Idealization XI Historical Studies On Abstraction And Idealization

IDEALIZATION XI: HISTORICAL STUDIES
ON ABSTRACTION AND IDEALIZATION

POZNAŃ STUDIES
IN THE PHILOSOPHY OF THE SCIENCES AND THE HUMANITIES
VOLUME 82
EDITORS
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IDEALIZATION XI:
HISTORICAL STUDIES
ON ABSTRACTION AND
IDEALIZATION
Edited by
Francesco Coniglione,
Roberto Poli
and
Robin Rollinger
Amsterdam - New York, NY 2004

The paper on which this book is printed meets the requirements of "ISO
9706:1994, Information and documentation - Paper for documents -
Requirements for permanence".
ISSN 0303-8157
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Printed in The Netherlands

CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
GENERAL PERSPECTIVES
Ignacio Angelelli, Adventures of Abstraction . . . . . . . . . . . . . . . . 11
Allan Bäck, What is Being qua Being? . . . . . . . . . . . . . . . . . . . 37
Francesco Coniglione, Between Abstraction and Idealization:
Scientific Practice and Philosophical Awareness. . . . . . . . . . . . 59
CASE STUDIES
Desmond Paul Henry, Anselm on Abstracts. . . . . . . . . . . . . . . . 113
Leen Spruit, Agent Intellect and Phantasms. On the Preliminaries
of Peripatetic Abstraction. . . . . . . . . . . . . . . . . . . . . . . . 125
Robin D. Rollinger, Hermann Lotze on Abstraction and Platonic Ideas . . 147
Roberto Poli, W.E. Johnson’s Determinable-Determinate Opposition
and his Theory of Abstraction. . . . . . . . . . . . . . . . . . . . . . 163
Maria van der Schaar, The Red of a Rose. On the Significance of
Stout’s Category of Abstract Particulars. . . . . . . . . . . . . . . . 197
Claire Ortiz Hill, Abstraction and Idealization in Edmund Husserl
and Georg Cantor prior to 1895. . . . . . . . . . . . . . . . . . . . 217
Guillermo E. Rosado Haddock, Idealization in Mathematics: Husserl
and Beyond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Andrzej Klawiter, Why Did Husserl not Become the Galileo of the
Science of Consciousness?. . . . . . . . . . . . . . . . . . . . . . . 253
Giovanni Camardi, Ideal Types and Scientific Theories. . . . . . . . 273

PREFACE
Discussions about abstraction are often vitiated by an original sin. When
Boethius introduced the term “abstractio”‚ into Latin, he was using it to trans-
late two different Greek words: the Aristotelean “aphairesis”‚ and the Platonic
“chorismos”. The former was used by Aristotle in the epistemological
procedure whereby he constituted, by means of “reduplication”, the field of
enquiry for scientific and philosophical reflection. The latter, which means
“separation”, referred instead to the universe of Platonic ideas. Thus, from the
very beginning, the important distinction between Aristotelean “abstraction”
and Platonic “idealization” was blurred by Boethius‚ use of a single term.
The distinction, however, was so important and so profound that it could
not be neglected, and it inevitably cropped up again in subsequent periods of
philosophical enquiry. Although many scholastic philosophers continued to
use Boethius’s‚ by now famous “abstractio”, they nevertheless asserted that
there were two different kinds of abstraction. Cajetan, for instance, distin-
guished between formal abstraction (the separation of form from substance)
and total abstraction (the separation of the universal from the particular).
Despite these ancient roots and after the great debate which characterised
the empirical and rationalistic tradition, interest in the problem has been absent
from the mainstream of mathematical logic and analytic philosophy. It seems
that there is a gap between the epistemological theorization, in which it is hard
to find new insights on the problem of abstraction, and the historical studies
concerning the development of philosophical thought, which present a more
fertile ground for it.
We have collected the papers of this volume with the awareness of the need
for building a bridge between historical research and theoretical speculation.
Accordingly the volume consists of both general overviews which sketch the
meaning and the fortunes of abstraction in science, philosophy and logic (the
first part) and historical case studies which focus on abstraction in particular
thinkers (the second part).
Francesco Coniglione
Roberto Poli
Robin Rollinger

Poznaƒ Studies in the Philosophy
of the Sciences and the Humanities
2004, Vol. 82, pp. 11-35
Ignacio Angelelli
ADVENTURES OF ABSTRACTION
Abstract. In view of the confusion that exists in the 20th century with regard to the meaning of
“abstraction” and “abstract” it is necessary to begin by recalling what is genuine abstraction and
by stating what should be expected from a theory of abstraction today (section 1). Since Boethius
and till the late 19th century abstraction enjoyed a peaceful life (except for some attacks from the
British empiricists), within logic, metaphysics and psychology, under the reign of the great mas-
ters of abstraction (2). Aside from neo-scholasticism, only a few individual authors carried the
torch of genuine abstraction in our century; for example: Husserl and Piaget; within modern logic:
Weyl, and especially Lorenzen. Probably because of Frege and Russell, abstraction disappeared
from the mainstream of modern logic and analytic philosophy (3). The void was filled by a prolif-
eration of pseudo-uses of the terms “abstraction” and “abstract”: the usurpers (4). The survival of
abstraction in modern logic (“modern abstraction”, Lorenzen) was unfortunately associated with
nominalism (5). Nominalism shuns the challenge of having to say something about the nature of
abstracta (6). But, thanks to nominalism, modern abstraction turns out to be immune to a recent
criticism (7). Signs of a renewed interest in abstraction are mentioned (8). The final reflection is
that philosophers have the right to reject abstraction, but then no pseudo-uses of the word should
be introduced (9).
1. Making a few preliminary points about abstraction1
The following conditions appear to be required in any discussion on abstrac-
tion.
1) The term “abstraction” should be used in the genuine sense. In the his-
tory of philosophy the word “abstraction” and cognate expressions (“abstract”,
etc.) have had a genuine meaning, according to which abstraction involves an
operation by which something is retained and something else is left out — to
use Locke’s words (1959, III, 1, 9). In the special sense relevant for philoso-
phy, the operation is intellectual, and the retaining and leaving out pertain to
1
The present essay offers extensions and revisions of my previous writings on abstraction,
starting with my (1979). The earlier papers, however, offer more details, and include references
not cited here.

12 Ignacio Angelelli
our mental consideration of things. This genuine meaning has been lost in the
mainstream of the logical and analytic tradition of the 20th century and, what
is worse, has been replaced by pseudo-uses.
2) The starting point of abstraction should be a set of (true) sentences. I
will assume that the operation of abstraction can be represented as an operation
on sentences — on sentences that one regards as true. Thus, in classical termi-
nology, the terminus a quo of abstraction is a collection of (true) sentences:
one decides to ignore, “hide”, abstract from, perhaps temporarily, some of
them.
3) A technique of abstraction should be provided. Leaving aside the possi-
bility of doing abstraction “at random”, any systematic abstraction should pro-
ceed according to some principle, reason or criterion. The latter, however,
cannot remain as a generic guidance, but must be broken down into precise in-
structions as to exactly which sentences to retain and which to leave out.
4) The nature of the abstracta should be clarified. The obvious immediate
output of abstraction is the set of sentences that are retained. However, senten-
ces are about objects, and abstraction affects our view, or the way we view,
those objects. The abstract view of an object resulting from the “hiding” of
true sentences about it is the remote output of abstraction, or the abstractum —
in the plural, the abstracta. Here the fundamental question arises: What is the
nature of the abstracta? In linguistic terms: What statements are true or false
about such abstracta?
These four points may serve as criteria to evaluate contributions to abstrac-
tion theory from different periods of the history of philosophy, past or present.
2. The great masters of abstraction
Berkeley writes:
It were an endless, as well as useless thing, to trace the Schoolmen, those great
masters of abstraction, through all the manifold inextricable labyrinths of error and
dispute, which their doctrine of abstract natures and notions seems to have led them
into (1964, Introduction, p. 17, my emphasis).
In this issue, like in many others, the so-called “modern” philosophers —
such as Berkeley — have had the regrettable effect of disconnecting philoso-
phical thought from the state which it had reached about 1600.
Undisturbed by the disconnection, the artisans of abstraction continued to
write, generally in Latin, well through the 20th century. One of their distin-
guished, recent representatives, Santiago Ramirez, devoted more than a hun-
dred pages, in Latin, to (genuine!) abstraction in his four volume treatise De
analogia (On analogy).
After a brief section on the notio abstractionis (the notion of abstraction),

Adventures of Abstraction 13
which is presented as a motion from a terminus a quo (a starting point) to-
wards a terminus ad quem (a terminal point), where the starting point is aliquid
compositum vel coniunctum (something composite or conjoined) and the ter-
minal point is the division or separation of what was united, Ramirez displays
an enormous variety of divisions of abstraction. As a matter of fact, the author
ominously announces that abstractio dicitur multipliciter (abstraction is said in
many ways). The distinctions and subdistinctions of the meanings of the word
abstractio are so exuberant that they require an extra folded sheet — a cartog-
raphy of abstraction!
It is tiresome to reach, through the scholastic labyrinths and distinctions
displayed by Ramirez, the kind of abstraction in which we are interested.
However, one positive fact remains: even in the many meanings foreign to our
concerns, the basic conception of abstraction as deletion or removal, today
forgotten, is preserved.
Ramirez begins with abstraction in ordine physico et corporali (in the
physical or corporeal realm). Here there is always a real separation eius quod
abstrahitur ab eo a quo abstrahitur: sicut navis abstrahitur a portu ad
navigandum... (of that which is abstracted and that from which it is abstracted,
as the ship is abstracted from the harbour, in order to navigate). Among the
scholastic authors, the examples are often amusing: abstractio tunice ab ha-
bente (abstraction of the tunic from him who has it, cf. Sebastianus de Arago-
nia 1992, p. 69), cum ab ave pennae avelluntur (when one plucks a bird, cf.
Signoriello 1893).
Next, Ramirez moves to less physical abstractions, namely to abstraction in
ordine psychologico (in the psychological realm). But again we stumble on a
new distinction. Psychologically, abstraction is said in two ways (dupliciter):
primo in linea vitalitatis et per ordinem ad idem subiectum vivens; secundo in
linea cognitionis et per ordinem ad idem obiectum (firstly in the order of life
and relative to the same living subject, secondly in the order of cognition and
relative to the same object, cf. p. 757). While in the first of these two orders (in
the order of life) we encounter strange things again (strange, at any rate, vis à
vis our expectations), such as death, abduction (p. 758), etc., in the second (or-
der of cognition) what we are looking for finally seems to appear.
However, a new subdivision is waiting for us. The abstraction in the order
of cognition is twofold: abstractio sensitiva and intellectiva. The first has to do
with the fact that one sense does not catch what other senses grasp in an object.
This is close, but not yet what we want.
When we reach the abstractio intellectiva we are finally in business — al-
though, alas, “business” means, in good scholasticism, still one more distinc-
tion, now the famous one between formal and total abstraction.
In total abstraction we recognize what has been usually called abstraction
in logic and philosophy, namely the familiar move from individuals to univer-

sals, from Peter and John to man, and from man and horse to animal, etc. The
word “total” refers to the universal regarded as a whole, of which the individu-
als, or inferior universals, are “parts”. One “abstracts from” the differences
between Peter and John, or from the differences between man and horse —
thus the universal concepts: man, animal, etc., emerge.
It is only with regard to formal abstraction that it becomes hard to under-
stand what was meant by the Schoolmen. I will mention two difficulties I have
encountered.
First, Thomas Aquinas, while generally indicating that in abstraction
something is retained and something is left out (to use again Locke’s terms),
says however that in formal abstraction both items remain, which seems to
mean that nothing is left out:
Respondeo dicendo quod duplex fit abstractio per intellectum. Una quidem, secun-
dum quod universale abstrahitur a particulari, ut animal ab homine; alia enim,
secundum quod forma abstrahitur a materia; sicut forma circuli abstrahitur per in-
tellectum ab omni materia sensibili. Inter has autem abstractiones haec est differen-
tia, quod in abstractione, quae fit secundum universale et particulare, non remanet
id, a quo fit abstractio; remota enim ab homine differentia rationali, non remanet in
intellectu homo, sed solum animal. In abstractione vero, quae attenditur secundum
formam et materiam, utrumque manet in intellectu; abstrahendo enim formam cir-
culi ab aere remanet seorsum in intellectu nostro et intellectus circuli et intellectus
aeris (1950, I 40, 3, respondeo, my emphasis).
An English translation:
I answer that, abstraction by the intellect is twofold — when the universal is ab-
stracted from the particular, as animal abstracted from man; and when the form is
abstracted from the matter, as the form of a circle is abstracted by the intellect from
any sensible matter. The difference between these two abstractions consists in the
fact that in the abstraction of the universal from the particular, that from which the
abstraction is made does not remain; for when the difference of rationality is re-
moved from man, the man no longer remains in the intellect, but animal alone re-
mains. But in the abstraction of the form from the matter, both the form and the
matter remain in the intellect; as, for instance, if we abstract the form of a circle
from brass, there remains in our intellect separately the understanding both of a cir-
cle, and of brass (my emphasis).
A second problem I find with formal abstraction is that it is not clear
whether the retained entity is an individual or a universal. If in considering a
circle made of brass, we ignore the bronze and just retain the circle, as Aquinas
wants, the question arises of whether this retained circle is an individual (acci-
dent) or a universal.
These two problems concerning abstractio formalis do not appear to be
clarified in the very large literature, primary and secondary, devoted to ab-
straction, that I have consulted. A precise, critical analysis of formal abstrac-
tion remains, therefore, as a challenge for critically minded historians of scho-

lastic thought.
Leaving aside this problem, let us evaluate the great masters’ performance
relative to the four criteria defined in section 1.
With regard to the first point, we find, for example, that Aquinas empha-
sizes that abstraction applies to what is united in reality. “What is united in re-
ality” can be reworded, linguistically, as “a true sentence”. Thus, the starting
point of abstraction is, or may be construed as a collection of sentences (ac-
cepted as true).
With regard to the second point (abstractive techniques), Aquinas’ doctrine
may be read as saying that to do abstraction is to “hide” some of the true sen-
tences while keeping other true sentences visible. Cromp tells us (1980, 4, p.
17) that Aquinas is quite emphatic about this: “le texte [Thomas Aquinas] ne
cesse de rappeler que l’abstraction consiste dans une non-considération” (the
text does not stop reminding us that abstraction consists in a non-considera-
tion). Crucial here is to keep in mind that to “hide” a sentence is not to deny it
or to regard it as false: abstrahentium non est mendacium (those who do ab-
straction are not liars, cf. Wyser 1947, p. 475).
Aside from this, however, in general one should not expect techniques of
abstraction prominently listed in the books of the great masters, since the ab-
stractive activity, in the scholastic framework, appears to be conceived as a
natural process, accomplished by the intellect (agens, possibile, etc.), rather
than as an operation that we must perform according to given rules.
Perhaps the closest to an abstractive technique in the scholastic tradition is
the neglected theory of reduplication, which mysteriously vanished from (non-
scholastic) logic textbooks in the 20th century.2
Reduplication is the linguistic trick by which we restrict our discourse, ini-
tially about {a, b, c...}, by attaching to the singular terms “a”, “b”, “c”... the
particle “qua-F” (or equivalents). Now we talk not about a, but about a-qua F;
for example not about Peter, but about Peter-qua-man. Of course the F is just
one out of infinitely many choices. Once F is fixed, however, lots of true
statements about a, b, c... are, in J. Lear’s felicitous phrase, “filtered out”, i.e.
hidden, or abstracted from.
The masters of abstraction do not seem to tell us much either with regard
to the nature of the abstracta in general, although they abundantly discuss
particular cases of objects that they regard as products of abstraction, such as
universals. Thus, through the particular case of universals we should expect to
gain some insight into the Aristotelian-scholastic conception of the nature of
abstracta in general. Consider, for example, the universal man (homo). Man
2
Reduplication has been finally dusted off by A. Bäck. Cf. his impressive volume, as well as
some of my publications.

“considered in itself” is, for the Aristotelian-scholastic authors, a collection of
other universals: {animal, rational}. From here we may perhaps generalize and
say that abstracta are, for those authors, collections of properties.
One difficulty to be kept in mind in evaluating the great masters’ under-
standing of the nature of universals, or abstracta in general, is the following.
Whatever intuitions the classical philosophers may have with respect to the
nature of universals, or equivalently, with respect to determining the true
statements that can be made about universals, the formulation of those intui-
tions will be hampered by the peculiarities of the Aristotelian (classical, pre-
Fregean) predication theory. The Organon is not too interested in sentences
such as “man is a universal” (or, to choose a good scholastic example, “man is
an abstract entity in absolute consideration”), and does not give instructions
about them.
3. Abstraction rejected from the logico-analytic mainstream
Abstraction is present, in different degrees, in most of the philosophers from
the modern period. In fact, an interesting idea developed: one entity taken as
representative of a class (Locke 1959, II, 11, 9). This is on the right track; for
instance, in moving from fractions to rationals one may choose a particular
fraction, say 1/2, as representative of the rational 1/2. Unfortunately, the con-
sequences of this notion were not rigorously pursued and the project remained
misleadingly half-way, unfinished, as if the consideration of the representative
involved an exemption from looking beyond the initial, (relatively) concrete
objects. To take the particular fraction 1/2 as representative (of the class of all
fractions that are equivalent to 1/2, with equivalence defined as having equal
cross-products) is tantamount to introducing a new entity in one’s domain of
discourse, and correspondingly a new singular term (“1/2 qua representative”).
Here, requirements (3) and (4) listed in section 1 of this paper must be satis-
fied: precise rules for the selection of statements, and some explanation on the
nature of the new entity must be offered. Aside from the self-defeating, mud-
dled comments on “the general triangle”, neither issue was satisfactorily ad-
dressed by Locke or subsequent philosophers.
In the past hundred years, abstraction has been present in individual
authors (Husserl, Piaget3
, and within modern logic: Peano, Weyl, Lorenzen),
but has been absent from the mainstream of mathematical logic and analytic
philosophy, probably as a consequence of Frege’s and Russell’s negative view
of it.
3
Cf. Battro (1966), articles on abstraction.

Frege speaks of abstraction in three senses (cf. my 1984): (i) the abstrac-
tion he calls “magical”, (ii) the common or ordinary abstraction, iii) the defini-
tion by abstraction. The latter phrase occurs only once in Frege’s writings, in a
letter to Russell, where Frege explains that the method of transforming a sym-
metric and transitive relation (similarity) into an identity (identity of the as-
pects in which the similar objects coincide) would perhaps be referred to by
Russell as “définition par abstraction” (in French). This is suggestive, but de-
ceitful: the phrase “definition by abstraction”, coined by Peano, corresponded,
for the Italian mathematician, to a genuine interest in abstraction, but this was
no longer the case in Russell, as we shall see. The notion of magical abstrac-
tion occurs within Frege’s vehement criticisms to the way in which the ab-
straction leading from a set to the number (cardinality) of the set had been
analyzed by mathematicians like Cantor. According to these authors, that ab-
straction leading, for instance, from the set of the Apostles to 12 consists in ig-
noring the peculiarities of each individual member of the set. Frege rightly
found this “impossible” – since the result would simply yield the concept
Apostle. Only magic could do it.4
Under the heading of common or ordinary
abstraction Frege deals with what can be regarded a simplistic version of the
classical account of abstraction. Here Frege had no major objections, but no
major enthusiasm either – it is as if he did not know what to do with that good,
old abstraction. In sum, it is possible to affirm, on the basis of a careful study
of the texts, that Frege’s final recommendation with regard to abstraction is not
to waste one’s time with it, and pass it on to psychology. Given Frege’s anti-
psychologist view, this amounts to a strong rejection.
In Russell, the abstraction story is different. As we see in The Principles,
Russell considers a style of defining called by Peano “definizione per astra-
zione”, that Russell finds bad enough to declare that it is not valid (end of sec-
tion 110) but at the same time good enough to try to improve it, and make it
into something acceptable. Peano – according to Russell – assumes that any
relation that is both symmetric and transitive (for instance, the similarity that
obtains between Mary and Peter insofar as both are human beings) allows us,
without further ado, to define, “by abstraction”, a new object that corresponds
to what is usually called a “common property” of the given objects (in the
given example, the property of being human). Russell first criticizes Peano for
not securing the uniqueness of that something common to the initial objects
(section 110); secondly, Russell objects to Peano’s lack of a proof of existence
of that common entity (section 210). Russell says that he secures existence by
a procedure called “principle of abstraction”, while uniqueness is trusted to a
nominal definition (section 110). This principle of abstraction will become,
4
Both Cantor and Frege were wrong however – cf. my (1984), §7, p. 468.

later on in Principia Mathematica, a beautiful theorem (72.66) – beautiful but
no less deceiving for anyone interested in genuine abstraction. To be sure,
theorem 72.66 does prove the existence of something common to Peter and
Mary, given that they stand in the symmetric and transitive relation: “x is
human y is human”. However, that common entity turns out to be, simply,
the equivalence class relative to the chosen equivalence relation. Anyone
expecting from theorem 72.66 an important and enlightening contribution to
the theory of abstraction will be quite disappointed. No operation of
abstraction is performed: all we do is move from human beings to a class
containing them. To reply that that class is an abstract entity would require an
explanation of the term “abstract”: if used in the pseudo-sense of “intangible”,
it is not interesting for our purposes; if a genuine abstraction is involved, it
should be properly explained and described. In sum, theorem 72.66 cannot be
called “principle of abstraction”. In fact, Russell himself corrects his
terminology later on, saying that rather than “principle of abstraction” one
should speak of a principle that dispenses with abstraction – a principle that
allows us to do well without worrying about abstraction (1956, p. 326).
4. The usurpers
Not only was abstraction exiled, but usurpers took its place in the mainstream
of logic and philosophical analysis in the 20th-century. There have been, or
rather there are, little usurpers and one big usurper.
4.1 The little usurpers
The little usurpers are uses of the term “abstraction” (“abstract”, etc.) in which
no reference is made to any properly abstractive process. There are two sorts
of little usurpers: (1) uses of the adjective “abstract” that tend to mean some of
the following: “non-accessible to the senses”, “intangible”, “neither spatial nor
temporal”, or “not possibly being located in space- time”. (2) Use of the noun
“abstraction” as tending to designate some notational operation which gener-
ates a term allegedly denoting an entity that is “abstract” in some of the just
described, pseudo-senses.
A venerable, but no less wrong, example is provided by H. Scholz. If, à la
Frege, the singular term 5 is deleted from the sentence “5 is prime”, we obtain
a concept-word: “( ) is prime”, which denotes a concept. This “punching out”
of singular terms has been called by H. Scholz die logistische Abstraktion-
stechnik (the logistic technique of abstraction, 1935, p. 15). I cannot imagine a
worse violation of the first or third conditions listed in section 1 of this paper.

In Feys and Fitch’s Dictionary three senses of “abstraction” are distingui-
shed: a syntactical, a semantical, and a real. Syntactical abstraction is the
symbolic operation of “obtaining a name of a function by operating with the λ-
operator on an associated form of the function” (40.3). The λ-operator is called
an “abstractor”, and the result of applying such an abstractor to a wff a: λxa, is
an abstract. For semantical abstraction Feys and Fitch refer to Church’s Intro-
duction, p. 22, where in fact we learn that “the passage from a form to an asso-
ciated function (for which the λ-notation provides a symbolism)” is an ab-
straction. Needless to say, both the syntactical and the semantical senses are
paradigms of pseudo-abstraction. In the final analysis, all they mean is that
some linguistic expressions are intended to signify entities that are called “ab-
stract” because they are intangible, not in space or time, etc.
The “real” abstraction mentioned by Feys and Fitch is described as “the
process of finding the function itself, given suitable information about the va-
lues of the function for all its arguments” (40.2). Here the word “process”
might hide some authentic, albeit undercover, abstractive activities, the consi-
deration of which probably stems from Feys’ scholastic background. At the
same time, however, I cannot fail to point out the slightly amusing nature of
the terminology chosen by Feys: “real abstraction”, in the tradition of the great
masters of abstraction, was rather applied to physical operations (cf. the exam-
ple quoted in section 2: plucking the feathers of a bird).
In Quine, the usurpation becomes even more systematic and prominent.
“Abstraction” is, for example, the title of an entire section (§24) of Quine’s
Mathematical Logic, where we learn, again, that “the forming of class names
by such prefixes (Quine refers to ordinary phrases as “the class of all elements
x such that” or a circumflex written on top of an x) will be called abstraction;
and the result... will be called an abstract”.
Particularly interesting, and revealing, are the following passages from
Quine 1961, pp. 117-119 (emphasis is mine, except for the word “inscrip-
tion”5
):
It may happen that a theory dealing with nothing but concrete individuals can con-
veniently be reconstrued as treating of universals, by the method of identifying in-
discernibles. Thus, consider a theory of bodies compared in point of length. The
values of the bound variables are physical objects, and the only predicate is “L”,
where “Lxy” means “x is longer than y”. Now where ~Lxy.~Lyx, anything that can
be truly said of x within this theory holds equally for y and vice versa. Hence it is
convenient to treat “~Lxy.~Lyx” as “x=y”. Such identification amounts to recon-
struing the values of our variables as universals, namely, lengths, instead of physi-
cal objects.
Another example of such identification of indiscernibles is obtainable in the theory
5
S. Pollard drew my attention to these significant texts.

of inscriptions, a formal syntax in which the values of the bound variables are con-
crete inscriptions. The important predicate here is “C”, where “Cxyz” means that x
consists of a part notationally like y followed by a part notationally like z. The con-
dition of interchangeability or indiscernibility in this theory proves to be notational
likeness, expressible thus:
(z)(w)(Cxzw Cyzw . Czxw Czyw . Czwx Czwy).
By treating this condition as “x=y”, we convert our theory of inscriptions into a the-
ory of notational forms, where the values of the variables are no longer individual
inscriptions, but the abstract notational shapes of inscriptions.
This method of abstracting universals is quite reconcilable with nominalism, the
philosophy according to which there are really no universals at all. For the univer-
sals may be regarded as entering here merely as a manner of speaking – through the
metaphorical use of the identity sign for what is really not identity but sameness of
length, in the one example, or notational likeness in the other example. In ab-
stracting universals by identification of indiscernibles, we do no more than rephrase
the same old system of particulars.
Unfortunately, though, this innocent kind of abstraction is inadequate to abstract-
ing any but mutually exclusive classes. For when a class is abstracted by this
method, what holds it together is the indistinguishability of its members by the
terms of the theory in question; so any overlapping of two such classes would fuse
them irretrievably into a single class.
Another bolder way of abstracting universals is by admitting into quantifiers, as
bound variables, letters which had hitherto been merely schematic letters involving
no ontological commitments. Thus if we extend truth-function theory by introduc-
ing quantifiers “(p)”, “(q)”, “(p)”, etc., we can then no longer dismiss the state-
ment letters as schematic. Instead we must view them as variables taking appropri-
ate entities as values, namely, propositions or, better, truth-values, as is evident
from the early pages of this essay. We come out with a theory involving universals,
or anyway abstract entities.
Here Quine observes that the quantifiers over sentential variables “happen
to be reconcilable with nominalism if we are working in an extensional
system”, so that “what seemed to be quantified discourse about propositions or
truth-values is thereby legitimised, from a nominalist point of view, as a figure
of speech”. However,
abstraction by binding schematic letters is not always thus easily reconcilable with
nominalism. If we bind the schematic letters of quantification theory, we achieve a
reification of universals which no device ... is adequate to explaining away. These
universals are entities whereof predicates may be thenceforward be regarded as
names.
Quine distinguishes an innocent abstraction and a bold abstraction. The
former, also called by him “the method of identification of indiscernibles” – of
which he gives two examples: lengths and notational forms – is practically the
abstraction method (Peano, Weyl, Lorenzen) to which I refer below (section
5). Quine finds this method, in spite of its innocence, “inadequate”. I fail to see
the force of this objection. One may simultaneously consider, although not

simultaneously perform, two different abstractions on the same domain of
initial objects; for instance, given the domain of fractions, one may consider
both an abstraction relative to the equivalence relation of having equal cross-
products and an abstraction relative to the equivalence relation of having even
numerators (“x’s numerator is even y’s numerator is even”). Thus, 2/4 and
7/14 become indiscernible within the former, but not within the latter abstrac-
tion.
With regard to Quine’s “bolder method of abstraction” I should say that it
may be bold — but it is not abstraction. No abstraction is involved in that me-
thod, except the fact of referring to, or naming entities that Quine calls “ab-
stract” probably because they are intangible or inaccessible to the senses.
I do not know of any reason to explain the proliferation of all these pseudo-
uses of “abstraction” and cognate words. They merely filled a void left by the
removal of genuine abstraction.
4.2 The big usurper: looking-around or circumspection
I have coined the designation “looking-around method” for the simple reason
that the phrase “looking around” is found in Carnap’s Meaning and necessity,
in connection with a procedure to be examined in this section. If a more “se-
rious” term is desired, the word “circumspection”, in its etymological sense, is
a perfect equivalent, for my purposes.6
The motivation for those who, following Carnap, practise this method is to
give a more exact sense to (“to explicate”) certain expressions from scientific
or ordinary language, such as: “the number of a concept F”, “the extension of a
concept F”, “the cardinality of a set S”, etc. Let us generally describe these ex-
pressions as being of the form “f(a)”, “f(b)”, etc.. For the purpose of attaining a
better definition, one pretends to forget the familiar meanings already asso-
ciated with such singular terms, so that, officially, the latter become temporar-
ily meaningless.
Next, instead of worrying immediately about the new meaning to be given
to “f(a)”, “f(b)”, etc., one will become concerned about identity conditions for
whatever entities are eventually assigned to these symbols. In fact, the formu-
lation of identity conditions will be the first step of our method, previous to the
assignment of any denotations to “f(a)”, “f(b)”, etc.. While pretending to ig-
nore, for example, the meaning of “the number of the concept F”, one stipu-
lates that for any meaning, the number of the concept F = the number of the
concept G iff F and G stand in the relation consisting in the one-one corre-
spondence between the objects that are F and those that are G. The chosen re-
6
This was suggested to me by Jaime Nubiola.

lation must be, as in the given example, symmetric and transitive, i.e. an
equivalence relation.
The second, and final, stage of the method consists, as Carnap tells us
(1960, p. 1), in looking around for entities that are suitable candidates to play
the role of denotata of our singular terms of the form “f(a)”, “f(b)” ... By “suit-
able” is meant that the assignment must be compatible with the identity condi-
tion stated in the first part of the method. One popular choice of a suitable en-
tity are the equivalence classes determined by the equivalence relation under
consideration; in this case, for instance, the singular term “f(a)” receives, as
denotation, the class of entities that stand to a in the given relation. But indefi-
nitely many other choices are equally suitable, and in principle, i.e. as far as
the method is concerned, there is no reason to choose one rather than another.
Equivalence classes somehow became the natural or first choice, but any other
type of “suitable” entity is equally entitled to be selected.
Circumspection is the method that Frege uses in two crucial points of his
theory: in the definition of number and in the definition of set (but, for a dif-
ferent opinion, cf. Thiel 1986, p. XLII). In The Principles, Russell transforms
Peano’s definitions by abstraction (ignoring their authentic abstractive compo-
nent) into the looking around method (continuing to call it, at least for some
time, “the principle of abstraction”). The method is applied by Russell and
subsequent authors, such as Carnap and Quine, in order to define – “explicate”
– several fundamental concepts: cardinal number, extension, intension, etc.
It is hard to understand, as already Husserl pointed out in his Philosophie
der Arithmetik in connection with the philosophical première of the method of-
fered by Frege, how such a procedure may have been regarded as progress in
logical theory. It is a frivolous, irrational method – one more hara-kiri of rea-
son (Weyl 1928, p. 41), since no reason or justification is provided for the
choice of entities (among the infinitely many that are “suitable”).
How can we regard circumspection as an usurper of abstraction’s place,
given that, according to the above description, the circumspection method has
nothing to do with abstraction?
The answer is quite simple: circumspection, at a certain point, either in
Peano himself or among his associates, began to be called “definition by ab-
straction”, a phrase originally coined by Peano with genuine abstraction in
mind. The semantical mutation (or usurpation) was facilitated by the fact that
Peano’s abstraction, as made fully clear, later on, by Lorenzen, needs as infra-
structure a previously defined equivalence relation (for instance: the equality
of cross-products of fractions, leading, by abstraction, to the rationals). Instead
of using the equivalence relation as an abstraction ladder, logicians employed
it as a platform for the circumspection method. A full disentanglement of the
complex circumstances and confusions that surrounded this tale of two cities
(abstraction and circumspection) is still an open task in the historiography of

modern logic.
While at no point do the rules of the circumspection method prescribe the
performance of any genuine abstraction, it must be observed that the fact that
all suitable candidates are “equal” can be construed as an abstraction. This is
true – but it only shows that the circumspection method badly needs a serious
overhaul. The abstraction involved in the fact that the differences among the
suitable candidates can be ignored (abstracted from), and that it does not really
matter which of the suitable candidates one finally chooses, should be explic-
itly highlighted and duly analyzed (just as Locke should have done with his
“representative” view of abstract objects). Without such a radical transforma-
tion of the looking-around method into an abstraction method, the former can-
not be called “abstraction” – neither “definition by abstraction” (as often done
in the 20th- century) nor “logical abstraction” (as Dummett proposes, cf. next
section).
Some authors, like Russell, intelligently realized that it was preposterous to
talk of abstraction where nobody plans to do abstraction, and corrected the
terminology (cf. section 3, above). Others, like Beth, unwisely kept the termi-
nology while inconsistently admitting that no abstraction is intended (1959, p.
94). Still others, the worst (in this respect) from a philosophical standpoint,
believed (in view of the fact that the favourite entities for the looking around
are the equivalence classes) that the method was really an analysis of the old
notion of abstraction in terms of the new set or class theory; so Reichenbach:
“the notion of class finds an important application in the interpretation of a
logical operation which traditionally has been called abstraction” (1956, sec-
tion 37).
4.3 The final blow: total semantic depletion
In Dummett’s (1991), the main theme in the discussion of Frege’s answer to
the question “What is number?” is abstraction. Philosophers of arithmetic are
classified by Dummett into “bad” and “good” depending on their using or not
using, respectively, abstraction in their definitions of number; this evaluation is
presented together with a strong attack on abstraction. Dummett thinks that
Frege is a champion of anti-abstractionism – I disagree: properly, Frege just
criticizes the misuse of abstraction by Cantor and others in their attempts to
define number. Finally, Dummett thinks that Frege does, after all, practise ab-
straction – but in a new, good sense of the term, which Dummett calls “logi-
cal” abstraction, in contrast with the bad, “psychological” abstraction. Dum-
mett characterises these two abstractions in the following manner:
Both types of abstraction aim at isolating what is in common between the members
of any set of objects each of which stands to each of the others in the relevant
equivalence relation: Frege’s logical method by identifying the common feature

with the maximal set of objects so related to one another and containing the given
objects; the spurious psychological operation by deleting in thought everything ex-
cept that common feature (pp. 167-8).
Frege’s “logical method” – so much praised by Dummett – is what I have
called the looking-around or circumspection method, a frivolous procedure
that cannot be referred to as “logical abstraction” or as abstraction of any sort
simply because there is no abstraction in it, except in the remote sense
mentioned in the preceding section. The method is not really interested in any
of the infinitely many, different particular types of objects that are “suitable”;
all that matters is their common suitability, i.e. the compatibility of the chosen
objects with the previously established condition (naturally, each author may
add extra requisites). But then of course this obscure desire for doing abstrac-
tion should be properly expressed, and the equivalence class could no longer
be put forward by the “circumspective” authors as the denotatum; rather the
equivalence class-qua-compatible with the previously established condition
should be the denotatum. Here, however, the denotatum of the singular term
“the equivalence class-qua-compatible with the previously established condi-
tion” is, and should be recognised, as quite a new entity, an abstractum in a
genuine sense. A rigorous handling of this abstractum and of the correspond-
ing abstraction would require, first, disassembling the looking-around tech-
nique, and secondly, reassembling it according to some abstraction method
(probably à la Lorenzen, cf. below, section 5).
Dummett’s phrase “logical abstraction” is a striking example of how the
word “abstraction” has lost, in contemporary philosophy, and especially in the
logico-analytic tradition, any connection with what is essential to abstraction:
“leaving out” something and “retaining” something.7
Dummett’s text quoted
above is perhaps a unique item in the collection of pseudo-uses of “abstrac-
tion” of the past century: no other text is known to me where the operation of
“deletion” (the “leaving out”) is explicitly excluded from the notion of ab-
straction.
Dummett’s study of Frege’s philosophy of mathematics represents a final
blow to the concept of abstraction, as well as the total semantical depletion of
the term.
4.4 Victims of the usurpers
Authors using the term abstraction (or related expressions) in a non-genuine
sense may be false prophets of abstraction (actively contributing to the usur-
7
Curiously, “logische Abstraktion” occurs in Kant-Jäsche (1800), Allgemeine Elementarlehre,
§ 15. But there abstraction is genuine: it means Absonderung (deleting, leaving out), cf. ibid. § 6.

pation) or mere victims of the false prophets (just misled by the false proph-
ets). Here are some instances of the latter.
The main victim has been H. Scholz, in his very important and equally ne-
glected Definitionen durch Abstraktion. The first part is a survey of what
Scholz calls “classical” abstraction theory (which is in fact medieval, since
Aristotle has little to do with it). Scholz believes that classical abstraction
should be replaced by “something better” (1935, § 1,1). Unfortunately, this
something better turns out to be the circumspection method.
Other examples of distinguished philosophers misled by the usurpers are
the following. Körner talks of “Frege’s principle of abstraction” and explicitly
commends the application of this terminology to the presentation of Frege’s
views: “the name given to the principle which is used to justify the definitions,
the principle of abstraction, has been well chosen”.8
G. Küng, in his study of
universals in recent philosophy writes: “The operation of constructing (or dis-
covering) the equivalence classes of things with equal concrete properties cor-
responds to what is traditionally called abstraction” (1967, p. 173). More re-
cently, Patzig (1983) describes Frege’s procedure in defining number as fol-
lows: “Frege schlägt hier einen Umweg ein, der als “Definition durch Ab-
straktion” folgenreich und berühmt geworden ist” (Frege recommends here an
approach that has become famous and fruitful as “definition by abstraction”).
5. Quiet survival (modern abstraction)
The negative view with regard to abstraction in the logico-analytic tradition
has one exception in the work of Paul Lorenzen, who has continued, in my
view, hints found in Peano and in H. Weyl. To be sure, the interest of these
scholars in genuine abstraction has been largely unnoticed.
Lorenzen’s main merit, with regard to abstraction, is that he was the first to
dare to overhaul the circumspection method in the direction indicated above
(section 4.2), turning it upside down, bringing out, and subjecting to a rigorous
analysis, the abstraction that obscurely underlies the “equivalence” of all “suit-
able” candidates.
The following is the key text for anyone interested in understanding the
confused interaction – started in Peano’s writings – between (genuine) ab-
straction and circumspection in the history of modern logic:
Dadurch [that is according to what is customary since Frege and Russell] soll die
Abstraktion auf die Einführung von “Klassen” zurückgeführt werden. Wir werden
8
Körner (1962), p. 38. Körner refers to Frege’s definition of number in Frege (1986), §§ 62-
69.

jedoch weiter unten sehen, dass die Klassen nichts anderes als ein spezieller Fall
von abstrakten Objekten sind (Lorenzen 1955, p. 101).
Thereby [that is according to what is customary since Frege and Russell] abstrac-
tion is to be reduced to the introduction of “classes”. However, we will see below
that classes are nothing else but a special kind of abstract objects.
First of all, it is remarkable that in this 1955 text both the noun “abstrac-
tion” and the adjective “abstract” are used in the genuine sense. This should be
a major surprise for readers familiar with modern logic. Moreover, the passage
is not an isolated, casual remark but belongs in a section titled: Abstraktion,
Relationen und Funktionen.
The great significance of the brief text may not be obvious for a reader un-
aware of the tensions between abstraction and circumspection since Peano’s
time.
In the first sentence Lorenzen points to the fact that in “Frege and Russell”
(I would more generally say: in the circumspection method at large) abstrac-
tion was reduced to classes (I would say more generally: abstraction was re-
placed by the looking-around method, where classes, equivalence classes that
is, happen to be the most popular among the “suitable” entities).
Thus, the first sentence means much more than what it says.
On the other hand, the second sentence says more than what I think is right.
That classes can be understood as abstract objects (again, genuine abstraction!)
is obvious to anyone who seriously reads, for instance, Frege’s definition of
Wertverlauf (Axiom V) in the light of Frege’s own comments in 1962, II, §
146.9
But I would not say that classes are “nothing else but a special kind of
abstract object”, since other – let us say “intuitive” – conceptions of class are
possible.
Having chased away the main usurper, and having restored genuine ab-
straction, Lorenzen moves to the description of a technique for abstraction,
conceived as an operation on sentences. This is as follows.
We start with a universe of discourse: a, b, c... on which an equivalence
relation, i.e. a relation (transitive and symmetric) is defined. At a certain point
we decide to do abstraction. This means that we decide to restrict our lan-
9
To be sure, Frege’s comments are not a full statement to the effect that his classes are
abstracta (in the genuine sense). All one can say is that Frege somehow realizes that one begins
with an equivalence relation defined for predicates (equiextensionality in his case), and that any
two predicates standing in that relation have etwas Gemeinsames (something in common), which
is the class. Unfortunately, this remarkable observation was never developed by Frege into a
theory of classes as abstracta, and Frege contented himself with the “circumspective” reading of
the “something in common” as some “suitable” entity (truth-values in this case).

guage, our discourse about the objects a, b, c..., in such a way that we will use
only predicates that are non-vacuously invariant10
with respect to the chosen
equivalence relation. The obvious effect of this linguistic decision is to re-
move from circulation a number of (true) sentences and to obtain that two
initial objects a, b, different before doing abstraction, become now indis-
cernible while doing abstraction.
In a large historical perspective, the novelty of all this is that while the
great masters of traditional abstraction trusted the entire operation to the natu-
ral mechanisms of the intellectus agens or possibilis, in the just described
modern abstraction it is our responsibility to perform the selection of sentences
to be hidden and of sentences to be retained, and we have precise instructions
on how to go about it. Thus, in the modern version, abstraction has been trans-
ferred from nature to art (technique).
Once genuine abstraction has been restored, and a technique for abstraction
as an operation on sentences has been explained (points (1), (2) and (3) from
the list given in section 1), it remains to discuss the most difficult issue,
namely what is the output of abstraction.
Indeed, we should begin with an even more fundamental question, namely
whether there is any output at all to the endeavours of modern abstraction. To
use Peano’s and Lorenzen’s favourite example, when we restrict our discourse
about fractions to statements that are invariant with respect to the equality of
cross-products, obviously the old domain of fractions (where for example 1/2
and 36/72 are so different) evaporates. We are not talking anymore about frac-
tions, since we do not want to say that, for instance, 1/2 is not further simplifi-
able. This is easy to accept, but the question comes next: Does a really new
domain emerge? H. Weyl encourages us to lean towards an affirmative an-
swer, when he writes that abstraction creates einen neuen Objektbereich (a
new domain of objects, 1928, p. 9).
If Weyl’s insight is accepted and followed up, there will be a justification
for introducing new singular terms – for the members of the alleged new do-
main. To continue the example of abstraction on fractions, we may write
“[1/2]”, or, with ordinary words: “the rational 1/2” to denote what is left (re-
tained, not abstracted from) the fraction 1/2. The rational 1/2 is quite different
from the fraction 1/2: for instance, the sentence “the rational 1/2 is not further
10
Cf. my (1979), § 3. My “non-vacuously” clause was intended to correspond to Lorenzen’s
condition (4.5) in (1962): “Wird A(ã) behauptet, so wird damit also A(a) behauptet” (If A(ã) is
affirmed, by that very fact is therefore A(a) affirmed). The significance of that clause will appear
below, in the examination of Siegwart’s critique (section 7). The usual formulation of invariance:
xy(A(x) x~y) A(y)), as an instruction on how to perform abstraction, has a loophole: it lets
in vacuously invariant predicates. This is of course against the intentions of anyone who wants to
do abstraction: the idea is to drop predicates, not to add predicates.

simplifiable” is not affirmed – it has been “hidden”.
Once we have introduced singular terms as designations for the abstracta,
the ontological question of what the entities they stand for are, i.e. what are the
abstracta, or its linguistic version of how the new singular terms should be
used, cannot be further postponed. This question has been clearly stated by
Lorenzen (1978, p. 41):
Man kann ein Wort an die Tafel schreiben, aber man kann keinen Begriff an die
Tafel schreiben. Ein Wort, so lautet die übliche Erklärung hierzu, ist etwas Kon-
kretes, ein Begriff aber etwas Abstraktes. Wer kritisch ist, wird sich mit dieser
Erklärung nicht zufriedengeben. Er wird fragen, was abstrakte Gegenstände
(kürzer: Abstrakta) seien. Oder schon etwas gewitzter: Wie wird das Wort “Ab-
straktum” verwendet? (emphasis mine).
It is possible to write a word on the blackboard, but a concept cannot be written on
the blackboard. A word, as the usual explanation goes, is concrete, while a concept
is abstract. Nobody who is critical can be satisfied with such an explanation, and
will ask: What are the abstract objects (briefly: the abstracta)? Or in a subtler way:
How is the term “abstract” used? (emphasis mine).
Lorenzen’s answer is, unfortunately, not as good as his question. In fact, it
would be more accurate to say that his answer ruins his question. Lorenzen
takes the output of abstraction to be a mere façon de parler (manner of speak-
ing). This is a consequence of a strong nominalism (in his lectures, Lorenzen
would write a class-expression on the blackboard and then would warn the
audience not to look for anything “behind” it). In the quoted passage two
questions are asked: one is: What are the abstract objects?, the other is: How is
the term “abstract” used? The nominalistic answer for the first question is:
“nothing”, and for the second question: “as a manner of speaking”.
Either because of the paralyzing effects of nominalism or for other reasons
(philosophers at a certain point in their careers may want to concentrate their
efforts on new topics), the fact is that a “correct description of the phenome-
na”11
pertaining to the nature of abstracta never became one of Lorenzen’s
leading concerns. The new singular terms for abstracta that Lorenzen in-
troduces are only apparent, they are just “abstractors”12
, i.e. reminders that we
are committed to abstraction in our discourse about the initial concreta. Thus,
11
Remembering Husserl’s impressive statement: “die Schwierigkeit liegt in den
Phaenomenen, ihrer richtigen Beschreibung, Analyse und Deutung” (the difficulty lies in the
phenomena, their correct description, analysis and interpretation, Husserl 1891, p.142).
12
(1973), III, §7: Abstraktoren: “Abstraktoren sind nicht Prädikatoren, die wir Gegenständen
zusprechen können, sondern lediglich Zeichen, die anzeigen, dass Aussagen in einere bestimmten
Weise verstanden werden sollen” (Abstractors are not predicators that we can ascribe to objects,
but only signs that indicate that sentences must be understood in a certain way).

the singular terms “[1/2]” or “the rational 1/2” do not denote any new entity
but only remind us that they (they? or just the old “1/2”?) can fill the blanks
only of predicates that are non-vacuously invariant. Alternatively, we may say
that the singular terms “[1/2]” or “the rational 1/2” are totally eliminable in fa-
vour of just “1/2” or of any other “m/n” such that 1.n = 2.m. Lorenzen’s fa-
vourite abstractor was a tilde “~”; thus, to indicate that a singular term “a” is
subject to abstraction, he would write “ã”.
This nominalistic approach is wrong. The very statement, in the preceding
paragraph, that the new singular terms are “just reminders” or “abstractors”
indicating that we are committed to abstraction “in our discourse about the
initial concreta”, reveals the incoherence. The truth is that “our discourse
about the initial concreta” is a matter of the past – now we are talking (or at
least trying to talk) about something else.
In spite of such nominalistic shortcomings, which hinder the proper study
of point (4) (in the abstraction program sketched in section (1) of this paper), it
must be said, in fairness, that Lorenzen’s contribution with regard to points
(1), (2), and (3) (restoration of genuine abstraction against circumspection, and
Entpsychologisierung of abstraction), remains outstanding.
6. On the difficulties of not being nominalists (with regard to abstracta)
There is also the other side of the coin: it is easy to complain about nominal-
ism but difficult to build something better, i.e. to produce a good answer to the
ultimate question: What is the output of abstraction? Here the phenomena – the
abstracta – are hard to describe.
It should be helpful to distinguish a merely practical and a theoretical
knowledge with regard to the abstracta. There are prestigious examples of the
existence of the former without the latter.
Consider the abstraction that takes place by restricting one’s language on
sets to properties that are non-vacuously invariant relative to bijection. One
may be quite confident that something definite is attained by that abstraction,
and one may give a name to it: “cardinal number”. Nevertheless, one may
content oneself and (worse) one’s readers, with a half-way notion and a half-
way account of the nature of the output of abstraction, describing the abstrac-
tum as “an arbitrary representative” (ein beliebiger Repräsentant) (Kamke
1962, § 8) of sets equivalent with respect to bijection. The author of the classi-
cal set-theory textbook I am referring to, Kamke, does not explain what is this
notion of an “arbitrary representative”. The notion is, as Locke’s “representa-
tive”, on the right track – but needs further analysis, without which the reader
wrongly believes that the universe of discourse continues to be the same: sets,
rather than quite new entities: abstracta. Kamke, the practising mathematician,

is not interested in illuminating the nature of such abstracta, or does not know
how to go about it, or is unaware of the issue. Still, on behalf of Kamke, it
must be said that he does better than his illustrious predecessor, the founder of
set-theory. In fact, Cantor’s definition of cardinal number (quoted in a footnote
to the last section of this paper) cannot be offered as a good example of even
merely practical knowledge of abstracta, since the technique of Cantor’s ab-
straction is on the wrong track from the very outset: he suggests that the ab-
straction leading from a set to its cardinality consists in ignoring the differ-
ences among the elements of the set (cf. my 1984).
By “theoretical” knowledge of the abstracta I mean that at least some hints
are offered with regard to the internal nature of abstracta. Peano (1958, appen-
dice) appears to say a bit about the interior of abstracta when he writes that
from the fraction 2/3 “otteniamo per astrazione l’idea del razionale 2/3, con-
servando tutte e sole le proprietà, o note, che esso ha comuni coi suoi eguali, e
sopprimendo quelle che lo distinguono” (we obtain by abstraction the idea of
the rational 2/3, keeping all and only the properties, or marks, that it has in
common with its equals, and suppressing those that differentiate it), emphasis
mine.
Is then the abstractum (in Peano’s example: the rational 2/3) a bundle of
properties – all those of the (relatively) concretum (the fraction) minus the
non-invariant ones? Or, by elaborating on Kamke’s “representative”, and con-
sidering that to take a concretum as representative of its equivalents is really to
take a slice of it, is the abstractum rather a mutilated concretum (amusingly, in
Peano’s example: “a fraction” of a fraction)?
7. A recent criticism of modern abstraction
Siegwart’s recent criticism of “constructive” abstraction – the derivation of a
contradiction – does not apply to modern abstraction (as I prefer to say instead
of “constructive abstraction”) as far as (my version of) Lorenzen is concerned.
Siegwart uses the predicate “x is identical to an abstractum” or “x is an abstract
entity”, which is only vacuously invariant in the initial domain of concreta –
hence beyond consideration.
Here one may appreciate both the convenience and the narrowness of
nominalism. While the non-consideration of predicates as “x is an abstract en-
tity” protects Lorenzen’s modern abstraction from Siegwart, it also reveals
Lorenzen’s self-inflicted, nominalistic unwillingness to recognize, in the ab-
stracta, anything beyond the mere façon de parler (manner of speaking). Not
having crossed the Rubicon – not having faced the problem of the ontology of
the abstracta – the nominalist has played safe, but has lost the opportunity of
conquering the abstract world.

Unfortunately, Lorenzen insisted on “reconstructing” (his favourite term)
the predicate used by Siegwart as an Anweisung (indication, reminder), “de-
constructing” its predicative content: “Freilich, Sätze wie ‘der Begriff ‘Revo-
lution’ ist ein abstractum’ (ein abstrakter Gegenstand) sind keine solche
Prädikationen, sonder wieder nur Anweisungen für den Gebrauch des Wortes
‘Begriff’” (“Surely, sentences as ‘the concept ‘revolution” is an abstractum
(an abstract object) are not such predications, but again only instructions for
the use of the word ‘concept’”) (1973, §7, p. 102). In 1977 I had the oppor-
tunity of asking Lorenzen what he thought about saying of an abstractum that
it is abstract — the question was rather dismissed, and my impression is that it
was regarded as a Sophisterei.
Certainly, the predicate “x is an abstract entity” is only vacuously invariant
and while doing abstraction should not be kept; but there is something else
aside from doing abstraction: to reflect on the abstraction that has been done.
Within this reflective context, the non-nominalist ontologist will try to recon-
struct predicates similar to “x is an abstract entity” as genuine Prädikationen,
not just as wieder nur Anweisungen (again only instructions). Of course, those
predicates should not be “eliminable”: being abstract should be true of the ab-
stract entity, not of the concretum from which the abstract entity was extracted
(thus, there should not be a reason for the derivation of Siegwart’s contradic-
tion).
Curiously, the suggested reconstruction points in the direction of some-
thing reminiscent of medieval semantics, where some statements about an es-
sence (nature, etc.) descend to the individuals (homo est animal), whereas
others do not (homo est species, which amounts to saying homo est abstrac-
tum). Analogously, in modern abstraction we would have statements F(ã) of
two sorts: some eliminable (that is, F(a) can be asserted), others not (when for
instance “F(x)”= “x is abstract”).
Incidentally, Siegwart appears to believe that there is, in addition to “con-
structive” (modern) abstraction, a classical theory13
(Lehre) of abstraction,
which he calls set-theoretical (1973, p. 251). He does not explain what he
means by “classical theory of abstraction” but his brief comment suggests that
he has in mind the usual move from members of equivalence classes to the
equivalence classes. If this is really so, Siegwart must be counted as one more
logician among the many who believe that circumspection deserves being
called “abstraction”. Once again, it must be emphasized that looking-around or
circumspection is not abstraction. As said above, circumspection does, obscu-
rely and unofficially, involve abstraction to the extent that none of the
13
Methode was Lorenzen’s preferred term with regard to abstraction, rather than Lehre or
Theorie.

particular chosen entities matter, as long as they are “suitable”, but this only
shows that circumspection must be overhauled in terms of modern abstraction,
as is done, for example, in Lorenzen (1955) (cf. the preceding section).
8. Current (real and only apparent) attempts to return to the foreground
Outside philosophy, genuine abstraction seems to be a topic increasingly at-
tractive for computer scientists. Even within the logico-analytic tradition there
are recent, albeit individual, examples: J. Lear (who practically reaches mod-
ern abstraction through the analysis of Aristotle’s reduplicative “qua” state-
ments) and N. Cocchiarella (who discusses Piaget’s “reflective abstraction”14
).
However, not all expressions of interest in abstraction correspond to an in-
terest in genuine abstraction. One example is W. Künne, who complains
(1982) that in analytic philosophy, while there is much discussion on “abstract
objects”, the question: “What is an abstract object?” is almost completely ig-
nored. Such an excellent and promising remark, unfortunately, does not appear
to be followed up in the rest of Künne’s work (for instance 1983, 1988), where
one misses the consideration of the essential feature of genuine abstraction
(“leaving out”, “retaining”).
9. Concluding reflection
From the great masters of scholasticism to the founder of set-theory abstrac-
tion has been regarded as a method, a way to reach such entities as universals15
or cardinals.16
In spite of these awesome historical credentials, and the fact that
to do abstraction seems to be such a simple matter, it may turn out that the
ghostly and elusive nature of the abstracta finally discourages us from trying to
14
Cocchiarella (1986), ch. II, introductory pages, and specially section 6: Ramified
constructive conceptualism. Cf. also Cocchiarella (1989), sections 7 and 8.
15
Eustachius (1616): Ut universalia cognosci possint, necesse est ut per operationem intel-
lectus abstrahantur a suis inferioribus (in order that the universals be knowable, it is necessary
that they be abstracted, by the operation of the intellect, from its inferiors), p. 30, quarta conclusio.
16
Cantor (1962): “Mächtigkeit oder Kardinalzahl von M nennen wir den Allgemeinbegriff,
welcher mit Hilfe unseres aktiven Denkvermögens dadurch aus der Menge M hervorgeht, dass von
der Beschaffenheit ihrer verschiedenen Elemente m und von der Ordnung ihres Gegebenseins
abstrahiert wird”. A slightly modified version of Russell’s translation (in 1956) is this: “We call
the power or cardinal number of M that general idea which, by means of our active faculty of
thought, emerges from the set M, by abstracting from the nature of its diverse elements m as well
as from the order in which they are given”. Interestingly, Cantor expresses this double abstraction
by writing a double bar on top of the name “M” of the set in question.

fulfil, properly, the fourth requirement mentioned at the beginning of this pa-
per, so that, short of joining the nominalists, no alternative is left but to forget
completely about abstraction in philosophy (meeting the fourth requirement
seems necessary for a philosophy of abstraction). However, if this happens,
one should not continue to use the terminology of “abstraction” in any of the
pseudo-senses (basically two: intangibility, looking-around), creating the false
impression (as in Dummett or Siegwart) that, after all, abstraction (“good” or
“logical” abstraction in Dummett, “classical set-theoretical abstraction” in
Siegwart) has a place in one’s theory.
Ignacio Angelelli
University of Texas, Austin
plac565@utxvms.cc.utexas.edu
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Poznaƒ Studies in the Philosophy
of the Sciences and the Humanities
2004, Vol. 82, pp. 37-58
Allan Bäck
WHAT IS BEING QUA BEING?
Abstract. I offer truth conditions for propositions about being qua being in Aristotle’s philosophy.
I show that in general Aristotle views expressions of the form “qua S” in “S qua S is P” (or “S is P
qua S”) as making a claim not about the subject “S”, but about the predication of “P” of “S”. I de-
velop necessary and sufficient truth conditions for propositions of the form “S qua S is P”. Finally,
I show how this analysis satisfactorily covers what Aristotle says about being qua being in the
Metaphysics.
There is a great deal of literature about what Aristotle conceives being qua
being to be. I have a great deal of difficulty in understanding this literature. Let
me defend my aporia. In brief, Aristotle announces in his Metaphysics that he
has discovered a new science of first philosophy, that is over and above the
particular sciences that deal with particular areas, like politics, physics, and
mathematics. He says that first philosophy deals with being qua being. As
many have pointed out, and as I shall review below, in some passages Aristotle
seems to identify first philosophy with theology, and hence being qua being
with the divine, while in other passages he refines his study of being qua being
to the study of substance, especially to individual substances. So there has
been a good deal of controversy, whether Aristotle considers being qua being
to be the divine, or God, or whether he considers it to point towards the con-
creteness or individuality of a singular thing. The problem that I have with this
debate is that it seems to presuppose that a prior issue has been settled. That is-
sue is how to determine the grounds on which the claims made by Aristotle
about being qua being are true.
Of course, it might be said that, if we knew what being qua being is, we
would have some notion, at least, of truth conditions for propositions about
being qua being. But. I would claim that it is neither sufficient nor necessary to
know what being qua being is (in a deep sense, as when Aristotle asks, what is
it? (to ti estin), in a search for real definitions) in order to have truth conditions
for propositions about being qua being. First, it is not sufficient since I may
know what being qua being is (not in the deep sense) without being able to de-
cide whether certain claims about being qua being are true. So even if I belie-

38 Allan Bäck
ved that being qua being is God, I still might be in doubt whether God created
the heavens or had a personal interest in souls with an active intellect. A notion
of God may afford no criteria by which I might assess these claims. Second, it
is not necessary to know what being qua being is in order to have truth condi-
tions for claims about it. Analogously, I may not know a solution to an equa-
tion, but I can still have, and may even be able to formulate, criteria according
to which claims about the solution may be evaluated. To be sure, I have to
have some notion of what the solution is, e.g. I have to know that it is a num-
ber. But it is not necessary to know what it is in a complete and fundamental
way. I take it that we, like Aristotle, may have a similar view of being qua
being. We have to have some notion of the field of our inquiry when we begin
our study of being qua being. Yet we need not know what being qua being is
in a fundamental way. Indeed, if we did, Aristotle would have had no need to
write much of his Metaphysics.
My project, then is to formulate truth conditions for propositions about
being qua being in Aristotle’s philosophy. The project will be a success if it
handles the text such that the propositions about being qua being affirmed by
Aristotle satisfy those conditions, while those denied by Aristotle do not sa-
tisfy them.
Two qualifications, or reminders, are in order here. First, in many passages
surely a lot of interpretation is required to extract direct and unambiguous
claims about being qua being. Suppose that the claims thus extracted conflict
with the truth conditions proposed. I submit that the success of such countere-
xamples varies inversely with the amount of interpretative effort needed to
extract them. Second, though it is uncharitable to suppose so, it is possible that
Aristotle is inconsistent at times in his claims about being qua being. So I
might be tempted to explain counterexamples away in this manner. Yet, in de-
ference to the first reminder, I admit that the success of my truth conditions
will vary inversely with the frequency and force of counterexamples.
The truth conditions that I shall propose will be for propositions of the
form, “being qua being is P”. It will turn out that they are easily applicable to
any proposition of the form, “S qua S is P”, or the form, “S kath hauto (per se)
is P”. Hence, many more examples in Aristotle’s writings will fall under these
conditions than might at first be supposed. One advantage of this extension is
that we shall be able to begin this project and to test its success on propositions
about subjects considerably less abstract than being qua being.
Let me position my project with respect to current scholarship. The pro-
blem that I shall be addressing does bear on the problem of there being two
different conceptions of first philosophy in Aristotle’s Metaphysics. These
conceptions may be called the ontological and the theological (cf. Patzig 1979,
pp. 33-4). I have claimed that before one confronts this issue one ought first to
try to formulate truth condions for propositions about being qua being. I shall

What is Being qua Being? 39
claim that, once these conditions are formulated, the problem of the subject
matter of the Metaphysics can be resolved. I shall indeed end by sketching out
such a resolution.
There is another problem about Aristotle’s task in the Metaphysics that too
has received a good deal of recent attention: the relation of Aristotle’s earlier
ontological views, especially those about the categories, with his “discovery”
of metaphysics. I shall not be addressing this problem here. Obviously, what I
have to say has some relevance to this; it would be helpful to know in such a
context what Aristotle’s task is in the Metaphysics. But my project of formula-
ting truth conditions, even if successful, would not suffice by itself to resolve
this issue.
What does Aristotle mean by “being qua being”? To put it in a spectacular
and inaccurate fashion, I shall argue that he does not mean anything by this
expression. More precisely, I shall suggest that “being qua being does” not re-
fer to anything different from that to which “being” refers. Instead, the “qua
being” phrase just restricts the sorts of predicates that may be predicated of
“being”.
I shall proceed as follows. First, I shall show that in general Aristotle views
expressions of the form “qua S” in “S qua S is P” (or “S is P qua S”) as making
a claim not about the subject “S”, but about the predication of “P” of “S”.1
Next, I shall develop necessary and sufficient truth conditions for propositions
of the form, “S qua S is P”. Finally, I shall show how this analysis satisfacto-
rily covers what Aristotle says about being qua being in the Metaphysics, and
draw some conclusions.
I.
In order to develop truth conditions for propositions about being qua being, we
need some instances of such propositions. Yet Aristotle does not offer many
statements where “being qua being” appears as subject or (simple) predicate.
The clearest cases concern his frequent assertions that “there is a science that
investigates being qua being” (Metaph. 1003a21-2; 1026a31; 1060b31-2),
which in medieval times came to be codified as “being qua being is the subject
of metaphysics” (e.g. Burleigh 1955, II.III.3, 76, pp. 19-32). Yet, as I shall di-
scuss further below, such statements do not tell us anything substantial about
being; rather they are remarks on a meta-level about a certain area of scientific
1
“Qua”, without italics, will represent the generic connective, of which “qua”, “insofar as”,
“inquantum”, and “ he ” are examples. Translations of Aristotle are from The Complete Works of
Aristotle, ed. J. Barnes (Princeton, 1984) and have been modified when indicated.

40 Allan Bäck
study. So such remarks do not provide very satisfactory data with which to
construct truth conditions.
Other, more substantial statements using the expression, “being qua
being”, may be gleaned from the text, but only with interpretation. Having
concluded that “it will belong to this (first philosophy) to consider being qua
being – both what it is and the attributes that belong to it qua being” (Metaph.
1026a31-2), Aristotle then begins Metaphysics Z with his famous claim: “that
which primarily is is the what, which indicates the substance of the thing”
(Metaph. 1028a13-5). So, we might paraphrase this assertion as “being qua
being is (primarily) substance”. Similarly, we might conclude, given Ari-
stotle’s identification of the science of being qua being with the science of the
immovable substance (Metaph. 1026a29-32; 1064a28-30), that “being qua
being is the immovable substance”. Still, clear, unproblematic examples of
Aristotle’s use of “being qua being” are scarce. Thus, it is reasonable to look at
how Aristotle uses other expressions isomorphic with “being qua being”.
In order to locate the parallels, note that the expression, “being qua being”
(to on he on) does not signify a process but a thing; literally it is “that which
is (being) qua being”. In this way, when Aristotle makes claims about being
(to on), it is best to take them substantively: what is (a) being? (a) being is (a)
substance. On this reading, parallels of “(a) being qua being” are suggested by
the text. When introducing his conception of first philosophy, Aristotle distin-
guishes things/beings qua being from things/beings qua quantitative and conti-
nuous, and from things/beings qua movable (Metaph. 1061a27-b11; 1064a29;
1025a26), which are the concerns of geometry and physics respectively. So
when Aristotle uses expressions like “the bronze qua movable” and “the
bronze qua bronze” (Ph. 201a28-9); “the doctor qua housebuilder and not qua
doctor” (191b4-6), he is using expressions that have the same grammatical
structure as his “being qua being”. As Aristotle does discuss various logical
properties of such expressions, I shall now turn to what he says.
A sentence like “being qua being is the subject of first philosophy” has the
form “S qua S is P”. Particularly in the Posterior Analytics such sentences are
grouped with those having the form, “S is P kath hauto”. In Posterior Anal-
ytics I.4, Aristotle gives something like a truth condition or definition for kath
hauto. He relates what is true kath hauto with what is true kata pantos (of
every instance), and with what is true katholou (universally). For Aristotle, the
expressions kata pantos, kath hauto, and katholou (in a demonstrative sense),
form a family hierarchy, where truth conditions for the former are contained in
the truth conditions for the latter.
P is said to belong to S kata pantos “if it does not hold in some cases and
not others, nor at some times and not at others” – i.e. if and only if every S is P
at all times, presumably, then, past, present, and future (An. Post. 73a28-9).
This sense of kata pantos is stronger than what is needed for the validity of

syllogisms; instead Aristotle claims that it is needed for good demonstrations.2
For the kata pantos, Aristotle gives examples of essential predication, like
“every man is animal” (An. Post. 73a29-30); accidental predications like
“every crow is black” were also commonly attributed to Aristotle (cf. Philopo-
nus 1905, 69, 22; Themistius 1900, 12, 13-4).
Aristotle often uses the expression “kath hauto”, and he himself distingui-
shes many senses in various texts. Of the four senses given in Posterior Anal-
ytics I.4, the first seems to be the basic sense.3
Aristotle says that “P” belongs
to “S” kath hauto if “P” is in the definition (ti estin) of “S” (An. Post. 73a34-
5). So, if “man” is defined as “rational animal”, “rational” and “animal” will
each belong to “man” kath hauto. To be in the definition of “S” seems to mean
not only to be an actual constituent of that definition, but also to be what is
implied by that definition. Thus, Aristotle says that “having its interior angles
equal to two right angles” belongs to triangle kath hauto (An. Post. 73b31-2).
So “substance” (a remote genus of man) and “risible” (a proprium of man) will
also belong to “man” kath hauto in this sense.
The other three senses of kath hauto given in this text seem derivative.
“Odd or even” belongs to “number” kath hauto in the first sense, and hence
“odd” by itself might be said to belong to “number” kath hauto.4
Thus Ari-
stotle says that “odd” belongs to “number” kath hauto since “number” is in the
definition of “odd” (An. Post. 73a39-40). In another sense, “P” belongs to “S”
kath hauto, if “S” does not have an underlying subject different from that of
“P”: such will be the case if “P” is implied by the definition of “S” (An. Post.
73b3-5). Finally, “P” belongs to “S” kath hauto, if there is an essential, causal
connection between “S” and “P”. E.g., “twinkling” belongs to “being distant”
kath hauto, since being distant causes, and, according to Aristotle, has as a
proprium, twinkling (An. Post. 73b13-6; 78a30-8). All these senses of “kath
hauto” seem to derive from the first, basic sense.
It is clear that if S is P kath hauto, then P is true of S kata pantos, as defi-
ned above. For everything that is S will then satisfy the definition of “S”, and
“P” is entailed by that definition. So one truth condition of “S is P kath hauto”
is “every S is P (always)”.
Another, suggested directly from the text is: “P” follows from the defini-
tion of “S”. But can the latter provide a clear truth condition? To use it, we
would have to know the real definitions not only of the “S” term, but also the
real definitions of their constituents (the higher genera and differentiae) and of
the other predicates (the propria and accidents). Even Aristotle should admit
2
Cf. Philoponus (1905), 58,26-59,14. Note that Aristotle also recognizes propositions true ut
nunc in the Prior Analytics.
3
W. D. Ross (ed. com.), Aristotle’s Prior and Posterior Analytics, p. 519.
4
Jonathan Barnes (trans. comm.), Aristotle’s Posterior Analytics (Oxford, 1975, p. 115).

42 Allan Bäck
ignorance of all that (An. Post. 78a28-30). In any case, our concern is with the
“S” term, “being”, which has no real definition, as there are no genera higher
than it.5
At best it would have “propria”, sc., terms convertible with it like
“one” and “good” (Metaph. 1003b22-6; 1005a3-11; 1053b25). So I fail to see
how the requirement that “P” be in the definition of “S” will offer an additio-
nal, functional truth condition, at least when the expression is “being kath
hauto”. I find “every S is P (always)” to be quite adequate, especially if the
kata pantos condition be taken strictly, to require that all S’s past, present, and
future be P. For, especially in view of the future S’s, for every S to be P, seems
to indicate some sort of natural necessity, so that we now would be able to
know now that every S is P.6
This “natural necessity” suggests that S is P be-
cause of the nature of S. So the kata pantos condition should suffice.
One problem with this account is that it runs together the kata pantos (in
the strong sense) and the kath hauto. Yet Aristotle seems to distinguish them,
at least according to the tradition: e.g. “every crow is black” is true kata pantos
but not kath hauto; it is not part of the definition of “crow” that it be black.
I have two points: (1) even for Aristotelians, the difference of “kata pan-
tos” and “kath hauto” is problematic. How can the truth of a universal claim,
holding for future cases too, be based on an accident (cf. Philoponus 1905, 58,
26)? Thus it is not clear that Aristotle can justifiably maintain the distinction of
“kata pantos” in this strong sense and “kath hauto”.7
(2) Even if I have omit-
ted one truth condition for “kath hauto”, I have at least one that is necessary.
So I err on the side of caution – not too big an error, given the difficulties of
actually using real definitions. And, as I shall urge, the simple, kata pantos
condition seems to suffice.
Aristotle then defines a demonstrative sense of “universal” (An. Post.
73b26-7). “P” belongs to “S” universally if (1) “P” is true of “S” kata pantos
(2) S is P kath hauto and (3) “S” is the primary subject for “P”. “S” is the pri-
mary subject for “P” if and only if “S” and “P” are on the same level of gene-
rality. As Posterior Analytics I.5 makes clear, “S” is not the primary subject
for “P” if “S” and “P” are not coextensive (An. Post. 74a33-b4).
5
As I shall note below, this feature would drop out for “being qua being in any case, as
Aristotle holds that “being” does not have a definition. On this see Routila (1969, p. 62).
6
The necessity here is not absolute in the modern sense, of holding in every possible world,
but holds only relative to this world. Such is the sort of necessity that Aristotle’s claims generally
have.
7
Indeed the text perhaps does not do so: Aristotle does seem to state that kata pantos and kath
hauto differ at 73b26-7, but perhaps he means that they have different criteria – one based on
singular things under the subject, the other on definitions of the species and genera, not that their
extensions differ. Or, perhaps, he did not intend for what holds kata pantos to be true at all times,
including the future. But I find neither of these suggestions compelling.

Now what makes all this discussion relevant to the project of this paper is
that in the course of defining commensurate universality Aristotle makes some
remarks on expressions of the form, “S qua S”:
I call universal whatever belongs to something both kata pantos and kath hauto and
qua itself. It is evident, therefore, that whatever is universal belongs from necessity
to its objects. (To belong kath hauto and qua itself are the same thing – e.g., point
and straight belong to line kath hauten (for they belong to it qua line), and two
right angles belong to triangle qua triangle (for the triangle is kath hauto equal to
two right angles) (An. Post. 73b26-32; modified translation).
Thus Aristotle asserts that the forms, “S is P kath hauto” and “S qua S is P”
are somehow equivalent. In the two examples he gives, he asserts the equiva-
lence both ways. So this text asserts at least the material equivalence of “kath
hauto” and “ he hauto” (qua itself). Moreover, the text above gives support to
the claim that “S qua S is P” and “S is P kath hauto” have the same meaning.
For Aristotle explains the use of one of these expressions by the other. So the
truth conditions for “kath hauto” will apply also to “ he hauto”.
What makes the situation more complex is that Aristotle seems to recogni-
ze both a demonstrative, commensurately universal sense, and a non-demon-
strative sense of “kath hauto” and “ he hauto”. In the examples just quoted,
Aristotle uses these expressions to stipulate that the subject (“triangle’) and the
predicate (“equal to two right angles”) be coextensive (An. Post. 74a30-8). But
in other cases (73a35): “triangle” and “line” he does not demand commensu-
rate universality. So Aristotle has both a demonstrative, coextensional sense
and a general non-demonstrative, non-coextensional sense of “kath hauto” and
“ he hauto”.
It is clear from Aristotle’s initial definition of “kath hauto” that he recogni-
zes the general sense. That Aristotle often uses it, and “ he hauto” in the de-
monstrative sense too can be supported further by considering more examples.
In the Posterior Analytics Aristotle is quite fond of the example that a triangle
has its interior angles equal to two right angles. As we have seen, this proposi-
tion is supposed to be true of the triangle kath hauto and he hauto, qua trian-
gle. Aristotle in various passages makes the following claims:
1) An isosceles qua triangle is equal to two right angles (74a36-b4).
2) A triangle qua triangle is equal to two right angles (73b30-1).
3) An isosceles qua triangle is equal to two right angles (74a16-7).
4) A triangle qua isosceles is equal to two right angles (85b9-13).
5) A triangle qua figure is equal to two right angles (73b33-4).
Aristotle claims that (1) and (2) are true, and (3) to (5) false. Now, since (2)
is true, (3) cannot be false for the reason that “being equal to two right angles”
is not in the definition of triangle. For then (2) would have to be false also. So,
“being equal to two right angles” is in the definition of “triangle” in the sense

Idealization XI Historical Studies On Abstraction And Idealization

Idealization XI Historical Studies On Abstraction And Idealization

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