God's Existence: Can It be Proven? A Logical Commentary on the Five Ways of Thomas Aquinas

61 Pages·2016·6.36 MB·English

by  Paul Weingartner

Paul Weingartner
METAPHYSICAL RESEARCH
Herausgegeben von I Edited by God's Existence.
Maria Elisabeth Reicher. Johanna Seibt
Can it be Proven?
Barry Smith. Daniel von Wachter
Band 101 Volume 10
A Logical Commentary on the
Five Ways of Thomas Aquinas
ontos
verlag
Preface
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The first task of this book is to show that the Five Ways of Thomas
Nationalbibliografie; detailed bibliographic data are available In the
Internet at http://dnb.d-nb.de. Aquinas can be presented in a form in which all five ways are logically
valid arguments. The first four ways as presented in the text by Thomas
m Aquinas need no essential additions or complementation to show the
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logical validity of the arguments. The fifth way however, is the exception
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Piscataway, NJ 08854-8042 Since the logical l'econstructions of the Five Ways offered here try to
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Second Way) that the arguments contain (additional) premises which are
not necessary to derive the conclusion. Nevertheless they have also been
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The second task of the book is to offer a detailed and critical discussion of
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the premises used in the arguments including important definitions used as
premises.
A further but no less important task of this book is to examine the two
preliminary questions of Thomas Aquinas, "Whether the existence of God
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The negative answer to the first preliminary question shows the non
triviality of the problem of the existence of God which is still more
underlined by the critical attitude of Thomas Aquinas to Anselm's
Ontological Argument. The answer to the second preliminary question
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Ways are like some scientific explanations which search for something
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unobservable as a cause for observable effects or which search for some
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This book is intended for philosophers, scientists, students of philosophy
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and educated people who are interested in the question of the existence of
N rI 0' ~"s book may be reproduced. sipred in reb'ieval systems or trnnsl1lin.d . God. However, for a deeper understanding, a modest knowledge of modem
op. , ' n1 I . . ofihnil1ll racord.ngorolhcrw,,,,
in any fonn or by nny mellns, ele<:tronl~. lI.och~'c • p ,otooopymS, mler . " r d e<:itieally for the logic (the equivalent of an introductory course with propositional and
wilhout wriUen pern,issioll from the Pubh.I1<lr. with the exccpllOn of "JOY ~uuonn ~~p ~~Ir::ns., of the work
purpo.eofbell18 .... tered nnd ."""utcd on n oomputer 'Y'wO!, for exc USIVC usc 0 • ,- predicate logic) and the history of philosophy (especially of the Ancient
and Medieval times) are needed. The logical validity of several of the Five
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This hardcover binding meets the International Library standard who have merely discussed one of the Five Ways in detail. In order to
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2
decide unambiguously whether the Five Ways can be interpreted as Contents
logically valid arguments, it is necessary to translate them into modem
logic and to show the validity of the proof. The interpretation and
1. The Five Ways: Preliminary Questions ........ " ............................................ 5
translation into modem logic has been done in two steps. The first step is a 1.1 Whether the Existence of God is Self-Evident? .................................................... 8
preliminary interpretation under the title "the premises and co~clusions of 1.1.1 Text: Thomas Aquinas' Answer ...................................................................... 8
the text" which attempts to concentrate all the relevant parts mto natural 1.1.2 Commentary on the Answer ........................................................................... 8
language and making them as precise as possible. The second step 1.1.2.1 Definition of Self-Evident Proposition .................................................... 9
1.1.2.2 Is God Exists Self-Evident to Us? .......................................................... 18
translates this into the symbolic language of First Order Predicate Logic.
l.r.3 Commentary to the Objections ..................................................................... 21
To read the book in the most efficient way, it is fitting to proceed by
1.1.3.1 Is the Knowledge of God Naturally Implanted in All? .......................... 21
chapters, as indicated. It is important to read the prelimina~ cha.pters firs.t, 1.1.3.2 Is the Existence of Primal Truth Self-Evident? ..................................... 21
since it is here that Thomas Aquinas explains the essential pomts of his 1.1.3.3 Thomas Aquinas' Commentary on the Ontological Argument ............. 21
understanding of man's access to the existence of God and to possible 1.2 Whether it Can Be Demonstrated that God Exists? ............................................ 29
demonstrations of it. 1.2.1 Text: Thomas Aquinas' Answer ....... " ........................................................... 29
1.2.2 Commentary on the Answer ......................................................................... 29
Acknowledgements: The author is indebted to Prof. Edgar Morscher for
1.2.2.1 Two Types of Demonstration ............................................................... ;29
several valuable suggestions concerning the reconstruction of the
1.2.2.2 The Cause as Necessary Condition for the Effects ............ , ................... 33
Arguments, to Mag. Albert Anglberger and Eva. Alteneder f~r the lay~ut 1.2.2.3 From God's Effects to His Existence ..................................................... 37
and typing of the manuscript, and to Kathryn Bishop for assistance with 1.2.2.4 Is it Necessary to Assume a Cause for the World (Universe) which is
the English text. Outside the World (Universe)? .......................................................................... 39
1.2.3 Commentary on the Objections .................................................................... 49
Salzburg, March 7, 2008 Paul Weingartner 1.2.3.1 Is the Existence of God an Article of Faith? ......................................... .49
1.2.3.2 Does One Need a Definition of God for Understanding His Existence?
............................................................................................................................ 50
1.2.3.3 Are God's Effects Proportional for a Demonstration of His Existence?52
2. The Five Ways .......................................................................................................... 53
2.1 The First Way ........................... , .......................................................................... 53
2.1.1 The Latin Text and its Translation into English ........................................... 53
2.1.2 The Premises and Conclusions of the Text... ................................................ 54
2.1.3 Reconstruction .............................................................................................. 55
2.1.4 Commentary """"".""".""""".".""." ... """ .. ""."""""""".""""." .. """ ....... 56
2.1.4.1 A Problem with Premise 6 ..................................................................... 56
2.1.4.2 Different Meanings of "Movetur" ......................................................... , 56
2.1.4.3 Definition of the First Mover ................................................................. 58
2.2 The Second Way "".".""""""" .. "" .." .".".""" ... "".""""" ... "".""."."""." .......... 61
2.2.1 The Latin Text and its Translation into English ........................................... 61
2.2.2 The Premises and Conclusions ofthe Text ................................................... 62
2.2.3 Reconstruction .............................................................................................. 62
2.2.4 Commentary .................................................................................................. 64
2.2.4.1 Irreflexivity of the Causal Relation ........................................................ 64
2.2.4.2 Cause as a Necessary Condition ............................................................ 64
2.2.4.3 Transitivity ............................................................................................. 66
2.2.4.4 Definition of the First Cause .................................................................. 68
2.2.4.5 Infinite Regress ...................................................................................... 71
2.3 The Third Way ..................................................................................................... 75
2.3.1 The Latin Text and its Translation into English .......................................... ~~
1. The Five Ways: Preliminary Questions
2.3.2 The premises and conclusions of the text .................................................... 7
2.3.3. Reconstruction............................................................................................. 7
2.3.4 Commentary ................................................................................................. 78
2.3.4.1 The Concept of Necessity Used in the Thi~d Way ....... :. ....................... 78 The existence of God is investigated in the second question (questio 2) of
2.3.4.2 The Concept of Necessity in Relation to Time ..................................... 79
the Summa Theologica of Thomas Aquinas. It is the beginning of the
2.3.4.3 Aristotle's view according to Thomas Aquinas ............ ; ....................... 82
treatise on God. The treatise on God is the first part of the Summa and
2.3.4.4 Thomas Aquina's Premises and Conclusions in the Third Way ........... ~~
2.4 The Fourth Way ................................................................................................ .. consists of 119 questions. The second part is concerned with the rational
2.4.1 The Latin.Text and its Tra~slation into English .......................................... ~~ creature's movement towards God and the third part deals with Christ, who
2.4.2 The Premises and ConclusIOns of the Text ................................................ .. as man, is our way to God.
2.4.3 Reconstruction .............................................................................................. 88 Question 1 (of part I) is preliminary to the whole Summa Theologica. It
2.4.4 Commentary ................................................................................................. 89
deals methodologically with theology as a kind of scientific approach.
2 4 4 1 Premise 2 ............................................................................................... 89
Question 2 is divided into three sections or articles. The first deals with the
2:4:4:2 The Definition which Connects Perfection with Being (Premise 5. of 92
question whether the existence of God is self-evident. The second article
2.4.3) .................................................................................... :. ........................... .
2.4.4.3 The Most Perfect Thing is the Cause of All Perfection ........................ ~~ asks whether the existence of God is demonstrable. Only after answering
2.5 The Fifth Way ................................................................................................... .. these preliminary questions does Thomas Aquinas offer his Five Ways as
2.5.1 The Latin Text and its Translation into English .......................................... 95 proofs for the existence of God. The following first part will prove that the
2.5.2 The Premises and Conclusions of the Text.. ................................................ ~~ consideration of these two preliminary questions is absolutely necessary
2.5.3 Reconstruction ............................................................................................. .
for a correct interpretation of the Five Ways.'
2.5.4 Commentary ...... ;. ......................................................................................... 9~
Before we begin with the first preliminary question of Thomas Aquinas it
2.5.4.1 Natural Beings Lacking Intelligence ..................................................... 9
2.5.4.2 The Question ofthe Validity of the Argument ........ :. ........................... 98 seems appropriate to recall some important assumptions which are
2.6 The Question of the Uniqueness in the Conclusions of the Five Ways .......... : ~~~ presupposed by the background of all medieval proofs of the existence of
2.6.1 First Question ........................... 1................................................................. 3 God:
2.6.2 Second Question ......................................................................................... 10
2.7 Commentary to the Objections ......................................................................... 107
(1) The first general assumption is that we live in a world or universe
References ................................................................................................................... 109
which has an ordered and rational structure and is governed by laws. And
that this ordered structure and the laws can be (at least in part) discovered
by human reason? .
(2) A second assumption is that the proofs of the existence of God
presuppose a realistic conception of truth, in the sense of the
Correspondence Theory of Truth. Thus empirical premises about the world
in the Five Ways are interpreted as true if and only if (iff) it is so in reality
(is a fact) as these premises say. A pragmatic or a coherence theory of truth
I Strangely enough the well known commentaries by Kenny and Bochenski also do
not COlUment on the (wo preceding articles with their important preliminaries.
2 Ct: Davis (1997, GRT) p. 8 f. also stresses these three assumptions in some different
wording. His third assumption on p. 9 is explicitly discussed and justified in the
second preliminary question by Thomas Aquinas Whether it can be demonstrated that
God exists?". In this sense it is not a presupposition in tbe preparation to his Five
Ways since Aquinas expJjcitly deals with it.
6
7
which replaces the correspondence to reality by an agreement to other
God. Observe here, however, that this is the concept of essence invented
propositions or by a utility status of the consequences would not be
by Aristotle and used by Thomas Aquinas. It is not a concept of essence as
sufficient. The important point here is that religious belief is also it is used in German Idealism or in Husserl.
interpreted in an objective way: a proposition which is religiously believed
A further reason for Christian understanding is that what we know and see
is true iff it corresponds to facts; even if these facts may be supernatural.
of God are his creatures wruch are the contingent outcome of his free will
and love but not the necessary outcome of his nature or essence. Therefore
(3) A third assumption is that the meaning of the word 'God' can be made
from looking at the creatures we cannot learn much about his essence.
clear to that extent to which it is needed in the proof for deriving the,
Even if it is admitted that Christ showed us something of God's essence,
conclusion. There is hardly any proposal for a proof of the existence of
this goes beyond philosophy and belongs to the domain of theology for
God by medieval philosophers or theologians in which it is claimed that which the holy scriptures are the foundation.
the proof uses a definition of the essence of God. And particularly Thomas
The general view connected with this third assumption is the meaning of
Aquinas stresses on several occasions (already in question 1, i.e. before the the word 'God' is that a proof of the existence of a First Cause or an id
preliminaries to the Five Ways in question 2) that we cannot know the quo maius cogitare non potest is a preamble provided by natural reason
essence of God in this life. Therefore the meaning of the word 'God' has to and philosophical thinking for theology.
be taken from what we know about the world and from our interpretation
It is then a further task of theology to show that this so characterised God
of the world as the effect of some intelligent cause. Consequently the Five
is identical with the God of revelation. But this goes beyond philosophical
Ways speak of a First Mover, First Cause, Necessary Being, Most Perfect reasoning.
Being, Being who directs the nonintelligent beings to their ends. And in
each of the five conclusions it is said that this is what we (humans)
understand when we speak of God. Also Anselm's conception of id quo
maius cogitare non potest, to which the most perfect being of the fourth
way is closest, is not a definition of the essence of God. Each of the five
conclusions can be understood as an existential assertion together with a
nominal definition or description. They have the following form: (1)
'There is a First Mover, a First Cause, a Necessary Being, a Most Perfect
Being, a Being who directs the non-intellectual beings to their end. (2) This
being (First Mover ... etc) we call God. Although this is a clarification of
the meaning of the word "God" or a nominal definition, it is not a
definition of God's essence. Since such a definition would have to be a
real definition according to Aquinas and Aristotle. A real definition is one
that (a) defines a species (definiendum) with the help of genus and
differentia specijica (definiens) and (b) is an essential and convertible
predication. Thus the species man is defined with the help of animal and
rationality and "man is a rational animal and rational animal is man" is a
convertible essential predication. Whereas man is a featherless biped is a
convertible but not essential predication. Such a real definition is first of all
necessarily true and secondly permits us to speak of the essence of the
species (man). Since God cannot be a species and moreover there is no
more general genus of God, we cannot have a definition of the essence of
8
9
1.1.2.1 Definition of Self-Evident Proposition
1.1 Whether the Existence of God is Self-Evident?
According to the text of Thomas Aquinas a proposition is self-evident if
the predicate is included in ~he essence of the subject. This is illustrated by
1.1.1 Text: Thomas Aquinas' Answer
the example: "Man is an animal" for animal is contained in (the essence) of
man. For the interpretation of this definition we shall be concerned with
I answer that, A thing can be self-evident in either of two ways; on the one
three questions:
hand self-evident in itself, though not to us; on the other, self-evident in
itself. and to us. A proposition is self-evident because the predicate is
1) Why is the essence (of the subject) mentioned?
included in the essence of the subject, as "Man is an animal", for animal is
2) What is the meaning of contained in in the context "animal" (in
contained in the essence of man. If, therefore the essence of the predicate
general: the genus) is contained in the (essence) of man (in general:
and subject be known to all, the proposition will be self-evident to all; ~ is
the species)?
clear with regard to the first principles of demonstration, the terms of which
3) Can the meaning of contained in (included) be interpreted by means
are common things that no one is ignorant of, such as being and non-being,
of modem logic or set theory?
whole and part, and such like. If, however, there are some to whom the
\
essence of the predicate and subject is unknown, the proposition will be
ad (1) Why is the ESSENCE mentioned?
self-evident in itself, but not to those who do not know the meaning of the
predicate and subject of the proposition. Therefore, it happens, as Boethius
The essence is mentioned because every real definition according to
says (Hebdom., the title o/which is: "Whether all that is, is good"), "that
Aristotle describes the essence of some species in its definiens.4
there are some mental concepts self-evident only to the learned, as that
As it has been said above (1.(3)), a real definition according to Aristotle
incorporeal substances are not in space." Therefore I sa~ that. this
and Thomas Aquinas is one that
proposition, "God exi,sts", of itself is self-evident, for the ~redlCate IS the
(a) defines a species (definiendum) with the help of a genus and differentia
same as the subject; because God is His own existence as Will be here-after
specifica (definiens) and
shown (Q. 3, A. 4). Now because we do not know the essence of God,. the
(b) is an essential and convertible predication. In this sense it characterises
proposition is not self-evident to us; but needs to be demonstrated by thmgs the essence of the species. In modem terms such a real definition has
that are more known to us, though less known in their nature namely, by the following properties: it is a proposition (not a rule); it is an
effects. 3 equivalence or an identity which is universal, affirmative and
necessarily true; it satisfies the criterion of eliminability (Le. the
definiendum is always replaceable by the definiens) it further satisfies
1.1.2 Commentary on the Answer the criterion of non-creativity which is weaker than the one of
Lesniewski used in modem logic: it forbids only to derive existential
statements from the definition which assert the existence of the
The text contains a definition of self-evident proposition and the distinction
definiendum or parts of the definiens.
between self-evident in itselfa nd self-evident/or us.
In the definition
4 And Thomas Aquinas considered many subtle details of Aristotle's theory of
definition in his commentary to the Posterior Analytics (see especially chs. 8, IO and
J (STh) I, 2, l. 13). Cf. also Weingartner (1991, NAT); (1976, WTh) II, 1 p.246 ff.
10
I I
=
man df rational animal ad (2) The Meaning of IS CONTAINED IN
the definiens rational animal describes the essence of (the species) man. The meaning of is contained in in the context "the genus is contained in (is
Now it is not difficult to see that "animal" is contained in the definiens part of) the species" is already discussed by Aristotle in his Metaphysics:7
"rational animal" which describes the essence of man. Or take the "Thus, we say that species are the parts of their genera .... the elements in
definition a proposition which serves as a definition are parts of a whole; in this sense
the genus is called a part of a species, though in another sense a species is
s
living thing = dfthing possessing growth, nutrition and propagation. part of its genus".
The scholastic commentators were familiar with this doctrine of Aristotle
Here the definiens "thing possessing growth, nutrition, and propagation" concerning the extensional and the intensional inclusion. Thus Thomas
describes the essence of living things. And similarly "groWth" ("nutrition", Aquinas writes in his Commentary to this passage:
"propagation") is contained in the definiens of living thing which describes "". species are said to be, parts of a genus. For a genus is divided into
the essence of living thing. In such cases the meaning of "contained in", as specie~, but not as 'a quantity is divided into quantitative parts. For a whole
"animal" is contained in the definiens "rational animal" of the definition of quantity is not in each one of its parts, but a genus is in each one of its
man, can be understood in two different ways: First syntactically in the species ... In a fourth sense parts mean those things which are placed in the
sense that the linguistic expression 'animal' is a part of the linguistic definition of anything, and these are parts of its intelligible structure; for
expression 'rational animal'. Secondly semantically in the sense that the example, animal and two-footed are parts of man.,,8
concept "animal" is part of the compound concept "rational animal". And A very clear understanding of these, two types of inclusion is due to
this latter sense will be elaborated more accurately now in the answer to DeMorgan. "In this9 the logicians have abandoned both Aristotle and the
the second question (see below ad (2». It is now easily understandable that laws of thought from which he drew the few clear words of his dictum: 'the
the definition of selj-evident is satisfied by the proposition "man is an genus is said to be part of the species; but in another point of view the
animal" or by the proposition "a living thing possesses propagation" since species is part of the genus'. All animal is in man, notion in notion: all man
the predicates animal and propagation are contained in the definiens of the is in animal, class in class. In the first, all the notion animal part of the
subject man and living thing. notion man: in the second, all the class man part of the class animal.,,10
These propositions are therefore selj-evident, not only in general but also
for us.6
no citations in Kant's writings although he certainly knew philosophers of the
5 This is the Aristotelian definition which is still basically correct on modern Antiquity and of the 17th century.
standards. Aristotle (Met) v, 25; 1023 b J 7 and 22-25.
6 It is interesting that Kant uses the same definiens for defining the analytic 8 Thomas Aquinas (CMA) V, L21, 1094 and 1096. The tirst part of Ihe question in
proposition (analytisches Urtei£): "the predicate is contained in the subject"- in modern terms: genus and species are like set and (special) subsets, but not like a heap
German: "das Prl1dikat B geh{lrt zum Subjekt A als etwas, was in diesem Begriffe and its parts. Since the whole heap is not in its parts, but the common characteristics of
A (in versteckter Weise) enthalten ist" (KRV, A6, BI0). It is quite likely that Kant the elements of a set are in (i.e. included) in the common characteristics of the
knew the definition of self-evident proposition of Thomas Aquinas, since the elements of each subset.
question 2 of the Five Ways was commented by many scholars in the time of 9 DeMorgan refers to a mistaken interpretation of W. Hamilton, but he defends
Enlightenment and later. To cite a forerunner was often obsolete in the time of Aristotle.
enlightenment, especially to cite somebody of the Medieval Time. There are almost 10 DeMorgan (1966, SPS) p. 201, note 2.
12
13
ad (3) CONTAINED IN Interpreted by Modern Logic.
Definition Dl is well known and will be called extensional inclusion. The
reason for calling it "extensional" can be explained by using the following
Before we begin with an interpretation using modern logic,(here especially definitions by abstraction:
by principles of the theory of classes), remarks need to be made for its
Ext/y) = df{x: xcy} with reference to the definiens, and
justification. The logical and conceptual background of the argumentation
Extiy) = df{x: xQ'} with reference to the definiendum.
of Thomas Aquinas is the Syllogistics of Aristotle with some extensions
Both understandings of extension have been used in the Middle Ages and
(especially concerning the logic of relations) which were developed in the
12
in Modern Times. The set theoretical inclusion as it is defined above will
Medieval Logic of the 12th and 13th century. It is well known that
Aristotle's Syllogistics is a part of the logic classes (more accurately: part therefore also be called extensional inclusion.
of the logic of the virtual theory of classes) or of the monadic part of The Observe that accidental properties are not suitable instances of D1 or D2
First Order Predicate Logic.11 since DI and D2 are used for an interpretation of what is a self-evident
Thus statements used by Aquinas such as "the predicate is contained in the principle of Thomas Aquinas. And his examples are only those which are
essential predications in the sense of Aristotle and Aquinas; for example
subject" or "the species is part of the genus" are understood by him as
man is animal i.e. species is included in genus. Even propia (pecularities)
having the form of an A-sentence of Syllogistics. And such an A-sentence
are not suitable instances. Consider musical ability, for example, which
of Syllogistics has the form of a set-theoretic inclusion in the theory of belongs to mankind but not to all men, or bipedalism which is
classes. Also if the genus is included in other classes (cf. definition D2) the characteristic of all men but not only to them, or those characteristics
interpretation is the same. The new intentional inclusion is explicitly wh.ich belong to mankind but only for a time, as in the case of committing
a sm, or even laughter, the capability of which belongs to mankind and all
defined (D2) and can therefore be explained with the help of the usual set
the time. The latter kind of proprium is even a convertible predication but
theoretical inclusion or with the help of the A-sentence.
it is not an essential predication according to Aristotle and Thomas
Aquinas.
The meaning of contained in (included in) in the context "the species is It is a more difficult question however to give a precise logical
part or contained in or included in the genus" can be easily interpreted as interpretation for the other type of inclusion, which is called intensional
the usual set theoretical inclusion: S {;;G where 'S', 'G' stand for 'species' inclusion. How is the genus included in the species? An answer which we
and 'genus' respectively. Its definition is: can grasp from the passages cited above, is this: all the characteristics
common to living things are included in (are part of) all the characteristics
Dl S {;; G +-+ df (Vx) (xeS ~ xeG) common to all men. So since growth, nutrition and propagation are
essential characteristics of living things they must be possessed by all men,
too. Or more general: the common characteristics of the genus are
contained in the common characteristics of the species. This idea can be
defined as the intensional inclusion, symbolised as G ~ S:
II See Hilbert-Ackermann (1959,GTL) §3 or Bochenski- Menne (1954, GRL) §27. D2 G ~ S H df(VZ) (G{;;Z ~ S {;;Z),
The stronger existential import ofthe Aristotelian A-sentence can be handled as an
additional premise in the four syllogistic modes darapti, fe/apton, bamalip andfcsapo.
The expression, virtual theory ofc lasses has been used by Quine to denote that
rudimentary theory of classes which can be developed in First Order Predicate Logic 12 ~or the hist0I?' of extension and intension see Frisch (1969, EeL). For a detailed
without using any axiom of Set Theory (in the sense of Zermelo-Fraenkel, Neumann log~cal elaboratIOn (also concerning views in the history of philosophy) see
Bemays-G{iqel or Quine). Weingartner (1976, WTh) II, 1 ch.3.4.
14
15
where G C Z, S C Z are the usual extensional (set theoretical) inclusions. In needed. 14 .Since the intensional relations of inclusion, intersection and
set theoretical terms: the genus is intensionally included in the species iff union and their respective extensional relations are interdefinable these
in al1 sets in which the genus is included the species is also included. "intensions" are reducible to "extensions" and vice versa. Thus they must
With the help of these definitions and PL 1 (Predicate Logic of First Order) not be confused with the ideas of Frege or Carnap concerning intensions.
or the so called virtual theory of classes, 13 one .can prove the following On the other hand these types of intensional relations are used in the
theorem which connects extensional and intensional inclusion: Middle Ages and in Leibniz and OeMorgan.
However, the form S C P is only one type of self-evident proposition
(VaVb) (a C b ~ b ~a) satisfYing the definition of Thomas Aquinas, since it is a universal
proposition of the form all A are B. Nevertheless it is an important type,
It says: a is extensionally included in b iff b is intensionally included in a. because it interprets the two examples given by the text: "'Man is an
animal' for animal is contained in the essence of man". It interprets also the
Proof: (l) a C b ~ b ~ a
second example of the text, an example of Boethius: '~Incorporal
(VZ)(a C b ~(b cZ ~a CZ)) by transitivity. Hence:
substances are not in space". Since the common characteristics of
a Cb ~(VZ)(b cZ ~a cZ) Apply 02. substances not in space are that they do not have (spatial) extension and
(2) (b ~ a ~ a Cb) these characteristics are (intensionally) included in the common
characteristics of incorporal substances. If the negations "not in space" or
b ~ a ~ VZ(b cZ ~a CZ) according to 02. Hence:
"not corporal" seem to make difficulties here, one simply can derive the
b ~ a ~ (b Cb ~a Cb) by instantiation. Thus: inclusion "Substances in space are corporal substances" from "Incorporal
substances are not in space" by contraposition. Accordingly, we may say
b Cb ~(b ~ a ~a Cb) and bCb is a theorem. Q.E.D.
that the common characteristics of corporal substances are included in the
common characteristics of the substances in space.
Using this interpretation we can now give a more precise definition of one
The question of the article however concerns the proposition "God exists",
type of self-evident propositions according to Thomas Aquinas: a
and this seems to be of a different type. But this proposition could be
proposition containing the subject term S and the predicate term P and
converted into the universal form: For all x, if x is God then x exists. This
having the form S C P is self-evident if and only if P ~ S. can be interpreted in two ways: in "x is God" God can be interpreted as a
It has been widely neglected that First Order Predicate Logic contains a property of the subject x, - which contradicts important other places of
theory of a special type of intension. This is a dual theory to the virtual Thomas Aquinas' writingsl5; further "x is God" can be interpreted as x is
theory of classes and can be obtained from it by introducing 02 and a identical with God - then the proposition is: "For all x: if x is identical with
God then x exists.".
definition for intensional intersection (and optionally also for intensional
union). But no definition of extension or intension by abstraction is
Applying our analysis to this proposition would lead to the following
interpretation: the proposition "For all x: if x is identical with God then x
14 The Theory is developed in Weingartner (1976, WTh) II I, p. 15 8ff and on the basis
13 This expression has been used by Quine to denote that rudimentary theory fo classes of a set theoretic Predicate Calculus in (1973, PCI). For the Occurrence of this kind of
which can be developed in First Order Predicate Logic without using any axiom of Set j~tension in the history oflogic cf. Weingartner (1976, BIG).
Theory (in the sense of Zermelo-Fraenkel, Neumann-Bernays-G5del or Quine). Cf. Thomas Aquinas (8Th) I, 3,3, ad 1.
16
17
exists" is self-evident if the common characteristics of existence or of
the Second Way the thing which is the First Cause is identified with God
existing thing are included in the characteristics of God. This is not
in a nominal definition. We could make a similar application to the other
unreasonable even if there are questions left open, for example, whether
four ways. In each way the thing which is the First Mover, the First Cause,
the "characteristics of God' is too weak and should be replaced by the
the necessary uncaused being, the highest value or that thing which directs
"essence ofG od. " This kind of interpretation can also be applied to what is
all things which do not have reason to their ends, is identified with God in
proved in the Five Ways: a first Mover a First Cause ... etc.: "For all x: if
a nominal definition. It is not a real definition - in the sense of Aristotle _
x is identical with the First Mover then x exists.". This proposition would
because it does not describe the essence. Therefore the text of Thomas
be self-evident if the notion of existence is contained in the notion of First
Aquinas is accordingly careful and says: "and this being we call God" or
Mover.
"and this everyone understands to be God" or "to which everyone gives the
name of God."
However, we may apply the interpretation of propositions of the type
Returning to the definition of existence with the belp of uniqueness we
" ... exists" as usual in modern logic. According to this understanding in a
may ask whether it is self-evident according to Thomas Aquinas to apply
proposition " ... exists", " ... " is not used as a referential name but as a
our analysis of intensional inclusion. In this sense the notion of uniqueness
description or descriptive phrase. Instead of saying "Wittgenstein exists"
should be contained in the notion of existence. This however is not
we may say "the philosopher who wrote the Tractatus Logico
sensible at first sight. In order to make the inclusion apparent we have to
Philosophicus exists" or "the philosopher Wittgenstein exists". The main
insert a principle or furtber premise of a certain type of ontology. It is the
point is that we do not use just a name but a description of the person or
principle that only individuals exist (actually) or in other words: everything
thing, which characterises him (it) as somebody who did this or that. There
that exists in reality is an indjvidual. And further it is a necessary condition
are several theories of description. One is relatively simple, though with
for individuals that there is at least one and at most one i.e. exactly one.
widespread applicability, that of Russell. 16 Putting the two principles together we have got a bridge which connects
According to this theory an existence statement is defined thus: existence with uniqueness:
E!(lx)(¢x) = df3bVx (¢x ~ x = b) (1) E! (lx)(r/Jx) -IN(lx)(r/Jx)
(2) IN (lx)(¢x) - UN(lx)(r/Jx)
In words: this thing with the property ¢ exists = dJ, there is at least and at where UN (lx)(r/Jx) = df 3bVx (r/Jx ~ x = b)
most one thing which has the property ¢. An application to the Second
(3) E! (lx)(r/Jx) - UN(lx)(¢x) from (1) and (2)
Way would be: this thing which is a First Cause exists is defined as: there
is at least and at most one thingl7 which is a First Cause. And at the end of
The respective intensional inclusions are then:
16 Cf. Whitehead-Russell (1927, PMt) § 14 and Introduction ch. III. The "widespread
applicability" of this theol)' (but also of all the usual other ones including those of (1') IN [;E!
Free Logic) has to be taken with care. As has been sho~n in Wein~artner \20?7,
RCN) it is best applicable only to the objects of ClasSIcal Mechamcs or SImIlar (2') UN [;IN
types of objects which are rigid and impenetrabh~ in a strong sense; ~u~ not to those
of Quantum Mechanics and not to those of SpecIal or General ~elattvlty.
17 Things of which there is (exists) at least one and at most one, I.e. exactly one are The question whether the uniqueness condition is also satisfied by objects of
called "unique". And the respective condition is called "uniqueness-condition". See
modem physics, i.e. objects of Quantum Theol)' and objects of the Theol)' of
the discussion on the uniqueness in the conclusion of the Five Ways below 2.6.
Relativity is discussed in Weingartner (2007, RCN).