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

2 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 genus Species-genus trees can be represented also as map-like partitions

6 From Species to Genera canary animal bird

7 From Species to Genera animal bird canary

8 Species Genera as Tree canary animal bird fish ostrich

9 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

11 Tree and Map/Partition

12 Alberti’s Grid c.1450

13 Coarse-grained Partition

14 Fine-Grained Partition

15 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

17 Question: what other sorts of partitions have this feature of transparency? the partitions of common sense (folk biology, folk physics, folk psychology...) Answer:

18 Aristotle the ontologist of common-sense reality Aristotle

19 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

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

21 The Empty Mask (Magritte) mama mouse milk Mount Washington

22 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

23 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

24 birds ostrich Natural categories have borderline cases sparrow

25... they have a kernel/penumbra structure kernel of focal instances penumbra of borderline cases

26 animal bird canary ostrich fish every cell in every common-sense partition is subject to this same kernel-penumbra structure:

27 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

28 but more:

29 in addition to objects (substances), which pertain to what a thing is at all times at which it exists: cow man rock planet

30 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

31 An accident = what holds of a substance per accidens

32 quid? substance quantum? quantity quale? quality ad quid? relation ubi? place quando? time in quo situ? status/context in quo habitu? habitus quid agit? action quid patitur? passion Nine Accidental Categories

33 = 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 substance species, genera animal instances frog

36 Common nouns pekinese mammal cat organism substance animal common nouns proper names

37 siamese mammal cat organism substance types animal tokens frog

38 Our clarification accidents to be divided into two large and essential distinct families of QUALITIES and PROCESSES

39 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 planet VW Golf

41 Quality universals pertain to how a thing is at some time at which it exists: red hot suntanned spinning Clintophobic Eurosceptic

42 Process universals reflect invariants in the spatiotemporal world taken as an atemporal whole football match course of disease exercise of function (course of) therapy

43 Processes and qualities, too, instantiate genera and species Thus process and quality universals form trees

44 Accidents: Species and instances quality color red scarlet R232, G54, B24 this individual accident of redness (this token redness – here, now)

45 substance one substantial category John, man nine accidental categories hunger, your hunger, being hungry your sun-tan your being taller than Mary accidents

46 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

47 substance Substances are the bearers of accidents accidents Bearers

48 substance Substances are the bearers of accidents accidents John = relations of inherence (one-sided existential dependence) Bearers hunger

49 s substance

50 Substance + Accident = State of Affairs setting into relief States of Affair s

51 instances Prototypicality among instances too albino frog

52 Aristotle 1.0 an ontology recognizing: substance tokens accident tokens substance types accident types

53 Not in a Subject Substantial In a Subject Accidental Said of a Subject Universal, 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)

54 Aristotle’s Ontological Square SubstantialAccidental 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 Universal Particular

55 Aristotle’s Ontological Square SubstantialAccidental 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 Universal Particular

56 Aristotle’s Ontological Square SubstantialAccidental 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 Universal Particular

57 Aristotle’s Ontological Square SubstantialAccidental 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 Universal Particular

58 Aristotle’s Ontological Square SubstantialAccidental 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 Universal Particular

59 Some philosophers accept only part of this ontology

60 Standard Predicate Logic – F(a), R(a,b)... SubstantialAccidental Attributes F, G, R Individuals a, b, c this, that Universal Particular

61 Bicategorial Nominalism SubstantialAccidental First substance this man this cat this ox First accident this headache this sun-tan this dread Universal Particular

62 Process Metaphysics SubstantialAccidental Events Processes “Everything is flux” Universal Particular

63 Aristotle 1.0 in fact however we need more than the ontological square What is missing from Aristotle 1.0 as an ontology of common-sense reality?

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

65 well, what about artefacts ?

66 Standard Aristotelian theory of artefacts: artefacts are mereological sums of substances

67 Positive and negative parts positive part negative part or hole (made of matter) (not made of matter)

68 quid? substance quantum? quantity quale? quality ad quid? relation ubi? place quando? time in quo situ? status/context in quo habitu? habitus quid agit? action quid patitur? passion Nine Accidental Categories

69 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)

70 What is missing from Aristotle? Gibson: affordances niches Barker:behavior settings

71 The metaphysics of holes

72 Aristotle 1.5 an ontology of substances + accidents + holes (and other entities not made of matter) + fiat and bona fide boundaries + artefacts and environments is true

73 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)

74 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 domains nonetheless

75 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

76 Both scientific partitions and common-sense partitions are based on reference-systems which have survived rigorous empirical tests

77 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?

78 Aristotle 2000

79 Universe/Periodic Table animal bird canary ostrich fish folk biology partition of DNA space

80 Universe/Periodic Table animal bird canary ostrich fish both are transparent partitions of one and the same reality

81 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)

83 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

84 Perspectivalism Different partitions may represent cuts through the same reality which are skew to each other

85 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

86 all express partitions which are transparent, at different levels of granularity, to the same reality beyond

87 Coarse-grained Partition what happens when a fringe instance arises ?

88 Coarse-grained Partition 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

90 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)

92 Edmund Husserl

93 Logical Investigations¸1900/01 the theory of part and whole the theory of dependence the theory of boundary, continuity and contact

94 Formal Ontology (term coined by Husserl) the theory of those ontological structures (such as part-whole, universal-particular) which apply to all domains whatsoever

95 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

97 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 tragically after 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 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

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 F  G stands for a property and so on

111 IFOMIS (Aristotelian) perspective Sparse theory of properties or better: non-Boolean theory of properties properties 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-instances process-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 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

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.

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.

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 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.

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."