Graham Clay. 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.

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 FORMALIZATIONP1. 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 & P6P2. 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 ANIMAIt 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 INTERPRETATIONIn “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 INTERPRETATIONThe 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 INTERPRETATIONSo 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 self-nutritive 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 INTERPRETATIONTherefore 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 WARDThe 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 414B19-415A13P1. 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 CLAIMThe 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.