Realist Ontologies a work in progress

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Realist Ontologies (a work in progress)
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• Paul Thom

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Contents
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Basic ontological relationships de
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if something is said of a subject both its
name and its definition are necessarily
predicated of the subject.Categories 5, 2a19ff.
Whenever one thing is predicated of another as of
a subject, all things said of what is predicated
will be said of the subject also.Categories 3,
1b10ff.
Axiom 1. If A de B and B de C, A de C
de said of a subject
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Basic ontological relationshipsin
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By in a subject I mean what is in something,
not as a part, and cannot exist separately from
what it is in. Categories 2, 1a24-25.
Colour is present in body, therefore also in an
individual body. Categories 5, 2b1ff.
Axiom 2. If A in B, for no X B in X
Axiom 3. If A in B and B de C, A in C
Axiom 4. If A de B and B in C, A in C
By Axiom 2 it is never the case that both A in B
and B in C. So Theorem 1 is vacuously true.
If A in B and B in A then for some X B in X but
by Axiom 2, if A in B then for no X B in X.
By Axiom 2, if A in A then not A in A.
Theorem 1. If A in B and B in C, A in C
Theorem 2. Not (A in A)
Theorem 3. If A in B, not (B in A)
in inheres in
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Basic ontological relationships ab
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Thus, for example, the grammarian gets his name
from grammar, the brave get theirs from
bravery. Categories 1, 1a14-15
Axiom 5. If A ab B, for no X B ab X
Axiom 6. If A in B, for no X B ab X
By Axiom 5 it is never the case that both A ab B
and B ab C. So Theorem 4 is vacuously true.
If A ab B and B ab A then for some X B ab X but
by Axiom 5, if A ab B then for no X B ab X.
By Axiom 5, if A ab A then not A ab A.
Theorem 4. If A ab B and B ab C, A ab C
Theorem 5. Not (A ab A)
Theorem 6. If A ab B, not (B ab A)
ab is from
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Basic ontological relationships ad
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We call relatives all such things as are said to
be just what they are, of or than other things,
or in some other way in relation to something
else. Categories 7, 6a36ff.
Definition. A sim B iff A ad B and (there is A
iff there is B)
Correlatives. Categories 7,.
ad
Correlatives
Axiom 7. If A ad B, B ad A
Axiom 8. If A ad B and B ad C, A C
sim is naturally simultaneous with
ad stands towards
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BeingsCategorial beings, denominatives
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Definition 1. cat(A) iff for no X A ab X
When things get their name from something, with a
difference of ending, they are called
paronymous. Categories 1, 1a12.
Definition 2. denom(A) iff for some X A ab X
Categorial beings
Denominatives
If A ab B then for no X B ab X Axiom 5, and
hence cat(B) Definition 2.
Theorem 7. If A ab B, cat(B)
denom is a paronym
cat is a categorial being
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BeingsIndividuals, universals
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Things that are individual and numerically one
are, without exception, not said of any
subject. Categories 2, 1b6-7.
Definition 3. indiv(A) iff for no X A de X
Definition 4. univ(A) iff for some X A de X
universals are what is predicated synonymously
of several things, which are what Aristotle calls
said of a substrate. Simplicius Categories
Commentary 45,3.
Universals
Individuals
indiv is an individual
univ is a universal
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BeingsSubstances, accidents
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It is a characteristic common to every substance
not to be in a subject. This is not, however,
peculiar to substance the differentia also is
not in a subject. Categories 5, 3a7-8 21-22.
Definition 5. subst(A) iff cat(A) and for no X A
in X
Definition 6. dif(A) iff for some X,Y A de X and
A ab Y and for no Z Y in Z
He calls accidents in a substrate because,
since they are not autonomous, they require
something else in which to exist. Simplicius,
Categories Commentary 44,28.
Definition 7. acc(A) iff for some X A in X
Substances
Accidents
If A in B then for no X B ab X Axiom 6, so
cat(B) Definition 2. If A in B then for no X B
in X Axiom 2. Thus, if A in B then cat(B) and
for no X B in X, i.e. subst(B) Definition 6.
For that which is in a substrate is in a
substance. Simplicius, Categories Commentary
46,24-30.
Axiom. If acc(A) and indiv(A) and A in B and A in
C, B C
Theorem 8. If A in B, subst(B)
dif is a differentia
subst is a substance
acc is an accident
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The Categories ontologyThe main theorem
A substance that which is called a substance
most strictly, primarily, and most of all is
that which is neither said of a subject nor in a
subject. Categories 5, 2a11-13.
Definition 8. subst1(A) if and only subst(A) and
indiv(A)
Animal is predicated of man and therefore also of
the individual man. Categories 5, 2a36ff.
Axiom 9. If subst(A) and not subst1(A) then for
some X A de X and subst1(X)
If acc(A) then for some B A in B Definition 7.
If A in B then subst(B) Theorem 8. If subst(B)
and not subst1(B) then for some X B de X and
subst1(X) Axiom 9. Therefore if acc(A) then for
some X A in X and subst1(X).
Theorem 9. If acc(A) then for some X A in X and
subst1(X)
The thesis follows from two lemmata that all
substances other than primary substances are
said-of a primary substance Axiom 9, and that
all accidents are present-in a primary substance
Theorem 9.
All the other things are either said of the
primary substances as subjects or present in them
as subjects. Categories 5, 2a34ff.
Theorem 10. If either subst(A) and not subst1(A),
or acc(A), then for some X subst1(X) and either
A de X or A in X
subst1 is a primary substance
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Derived ontological relationshipsNon-relatives
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Definition 9. A was B iff for some X,Y A ab X
and X de Y and B ab Y
All moving things are changing is not to be
taken to mean that this is so absolutely, but
rather while the substance of the moving thing is
moving. Avicenna, Pointers 265.
Definition 10. A est B iff for some X A ab X and
X in B
Footed two-footed are differentiae of
animal. Categories 3, 1b19
Axiom 10. If A was B and B de C, A de C
was wasfi predication (Avicenna)
est is substantially
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Derived ontological relationships Relatives
Definition 11. coltA,Bgt iff for some C,D C ab A,
D ab B and C ad D
A relation is an accident that is truly and
simply one, not requiring a multitude in which it
is, but just one thing only in respect of
another. A correlation is a pair of relations
mutually respecting one another. Kilwardby, De
Natura Relationis 1619-24
Correlation
co
paternity is not of filiation but father is
stands to son and son to father. A relation,
then, is not relative except through the
concrete being which it is in. And so a
relation is relative per accidens. Kilwardby, De
Natura Relationis 522-28.
ad
Correlatives
Definition 12. co1ltA,Bgt iff for some C,D
indiv(ltC,Dgt) and ltC,Dgt de ltA,Bgt and co,ltC, Dgt
co1 are an individual correlation
co are a correlation
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The Trinity Sabelliuss model
God, Father, Son, Holy Spirit
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The Trinity Arriuss model
Father
Son
God
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The Trinity Philoponuss model
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The godhead and substance that is in the adorable
Trinity is one not in reality but only in mind
and abstraction. In this way God is understood as
one, but there are three substances of God, with
the substances and natures being divided in the
hypostases. Thus the Father is another God, the
Son another God, and the Holy Ghost another
God. Michael the Syrian, quoting Philoponus
Father
Son
Spirit
God
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The Trinity Augustines logical model
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There is at least no doubt that God is
substanceDe Trinitate V.i.3
The Holy Spirit insofar as he is properly or
peculiarly called the Holy Spirit is so called
relationship-wise, being referred to both Father
and Son, since the Holy Spirit is the Spirit of
the Father and of the Son. De Trinitate V.iii.12.
Father
Son
every being that is called something by way of
relationship is also something besides the
relationship thus a master is also a man, and a
slave is a man . So if the Father is not also
something with reference to himself, there is
absolutely nothing there to be talked of with
reference to something else. De Trinitate
VII.i.2 the substance of the Father and the Son
is one. De Trinitate V.i.4.
Spirit
God
Thus the Father is God and the Son is God and the
Holy Spirit is God, and no-one denies that this
is said substance-wise . De Trinitate V.ii.9
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The Trinity Augustines psychological model
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They are two somethings, lover and love and
these are called two things relatively to one
another. So then, insofar as they are referred
to each other, they are two but both together
are one spirit, they are each mind and both
together one mind.De Trinitate IX.i.2.
Love
Knowledge
Now the mind cannot love itself unless it also
knows itself. De Trinitate IX.i.3.
Thus mind is of course in itself. Since it is
called mind with reference to itself, though it
is called knowing or known or knowable relative
to its knowledge also as loving and loved or
lovable it is referred to the love it loves
itself with. De Trinitate IX.i.8.
Lover, Beloved
Knower, Known
Mind
The psychological model is significantly
different from the logical model.
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The Trinity Gilbert of Poitiers model
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those same ones, whose essence is none other
than simple, are different by different
properties. Gilbert of Poitiers, De Trinitate
I,3,34.
Paternity
Filiation
But some people of little understanding, hearing
that God is simple, take Him and any of the
diversity of names said of Him (such as God,
one, eternal, Person, principle,
author, father, Son, Connection and
others like this) to be of the same nature and
ratio, so that God is both the essence by which
He is said to be and the unity by which he is
said to be one and the eternity by which He is
said to be eternal and similarly for the others,
and likewise the Father himself is
paternity. Gilbert of Poitiers, De Trinitate
II,2,prol.
Father
Son
Divinity
Spiration
Spirit
God
By defs.1 and 5, if God is a substance, there is
no X such that God ab X. To meet this difficulty,
we could alter def. 1 and 2 thus Def.1a. cat(A)
iff for all X if A ab X, AX. Def.2a. denom(A)
iff for some X A ab X and A?X. Axiom 5 would be
dropped, and Theorems 4-7 would be modified
accordingly. This allows that A ab A, in which
case there would be a purely conceptual
distinction between a being and its abstract.
With these changes Theorem 10 is still provable.
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The Trinity Bonaventures model
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Respects and relations (properly speaking) are in
the Persons because the Persons are related and
distinguished according to them, but they are
not in the essence because it is not related or
distinguished. However, they are in the divine
essence (speaking commonly and improperly) as
everything that is the divine essence or
everything that is in either the divine essence
or in a Person may be said to be in the divine
essence. Bonaventure, Sent. I d.33 art. unic. q.1
to abstract something from something is
ambiguous. One sort of abstraction has its origin
in the nature of the thing (and it is thus that
the universal is abstracted from the particular,
and form from matter, in both of which cases
there is composition and diversity). Another sort
of abstraction has its origin in our
understanding. For our understanding, when it
understands something complete, of necessity
understands it in two ways or under a double
concept, namely by way of what-is and of
that-from-which. In God, what-is and that
from-which it is are the same.Bonaventure, Sent.
d.33 q.1 d.23 a.1 q.3
Filiation
Paternity
Father
Son
Divinity
Spiration
Spirit
God
According to comparison to the subject, a
relation crosses over ...
according to comparison to the object or
terminus it remains Bonaventure, Sent. I d.27
part 1 q.3
The true and common opinion which mastersnow
hold in common is that the properties are the
persons and are in the persons, and yet differ
from the persons in some way.Bonaventure, Sent.
I d.33 q.1
By Theorem 5 its not possible that A ab A. This
becomes possible if we adopt Def.1a. cat(A) iff
for no X A ab X and A?X. Def.2a. denom(A) iff
for some X A ab X and A?X .
By Theorem 2 its not possible that A in A. This
becomes possible if we adopt Def.7a. acc(A) iff
for some X A in X and A?X.With these changes
Theorem 10 is still provable.
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The divine attributesAquinass model
Earthly wisdom
Earthly goodness
likeness
likeness
Divine perfection
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The divine attributesScotuss model
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Wisdom
Goodness
de
de
Individual divine perfection
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The divine attributesOckhams model