REVISITING THE SERIES ARGUMENTS
Graham Clay
THE NICOMACHEAN ETHICS PASSAGE
Those who introduced this view did not posit Ideas
in cases in which they recognized the prior and the
posterior—this is the very reason they did not
establish an Idea of numbers; but something is called
good both in [the category of] what it is [substance,
or ousia] and in that of quality and in that of
relation, and that which is per se, i.e. substance, is
prior in nature to the relative (for the latter seems like
an offshoot co-incident with what is); so that there
could not be some common Idea set over all these
goods. (1096a17-23)
MY FORMALIZATION
P1. Series S are lists whose elements are ordered according to their priority: (x1, x2, ...).
P2. For all series S (x1, x2, ...), there is no Form for the elements of the series (x1, x2, ...).
P3. (Some) substances, qualities, and relatives are (properly) called goods (that is, they are goods).
P4. Substances are ontologically prior to qualities which are ontologically prior to relatives.
C1. There is a series A whose elements are the substances, the qualities, and the relatives where all of the
substances are ontologically prior to all of the qualities which are ontologically prior to all of the relatives. A takes
the following form: (s1, s2, ..., sn, q1, q2, ..., qn, r1, r2, ..., rn), where sx is a substance, qn is a quality, and rx is a
relative. [from P1 and P4]
P5. Series with more than two elements can always be decomposed into other series wherein all of the elements of the
other series maintain the same ordering relations relative to one another as they did in the original series.
C2. A can be decomposed into two series that maintain the ontological priority of its members: A1 (s1, s2, ..., sn,
q1, q2, ..., qn), A2 (q1, q2, ..., qn, r1, r2, ..., rn). [from C1 and P5]
P6. If the elements of category C1 and the elements of another category C2 form a series P where all of the
elements of C1 are ontologically prior to all of the elements of C2, and (some of) the elements of both C1 and
C2 are (properly) called goods, then those goods of C1 and C2 form the series of goods G where the goods of
C1 are normatively prior to (better than) the goods of C2.
C3. Substances, qualities, and relatives form a series of goods G where all of the substances are normatively prior
to (better than) all of the qualities which are normatively prior to (better than) all of the relatives: (gs1, gs2, ..., gsn,
gq1, gq2, ..., gqn, gr1, gr2, ..., grn). [from P3, double application of P6, and P7]
C4. There is no Form for the elements of series G. [from P2 and C2]
C5. But the members of G are the goods, so there is no Form of Goodness. [from C3 and the definition of the FOG]
PROBLEMATIC PREMISES: P2 & P6
P2. For all series S (x1, x2, ...), there is no Form for the elements of the series (x1, x2, ...).
Aristotle indicates that the Platonists endorse P2 because they think that the numbers have no Form over
them. He seems to think that given that the numbers have no Form over them and they are a series, the
same is true of other series. But there are reasons to think that the numbers are different from other things
arranged in series. For instance, unlike, say, the white things, the numbers have no qualitative content so it is
hard to make sense of any sort of resemblance relation that they are supposed to stand to any Form that
might be over them.
P6. If the elements of category C1 and the elements of another category C2 form a series P where
all of the elements of C1 are ontologically prior to all of the elements of C2, and (some of) the
elements of both C1 and C2 are (properly) called goods, then those goods of C1 and C2 form the
series of goods G where the goods of C1 are normatively prior to (better than) the goods of C2.
The worry about this premise is that normative priority (the better/worse than relation) does not seem to
track ontological priority. For example, though the substance worm is ontologically prior to the quality of
being close to God, most people would think that being close to God is better than being a worm. Indeed, it
seems to be because the qualities of the substance worm are what they are that we think of it as a lesser
being. Being small, weak, non-thinking, and so on (all lesser qualities) seem to trump its status as a
substance.
THE EUDEMIAN ETHICS PASSAGE
Further, in things having an earlier and a later, there
is no common element beyond, and, further,
separable from, them, for then there would be
something prior to the first; for the common and
separable element would be prior, because with its
destruction the first would be destroyed as well; e.g.
if the double is the first of the multiples, then the
universal multiple cannot be separable, for it would
be prior to the double... (1218a1-15)
SHIELDS’ FORMALIZATION
P1. If a, b, c, ..., n φ things are arranged as prior and posterior, then there is a φ set over them all only if
(1) it itself is φ and (2) it is the most φ thing. [self-predication and paradigm principle]
P2. If φ is itself φ, then φ is a member of the series of φ things.
C1. So if a, b, c, ..., n φ things are arranged as prior and posterior and there is a φ set over them, then
φ is a member of the series prior to the first member of the series (φ, a, b, c, ..., n). [from P1 and P2]
P3. For all sets of φ things, there is some φ set over them. [one over many]
C2. So if φ is a member of the series prior to the first member of the series, then there is a φ* over
the series φ, a, b, c, ..., n. [from C1 and P3]
P4. If φ* is itself φ and it is the most φ thing, then it is a member of the series φ, a, b, c, ..., n prior to the
first member of the series: (φ*, φ, a, b, c, ..., n).
P5. By repeated application of 6, there is no first member of the series, for there is always some new φ
that can be generated and placed prior to the first member of the series.
C3. So there is a Form φ set over the series of φ things only if there is no first member of the
series.
P6. Every series has a first member.
P7. The categories of being form a series of good things arranged as prior and posterior.
C4. So there is no Form of Goodness itself set over the categories of being. [from C3, P6, and P7]
A SECOND THIRD MAN?
This argument seems to be a good interpretation of the passage from the Eudemian Ethics on 1218a-15, the passage from
the Metaphysics found on 999a6-10, and that part of the passage from the Nicomachean Ethics on 1096a17-23 mentioning
numbers. And it has no (seemingly) bad premises like the prior formulation. However, it is eerily similar to the third man
argument. Here is Shields’ version of the third man argument (TMA) for comparison:
P1. There is a set G of all of the good sensible particulars {g1 , g2 , g3, …, gn}.
P2. For all sets of φ things, there is some Form of φ set over them. [one over many]
C1. Hence, there is a Form, Goodness, over that set G of good things. Call it FoG1. [from P1 and P2]
P3. If there is a Form of φ, this Form is φ. [self-predication]
C2. Hence, the Form of Goodness is itself good. [from C1 and P4]
P4. No Form partakes of itself. [non-self-partaking]
C3. Hence, the Form of Goodness is not a member of G. [from P1, C2, and P6]
P5. But there is a set H {FoG1 , g1 , g2 , g3, …, gn}.
C4. So there is also a Form, call it FoG2, over H. [from P2 and P8]
P6. But then it is not the case that there is just one Form corresponding to good things; instead, repeating this argument ad
infinitum, we can say that there are infinite Forms of Goodness.
C5. So the Form of Goodness is not unique and so goodness is not univocal. [from P6]
While this version of the TMA challenges the claim that the Forms are unique (and so challenges the univocity of
goodness) and the prior argument challenges the paradigmatic status of the Forms, both generate their conclusions by
combining one over many and self-predication to generate problematic infinite regresses. The main difference is that the
series argument relies on the particular ordering of the set in question and the TMA does not.
A PASSAGE FROM DE ANIMA
It is now evident that a single definition can be given of soul only in the
same sense as one can be given of figure. For, as in that case there is no
figure apart from triangle and those that follow in order, so here there is
no soul apart from the forms of soul just enumerated. It is true that a
common definition can be given for figure which will fit all figures
without expressing the peculiar nature of any figure. So here in the case
of soul and its specific forms. Hence it is absurd in this and similar cases
to look for a common definition which will not express the peculiar
nature of anything that is and will not apply to the appropriate
indivisible species, while at the same time omitting to look for an
account which will. The cases of figure and soul are exactly parallel; for
the particulars subsumed under the common name in both cases—
figures and living beings—constitute a series, each successive term of
which potentially contains its predecessor, e.g. the square the triangle, the
sensory power the self-nutritive... Why the terms are related in this serial
way must form the subject of examination... It is evident that the way to
give the most adequate definition of soul is to seek in the case of each of
its forms for the most appropriate definition. (414b19-415a13)
WARD’S INTERPRETATION
In “Souls and Figures: Defining the Soul in De anima ii 3,” Julie Ward argues that Aristotle’s
intention in this passage is to show that an account of soul can be generated that is just like the
account of figure, as he states that the former can be given in the “same sense” that the latter
can. However, she argues that Aristotle fails, despite his intentions, due to souls not being like
figures.
First, what sort of accounts are options?
1. Synonymy: ‘Figure’ and ‘soul’ could each signify a single thing with a definable essence. This
would require all figures and souls to belong to the same genera and species.
2. Core-dependent Homonymy: ‘Figure’ and ‘soul’ could each signify something different
depending on the species of figure or soul in question but all such species would share some
common primary attribute(s).
3. Homonymy: ‘Figure’ and ‘soul’ could each signify things with only a name in common.
Ward thinks that a variant of account type 2 can be given for figures.
WARD’S INTERPRETATION
The series of figures…
…
With their core revealed…
…
Ward argues that Aristotle holds that, in the case of the series of figures, the prior (the triangle)
exists potentially in the posterior (the square, the pentagon, etc). That is, each figure after the
triangle in the series of figures can be analyzed into some number of triangles. This is true as
well of lines: on 1019a8-11, Aristotle states that the half line exists potentially prior to the
whole line, though in actuality the whole line is prior since it must be dissolved for there to be a
half line.
WARD’S INTERPRETATION
So in the case of figures, we have a case of core-dependent homonymy where all the figures are
posterior in potentiality to the triangle, as they all can be divided into triangles, but prior in
actuality, as they are not constituted by triangles (the triangles are conceptual parts, not
metaphysical parts that are ontologically prior). What about souls?
Aristotle states in the above passage that for souls the self-nutritive power is to the sensory
power just like the triangle is to the square for figures. But Ward thinks that the analogy stops
there, for she believes that one cannot say that the self-nutritive power is present in the sensitive
soul only potentially and not actually. Aristotle claims that the sensitive soul has the power of
self-nutrition, like all souls, and the sensory power. That is, it would appear that all souls actually
have the self-nutritive power; unlike the triangle, the self-nutritive power is not a mere
potentiality. Furthermore, the powers beyond the self-nutritive power do not have the selfnutritive power as parts, conceptual or otherwise, as higher souls are not homogenous like
figures.
Ward’s Conclusion: Soul is not an actual unity with potential parts like figure. “The serial
account of soul does not reveal a unifying nature or cause of the various distinct capacities…
So, while Aristotle aims to demonstrate that soul is a unifying principle of organization of the
living thing, the goal eludes his grasp.”
WARD’S INTERPRETATION
Therefore the plurality of parts option mentioned on 414b19-415a13 is a no-go:
We must be careful not to ignore the question whether soul can be defined
in a single account, as is the case with animal, or whether we must not give
a separate account for each sort of it, as we do for horse, dog, man, god
(in the latter case the universal animal—and so too every other common
predicate—is either nothing or posterior). Further, if what exists is not a
plurality of souls, but a plurality of parts of one soul, which ought we to
investigate first, the whole soul or its parts? (403b5-11)
And now Aristotle will seek an account for each species of soul (and so we have homonymy
after all):
It is evident that the way to give the most adequate definition of soul is to
seek in the case of each of its forms for the most appropriate definition.
(415a12-13)
SOME WORRIES FOR WARD
The homogeneity point aside, Ward’s argument relies entirely on the parts of soul being actual
and not potential. It is because all souls actually have the power of self-nutrition that she claims
that there is a disanalogy with the figures. But there are some textual and terminological reasons
to doubt this claim.
Right after 415a12-13, at the start of De anima ii 4, Aristotle states the following:
But if we are to express what each [form of soul] is, viz. what the thinking power
is, or the perceptive, or the nutritive, we must go farther back and first give an
account of thinking or perceiving; for activities and actions are prior in definition
to potentialities. (415a16-20)
Here Aristotle labels the powers of souls potentialities. In fact, the Greek term for ‘power’ is the
same as that for ‘potentiality’: dunamis. So all of the parts of the (thinking) soul are potential in
this sense. They exist in the whole (the thinking soul) but are not (always) actual.
SOME WORRIES FOR WARD CONTD.
Likewise, in the Physics, Aristotle makes the following claim in discussing Zeno’s paradoxes:
Hence Zeno’s reasoning is false when he argues that there is no part of the millet
that does not make a sound; for there is no reason why any such part should not
in any length of time fail to move the air that the whole bushel moves in falling.
In fact it does not of itself move even such a quantity of the air as it would move
if this part were by itself; for no part exists otherwise than potentially in the
whole. (250a19-24)
Thus it is not clear if Ward can maintain that there is not an analogy between the case of figure
and the case of soul because the powers of the soul are actual parts and not potential parts of
the soul. There is reason to think that Aristotle thought that all powers are potential in some
sense and, more generally, that all parts are potential.
SOME WORRIES FOR WARD CONTD.
And although Ward is right that Aristotle states at 415a12-13 that he must investigate each soul
and seek a definition of each, he also indicates that the fact that souls stand in a series is
significant.
At 415a14-16, he states that “It is necessary for the student of these forms of soul first to find a
definition of each, expressive of what it is, and then to investigate its derivative properties, &c”
(emphasis mine). Simply because he turns to examine the different kinds of soul after the main
De anima passage does not mean that he does not think that there could be a fruitful
investigation into the best account of serial relationship that souls stand in.
Likewise, Aristotle states in the main passage (414b34-415a1) that the series is to be investigated
in itself: “Why the terms are related in this serial way must form the subject of examination.”
So it appears that there are good reasons to think that Aristotle holds that a core-dependent
homonymous account can be given for soul.
MY FORMALIZATION OF 414B19-415A13
P1. There are things of the same genus subsumed under the name ‘soul’, namely souls.
P2. Souls are arranged in a series with the (species of) thinking soul first, the (species of) moving soul second, the (species
of) sensing soul third, and the (species of) self-nutritive or feeling soul fourth.
P3. A definition can be given for ‘soul’ that fits all souls in the series of souls but that does not fit any particular soul better
than any other.
P4. If a definition can be given for ‘soul’, then an account of soul can be given.
C1. An account of soul can be given. [from P3 and P4]
P5. All souls share the self-nutritive power, fewer share the sensory power, fewer still share the power of locomotion, and
even fewer still share the power of thought.
MY FORMALIZATION CONTD.
P1. There are things of the same genus subsumed under the name ‘soul’, namely souls.
P2. Souls are arranged in a series with the (species of) thinking soul first, the (species of) moving soul
second, the (species of) sensing soul third, and the (species of) self-nutritive or feeling soul fourth.
P3. A definition can be given for ‘soul’ that fits all souls in the series of souls but that does not fit any
particular soul better than any other.
P4. If a definition can be given for ‘soul’, then an account of soul can be given.
C1. An account of soul can be given. [from P3 and P4]
P5. All souls share the self-nutritive power, fewer share the sensory power, fewer still share the power of
locomotion, and even fewer still share the power of thought.
P6. Amongst those souls that share a particular power, there is variation in the extent and way in which the
different species of soul have the power.
P7. If P5 and P6, then the two tasks of any account of soul are 1) to explain the cause and nature of the
serial relationship between souls and 2) to explain the cause and nature of the variation internal to each set
of souls under a particular power.
C2. The two tasks of any account of soul are 1) to explain the cause and nature of the serial
relationship between souls and 2) to explain the cause and nature of the variation internal to each set
of souls under a particular power. [from P5, P6, P7]
P8. If C2, then the best account of soul is the one that best completes these tasks.
C3. The best account of soul is the one that best completes these tasks.[from C2 and P8]
THE CLAIM
The core upon which the homonymy of ‘soul’ depends is the
power of self-nutrition.
Although ‘soul’ can refer to any of the various species of soul, all uses of ‘soul’
appeal to ‘soul’ taken in its core sense. That is, any reference to a soul must allude to
the power of self-nutrition that is at the core of all souls, though references to those
souls that only have a self-nutritive power mustn’t allude to any other power. This
asymmetry is required of any terms that are homonymous in a core-dependent way.
THE RELEVANCE OF THE FOREGOING
C2. The two tasks of any account of soul are 1) to explain the cause and nature of the serial
relationship between souls and 2) to explain the cause and nature of the variation internal to each set
of souls under a particular power. [from P5, P6, P7]
P8. If C2, then the best account of soul is the one that best completes these tasks.
C3. The best account of soul is the one that best completes these tasks.[from C2 and P8]
The relevance of these conclusions: Plato’s theory of Forms is not the best account of soul
because it does not complete either of these tasks. A single Form can neither explain the cause
and nature of a serial relationship nor explain the cause and nature of the variation of each
species of soul.
Thus this series argument is a positive argument of Aristotle’s. Whatever the other issues that
he thinks that the theory of Forms has, this argument is his attempt to show by contrast that
Plato’s account does not remotely do what it needs to do to be the best account of soul.
Aristotle’s account, at the very least, attempts to complete the tasks that it is faced with. But
does it generalize beyond the case of the soul?
GENERALIZING
The core upon which the homonymy of ‘being’ depends is
substance.
Although ‘being’ can refer to any of the various species of being (beings belonging to
the latter nine categories), all uses of ‘being’ appeal to ‘being’ taken in its core sense.
That is, any reference to a being must allude to substance that is at the core of all
beings, though references to those beings that are (primary) substances mustn’t allude
to any other species of beings.
GENERALIZING CONTD.
Some relevant quotes to motivate this picture:
As, then, there is one science of all healthy things, the same applies in the other cases also. For not only
in the case of things which have one common notion does the investigation belong to one science, but
also in the case of things which are related to one common nature; for even these in a sense have one
common notion. It is clear then that it is the work of one science to study all things that are, qua being.
(1003b11-16)
It is obvious then from these considerations too that it belongs to one science to being qua being. For
all things are either contraries or composed of contraries, and unity and plurality are the starting-point
of all contraries. And these belong to one science, whether they have or have no one common notion.
Probably they have not; yet even if ‘one’ has several meanings, the other meanings will be related to the
primary meaning—and similarly in the case of the contraries.—And if being or unity is not a universal
and the same in every instance, or is not separable from the particular instances (as in fact it probably is
not; the unity is in some cases that of common reference, in some cases that of serial
succession)…(1005a1-11)
GENERALIZING CONTD.
Actuality means the existence of the thing, not in the way which we express by ‘potentiality’; we say
that potentially, for instance, a statue of Hermes is in the block of wood and the half-line is in the
whole, because it might be separated out, and even the man who is not studying we call a man of
science, if he is capable of studying. Otherwise, actually. Our meaning can be seen in particular cases
by induction, and we must not seek a definition of everything but be content to grasp the analogy,—
that as that which is building is to that which is capable of building, so is the waking to the sleeping,
and that which is seeing to that which has its eyes shut but has sight, and that which is shaped out of
the matter to the matter, and that which has been wrought to the unwrought. Let be actuality be
defined by one member of this antithesis, and the potential by the other. But all things are not said in
the same sense to exist actually, but only by analogy—as A is in B or to B, C is in D or to D; for some are
as movement to potentiality, and the others as substance to some sort of matter. (1048a30-b9)
GENERALIZING CONTD.
Would this kind of account work for ‘good’?
If it works for ‘being’, I cannot see why it would not. If it does, then
univocity in Plato’s sense is lost, though the complete account of the
core-dependent homonymy of ‘good’ could still perhaps allow for
commensurability and cross-species comparisons.