It would seem that the second set of arguments is already as exhaustive as the first, so it is surprising that Plato continues further with more arguments about the relation of the One to other things. This breaks the symmetry of his arguments, causing section (1b) to be much longer than all the others. This extension of the second set of arguments was drawn upon heavily by Neoplatonists, to the point that one might suspect that this were a later addition to the dialogue, except there are no critical grounds for this supposition. Besides, there are many other parts of the Parmenides cited by the Neoplatonists. Accepting this section as authentic, it is here that we might expect to find clues to how, if at all, Plato proposes to find a way out of these antinomies.
This section is distinct from the others in that it is not truly antinomic, for Parmenides here asserts different senses or times in which the One is older or younger, greater or lesser. A further distinction is that it is concerned with affections
(pathemata) of the One, i.e., things that befall the One or are accidental to the One. This is important because, even if we do not know whether the One exists, there undeniably exist some accidents or relations to the One, such as our thoughts or opinions about the One.
Our starting point is the question of time or priority of the One in relation to Others. We know that the Others are more numerous than the One, so the One must be first in being for it is impossible for there to be more than One unless there is One. One might object that all could come into being simultaneously, yet we must recall that the One is not just one object,
but Unity as such, for we have collapsed the distinction between concept and concrete object, in the absence of participation. Thus Unity must be prior to more numerous things, though now we might say priority in thought or reason is being confounded with priority in time. Yet if we are operating under the hypothesis that the One is, and that there is no participation, then the One as concrete existent and the One as concept are one and the same. Since it has only one kind of existence both as concept and as object, its priority in being must entail priority in existence (i.e., as act), which entails priority in time. So the one is first in time, and thus older than the Others, and the Others are later in time, and thus younger than the One.
Parmenides reminds Aristoteles that the One was shown to have parts (2.2.2); by invoking this conclusion, we are still operating within the second set of arguments (1b). If the One has parts, it must have a beginning, middle and end. The beginning comes into being first, and then the other parts, until we reach the end. It might be said that we are confounding spatial parts with parts of time. Yet all that we have specified about the One is that it has parts in the sense of numerous constituents (2.2.2) and all we mean by beginning and end is that the One has limits or extremes (2.2.4), and the middle
is whatever is within those limits. Parmenides introduces the additional premise, accepted by Aristoteles, that the One (and presumably anything else) cannot come into being contrary to its own nature. Since its nature is to consist of parts, it must come into being in parts. A further implied supposition is a non-saltational view of time. If we portray the passage from non-being into being as crossing a line, it is evident that anything consisting of parts will require some finite passage of time to come fully into being. It could not come into being all in the same instant unless every part was a monad, which is contrary to the notion of part. If the thing is to come into being according to its own nature, i.e., not first as something else and then transforming into its present nature, then first one of its limits or extremes must come into being, then all that intervenes, and finally its other extreme. Whichever extreme comes into being first we here call the beginning.
The parts of the One are Others, for they are not the One as such. Thus, as the One comes into being by parts, first the Other that is the beginning of the One, then the Others that are the middle of the One, and finally the Other that is the end of the One, come into being successively. The One is becoming, but we cannot say it has become
or come into being until it is whole, with beginning, middle, and end. By that point, all of the Others that are its parts will have become
or come into being already, with the last Other (the end) simultaneous with the One. Thus the Others have become
before the One has become.
This is an apparent contradiction with the One being older or prior to the Others, for now we find the One is younger than Others (at least those which are its parts) and Others are older than the One.
Yet, by further consideration, each of these parts is itself a whole, and thus also One. So the One comes into being together with each part. Thus the One is the same age as each part, and so is the same age as these Others. Though Parmenides does not do so, we might carry this argument recursively with the previous, treating each part as a whole with beginning, middle and end, again finding parts that are older than this Unity, though the difference in age is smaller since this unit or whole is smaller and takes less time to come into being. Each of these parts in turn can be treated as wholes, and with each iteration, the difference in ages gets smaller. Using modern notions of infinite sequences, we can see that this converges, so that the limit of the difference in ages between Unity and parts is zero as this sequence goes to infinity.
So far the results are antinomic, and so far we have considered the priority in which the One is
(esti) and has come into being
(gegonos) with respect to the Others. By different considerations, we have found that:
isand
has come into beingbefore the Others, for the existence of the more numerous requires prior existence of One.
isand
has come into beingafter Others, for its parts must come into being before it is whole or a unity.
isand
has come into beingat the same time as Others, for each of its parts is a Unity.
What then, if we consider becoming
rather than has become
or being
? Again, different considerations are made. First, we consider that, if one thing at first is older or younger than another, then it cannot become older or younger to a greater degree as time passes, for equal time or age is added to both things, leaving the difference in ages the same as at first. So it would follow that the One is not becoming any older or younger than the Others than it was at first.
A second consideration is that the One, being older than the Others, has come into being for a greater time than the Others. Thus it has become
for a greater time than the Others have become.
If we add equal time to two unequal times, the proportion of times (i.e., the ratio of the greatnesses of the times that each thing has become
) will change over time, so that the relative difference in times will decrease. Thus the older thing will become less older, which is to say younger
in a relative sense, as time passes, and the younger thing will be less younger, and thus older
in a relative sense. Insofar as the One is at first older than the Others, it becomes younger in relation to the Others over time. Insofar as, by another consideration, the One is at first younger than the Others, it becomes older in relation to the Others over time. In each case, the reverse holds for the Others. At no point does one thing overtake the other thing in age, so, for example, if the One at first is older than the Others, at no point can it be said that it is younger
than the Others, though it is always becoming younger
in relation to the Others. The same holds for all other cases, so that which is older
is becoming younger
and that which is younger
is becoming older.
Since becoming younger
in all these cases never results in being younger,
at no point can it be said that the thing becoming younger has become younger,
for that implies being younger.
Similarly, the thing becoming older can never be said to have become older.
So far we have another pair of antinomies, this time in terms of becoming older
and becoming younger.
Theses 1 and 2 consider older
and younger
in terms of absolute duration, while theses 3 and 4 consider them in terms of proportionate or relative duration. This yields the antinomies of 1-3 and 2-4.
Parm.: But since the One partakes of time, and partakes of becoming older and younger, must it not also partake of the past, the present, and the future?
Arist.: Of course it must.
Parm. Then the One was and is and will be, and was becoming and is becoming and will become?
Once it is acknowledged that the One is a subject of becoming and that it is situated in time, it is necessary that there is and was and will be something which is in relation to the One, for becoming supposes something Other than the One, to which it stands in relation in the past, present and future. Anything posited in relation to the One in some sense pertains to the One (for relation is an accident in the most generic sense, and pertaining is the defining characteristic of accidents).
In particular, we have at this moment opinion and knowledge and perception of the one,
an indubitable fact that is consistent with the hypothesis that the One exists. Moreover, the existence of these mental apprehensions implies that there is name and expression
for the One, i.e., that it can be named and expressed no less than other things.
Parm.: If the One is both one and many, as we have described, and is neither one nor many, and participates in time, must it not, in as far as it is one, at times partake of being, and in as far as it is not one, at times not partake of being?
Arist.: Certainly.
Parm.: But can it partake of being when not partaking of being, or not partake of being when partaking of being?
Arist.: Impossible.
Parm.: Then the One partakes and does not partake of being at different times, for that is the only way in which it can partake and not partake of the same.
This is a pivotal passage, as for once Parmenides does not merely accept a paradox, but attempts to resolve it. Another break from form is the invocation of arguments from outside the current set (1b). We are synthesizing the first set (1a), in which the One is neither one nor many, and the second set (1b), in which the One is both one and many. Since both sets of arguments assume the hypothesis, The One is,
we cannot take the One is neither one nor many
as implying absolutely, The One is not.
The proposed resolution, in light of the One’s participation in time, is that sometimes the One partakes of being and sometimes it does not.
Our interpretation would make the first two sets of arguments meaningful, and not just pointless sets of antinomies. Recall that (1a) concluded from the antinomies of impredicability that the One could not partake of being, and so could not be one, and nor could there be any name, expression, opinion or knowledge of it. Yet Parmenides and Aristoteles agreed that all this could not be true of the One, implying that they do not accept all the conclusions of (1a). Indeed, we now see that it is incontestable that opinions and thoughts about the One exist, which is contradictory of the absolute non-existence of the One. We now synthesize (1a) and (1b) by appeal to the One’s participation in time, so that it may come into and out of being at different times.
As the One comes into being it becomes one, and as it ceases to be it becomes many.
Parm.: And as it becomes one and many, must it not inevitably experience separation and aggregation?
Arist.: Inevitably.
Parm.: And whenever it becomes like and unlike it must be assimilated and dissimilated?
Arist.: Yes.
Parm.: And when it becomes greater or less or equal it must grow or diminish or be equalized?
All the other paradoxes of predication are resolved by making the One subject to change over time. Yet change or movement itself creates a new paradox, if we consider how the One goes from motion to rest or from rest to motion.
And when being in motion it rests, and when being at rest it changes to motion, it can surely be in no time at all?
How can it?
But that a thing which is previously at rest should be afterwards in motion, or previously in motion and afterwards at rest, without experiencing change, is impossible.
Impossible.
And surely there cannot be a time in which a thing can be at once neither in motion nor at rest?
There cannot.
But neither can it change without changing.
True.
When then does it change; for it cannot change either when at rest, or when in motion, or when in time?
It cannot.
And does this strange thing in which it is at the time of changing really exist?
What thing?
The moment. For the moment seems to imply a something out of which change takes place into either of two states; for the change is not from the state of rest as such, nor, from the state of motion as such; but there is this curious nature, which we call the moment lying between rest and motion, not being in any time; and into this and out of this what is in motion changes into rest, and what is at rest into motion.
So it appears.
And the One then, since it is at rest and also in motion, will change to either, for only in this way can it be in both. And in changing it changes in a moment, and when it is changing it will be in no time, and will not then be either in motion or at rest.
Here time is taken as implying duration, as Aristotle would later teach more explicitly, while a moment is an infinitesimal limit between times.
And it will be in the same case in relation to the other changes, when it passes from being into cessation of being, or from not-being into becoming-then it passes between certain states of motion and rest, and, neither is nor is not, nor becomes nor is destroyed.
Very true.
And on the same principle, in the passage from one to many and from many to one, the one is neither one nor many, neither separated nor aggregated; and in the passage from like to unlike, and from unlike to like, it is neither like nor unlike, neither in a state of assimilation nor of dissimilation; and in the passage from small to great and equal and back again, it will be neither small nor great, nor equal, nor in a state of increase, or diminution, or equalization.
The impredicability of all things is explained by appeal to moments of transition between one and the other predicate, where neither applies, for in this limit or instant there is neither being nor non-being.
All these, then, are the affections of the One, if the One has being.
It may seems strange that these impredications are called affections
as though they were positive entities, yet in fact this notion encompasses anything that befalls a subject, in this case the One. The state of not having various predicates is something that happens to the One as it passes into or out of being, or from one kind of being to another.
This passage indicates Plato’s true opinion on the matter, namely that the One exists, and all things and their contraries are predicable of it, but at different times, and these all are impredicable of it in the instants of transition. He does not here offer a theory of participation, but has created an opening in which one might be developed, as he would do later in the Sophist and the Statesman. We have established that it is not a contradiction for a thing and its contrary to be predicated of the same thing, if these predications occur at different times, and for these same predicates to be inapplicable to the subject at instants of transition. Time and becoming are critical to the eventual establishment of a theory of predication and participation, breaking free of the static ontology of being.
The dialogue apparently returns to form with a third set of arguments, now considering what happens to the Others, i.e., the complement of the One, on the supposition that the One exists. First, we consider what are the positive affections or accidents of the Others, starting with the fact that the Others, being other than the One, are not One.
Nor are the others altogether without the One, but in a certain way they participate (metechei) in the One.
Plato’s word for participate
is better translated as partake,
so that the Others partake of unity, standing in some positive relationship with the One. The precise definition of this relationship is the problem of participation, which Parmenides now addresses directly for the first time in the second part of the dialogue.
The Others have parts, for if they had no parts they would be simply One, i.e., they would be unities or monads. Parts have a relation to the whole. And a whole must necessarily be one made up of many; and the parts will be parts of the one, for each of the parts is not a part of many, but of a whole.
A part must be part of something, so it is part of a plurality (many) or part of a unity (whole). Consider the alternative:
If anything is a part of many, and is itself one of the many, it will be a part of itself, [157d] which is impossible, and of each one of the others, if it is a part of all. For if it is not a part of some particular one, it will be a part of the rest, with the exception of that one, and thus it will not be a part of each one, and not being a part of each one, it will not be a part of any one of the many. But that which belongs to none cannot belong, whether as a part or as anything else, to all those things to none of which it belongs.
If P is a part of the many, then it must be a part of itself, for it is among the many, since the notion of part entails multiplicity, i.e., a part cannot be solitary. For the rest of the argument, we must appreciate that the Greek term ekastos, translated as each,
or each one,
is closer in meaning to severally,
entailing plurality, so the English idiom each one
would be practically a contradiction. For something to be a part of many, the relation being a part
must apply severally to the many. If there is one of the many of which it is not a part, then it is not a part severally of the many. This is why Plato says that if there is even one exception among the many of which something is not a part, then it cannot be said to be a part of each
of the many. Since there is definitely one such exception, namely P itself, then P is not part of each of the many. This elides to the inference is not part of any of the many. We may write this symbolically as:
¬[m1(P) ∧ m2(P) ∧ … mi(P)]
¬m1(P) ∧ m2(P) ∧ … mi(P)
The second statement is obviously not a consequence of the first, so is Parmenides here just offering some sophistry, exploiting the syntactic use of the negative in Greek? A more literal rendering of the Greek, rather than not be a part of any one of the many,
would read is [a part of] not one of the many.
That is to say it is part of none of the many. How does this follow from a single exception?
Darren Gardner (2019) offers a solution, explaining that the many
is not a whole with discrete parts, but a mass
many, since we are operating on the hypothesis that it lacks wholeness.
This is true because themanyis not in any way a whole with discrete parts (it cannot be in any way unified as any kind ofone), but only a massmany.As such, amanythat entirely lacks wholeness requires its parts to be understood either distributively across it, or not at all. [Darren Gardner.Plato's Parmenides and The Knowable Many: Cosmos as Discursive Order in Hypothesis 3.Etudes Platoniciennes (2019), 15.
The contradiction that results from this hypothesis will show that in fact we cannot speak of the many without the One, and thus the many in some sense partake of the One. Before completing this proof by contradiction, let us further clarify what is meant by a mass
many, considering Gardner’s example: Moisture is part of clouds.
In order for this to be true in the sense intended, moisture must be part of all clouds without exception. This is because we are not considering clouds
as a whole set composed of individual clouds, for we are supposing that there is no wholeness in this many.
That is to say, we are considering clouds
abstracted from discrete number. Thus when we make moisture
a part of clouds,
this statement, to be true, cannot depend on discrete number. Being part of only these three clouds
would not satisfy the requirement of abstraction from discrete number, so this would not suffice to make Moisture is part of clouds
true in the intended sense of that statement.
Returning to the proof, we have shown so far that a part of he many, not being a part of the many necessarily must be part of all of the many, without exception. Yet it is impossible for it to be a part of itself, which is one of the many, so it in fact cannot be part of all of the many, and thus not part of each, i.e., it cannot be part of the many distributively, and thus it is not part of any of the many (considered as a mass many
), or rendering the Greek more literally, it is part of not one of the many.
But being a part of not one of the many, it cannot be a part or anything else whatsoever of those things of none of which it is anything.
This last statement is difficult to parse, but essentially we are inferring that, if P is not a part of any one of the many, it cannot be anything else of the many. For if it is not part of the many, it does not partake of them in anyway; it has nothing of them. Inversely, there is nothing of P to be found in the many. From this we infer that there can be no positive ontological relationship between P and the many, for their conceptual intersection is null.
Then the part is not a part of the many, nor of all, but is of a certain single form, which we call a whole, being one perfect unity framed out of all—of this the part will be a part.
The supposition that there is no unity in the many resulted in the contradiction that anything posited as a part would fail to be a part of the many (conceived without unity). This contradiction forces us to conclude either that the many are without parts, which is absurd, or that the supposition is false. Taking the latter course, Plato now offers a means by which the may Others can be unified by a single principle, and thus introduces the Form for the first time in the second part of the dialogue. This introduction suggests that the dialogue is no longer purely antinomic, but is offering some positive resolutions. As Gardner says, the dialogue began by arguing about the One abstracted from the existence of Forms, but only now are Forms introduced as they have become necessary to resolve a perplexity not about the One as such, but about the Others in relation to the One. Parmenides advised Socrates in the first part of the dialogue that it is necessary to consider not only the One but also what will be the consequences to the many in relation to themselves and to the One,
and now this is being realized, further showing that the dialogue has a definite plan, and is not a pure collection of paradoxes.
Given that anything with parts must partake of wholeness, which is a Unity, it follows that the Others, if they have parts, must likewise partake of wholeness and Unity. We have earlier established that the Others must have parts, for otherwise they would be simply Unity. Thus the Others must partake of Unity, though they are not identical with it, for their parts combine into a complete whole.
Not only are the Others collectively a whole, but each part must partake of the one, for if each of them is a part, this presumably indicates each to be one, distinguished from the rest,
if indeed it is each.
Again, this dense argument, which I have attempted to render literally, rests on an aspect of the Greek notion of each.
Here we focus not on its distributive aspect, but on its distinction from others in its class or group. If the parts were completely indistinguishable, we could not identify them as each.
Thus a particular each
must be distinguishable from the rest, and in that way enumerable as one.
Having already posited that we are speaking of parts of the Others, it follows that these parts are Other than the One, i.e., they are not identical with the One. Thus partaking or participating in the One must mean something different from simply being the One in the sense of identity. The whole, being a unity of all its parts, partakes of the One, and the parts, partaking of wholeness, thereby also partake of the One. For the whole
will be one whole, of which the parts will be parts; and each part
will be one part of the whole which is the whole of the part.
This partaking can be expressed syntactically by using one
as an adjective or as a predicate, but we have yet to express what this means ontologically.
Since things which partake of the One are Other than the One, they must be many (i.e., more than one), not one. We exclude the non-one of zero, for that would be nothing. We may go further and argue that those things that partake of the one, whether as a part or as a whole, must be infinite in number.
Let us look at the matter thus: Is it not a fact that in partaking of the one they are not one, and do not partake of the one at the very time when they are partaking of it?
Clearly.
They do so then as multitudes in which the one is not present?
Very true. And if we were to abstract from them in idea the very smallest fraction, must not that least fraction, if it does not partake of the one, be a multitude and not one?
Some of this argument will be reflected at the very end of the dialogue, where it is asserted that without the One, nothing else can exist, for the many cannot be conceived without the One. Here it is asserted that the many, by virtue of partaking of the One, are not identical with the One, and thus in another sense do not partake of the One. They must partake of the One, not as themselves being one, but as multitudes in which the one is not present. It would be vacuous for the One to participate in itself; that is not what we are saying when we say the many partake of the One. In the latter statement, the subject many
is considered not to contain the concept of Unity. The absence of unity in the concept many
is a necessary precondition of our statement, The many partake of the One.
This accentuates the paradox of participation: how is it that many,
in which there is no unity, can partake of unity?
A tiny fraction of some part of the many may be conceived at least in thought, even if that part is not really divisible, and that fraction, since it does not partake of one, must itself be many, and we could take each of these many and proceed dividing in thought ad infinitum. Thus the many, considered in itself and devoid of unity, is infinite in number.
Yet when each part becomes a part, it is thereby distinguished from the rest, and bounded or limited in relation to other parts and to the whole. Considered in this way, the part is finite or limited. Thus the Others are infinite or limitless when considered in their own nature (i.e., as many, devoid of unity), but they are finite or limited when considered in relation to each other and to the whole. Partaking or participation, it would seem, is relational.
Recall the paradox of X participates in the One.
Insofar as X is participating, it is not identical with the One. Indeed, it participates insofar as it is non-One. It participates in Oneness to the extent that it does not participate in Oneness. We resolve this by noting that X, i.e., the Other, does not participate in Oneness in the sense that One is not conceptually contained in the Others or Many. They are non-overlapping concepts. Yet the Others or Many do have unity when considered in relation to each other or to themselves as a whole. If the Others or Many exist, then Unity must also exist, not because unity is contained in the notion of many, but because the relations among the many require the existence of unity. We might say that the participation of the Many in the One is an extrinsic, yet ontologically necessary, relation. The Many do not partake of the One in the sense of having Unity inside
the concept of Many, but in the sense of standing in some positive relationship to Unity (i.e., as opposed to a merely negative relationship of impredicability). This may still not be perfectly lucid, but we are at least moving toward coherence.
Returning to the present argument, we develop another set of antinomies. Insofar as all the Others partake of limit, they are affected in the same way. Insofar as they are both limited and unlimited, they are affected in opposite ways, and opposites are unlike. Thus, in regard to either of their affections (limited or unlimited), they will be like themselves and each other, but considered in reference to both of them together, they are opposed and unlike each other. Note again that this is not a true paradox, since we are explicitly operating under two different considerations. By similar reasoning, they are the same and different, in motion and at rest, and so on for every other affection discussed.
The key to resolving these antinomies is whether we consider a thing intrinsically (in its nature) or in relation to other things. This dual consideration yields a pair of opposing affections or predications. We may say a thing is like itself and others of its kind insofar as we regard it (and its brethren) intrinsically, and it is unlike itself and others of its kind insofar as we contrast intrinsic characteristics with relational characteristics.
Suppose, now, that we leave the further discussion of these matters as evident, and consider again upon the hypothesis that the One is, whether opposite of all this is or is not equally true of the others.
Introducing this fourth set of arguments, Parmenides invites us to consider that the conclusions of both (1c) and (1d) may be true. This is not mere sophistry, arguing both sides to show one’s ingenuity. As we have seen with the internal antinomies of (1c), it is quite possible to accept both sides if one accounts for distinctness of considerations. It seems that Plato really believes that (1c) and (1d) are both true, given the hypothesis that the One is. We agree with Gardner that this fourth set of conclusions is complementary to the third, for if the Others cannot participate in the One, neither can anything else be predicated of them, i.e., we cannot discourse about the Others at all. Thus participation in Unity is necessary in order for the Others to be objects of discourse, and the third set of conclusions is to be preferred, since it is incontrovertible that we can discourse about the Others.
The One and the Others are separate (choris) from each other, for nothing else exists besides the One and the Others, hence there can be nothing in which both the One and Others are. If there is no same thing in which both the One and the Others, then the Others are separated or apart from each other.
The One, being separated from the Others, cannot be in the Others, certainly not as a whole. Nor can part of the One be in the Others, for the One as such has no parts. If neither the whole of the One nor part of the One can be in the Others, then it would seem that the Others cannot partake of the One in any way. Thus the Others are not one in any sense.
Yet neither can the Others be many, for then each would be a part of the whole, but the Others do not participate in the One, and hence cannot be part of a whole, for the whole has unity, as does each part. Note that this part of the argument appeals to (1c), again showing complementarity of (1c) and (1d). Nor can they be two or three or any other number, for all numbers partake of unity.
The lack of number in the Others implies that they are neither like nor unlike the One, nor is there likeness or unlikeness in them, for any of these conditions would require there to be two opposing elements in them, and to be two would involve partaking of unity. Likewise, lack of participation in number precludes the Others from having affections such as being the same, other, in motion, at rest, becoming, being destroyed, greater, less, or equal. It was shown in the extension to (1b) that all these things are affections of the One, either when it is united as itself or dissimilated into many, so they are all affections of things with number. Here the converse is asserted, i.e., that anything with these affections must partake of number. Equality involves the notion of even number, i.e., division of a whole into two parts, while inequality involves the odd. Sameness is a special case of likeness, so it involves the notion of two. Rest, remaining in the same place, involves sameness, and therefore two. Motion involves dissimilarity of place, so a comparison of two is required. The same holds for any other kind of change, including becoming and being destroyed, increase and diminution. Thus none of these things is predicable of the Others.
Therefore if One is, the One is all things, and it is also nothing, both in relation to itself and to the Others.
This summarizes the conclusions of all four sets of arguments under they hypothesis that the One is. Everything is predicable of the One and nothing is predicable of the One. Likewise everything is predicable of the Others and nothing is predicable of the Others. Yet we have seen, on closer inspection, that things are predicable of the One insofar as it is One and are impredicable of the One insofar as the One can become other than one or it is considered in relation to things other than itself. Likewise, the Others partake of the One insofar as they are considered in relation to the One, and in that same degree the One overlaps with or shares its being with the Others, not considered in itself, but considered in relation to the Others. The Others, considered strictly as Other, rather than as many, do not partake of Unity and therefore do no partake of any of the predicates, for these presuppose One or two or some higher number.
Thus the conclusions of the first four sets of arguments are not truly contradictory. They can be resolved by a notion of participation or partaking, in which we can consider a being not in its nature (i.e., by its definition), but in relation to things other than itself. This is why the notion of Others is most apt for fleshing out the nature of participation, and why in the last set of arguments we needed to abstract Otherness even from plurality.
As we shall see, Plato actually favors the hypothesis The One is
over The One is not,
for the latter results in an impossible contradiction of our actual ability to discourse about objects. Thus the conclusions outlined above represent the real conclusions of the overall dialogue, and are the basis for justifying a notion of participation, which will be elaborated further in the Sophist and the Statesman.
The dialogue now moves to the counter-hypothesis, The One is not,
to see if the antinomies of the first four sets of arguments might be avoided on that assumption. As Parmenides noted at the outset, it is not enough to find contradictions assuming a hypothesis, for one should also examine whether similar or worse contradictions may arise from the counter-hypothesis.
The fifth set of arguments may seem poorly posed to modern readers, especially if they are accustomed to the conventions of modern logic. Following Russell (who in turn was informed by Frege and Boole, who handled this matter differently), modern predicate logic considers the predicate statement P(x) to be false if x does not exist, for non-existence is taken as implying that x is nothing, nullity, or an empty class, indistinguishable from any other nullity. Thus we should regard the statement A unicorn is single-horned,
as false, and the same for any other statement involving unicorns.
This interpretation seems to confuse existence with discourse about concepts, so some clarifications are in order. First, when we say x is P
or P(x), we may interpret this either as assuming or not assuming that x exists. Russell adopted the latter convention, known as aspect theory
when applied to interpreting the copula is,
so x is P
means x is existent as P.
Plato, on the other hand, likely held the copulative theory,
in which x is P,
need not imply that x exists. The copulative theory allows us to discourse about things even while explicitly supposing that they do not exist. According to Russell, this would be nonsense, for supposing non-existence of x is to suppose that x is nothing, and nothing can be predicated of nothing. To answer this, we must clarify what is meant by non-existence or not being.
Naturally, we can discourse about things that do not exist in the physical universe or are of doubtful existence. Physicists discussed black holes and their properties before it was certain they really existed. English majors may discuss Hamlet and other fictional characters and their qualities. When discussing history, we are discussing events and persons no longer occurrent and therefore non-existent. If one wishes to define a logical calculus that only deals with physical existents, and treats everything else as a nullity, one may do so, but such a calculus falls short of being a comprehensive logic, as its domain does not encompass all rational thought. Clearly, a universe of discourse, consisting of concepts or thought-objects, can be much broader than the physical universe.
Nonetheless, when we posit an object, we are supposing it to exist at least as an object of discourse, so it is existent in the universe of rational discourse, as long as its conception does not involve any intrinsic contradiction that would make it incoherent to conceive. There is nothing intrinsically illogical about a horse with a horn on its head like a narwhal, so a unicorn is a valid concept and we may discuss unicorns based on their definition, inferring that they have four legs, given that they are horses in all respects except for their horn and any other defining characteristics we choose. When we say a unicorn has four legs, we may mean, A unicorn, if it were to exist (in some reality, not just in thought), would have four legs, given the definition of a unicorn as a single-horned horse.
Yet this broadened universe of discourse, accepting any concept with intelligible content, does not save us from the apparent absurdity of asserting, A unicorn, supposing it does not exist, has four legs.
To explicitly suppose something not to exist would entail it cannot have four legs or anything else, at least not in the same mode of reality in which it does not exist. This last qualification is the crux of the matter, for perhaps we are supposing that a unicorn does not actually exist in the cosmos, but it may nonetheless have four legs in some other sense (conceptually, in thought). That is to say, our denial of its existence need not imply a denial of our ability to discourse about it, considered in itself or in relation to other concepts or objects.
Parmenides effectively argues for the copulative theory, considering that when we say The One is not,
that means something different than saying The not-One is not.
That is to say, the subject the One
is something which is known,
it is not a mere conceptual nothing, or else both statements would mean the same thing, when in fact they are opposed. That which is said not to be is known to be something all the same, and is distinguished from other things.
The meaning of the One
is something other than all the Others.
So, even supposing that the One does not exist, its positive conceptual content implies that there can nonetheless be knowledge or understanding of the One, or we could not even know what the expression if the One is not
means. Second, it is different from the Others, or else it would not be possible for the Others to exist on the assumption that the One is not, and clearly something exists, so if not the One, then the Others. Thus, even on the assumption that the One does not exist, difference (from the Others) is predicable of the One.
We would be unable to indicate the One or distinguish it from the Others unless it could also accept predicates such as this,
that,
or some(thing),
identifying it as a particular concept among others. Thus even on the assumption that the One does not exist, many things are predicable of the One, if other things are allowed to exist. This further indicates that simple being and participation are distinct. Moreover, when we say The One is not,
we mean that One of which we are speaking, and not something else. If neither that
nor One
were the subject that is not, we would indeed not be able to predicate anything of such a subject, or even speak of such a subject. If, on the other hand, that One
is the subject that is not, then the predicates mentioned pertain to that subject.
Note that Plato allows us to speak of something as a particular or determinate something, that,
even on the assumption of its non-existence. This is contrary to the convention of Frege, followed by modern mathematical logic, where the particular quantifiers some
or one
are expressed as an existential there exists
operator. In modern logic, quantification is thereby confounded with existence, and this is done only for particular, but not universal, quantifiers. Whatever the benefits of such a convention, we cannot escape the fact that, in the abstract, it is perfectly intelligible to speak of some
or this
with regard to concepts, without supposing their manifestation as existents in the cosmos.
Perhaps we could suppose, per impossibile, that nothing exists, neither the One nor any Others. In that case, there may indeed be nothing predicable of the One, since there is nothing in which it might participate, but then no discourse at all is possible.
Given that the One is the subject of non-existence and that Others may exist, the One may possess unlikeness in relation to the Others, being different from them. For the Others cannot be unlike the One without the One being unlike the Others. Moreover, the One must possess likeness to itself. If the One possesses unlikeness to the One, our argument will not be concerned with that which is of the nature of the one, and our hypothesis will not relate to the One, but to something other than One.
Since this is contrary to the hypothesis, it must be admitted that the One is not unlike the One, i.e., it is like the One.
If the One does not exist, it cannot be equal to the Others, for the Others exist and the One does not, so they are not equal in that respect. Further, if they were equal, the One would be like the Others, but it was just shown to be unlike the Others. Thus the One is unequal to the Others, and the One partakes of inequality. Inequality is composed of greatness and smallness, so the One partakes of greatness and smallness. Greatness and smallness are separated by equality, which is between them. Thus whatever has greatness and smallness will have equality, so the One partakes of equality.
These arguments play fast and loose with two-place predicates: inequality, equality, etc. In Plato's treatment, each of the two terms may be considered a subject of the predicate. E(x, y), may be read as x is equal to y AND y is equal to x.
This, in Plato's reading, justifies saying x is an equal,
and y is an equal.
Exploiting the negativity of inequality, we can make the non-existent One a term of that predicate, contrasting it with an existent term (the Others). Then, exploiting the reciprocity of inequality and rendering each term as a subject, we conclude that the Others are unequal
and the One is unequal.
Then the relation of inequality, considered in itself, partakes of concepts of greatness and smallness. Thus whatever partakes of inequality partakes of these concepts as well. Inequality requires a distinction between greatness and smallness, and Plato says it is impossible to distinguish them without the notion of equality. How could we know one thing is greater and another is smaller unless we also had the concept of equality?
To know that one thing is greater than another, we must be able to show that the full extent of the smaller thing is commensurate with some part of the greater thing, with something of the greater thing left over. Thus we must show the equality of the smaller thing with a part of the greater thing. It follows, then, that anything which partakes of smaller
also partakes of equality (with a part of the greater). Likewise, that which partakes of greater
can partake of equality, via one of its parts. It is not necessary for the whole of the greater to be equal to anything, for partaking is not the same as simply being; participation is not identity.
More stunningly, Parmenides infers that the One partakes of being (ousia). The One must be in the conditions described above, otherwise, we would not be speaking the truth in saying the One is not. If we speak the truth, then we say that which is. By speaking the truth about the One that is not, the One that is not is
in some sense. If the One is to continue in not-being, then the being of its not-being must persist. Here being
refers to the truth-making reality behind our true statement, The One is not.
If it is true that the One is not,
then it is really the case that the One is not, and the reality of the One’s not-being is what we call the being
of its not-being. This being of not-being
must persist as long as the One is not, so it performs the function of substantial being (ousia).
The mirror situation for an existent being that remains existing is that its not-being
persists in non-existence. That is, the not-being of its not-being perfects its being. The being of being,
i.e., the reality making it true that something exists, and the not-being of not-being,
i.e., the reality making it false that something does not exist, are most truly asserted when being
itself is taken as the subject. For being as such most certainly partakes of the being of being; it cannot be otherwise, as it is necessary that being is
(considered conceptually). (This is different from other subjects, for which it is conceivable either for them to be or not to be.) Likewise for being, it is certain that its not-being is not, so the not-being of its not-being
can be asserted with absolute certainty. So any existent, i.e., anything that partakes of being, will have both the being of being
and the not-being of not-being.
In short, every non-existent partakes of being (i.e., the being of its not-being), and every existent partakes of not-being (the not-being of its not-being). This counterintuitive result is possible because Plato considers being
in a broader sense than positive physical existence. The being
of not-being is a truthmaking reality or state of affairs. In order for it to be true that the One is not,
there must be some definite state of affairs posited. Asserting the non-existence (in the cosmos) of some intelligible object adds positive knowledge to our description of reality (the cosmos), so the non-existence of that object is itself an object of discourse, and thus has being
insofar as the state of affairs with the object being non-existent is a real state of affairs. Reality would be different if it were not the case that the object in question is non-existent, so the object’s non-existence does inform being. A non-trivial example of this is the non-existence of magnetic monopoles, which informs the laws of electrodynamics.
Given that being may be predicated of the One that is not, and more trivially that not being
can be predicated of the One that is not, we find that both being and not-being are predicated of the non-existent One. Yet being and not-being are two different states, so the non-existent One must be susceptible to change or movement, but of what sort? It cannot be susceptible to change of place, for the One that is not is nowhere among what is. Nor can it turn in the same spot, for the same
is, so the One that is not cannot be in contact with the same. The One, regardless of whether it is or is not, cannot be changed into something other than itself, for then we would no longer be speaking of the One, i.e., the uniqueness of the one forbids any alteration. This exhausts the possible kinds of movement (save increase and decrease, which are obviously incompatible with being one), so the one is not capable of movement after all. Thus the One is at rest, though by the earlier consideration it must be moved (between being and not-being), and thus must undergo a change. Thus the One that is not is altered and not altered (at rest).
The One is changed or altered insofar as it can go between being and not-being, but it is unchangeable insofar as it cannot be changed without ceasing to be One. This is a similar paradox as we had seen in the extension to (1b), where we arrived at contrary conclusions depending on whether we required the One to be one or allowed it in a sense to be non-one, i.e., to be in relation to something Other than One. That was resolved by allowing that the One can pass into and out of being over time. Here, however, it is less clear that we have really identified two distinct states of the One, for we have (1) its not-being, and (2) the being of its not-being. Yet The One is not
and It is true that the One is not
are not identical statements, even in modern logic. The first statement focuses on the One’s lack of existence, and the second focuses on the positive state of affairs implied by this lack of existence. Non-existence is purely a negation in (1), but in (2) we consider how the non-existence of the One informs our knowledge of positive reality. Thus the One, though it is not, is in some relation to being, contributing to our description of the cosmos that is. If we make the same resolution, that the One cannot be and not be at the same time, then it may change into or out of being. This argument works only if we do not allow there to be more than one sense of being.
We may identify three possible problems with this argument: (1) the explicit use of being as a predicate; (2) the possibility that being
has different senses in The One is not
and being of not-being
; (3) the change of the One between being and not-being, even if admitted, should be harmonized with the claim that the One cannot be changed into something other than One. First, the term einai, translated as being,
is the infinitive verb form of to be,
so its significance is closer to the Thomist esse, to exist.
Even this, however, runs into Kant’s criticism that existence is not a predicate, in the sense that it adds no conceptual content to whatever subject it is applied. Further criticism is that existence is too ill-defined to add any conceptual content. Yet if existence is too poorly defined, we should not admit it in discourse, and would not be able to speak of the truth or falsity of anything. The difficulty is that existence is not really a concept, but something more primitive and ineffable, yet foundational to all logic and ontology. Plato accepts that everyone has an intuitive sense of what it means to be,
and a supportive notion of a cosmos containing all that is (thereby defining the reality in which things are). Thus to be
is to be in the cosmos, which is a valid predicate, albeit ultimately circular, since the cosmos consists of all that is, but at some point we cannot regress to more primitive definitions. The One is not in the cosmos, and its absence from the cosmos is a fact contributing to our description of the cosmos. Since the One is both absent from the cosmos and a contributor to our description of the cosmos, it is not in the cosmos yet it is in relation to the cosmos. Yet since the cosmos contains all that exists, it cannot be outside the cosmos, but is utterly non-existent. If we consider it contradictory for a non-existent thing to be in relation to the cosmos, we would have to admit that it can come into and out of existence. This would not be contradictory if we admit that being in relation
is different from simple existence, but we are again abstracting from the possibility of participation, yielding an antinomy.
The conclusion that the One can come into and out of existence was derived from the ability of the non-existent One to be in relation to the cosmos (i.e., the set of all existents). The unchangeability of the One was derived directly from its non-existence considered as such. Again we have an antinomy derived from considering a thing intrinsically and again in relation, and only some notion of participation might help us resolve it.
Parmenides takes both halves of the antinomy at face value:
And must not that which is altered become other than it previously was, and lose its former state and be destroyed; but that which is not altered can neither come into being nor be destroyed?
Very true.
And the one that is not, being altered, becomes and is destroyed; and not being altered, neither becomes nor is destroyed; and so the one that is not becomes and is destroyed, and neither becomes nor is destroyed?
Without participation, the One that is not is both alterable and unalterable, becoming and not becoming, destroyed and not destroyed. Thus the assumption that the One is not yields antinomies no less formidable than those under the hypothesis that the One is, and we cannot evade the need for participation by supposing that the One is not.
The sixth set of arguments is less controversial from the perspective of modern logic, as we consider The One is not
to imply that the One does not partake of being in any way, so nothing can be predicated of it. That which has no being cannot assume or lose being, i.e., it cannot become or be destroyed. From that it follows that it cannot be altered, since that is a mode of becoming or passing away. Nor can it be moved, for that is a form of change or alteration. Nor can it stand at rest, for it would have to be somewhere or in some spot. Nor can any existing thing be attributed to it, for then it would partake of being. It cannot have smallness, nor greatness, nor equality. It cannot have likeness nor difference, in relation to itself or others.
Yet, by reciprocity or inversion of the relations like, unlike, same, different,
it follows that existent things cannot be like or unlike, the same or different with respect to the One. The One, as non-existent, cannot stand in relation to anything. It cannot be past present or future. Nor can knowledge, or opinion, or perception, or name, or any other thing that is, have any concern with it?
The consideration of non-existence as implying thorough impredicability has far-reaching implications, for it limits even what can be predicated of existents. More strikingly, it eliminates the One as an object of discourse. If it is thoroughly non-existent, we cannot have a thought about it, opine about it, or even give it a name. For what can the One
mean, on this assumption, besides nothing? These implications show that the modern logical treatment of non-existence is problematic, insofar as modern logic pretends to be a comprehensive account of rational discourse, rather than a calculus with more limited domain.
Sections (2a) and (2b) have the same conclusions as (1a) and (1b), namely that everything is predicable and impredicable of the One. These paradoxes arise regardless of whether we suppose the One to exist or not to exist. In (1b)-(1c), participation was offered as a resolution. It remains to be seen if any solution can be offered on the assumption that the One is not.
Once again the key is to consider the implications regarding the Others. Under the first hypothesis (The One is
), two sets of arguments yielded conclusions that various contraries are predicable of the Others (1c), or that these contraries are impredicable of the Others (1d). In these final two sets of arguments, we will find mixed conclusions, where some pairs of contraries are predicable and other pairs are impredicable. The critical pairs are one and many, likeness and unlikeness.
The Others must surely be; for if they, like the One, were not, we could not be now speaking of them.
The alternative would be for nothing at all to exist, in which case we could not be speaking of anything.
But to speak of the Others implies difference—the termsotheranddifferentare synonymous? [also Lat. alter]
True.
Other means other than [an]other, and different, different from the different?
Certainly.
And what can that be?—for if the One is not, they will not be other than the One.
They will not.
Then they will be other than each other; for the only remaining alternative is that they are other than nothing.
Since the One is not, there can be no unity in the Others, so any part of them is a plurality, and the same for this plurality of sub-parts, ad infinitum, so every part of them is infinite in number. This repeats an argument made in (1c). Here, however, we go further.
And in such particles the Others will be other than one another, if Others are, and the One is not?
Exactly.
And will there not be many particles, each appearing to be one, but not being one, if One is not?
True.
And it would seem that number can be predicated of them if each of them appears to be one, though it is really many?
Likewise, there would only be the appearance, though not the reality, of odd and even among them, and a least among them, though this least would seem large compared to the fractions it contains.
And each particle will be imagined to be equal to the many and little; for it could not have appeared to pass from the greater to the less without having appeared to arrive at the middle; and thus would arise the appearance of equality.
Yes.
And having neither beginning, middle, nor end, each separate particle yet appears to have a limit in relation to itself and other.
How so?
Because, when a person conceives of any one of these as such, prior to the beginning another beginning appears, and there is another end, remaining after the end, and in the middle truer middles within but smaller, because no unity can be conceived of any of them, since the one is not.
The absence of unity implies the absence of wholeness and the absence of enumerable parts. Moreover, the infinite divisibility of particles of the Others precludes there from being a definite beginning or end; the particles are akin to fractals in this respect. This divisibility extends even to the interior, for we cannot admit any definite enumerable part that could be treated as a whole, for the One is not.
So this thoroughly fractured being, nonetheless, has the appearance of unity.
And such being when seen indistinctly and at a distance, appears to be one; but when seen near and with keen intellect, every single thing appears to be infinite, since it is deprived of the One, which is not?
Nothing more certain.
Then each of the Others must appear to be infinite and finite, and one and many, if Others than the One exist and not the One.
From a distance, insofar as they appear to be a unity, the Others appear to be alike, i.e., in the same state.
But when you approach them, they appear to be many and different; and because of the appearance of the difference, different in kind from, and unlike, themselves?
True.
And so must the particles appear to be like and unlike themselves and each other.
Certainly.
And must they not be the same and yet different from one another, and in contact with themselves, although they are separated, and having every sort of motion, and every sort of rest, and becoming and being destroyed, and in neither state, and the like, all which things may be easily enumerated, if the One is not and the many are?
Most true.
This is a softer antinomy, for the Others only appear to be like and unlike themselves and each other. The same must hold for the other antinomies derived from this. Moreover, there is an asymmetry, for we have supposed that the One is not, thus the likeness of the Others is an illusion while their unlikeness would seem to be more fundamental. Yet the infinite regress in division would make likeness and unlikeness alternate, for each successive division of particles appears to consist of units, which on closer inspection are fractured into smaller particles, and so on. Thus we never arrive at a fundamental reality, and both likeness and unlikeness are appearances.
Starting again, we assume once more that the One is not, so the Others cannot be one, but now we also consider that the Others cannot be many, for the many contain the one. Here we are considering an enumerable many, and the Others cannot be many in this sense if the One is not. Yet if they are neither one nor many, we have exhausted all possibilities of number save nothing. But if no one of them is one, all of them are nought, and therefore they will not be many.
Not only are the Others neither one nor many, but they cannot even appear to be one or many. Because the others have no sort or manner or way of communion with
any sort of not-being, nor can anything which is not, be connected
with any of the others; for that which is not has no parts.
Here we are considering that non-existents such as the One are incapable of any sort of partaking or participation or relation with respect to existents, nor can the latter partake of non-existents by an inverse or reciprocal relation. On the assumption of that incapacity (such as is supposed by modern logic), it should not be possible for the Others to even appear to be one or many, for that would require having some relation to a non-existent. The same holds for even having an opinion or thought about the non-existent One in relation to the Others.
Then if One does not exist, none of the Others will be conceived (doxasai) of in any way whatsoever in any way whatsoever as related to the Others.
We should not be able even to imagine such a relation, as it should be unthinkable.
Then if One does not exist, none of the Others will be conceived as being one or as being many, either; for it is impossible to conceive of many without one.
In turn this implies the impossibility of the Others being conceived as like or unlike, same or different, in contact or separate, nor any of the other things we were saying they appeared to be.
This last is in reference to (2c). Now we are saying:
The Others neither are nor appear to be any of these, if the One does not exist.
Then if we were to say in a word,if the one is not, nothing is,should we be right?
Most assuredly.
This radical conclusion was derived by supposing that non-existence of an object abolishes that object from the universe of discourse, so that nothing can be thought of it or predicated of it. The implication, when that object is the One, is that the Others cannot have any predications or even appear to have them, which is possible only if the Others do not exist, in which case nothing exists. This is so manifestly untrue that it is clear that we must reject either the supposition that the One is not, or that non-existence entails inconceivability.
Then let us say that, and we may add, as it appears, that whether the One is or is not, the One and the Others in relation to themselves and to each other all in every way are and are not and appear and do not appear.
Very true.
Taken on its own, this conclusion of the dialogue would seem purely paradoxical, making no choice between hypothesis and counter-hypothesis, and merely accepting all the antinomies resulting from comparison of the eight sets of arguments. Yet we have seen earlier that there are deliberate asymmetries among the arguments, and this conclusion itself explicitly refers to one of them, namely the conclusion of (2d). For let us say that
refers to: if the One is not, then nothing is.
The acceptance of the conclusion of (2d) as valid implies acceptance that non-existence entails inconceivability. Since the conclusion is manifestly false, we must reject either the hypothesis that the One is not or that non-existence entails inconceivability. Yet Parmenides reminds us that rejecting hypothesis (2) will not save us, for similar paradoxes resulted under hypothesis (1), the One is.
Back in (1a), it was concluded that the One could not partake of being, so there could not be any name, expression or knowledge of it. Under hypothesis (1) or (2), the problem is supposing that non-being abolishes an object from the universe of discourse, and thus from predicability. The only way out of these intractable paradoxes is by admitting a notion of partaking or participation in being that is distinct from existential identity. The doctrine of participation or forms is not to be blamed for the surrounding paradoxes of being. On the contrary, the only way out of these paradoxes is to somehow distinguish conceptual content from simple existence, and this is what the doctrine of participation attempts. The numerical difficulties engendered by consideration of forms and their instantiations are not circumvented by abolishing forms, for we see they would persist in full force if participation were denied.
A basic notion of participation or partaking is articulated in the extension to (1b) and in (1c). X may partake of Y without X being identical to Y or being a subset of Y, for X may stand in some relation to some Y that is altogether other than X. It may even stand in relation to a Y that does not correspond to any actual existent, yet has positive conceptual content. For telling us what X is not can add to the definition of X, as is commonplace in mathematical definitions, for example. Thus a truly comprehensive logic and corresponding ontology ought to account for such possibilities.
Having dispensed with the most formidable objections to the doctrine of Forms, including some that would not be admissible under the self-imposed constraints of modern logic, Plato is now free to articulate this doctrine more clearly and indicate how it is a better solution to the paradoxes described. This work, as undertaken in the Sophist and the Statesman, will be discussed in a future essay.
Direct quotations from the dialogue are taken from the translation by Benjamin Jowett (esp. in the first part) and from the H.N. Fowler (1925) translation at Perseus Digital Library (Tufts University, Medford, Mass.). Capitalization and word substitutions have been made by the author for clarity and closer agreement with the argument made in Greek.
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