mammal

organism

substancetypes

animal

tokens

Our clarification

accidents to be divided into

two large and essential distinct families of

QUALITIES

PROCESSES

There are universals

both among substances (man, mammal)

and among qualities (hot, red)

and among processes (run, movement)

There are universals also among spatial regions (triangle, room, cockpit)

and among spatio-temporal regions (orbit)

Substance universals

pertain to what a thing is at all times at which it exists:

cow man rock planetVW Golf

Quality universals

pertain to how a thing is at some time at which it exists:

red hot suntanned spinningClintophobic Eurosceptic

Process universals

reflect invariants in the spatiotemporal world taken as an atemporal whole

football match

course of disease

exercise of function

(course of) therapy

Processes and qualities, too, instantiate genera and species

Thus process and quality universals form trees

Accidents: Species and instances

quality

scarlet

R232, G54, B24

this individual accident of redness (this token redness – here, now)

substance

one substantial categoryJohn, man

nine accidental categorieshunger, your hunger, being hungryyour sun-tanyour being taller than Mary

accidents

substance

place (in the Lyceum)

time (yesterday)

position (is sitting)

possession (has shoes on)

action (cuts)

passion (is cut)

quantity (two feet long)

quality (white)

relation (taller than)

accidents

substance

accidentsBearers

substance

accidents

John = relations of inherence(one-sided existential dependence)

Bearers

hunger

substance

Substance + Accident = State of Affairs

setting into relief

States of Affair

instances

Prototypicality among instances too

albino frogalbino frog

Aristotle 1.0

an ontology recognizing:substance tokensaccident tokenssubstance typesaccident types

Not in a SubjectSubstantial

In a SubjectAccidental

Said of a SubjectUniversal, General,Type

Second Substances

man, horse, mammal

Non-substantial Universals

whiteness, knowledge

Not said of a Subject Particular, Individual,Token

First Substances

this individual man, this horse this mind, this body

Individual Accidents

this individual whiteness, knowledge of grammar

Aristotle’s Ontological Square (full)

Aristotle’s Ontological Square

Substantial Accidental

Second substance

Second accident

headache

sun-tan

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Second substance

Second accident

headache

sun-tan

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Second substance

Second accident

headache

sun-tan

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Second substance

Second accident

headache

sun-tan

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Second substance

Second accident

headache

sun-tan

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Some philosophers

accept only part of this ontology

Standard Predicate Logic – F(a), R(a,b) ...

Attributes

F, G, R

Individuals

a, b, c

this, that

Bicategorial Nominalism

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Process Metaphysics

Events

Processes“Everything is flux”

Aristotle 1.0

in fact however we need more than the ontological square

What is missing from Aristotle 1.0 asan ontology of common-sense reality?

Is everything in common-sense reality either a substance or an accident?

well, what about artefacts ?

Standard Aristotelian theory of artefacts:

artefacts are mereological sums of substances

Positive and negative parts

positivepart

negativepartor hole

(made of matter)

(not made of matter)

quid? substance quantum? quantity quale? qualityad quid? relationubi? placequando? timein quo situ? status/contextin quo habitu? habitusquid agit? actionquid patitur? passion

Nine Accidental Categories

Places

For Aristotle the place of a substance is the interior boundary of the surrounding body

(for example the interior boundary of the surrounding water where it meets a fish’s skin)

What is missing from Aristotle?

Gibson: affordancesniches

Barker: behavior settings

The metaphysics of holes

Aristotle 1.5

an ontology ofsubstances + accidents+ holes (and other entities not made of matter)+ fiat and bona fide boundaries+ artefacts and environments

is true

folk biology

Aristotelian folk biology, folk physics, folk psychology, etc., are true of the common-sense world as it currently exists

(they have nothing to offer regarding its pre-history, its long term evolution, its position in the cosmos)

reference vs. theory

They have not much to offer, either, by way of good explanatory theories of the entities in their respective domains,

but they are transparent to those domainsnonetheless

reference realism vs. theory realism

this distinction applied not only to science (against T. S. Kuhn et al.) but also to common sense (against sceptics of various stripes)

the sun exists, and has existed for a long time – the very same object

Both scientific partitions and common-sense partitions

are based on reference-systems which have survived rigorous empirical tests

The $64000 Question

How do those parts and dimensions of reality which we call the common-sense world

... relate to those parts and dimensions of reality which are studied by science?

Aristotle 2000

Universe/Periodic Table

animal

canary

ostrich

fishfolk biology

partition of DNA space

Universe/Periodic Table

animal

canary

ostrich

both are transparent partitions of one and the same reality

many transparent partitions

at different levels of granularity

will operate with species-genus hierarchies

and with an ontology of substances (objects) and accidents (attributes, processes)

along the lines described by Aristotle

relative hylomorphism

substances and accidents reappear in the microscopic and macroscopic worlds of e.g. molecular biology and astronomy

(Aristotelian ontological zooming)

we do not assert

that every level of granularity is structured in substance-accident form -- perhaps there are pure process levels, perhaps there are levels structured as fields

Perspectivalism

PerspectivalismDifferent partitions may represent cuts through the same reality which are skew to each other

An organism is a totality of molecules

An organism is a totality of cells

An organism is a single unitary substance

... all of these express veridical partitions

An organism is a totality of atoms

all express partitions which are transparent,

at different levels of granularity,

to the same reality beyond

what happens when a fringe instance arises ?

Aristotle 1.0: you shrug your shoulders

Aristotle 2000:you go out to find a finer grained partition which will recognize the phenomenon in question as prototypical

The advance of science

is not an advance away from Aristotle towards something better.

Provided Aristotle is interpreted aright, it is a rigorous demonstration of the correctness of his ontological approach

The Empty Mask (Magritte)

Edmund Husserl

Logical Investigations¸1900/01

the theory of part and whole

the theory of dependence

the theory of boundary, continuity and contact

Formal Ontology

(term coined by Husserl)

the theory of those ontological structures

(such as part-whole, universal-particular)

which apply to all domains whatsoever

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

Leibnizian idea of a universal characteristic

…symbols are a good thing

Warning

don’t confuse Logical with Ontological Form

Russell

Part-whole is not a logical relation

for Frege, Russell, Lesniewski,

Wittgenstein, Quine

Logic is a ‘Zoology of Facts’

Formal theories are theories of reality

with one intended interpretation

= the world

tragicallyafter starting off on the right road

Logic took a wrong turn

Tarski, Carnap, Putnam, Sowa, Gruber:

Forget reality!

Lose yourself in ‘models’!

IFOMIS Ontology

is an ontology of reality

Standard Information Systems Ontologies

are ontologies of mere 'models'

Standard Information Systems Ontologies:

programming real ontology into computers is hard

therefore: we will simplify ontology

and not care about reality at all

IFOMIS Strategy

get real ontology right first

and then investigate ways in which this real ontology can be translated into computer-

useable form later

NOT ALLOW ISSUES OF COMPUTER-TRACTABILITY TO DETERMINE THE

CONTENT OF ONTOLOGY

First order logic

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 propertya stands for an individual

properties belong to Platonic realm of forms

orproperties are sets of individuals for which

F(a) is true

Fantology

The forms F(a) and R(a,b) [on either of these understandings] are the basic clue to ontology

(Confusion of logical form and ontological form)

For the fantologist

“(F(a)”, “R(a,b)” … is the language for ontology

The fantologist sees reality as being made up of atoms plus abstract (1- and n-place) ‘properties’ or ‘attributes’

Booleanism

if F stands for a property and G stands for a property

F&G stands for a property

FvG stands for a property

not-F stands for a property

FG stands for a property

and so on

IFOMIS (Aristotelian) perspective

Sparse theory of propertiesor better: non-Boolean theory of

propertiesproperties come in two forms: as types

and as tokens (accidents)or better: do not use the word property at

all, talk rather of quality-universals and quality-instancesprocess-universals and process-instances

IFOMIS syntax

variables x, y, z … range of

universals and particulars

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)

What about sets?

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.

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.

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. This 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 therefore timeless (they don't 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; English is the only intrinsically understandable language.)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.

1 From Aristotle to Analytic Metaphysics – From Frege to Tarski: A Critical Introduction to...

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