1 From Aristotle to Analytic Metaphysics – From Frege to Tarski: A Critical Introduction to Ontology and First-Order Logic Barry Smith

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From Aristotle to Analytic Metaphysics – From Frege to Tarski:

A Critical Introduction to Ontology and First-Order Logic

Barry Smith

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Aristotle

author of The Categories

Aristotle

3

From Species to Genera

canary

animal

bird

4

Species Genera as Tree

canary

animal

bird fish

ostrich

5

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

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canary

animal

bird

7


animal

bird

canarycanary

8

Species Genera as Tree

canary

animal

bird fish

ostrich

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Species-Genera as Map/Partition

animal

bird

canary

ostrich

fish

canary

10

If Aristotelian realism is right,

then such partitions are transparent to the reality beyond

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Tree and Map/Partition

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Alberti’s Grid

c.1450

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Coarse-grained Partition

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Fine-Grained Partition

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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)

16

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

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Question:

what other sorts of partitions have this feature of transparency?

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

Answer:

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Aristotle

the ontologist of common-sense reality

Aristotle

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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

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

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The Empty Mask (Magritte)

mama

mouse

milk

Mount Washington

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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

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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

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birds

ostrich

Natural categories have borderline cases

sparrow

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... they have a kernel/penumbra structure

kernel of focal

instances

penumbra of borderline cases

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animal

bird

canary

ostrich

fish

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

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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

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but more:

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in addition to objects (substances),

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

cow man rock planet

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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

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An accident

= what holds of a substance per accidens

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

Nine Accidental Categories

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= relations of inherence(one-sided existential dependence)

John

hunger

Substances are the bearers of accidents

34

Both substances and accidents

instantiate universals at higher and lower levels of generality

35

siamese

mammal

cat

organism

substancespecies, genera

animal

instances

frog

36

Common nouns

pekinese

mammal

cat

organism

substance

animal

common nouns

proper names

37

siamese

mammal

cat

organism

substancetypes

animal

tokens

frog

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Our clarification

accidents to be divided into

two large and essential distinct families of

QUALITIES

and

PROCESSES

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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)

40

Substance universals

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

cow man rock planetVW Golf

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Quality universals

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

red hot suntanned spinningClintophobic Eurosceptic

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Process universals

reflect invariants in the spatiotemporal world taken as an atemporal whole

football match

course of disease

exercise of function

(course of) therapy

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Processes and qualities, too, instantiate genera and species

Thus process and quality universals form trees

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Accidents: Species and instances

quality

color

red

scarlet

R232, G54, B24

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

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substance

one substantial categoryJohn, man

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

accidents

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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)

John

accidents

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substance


accidentsBearers

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substance


accidents

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

Bearers

hunger

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s

substance

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Substance + Accident = State of Affairs

setting into relief

States of Affair

s

51

instances

Prototypicality among instances too

albino frogalbino frog

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Aristotle 1.0

an ontology recognizing:substance tokensaccident tokenssubstance typesaccident types

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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)

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Aristotle’s Ontological Square

Substantial Accidental

Second substance

man

cat

ox

Second accident

headache

sun-tan

dread

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

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Second substance

man

cat

ox

Second accident

headache

sun-tan

dread

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

56



Second substance

man

cat

ox

Second accident

headache

sun-tan

dread

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

57



Second substance

man

cat

ox

Second accident

headache

sun-tan

dread

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

58



Second substance

man

cat

ox

Second accident

headache

sun-tan

dread

First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

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Some philosophers

accept only part of this ontology

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


Attributes

F, G, R

Individuals

a, b, c

this, that

Uni

vers

alP

artic

ular

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Bicategorial Nominalism


First substance

this man

this cat

this ox

First accident

this headache

this sun-tan

this dread

Uni

vers

alP

artic

ular

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Process Metaphysics


Events

Processes“Everything is flux”

Uni

vers

alP

artic

ular

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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?

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

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well, what about artefacts ?

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Standard Aristotelian theory of artefacts:

artefacts are mereological sums of substances

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Positive and negative parts

positivepart

negativepartor hole

(made of matter)

(not made of matter)

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

Nine Accidental Categories

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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)

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What is missing from Aristotle?

Gibson: affordancesniches

Barker: behavior settings

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The metaphysics of holes

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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

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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)

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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

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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

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Both scientific partitions and common-sense partitions

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

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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?

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Aristotle 2000

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Universe/Periodic Table

animal

bird

canary

ostrich

fishfolk biology

partition of DNA space

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Universe/Periodic Table

animal

bird

canary

ostrich

fish

both are transparent partitions of one and the same reality

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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

82

relative hylomorphism

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

(Aristotelian ontological zooming)

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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

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Perspectivalism

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

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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

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all express partitions which are transparent,

at different levels of granularity,

to the same reality beyond

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what happens when a fringe instance arises ?

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what happens when a fringe instance arises ?

Aristotle 1.0: you shrug your shoulders

89

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

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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

91

The Empty Mask (Magritte)

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Edmund Husserl

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Logical Investigations¸1900/01

the theory of part and whole

the theory of dependence

the theory of boundary, continuity and contact

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Formal Ontology

(term coined by Husserl)

the theory of those ontological structures

(such as part-whole, universal-particular)

which apply to all domains whatsoever

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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

96

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

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Formal Ontology and Symbolic Logic

Great advances of Frege, Russell, Wittgenstein

Leibnizian idea of a universal characteristic

…symbols are a good thing

98

Warning

don’t confuse Logical with Ontological Form

Russell

Part-whole is not a logical relation

99

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

100

Logic took a wrong turn

101

Logic took a wrong turn

102

Tarski, Carnap, Putnam, Sowa, Gruber:

Forget reality!

Lose yourself in ‘models’!

103

IFOMIS Ontology

is an ontology of reality

Standard Information Systems Ontologies

are ontologies of mere 'models'

104

Standard Information Systems Ontologies:

programming real ontology into computers is hard

therefore: we will simplify ontology

and not care about reality at all

105

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

106

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

107

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

108

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)

109

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’

110

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

FG stands for a property

and so on

111

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

112

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)

113

What about sets?

114

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.

115

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.

116

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.

117

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.

118

Set theory

allows unrestricted (Boolean) combinations

therefore gives as far more objects than we need

{all red things, the number 6}

119

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.

120

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?

121

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.

122

Other problems

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

123

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?

124

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:

125

1.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; indeed standard mathematical oppositions, such as that between a dense and a continuous series, here find no application.

126

2. The set-theoretical construction of the continuum is predicated on the highly questionable thesis that out of unextended building blocks an extended whole can somehow be constructed. The experienced continuum, in contrast, is organized not in such a way that it would be built up out of particles or atoms, but rather in such a way that 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.

127

3. 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).

128

4. The application of set theory to a subject-matter 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.

129

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

130

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

131

Mereology

allows a nicer treatment of both plurals and mass nouns than set theory.

132

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

133

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."

Documents

1 From Aristotle to Analytic Metaphysics – From Frege to Tarski: A Critical Introduction to Ontology and First-Order Logic Barry Smith