## 1. Overview of the Dialogue




Plato's *Parmenides* consists in a critical examination
of the theory of forms, a set of metaphysical and epistemological
doctrines articulated and defended by the character Socrates in the
dialogues of Plato's middle period (principally *Phaedo*,
*Republic* II-X, *Symposium*). According to this
theory, there is a single, eternal, unchanging, indivisible, and
non-sensible form corresponding to every predicate or property.
The theoretical function of these forms is to explain why things
(particularly, sensible things) have the properties they do.
Thus, it is by virtue of being in some way related to (i.e., by
participating in, or partaking of) the form of beauty that beautiful
things (other than beauty) are beautiful, it is by virtue of partaking
of the form of largeness that large things are large, and so on.
Fundamental to this theory is the claim that forms are separate from
(at least in the sense of being not identical to) the things that
partake of them.




In the metaphysics of his middle period, Plato does not provide a
theory of the nature of the partaking relation. But in the
*Parmenides*, Plato considers two accounts of the partaking
relation. According to the first "Pie Model" account,
for *X* to partake of *Y* is for the whole or a part
of *Y* to be in *X* (as a part of *X*). According
to the second "Paradigmatistic" account, for *X* to
partake of *Y* is for *X* to resemble *Y*. In the
first part of the dialogue, Plato sets out reasons for thinking that,
on either of these accounts of partaking, the theory of forms is
internally inconsistent.




Immediately following these criticisms, Plato describes a general
method of training designed to save the forms. The method
consists of a series of eight Deductions (with an Appendix to the first
two) focusing on consequences that may be derived from positing the
being of a particular form and consequences that may be derived from
positing the non-being of that form.




In the second part of the dialogue, Plato instantiates this method,
taking the form, the one, as his example. Plato shows, in
particular, that whether the one is or is not, the one (and also things
other than the one: the others) have *none* of a series of pairs
of contrary properties (whole/divided, in motion/at rest,
same/different, like/unlike, equal/unequal, older/younger). Plato
also shows that, whether the one is or is not, the one (and also the
others) have (or, at least, appear to have) *all* of these
contrary properties.


## 2. The Introductory Section: Zeno's Argument 126a-128e




The dialogue's narrator is Cephalus, who has just arrived in
Athens after a trip from his home in Clazomenae. Cephalus runs
into Plato's half brothers, Adeimantus and Glaucon, and asks them
to confirm the existence of someone who has completely memorized a
conversation that Parmenides and Zeno once had with Socrates.
Adeimantus confirms that his half brother, Antiphon, can recite the
conversation *in toto*, having heard it from a friend of
Zeno's, Pythodorus, in whose house the conversation took
place. Cephalus, Adeimantus, and Glaucon then pay a visit to
Antiphon, who, after a bit of prodding, agrees to replay the
conversation. As most scholars agree, the conversation putatively
recorded by Pythodorus, passed on to Antiphon, and then recounted to
Cephalus, is almost certainly fictitious. This hypothesis partly
explains why Plato chose to frame the dialogue at third
remove.




As Antiphon tells the story, the noted Eleatic philosophers, Parmenides
(then around 65 years old) and Zeno (then around 40 years old), have
come to Athens for the Great Panathenaea. Having heard of their
arrival, a youthful Socrates (then around 20 years old) and some
friends of his have come to Pythodorus's house to listen to Zeno
read from his book. At the end of Zeno's performance,
Pythodorus, Parmenides, and Aristotle, who have been waiting outside
the house, return and witness an exchange between Zeno and
Socrates.




The exchange begins with Socrates' summary of the general
structure of one of Zeno's arguments:


1. If the things that are are many, then they are both like and
unlike.
2. Nothing can be both like and unlike.


So,


3. The things that are are not many.




Zeno then explains that he intends this argument as a kind of
defense of Parmenides' monism: just as others have argued that
monism leads to absurd results (Zeno may be thinking here of the sorts
of absurdities mentioned by Plato at *Sophist* 244b-245e), so
pluralism suffers consequences that are, if anything, even more
absurd.


## 3. Socrates' Speech: The Theory of Forms 128e-130a




Socrates then objects to Zeno's argument by denying premise
(2). His lengthy objection depends on the theory of forms
articulated in Plato's middle period
dialogues.




One of the main principles of this theory is Causality
(*Phaedo* 100c4-6, 100d7-8, 100e5-6,
101b4-6, 101c4-5):



> 
> 
> (Causality) Things that are *F*
> (other than the *F*) are *F* by virtue of partaking of
> the *F*.




Another crucial principle is Separation (*Phaedo* 75c11-d2,
100b6-7; *Republic* 476b10, 480a11):



> 
> 
> (Separation) The *F* is itself by itself, at
> least in the sense of being separate from, and hence not identical
> with, the things that partake of it.




According to Separation, like things are like by partaking of a
separate form of likeness and unlike things are unlike by partaking of
a separate form of unlikeness. Although the properties of being
like and being unlike are contraries, they are not
contradictories. As Socrates emphasizes, it is possible for
sensible things to partake of both likeness and unlikeness, and hence
be both like and unlike.




Socrates does not explain how this is supposed to be possible, but
we can extract an explanation from what he says later in his speech
about the properties of being one and being many, and from what the
*Phaedo* tells us about the properties of being tall and
short. At *Parmenides* 129c, Socrates claims that he
himself is one (being one among the seven persons who are present) and
many (in having many parts: right/left, front/back, upper/lower).
At *Phaedo* 102b ff., Socrates points out that Simmias is taller
than Socrates (and hence tall), but that Simmias is also shorter than
Phaedo (and hence short). Thus, Simmias is both tall and
short. Clearly Socrates envisages the possibility of sensible
things being both like in one way and unlike in another. For
example, Socrates is like Plato (in that each is a philosopher) and
unlike Meletus (in that one, but not the other, is a poet). Being
like Plato, Socrates is like. Being unlike Meletus, Socrates is
unlike. Hence Socrates is both like and unlike. The general
principle here is Impurity-S:



> 
> 
> (Impurity-S) Sensible things are *impure*
> inasmuch as they can (and, in fact, often do) have contrary properties.
> 




It follows from Impurity-S that premise (2) of Zeno's argument
is false. But Socrates is willing to grant that Zeno is not
entirely mistaken. He insists that although it is far from
surprising to be told that sensible things have contrary properties, he
would be *astonished* to learn that *forms* have contrary
properties. Socrates, then, holds that forms are *pure*,
in the following sense:



> 
> 
> (Purity-F) Forms
> cannot have contrary properties.




According to Purity-F, not only is it the case that the one cannot
be both one and many and the like cannot be both like and unlike, but
it is also the case that the one cannot be both like and unlike and the
like cannot be both one and many. (Notice that Purity-F and
Impurity-S together entail Non-Identity, the claim that no form is
identical to any sensible thing. A similar form of argument
appears at *Phaedo* 74b-c, a passage in which Socrates argues
that whereas sensible things that are equal are also unequal, the equal
is not unequal, and hence the equal is not identical to any equal
sensible thing.)




Socrates' speech therefore articulates some of the basic elements
of the theory of forms, namely Causality and Separation, along with
Impurity-S and Purity-F. The theory to which Socrates alludes is
richer than his brief description of it suggests. As the dialogue
proceeds, the interlocutors invoke a number of additional middle period
principles. One principle that plays an important role in the
sequel is One-over-Many (see *Republic* 596a6-7):



> 
> 
> (One-over-Many) For any
> plurality of *F* things, there is a form of *F*-ness by
> virtue of partaking of which each member of the plurality is
> *F*.




It is from this principle that Plato infers, by means of the Third
Bed Argument (*Republic* 597c1-d3), the principle of
Uniqueness:



> 
> 
> (Uniqueness)
> For any property *F*, there is exactly one form of *F*-ness.




(Notice that the conjunction of One-over-Many and Uniqueness entails
Causality.) The sequel also depends on yet another important
middle period principle, Self-Predication:



> 
> 
> (Self-Predication) For
> any property *F*, the *F* is *F*.




For example, the beautiful is beautiful (*Euthydemus* 301b5,
*Cratylus* 439d5-6, *Hippias Major*
292e6-7). It is clear that the point generalizes to all
properties, including the property of being large (see *Phaedo*
100c4-6, 102e5-6). (Notice that Purity-F and
Self-Predication together entail the principle that the *F*
cannot possess the property that is contrary to *F*, instances
of which appear throughout the middle dialogues (*Hippias
Major* 291d1-3, *Phaedo* 102e5-6) and in
Socrates' speech (*Parmenides* 129b1-3, 129b6-c1).)
And finally, the sequel also repeatedly refers to the principle of
Oneness:



> 
> 
> (Oneness) Each form is
> one.




Oneness and Uniqueness are different principles: to say that a form
is *one* is to say not that it is *unique*, but rather
that it is something that can be counted. For Plato infers the
oneness of the beautiful from the fact that the beautiful and the ugly
are two, the oneness of the large from the fact that the large and the
small are two, and so on (*Republic* 475e9-476a6,
524b3-9). Thus the sense in which each form is one is similar to
the sense in which Socrates is one, in being one among many
(*Parmenides* 129c8-d2).


## 4. Problems for the Theory of Forms 130a-134e




At the conclusion of Socrates' speech, Parmenides articulates
six different lines of criticism directed at the theory of forms.


### 4.1 The Extent of the Forms 130a-e




Parmenides begins by questioning Socrates' initial acceptance
of the claim that there is a separate form corresponding to
*every* predicate or property. Socrates expresses
confidence in the existence of separate forms of justice, beauty,
goodness, and every form of that sort, uncertainty about the existence
of separate forms of humanity, fire, and water, and outright skepticism
about the existence of separate forms for hair, mud, and
dirt.




It is unclear *why* Socrates finds himself in doubt about the
existence of forms for natural kinds (such as humans and water) and
stuffs or mixtures (such as hair and mud). For some commentators,
Socrates simply makes "a wrong admission" as a result of
his youth and inexperience (Allen 1997, 124; see also Sayre 1996,
74). After all, Plato alludes to a form of bee at *Meno* 72b-c,
a form of shuttle at *Cratylus* 389d, and forms of bed and of
table at *Republic* 596b. Although shuttles, tables, and beds
are artifacts, and hence perhaps relevantly different from natural
kinds, such as human beings and water, there seems no reason to think
that humans differ from bees in regard to whether they have
corresponding forms. However, it is difficult to understand why
*Plato* would pen a conversation in which a character who
embodies his own middle period theory would admit something he has no
good reason to admit.




One possibility (see Gill 1996, 22) is that Plato is alluding to the
middle period thesis that only certain types of properties summon the
understanding to think about forms. For example, in the
*Republic*, *Socrates* claims that whereas "the
soul isn't compelled to ask the understanding what a finger is,
since sight doesn't suggest to it that a finger is at the same
time the opposite of a finger" (523d2-6), the soul is compelled
to ask what the large is and what the small is when sight suggests to
it that the index finger is both big (relative to the pinky) and small
(relative to the middle finger). If forms were merely posited to
explain the compresence of contrary properties in sensible things, then
there would be no need to posit a form corresponding to properties
(such as water and dirt) that have no contraries. However, this
is unlikely to be the source of Socrates' worry here, for the
*Republic* passage does not discuss metaphysical reasons for
positing the existence of forms, but rather discusses the psychological
and epistemic question of what prompts the soul to think of forms that
have already been posited.




Another option (Rickless 2007, 54-55; see also Miller 1986, 46) is that
Plato means us to recognize a tension between Self-Predication and
Separation (or Non-Identity) in the theory of forms. On the one
hand, the fact that justice is just, beauty beautiful, and goodness
good does not suggest that justice, beauty, and goodness are concrete,
sensible things. That is, Self-Predication gives us no reason to
deny that justice, beauty, and goodness are separate forms, numerically
distinct from sensible things. By contrast, if there are forms
for human and mud, then Self-Predication requires that the human be a
human being and the mud be muddy. It is difficult to see how
human things and muddy things could be non-sensible. So
Self-Predication gives us at least some reason to deny that there is a
form for human and mud that is distinct from every sensible
thing. This interpretation meshes well with Socrates'
remark that hair, mud, and dirt "are in fact just what we
see" and that it is for this reason that "it's too
outlandish to think there is a form for them"
(*Parmenides* 130d3-5).


### 4.2 The Whole-Part Dilemma 130e-131e




After leading Socrates to worry about whether there is indeed a form
corresponding to every property, Parmenides derives a number of
absurdities from the result of combining the theory of forms with a
particular conception of the partaking relation, the Pie Model.
According to the Pie Model, participants literally get a share of the
forms of which they partake, in a way analogous to the way in which
those who partake of a pie literally get a share of the pie. The
Pie Model comes in two versions: according to the Whole Pie Model, for
*X* to partake of *Y* is for *X* to get the whole
of *Y* as its share of *Y* (i.e., for the whole
of *Y* to be in *X*); according to the Piece-of-Pie
Model, for *X* to partake of *Y* is for *X* to
get a (proper) part of *Y* as its share of *Y* (i.e.,
for a (proper) part of *Y* to be in *X*). What
Parmenides goes on to argue is that the theory of forms is internally
inconsistent on either version of the Pie Model.




Suppose, first, that partaking conforms to the Whole Pie Model. Now
imagine that there are at one time three sensible *F*
things, *A*, *B*, and *C*, each separate from
each of the others. According to Causality, each
of *A*, *B*, *C* is *F* by virtue of
partaking of the *F*, and hence each
of *A*, *B*, *C* partakes of
the *F*. Given the Whole Pie Model, it follows that
the *F* is, as a whole and at a single time, in each
of *A*, *B*, *C*. Thus, the whole of
the *F* is in *A*, the whole of the *F* is
in *B*, and the whole of the *F* is
in *C*. If *A*, *B*, and *C* are in
separate places, then Causality and the Whole Pie Model together
require that one and the same form be, as a whole, in separate places
at the same time. Parmenides concludes that, on this picture, the
relevant form would be "separate from itself"
(*Parmenides* 131b2).




On some interpretations (Meinwald 1991, 13-14; Allen 1997, 130;
Rickless 2007, 57-58), Plato thinks of the claim that a form is
separate from itself as an absurdity in itself. On other
interpretations (Teloh 1981, 155; Miller 1986, 48; Sayre 1996, 76),
Plato does not treat this result as absurd in itself. Absurdity
only arises when this result is combined with the further thought that
nothing that is separate from itself could be a single thing. (In
this case, the same form would have to be three things, rather than one
thing.) For the claim that the relevant form is not one
contradicts Oneness, the claim that every form is one.




Socrates tries to avoid the relevant absurdity, however it is
understood, by supposing that a form is like a day, in the following
sense: just as a day can be in many separate places at the same time
without being separate from itself, so a form can be in many separate
places at the same time without being separate from itself.
Parmenides does not think much of Socrates' suggestion. He
immediately counters that Socrates' day is like a sail: a day can
be in many separate places at the same time only inasmuch as different
parts of it are over the separate places, just as a sail can cover many
separate people only inasmuch as each person is covered by a different
part of the sail. Parmenides' point, then, is that the only
way to make sense of Socrates' day analogy is to reduce it to the
Piece-of-Pie Model, the very model against which Parmenides goes on to
argue.




A number of scholars have balked at Parmenides' assimilation of
the day analogy to the sail analogy (Cherniss 1932, 135; Peck 1953,
132; Crombie 1963, 330-331; Sprague 1967, 96; Miller 1986, 49-50; Sayre
1996, 76). They have assumed that, whether a day is thought of as
a time-interval or as the sun's rays (the light of day), it is in
fact possible for one and the same day to be in separate places at the
same time. However, it does not in fact make sense to suppose
that a time-interval is in separate places *at the same time*
(Rickless 2007, 58). And it is not in fact true that the same
packet of rays shines on the separate places bathed by the light of
day; rather, different packets of rays shine on different places
(Panagiotou 1987, 18). Moreover, it makes little sense to suppose
that Plato would introduce a way out of the dilemma he himself has
constructed without explicitly alerting his readers to that fact.
To suppose otherwise would be to defend a particularly esoteric reading
of Plato's intentions.




Having assimilated Socrates' day analogy to the Piece-of-Pie Model,
Parmenides turns to a criticism of this second version of the Pie
Model. Suppose, then, the same three sensible *F*
things--*A*, *B*, and *C*--in separate
places at the same time. According to Causality, *A*,
*B*, and *C* are *F* by virtue of partaking of
the *F*, and hence *A*, *B*, and *C*
partake of the *F*. Given the Piece-of-Pie Model, there is a
part of the *F* in each of *A*, *B*, and
*C*. If the same absurdity generated from the Whole Pie Model
is to be avoided, we must suppose that the part of the *F* that
is in *A* is numerically distinct from the part of the
*F* that is in *B* and from the part of the *F*
that is in *C*, and also that the part of the *F* that
is in *B* must be numerically distinct from the part of the
*F* that is in *C*. (Otherwise we would have the same
part of the *F* existing, as a whole, in separate places at the
same time; and hence we would have something that is separate from
itself.) Thus the *F* must have numerically distinct parts, and
must therefore be divided (or, at least, divisible). Parmenides
concludes from this that the *F* cannot be one, a conclusion
that clearly contradicts Oneness.




Although many commentators take it for granted that Parmenides'
conclusion follows from the principle that divided (or divisible)
things automatically lose their unity, this supposition makes little
sense in the wake of Socrates' speech. There Socrates
insisted that he himself is one (in being one among many) even though
he has many parts (front and back, upper and lower, and so on).
So Socrates does not suppose that it is true *in general* that a
thing with parts cannot be one. The hypothesis that makes most
sense of Socrates' admission at the end of Parmenides'
criticism of the Piece-of-Pie Model that the *F* cannot be one is that
Socrates is antecedently committed to Purity-F. For anything that
has many parts is *ipso facto* many (just as Socrates'
having many parts is sufficient for his being many), and yet, by
Purity-F, no form can have contrary properties. Given that the
property of being one and the property of being many are contraries, it
follows from Purity-F and the claim that the *F* is many that the *F*
cannot be one (Rickless 2007, 59-60).




The upshot of the Whole-Part Dilemma is that absurdity or
inconsistency follows from the theory of forms on either of the two
possible versions of the Pie Model conception of
partaking. If *X*'s partaking of *Y* amounts to the
whole of *Y* being in *X* (the Whole Pie model), then
Causality conjoined with the existence of sensible things in separate
places at the same time entails the absurd conclusion that forms are
separate from themselves; but if *X*'s partaking of *Y*
amounts to a part of *Y* being in *X* (the Piece-of-Pie
Model), then Causality and Purity-F (along with the claim that having
many parts is sufficient for being many and the claim that being one
and being many are contrary properties) are inconsistent with
Oneness.




At the conclusion of the Whole-Part Dilemma, Parmenides extracts four
more absurdities from the result of combining Causality with the
Piece-of-Pie model:


1. By Causality, every *F* thing (other than the *F*)
is *F* by partaking of the *F*. But, by the Piece-of-Pie
Model, for *X* to partake of *Y* is for *X* to
get a part of *Y*. Thus, every *F* thing (other than the
*F*) is *F* by getting a part of the *F*. Now
let *F* be the property of being large. In that case, every
large thing (other than the large) is large by getting a part of the
large. But, Parmenides assumes, if *X* is a part of *Y*,
then *X* is smaller than *Y* (and *Y* is larger
than *X*), and hence *X* is (in some way) small. So
every large thing (other than the large) is large by getting something
small. But this is absurd: as Socrates himself emphasizes
at *Phaedo* 101b, it would be monstrous to say that something
is made larger by something small. This is an instance of a general
claim we might call "No Causation by Contraries":



> 
> 
> (No Causation by Contraries) For any property *F*, nothing that
> is *F* could make something possess a property that is contrary
> to the property of being *F*.
2. The result of combining Causality with the Piece-of-Pie Model
entails that equal things (other than the equal) are equal by getting a
part of the equal. Given that any part of *X* must be smaller than
*X* (see above), it follows that equal things (other than the
equal) are equal by getting something that is smaller than the
equal. But, Parmenides assumes, if *X* is smaller
than *Y*, then *X* is unequal to *Y*, and
hence *X* is (in some way) unequal. So every equal thing (other
than the equal) is equal by getting something unequal. But, again by
No Causation by Contraries, this result is absurd: nothing that is
unequal could make something be equal.
3. The result of combining Causality with the Piece-of-Pie Model
entails that small things (other than the small) are small by getting
a part of the small. This result entails that if there are any small
things (as indeed there are), then the small must have parts. But
if *X* is a part of *Y*, then *Y* is larger
than *X* (see above), and hence *Y* is (in some way)
large. Consequently, the small must be large. But, by
Self-Predication, the small is small. So the small is both large and
small. But this result contradicts Purity-F, according to which the
small cannot have contrary properties, and hence cannot be both large
and small.
4. As before, the result of combining Causality with the
Piece-of-Pie Model entails that small things (other than the small)
are small by getting a part of the small. But, Parmenides assumes,
for *X* to get *Y* is just for *Y* to be added
to *X*. It follows that small things (other than the small) are
small by having a part of the small added to them. But this is absurd:
it is impossible to make something small by *adding* something
to it.




These four quick arguments show that the result of combining Causality
with the Piece-of-Pie Model does not sit well with other aspects of
the theory of forms, in particular No Causation by Contraries (1 and
2), and the conjunction of Purity-F and Self-Predication (3).


### 4.3 The Third Man Argument 132a-b




Plato never refers to any argument as the "Third
Man". The moniker derives from Aristotle, who in various
places (e.g., *Metaphysics* 990b17 = 1079a13, 1039a2;
*Sophistical Refutations* 178b36 ff.) discusses (something akin
to) the argument at *Parmenides* 132a-b in these terms.




Parmenides sets up the argument by pointing out that, according to the
theory of forms, Oneness is supposed to follow from
One-over-Many. (Some, e.g., Fine (1993, 204), claim that Plato
means the sentence "each form is one" to express
Uniqueness, not Oneness. But this is certainly not what a
relevantly similar sentence expresses at *Republic* 476a2-6 and
524b7-11--see above.) From the existence of a plurality of *F*
things and the fact that, for any such plurality *P*, there is a form of
*F*-ness by virtue of partaking of which each member
of *P* is *F*, it follows that there is one form
"over" the many members of *P* (in the sense of
being that by virtue of partaking of which each member of *P*
is *F*). And given that anything that is one "over"
many is (in some sense) one, it follows that any form that has
participants is one.




There is a vast literature on the Third Man argument, initiated by the
groundbreaking analysis of the reasoning in Vlastos (1954). (See,
among others, Sellars (1955), Vlastos (1955), Geach (1956), Vlastos
(1956), Cherniss (1957), Peck (1962), Moravcsik (1963), Strang (1963),
Vlastos (1969), Cohen (1971), Teloh and Louzecky (1972), Peterson
(1973), Goldstein and Mannick (1978), Mann (1979), Mates (1979),
Pickering (1981), Teloh (1981, 158-167), Waterlow (1982), Prior
(1985, 64-75), Curd (1986), Sharvy (1986), Penner (1987,
251-299), Scaltsas (1989), Malcolm (1991, 47-53), Meinwald
(1992), Scaltsas (1992), Fine (1993, 203-241), McCabe (1994,
84-87), Schweizer (1994), Frances (1996), Allen (1997,
152-167), Hunt (1997), Rickless (1998), Pelletier and Zalta
(2000), and Rickless (2007, 64-75).) Most commentators agree
that the reasoning relies on at least three principles: One-over-Many,
Self-Predication, and Non-Identity (about which more anon). (Allen
(1997, 163) accepts that the reasoning relies on the claim that the
large is large--an instance of Self-Predication, but denies that
the argument, when generalized to forms other than the large, relies
on Self-Predication.) They also agree that the reasoning generates an
infinite regress of forms of largeness, and that the argument could be
generalized to generate an infinite regress of forms corresponding to
any predicate. But commentators differ over why Plato takes the
regress to be vicious or problematic, and what Plato would have
recommended as a way of avoiding the absurdity generated by the
reasoning.




Parmenides generates the infinite regress as follows. Consider a
plurality of large things, *A*, *B*, and *C*. By
One-over-Many, there is a form of largeness (call it
"*L1*") by virtue of partaking of
which *A*, *B*, and *C* are large. By
Self-Predication, *L1* is large. So there is now a new
plurality of large things, *A*, *B*,
*C*, and *L1*. Thus, by One-over-Many, there is a form
of largeness (call it "*L2*") by virtue of
partaking of which *A*, *B*, *C*, and
*L1* are large. Hence *L1* partakes of *L2*. At this point,
Parmenides assumes something like the following Non-Identity
assumption:



> 
> 
> (Non-Identity)
> No form is identical to anything that partakes of it.




(Notice that Non-Identity follows directly from Separation.) From the
fact that *L1* partakes of *L2*, Non-Identity entails
that *L2* is numerically distinct from *L1*. Thus, there
must be at least two forms of largeness, *L1*
and *L2*. But this is not all. By Self-Predication, *L2*
is large. So there is now a new plurality of large
things, *A*, *B*, *C*, *L1*,
and *L2*. Thus, by One-over-Many, there is a form of largeness
(call it "*L3*") by virtue of partaking of
which *A*, *B*, *C*, *L1*, and *L2*
are large. Hence *L1* and *L2* both partake
of *L3*. But then, by Non-Identity, *L3* is numerically
distinct from both *L1* and *L2*. Thus, there must be at
least three forms of largeness, *L1*, *L2*,
and *L3*. Repetition of this reasoning, based on One-over-Many,
Self-Predication, and Non-Identity, then generates an infinite
hierarchy of forms of largeness, with each form partaking of every
form that lies above it in the hierarchy. (That is, for every m and n
such that m<n, *Lm* partakes of *Ln*.)




In what way does the existence of an infinite regress of forms
represent a problem for the theory of forms? One answer to this
question (see Vlastos (1954, 328, fn. 12; 1955), Goldstein and Mannick
(1978), Penner (1987, 279-282), and Fine (1993, 204)) is that
the nature of the problem is fundamentally epistemic. On this view,
the theory of forms includes the thesis that, for any
property *F*, the primary function of the *F* is to
explain the *F*-ness of *F* things, and hence to make it
possible for humans to apprehend and know things as
*F*. But, so the story goes, Plato assumes that an infinite
regress of forms of *F*-ness, each of which explains
the *F*-ness of the forms of
*F*-ness below it in the hierarchy, cannot explain the *F*-ness of the
original plurality of *F* things: explanation must come to an end
somewhere. Although this interpretation makes sense of the
epistemic language that Plato sprinkles throughout the Third Man
passage, it does not make sense of the fact that Parmenides sets up the
argument by pointing out that the oneness of the large follows from its
being one "over" many large things (see above). So it
is unlikely that the epistemic reading of the Third Man is what Plato
had in mind.




Other scholars claim, quite correctly, that the existence of infinitely
many forms (indeed, the existence of so much as two forms)
corresponding to any predicate is inconsistent with Uniqueness.
And, indeed, this result appears to be at least part of what the Third
Man argument is designed to uncover. But Plato seems to be
looking to establish more than this. For in the last sentence of
the relevant passage, Parmenides announces that the argument shows that
each form is no longer one, but infinitely many. Although most
commentators gloss this comment as the claim that there is no longer
one form per predicate, but rather infinitely many, this is not what
the sentence actually says. What the sentence suggests is that
the existence of infinitely many forms of largeness conflicts with
Oneness.




One way to make sense of this claim is by way of the following chain of
reasoning. As we've seen, One-over-Many, Self-Predication,
and Non-Identity together generate an infinite hierarchy of forms of
largeness, each of which partakes of the forms above it in the
hierarchy. Thus, *L1* partakes of infinitely many forms, *L2*
partakes of infinitely many forms, *L3* partakes of infinitely many
forms, and so on. Now there are passages in which Plato appears
to assume that forms are as many as the predicates that can be truly
applied to them (see *Philebus* 14c8-d3, and Rickless (2007,
71)). And if we assume that Parmenides is still working with the
Piece-of-Pie model of partaking, then the fact that a form partakes of
infinitely many forms entails that it has infinitely many parts, and
hence is itself infinitely many. So from the existence of an
infinite regress of forms and from what appear to be dialectically
appropriate assumptions, it is possible to argue that each form in the
hierarchy is infinitely many. Given that the property of being
one and the property of being many are contraries, it then follows
directly from Purity-F that each form in the hierarchy is not
one. This interpretation explains why Parmenides announces at the
end of the argument that each form is no longer one, but infinitely
many (see Rickless (2007, 64-75)).




Many commentators think that the fundamental inconsistency revealed by
the Third Man argument rests with the combination of One-over-Many,
Self-Predication, and Non-Identity. For them, the Third Man
requires that Plato give up at least one of these principles. But
on the interpretation that best explains the set-up and final sentence
of the passage, Plato need not give up any of these principles in order
to avoid inconsistency: he can simply abandon Purity-F (and perhaps
also Uniqueness).


### 4.4 Forms as Thoughts 132b-c




At the conclusion of the Third Man argument, Socrates suggests that
it might be possible to avoid all previous inconsistencies at the heart
of the theory of forms by supposing that forms are thoughts that reside
only in minds. In what appears to be a severely truncated
argument, Parmenides provides two sets of reasons for thinking that
this suggestion will not avoid absurdity either. (Allen (1997,
174) argues that Parmenides only provides a single argument here, one
that most would identify as the second of two.)




Parmenides' first argument appears to have the following
structure. First, all thoughts have intentional objects: every thought
is *of* something rather than nothing. Second, the object of
any thought *T* is something that *T* thinks to be one
over all the instances. But anything that is thought to be one over
all the instances is a form. Parmenides concludes that the intentional
object of every thought is a form, and hence if every form is a
thought then every form is a thought of a form. Although Parmenides
does not make this explicit, it is plain that if every form is
numerically distinct from the form of which it is the intentional
object, then (thanks to Self-Predication and Non-Identity) an infinite
regress of forms beckons (see Rickless (2007, 75-79), and also
Gill (1996, 40) and Sayre (1996, 84)). Again, there is nothing to
suggest that Plato finds the existence of an infinite regress
problematic in itself. Rather, the existence of a regress threatens
Uniqueness, and, when combined in the appropriate way with Purity-F,
threatens Oneness. The reasoning that leads to conflict with Oneness
is parallel to the relevantly similar portion of the Third Man
argument (see section 4.3 above, and Rickless (2007, 79-80) for
details).




As if this weren't bad enough, Parmenides goes on to derive a
further absurdity from the result of combining the proposal that forms
are thoughts with the Pie Model conception of partaking. Assuming
that thoughts do not have parts, the only way for an object to partake
of a thought in accordance with the Pie Model is for the object to get
the thought as a whole. So if forms are thoughts, then according
to the Pie Model everything is composed of thoughts, and hence all
things think. But, Parmenides assumes, this panpsychist thesis is
absurd. Parmenides considers a way of avoiding this absurdity
that depends on assuming that something's having a thought as a
part does not entail that it is a thinking thing. But, argues
Parmenides, the only way to make sense of this proposal is to assume
that thoughts are unthinking, an assumption that is also absurd in
itself.


### 4.5 The Likeness Regress 132c-133a




At the conclusion of Parmenides' criticism of Socrates'
suggestion that forms might be thoughts, Socrates tries a completely
different tack: he suggests that forms are patterns set in nature
(*paradeigmata*) and that partaking of a form amounts to being
like it (call the combination of these claims
"Paradigmatism"). Paradigmatism is incompatible both
with the proposal that forms are thoughts and with the Pie Model
conception of partaking. The idea that forms are patterns that
serve as models for their participants is not new, for it appears in
various places in the dialogues of Plato's middle period (see
*Republic* 472b7-c7, 510a ff., 597a4-5,
596b6-8--and also
*Timaeus* 29c1-2, 48e5-49a1, 50c4-6).




Most commentators agree that Parmenides' criticism of
Socrates' Paradigmatism depends (at least in part) on the
construction of an infinite regress. But scholars are divided
over the identity and structure of the regress: whereas some see the
reasoning as basically indistinguishable from the Third Man argument (e.g.,
Owen (1953), Vlastos (1954), Cherniss (1957), Hathaway (1973), Lee
(1973), Teloh (1981, 166), Spellman (1983), Prior (1985, 71-75), and
Fine (1993, 211-215)), others see the reasoning as generating a regress
of forms of likeness (e.g., McCabe (1994, 87-89), Schofield (1996),
Allen (1997, 179-186), and Rickless (2007, 80-85)). (The main
textual and thematic reasons for preferring the latter reading to the
former are clearly described in Schofield (1996)--see also Gill
(1996, 44-45). For a rejoinder to Schofield, see Scolnicov (2003,
67-68).)




On the first view, the regress arises as follows. Consider a plurality
of *F* things, *A*, *B*, and *C*. By
One-over-Many, each of *A*,
*B*, *C* is *F* by virtue of partaking of a form
of *F*-ness (say, *F1*). By
Self-Predication, *F1*
is *F*. Hence *A*, *B*, *C*
and *F1* are all *F*, and each is like the others in
being *F*. Now consider the new plurality of *F*
things, *A*, *B*, *C*, and *F1*. By
One-over-Many, each of
*A*, *B*, *C*, and *F1* is *F* by
virtue of partaking of a form of *F*-ness (say,
*F2*). By Non-Identity, *F2* is numerically distinct
from *F1*. By Self-Predication, *F2*
is *F*. Hence *A*, *B*, *C*, *F1*,
and *F2* are all
*F*, and each is like the others in being *F*. Now
consider the new plurality of *F*
things, *A*, *B*, *C*, *F1*,
and *F2*. By One-over-Many, each
of *A*, *B*, *C*, *F1*, and *F2*
is *F* by virtue of partaking of a form of
*F*-ness (say, *F3*). By Non-Identity, *F3* is
numerically distinct from both *F1* and *F2*. We
therefore have three distinct forms of
*F*-ness. Repetition of the same pattern of reasoning then
establishes the existence of an infinite regress of forms of
*F*-ness. This reasoning is homologous to the Third Man argument
inasmuch as both arguments rely in the very same way on One-over-Many,
Self-Predication, and Non-Identity.




On the second view, the regress arises differently. In particular, the
reasoning relies explicitly on Paradigmatism and on an assumption that
Parmenides emphasizes as he is setting up his criticism, namely that
the relation of likeness is symmetrical: if *X* is
like *Y*, then *Y* is like *X*
(*Parmenides* 132d5-7). Consider two things, *A*
and *B*, that both have the property of being *F*.
Given that there is a property that *A* and *B* both
share, it follows that
*A* is like *B* and that *B* is
like *A*. Thus, *A* is like something (and hence, in
some way, like) and *B* is like something (and hence, in some
way, like). But, by One-over-Many, *A* and *B* are both
like by virtue of partaking of a form of likeness
(say, *L1*). Now assume for *reductio* that something is
like *L1* or *L1* is like something. Clearly
if *L1* is like something, then *L1* is (in some way)
like. And if something is like *L1*, then, by symmetry of
likeness, *L1* is like it, and hence again *L1* is (in
some way) like. So, whether something is like *L1*
or *L1* is like something,
*L1* is like. Now, by One-over-Many, *L1* is like by
virtue of partaking of a form of likeness (say, *L2*), and
hence *L1* partakes of
*L2*. By Non-Identity, *L2* is numerically distinct
from *L1*. But also, by Paradigmatism, *L1* is
like *L2*, and hence, by symmetry of likeness, *L2* is
like *L1*. So *L2* is (in some way) like. By
One-over-Many, then, *L1* and *L2* are like by virtue of
partaking of a form of likeness (say, *L3*), and
hence *L1* and *L2* both partake of
*L3*. By Non-Identity, *L3* is numerically distinct from
both *L1* and
*L2*. This gives us three distinct forms of likeness.
Repetition of the same pattern of reasoning then establishes the
existence of an infinite regress of forms of likeness. Taking for
granted that the existence of such a regress is in some way absurd or
problematic, Parmenides infers that the assumption for
*reductio* is false, i.e., that nothing is like *L1*
and *L1* is like nothing. But this result is itself
unacceptable. For we already have it that *A* and *B*
are like by virtue of partaking of *L1*, and hence
that *A* and *B* partake of *L1*. By
Paradigmatism, *A* and *B* are like *L1*, and
hence something is like *L1*. Moreover, by symmetry of
likeness, *L1* is like *A* and *B*, and
hence *L1* is like something. Thus the assumption
for *reductio* is true. Contradiction. Notice that, on the
second view, the reasoning leading to the relevant regress
is *not* homologous to the Third Man argument: instead of
deriving the claim that each form of likeness is like from
Self-Predication, Parmenides infers it from the conjunction of
Paradigmatism and the symmetry of likeness.




On either interpretation of the identity and structure of the relevant
regress, it is as yet unclear why Parmenides finds the regress
problematic. It is reasonable to assume that Parmenides'
reason for finding the likeness regress problematic is the same as his
reason for finding the largeness regress problematic in the Third Man
argument. The same three options canvassed in section 4.3 are
available. Some contend that the regress is epistemic and vicious
by its very nature, others that the regress conflicts with Uniqueness,
and yet others that the regress leads to the claim that each form in
the relevant infinite hierarchy is many, and hence, by Oneness and
Purity-F, both one and not one.


### 4.6 The Greatest Difficulty 133a-134e




At the conclusion of the Likeness Regress, Parmenides raises what he
characterizes as the greatest difficulty for the theory of forms.
This difficulty takes the form of two arguments, the first designed to
show that, if the forms are as Socrates has described them, they cannot
be known by human beings, the second designed to show that, if the
forms are as Socrates has described them, then the gods cannot know
human affairs. Both of these conclusions, if true, would be
devastating to the theory of forms. For, first, in the middle
dialogues, Plato takes for granted that humans can know at least some
forms (see *Meno* 76a6-7 and *Phaedo* 74b2-3) and
sketches a method (i.e., dialectic) that is designed to provide humans
with knowledge of the forms (*Republic* 534b3-c5); and, second,
as Socrates himself accepts, it would be "shocking" (134c4)
and "astonishing" (134d8) to be told that the gods cannot
know human affairs.




Commentators differ over the proper way to reconstruct and evaluate the
two arguments. With respect to the first argument, some scholars
(such as Lewis (1979)) claim that the argument is invalid, some (such
as Peterson (1981)) that there are two different valid ways of
reconstructing it, others (such as Yi and Bae (1998), and Rickless
(2007, 86-90)) that there is a single way to reconstruct the argument,
one on which it comes out valid. The second argument is usually
thought to be largely homologous to the first.




The first argument begins with the assumption (call it P1) that nothing
that is itself by itself is in (or among) humans (*Parmenides*
133c3-6). This assumption reflects a particular understanding of
what Separation requires, a conception that is emphasized in the
dialogues of the middle period (see *Symposium* 211a8-b1 and
*Timaeus* 52a1-3--for discussion, see Rickless (2007,
19-20)). Parmenides then adds, and provides instances of, two
further premises, P2 and P3:



> 
> 
> 
> (P2) If *X* is a form and *X* is what it is
> in relation to *Y*, then *Y* is a form.
> 
> 
> 
> 
> (P3) If *X* is in humans and *X* is what it
> is in relation to *Y*, then *Y* is in humans.
> 
> 
> 




For many commentators, P2 states that forms are related to other
forms, but not to sensible things, and P3 states that sensible things
are related to other sensible things, but not to forms (see, e.g.,
Ryle (1939), Cherniss (1944, 282 ff.), Chen (1944), Runciman (1959),
Schipper (1965, 15), Matthews (1972), Weingartner (1973,
185-187), Fujisawa (1974, 30 ff.), Shiner (1974, 24 and 31),
McCabe (1994, 90-94), Gill (1996, 45-48), Sayre (1996,
88-91), and Allen (1997, 193-203)). But there are good
reasons for thinking that this interpretation is incorrect (see
Forrester (1974), Lewis (1979), Peterson (1981), Yi and Bae (1998),
and Rickless (2007, 88)). Plato's formulation of P2 and P3
presupposes a distinction between two ways of being: relative being
and absolute being. Something has relative being if it is impossible
to describe its nature without mentioning something else to which it
is related. Something has absolute being if it does not have relative
being. The point of P2 is that it is in relation to another form that
any form with merely relative being is defined. The point of P3 is
that it is in relation to another sensible thing that any sensible
thing with merely relative being is defined. This reading is confirmed
by Parmenides' illustrations of P2 and P3. Mastery itself, he
says, is what it is in relation to slavery itself, but it is not the
case that mastery itself is what it is in relation to a human
slave. Similarly, slavery itself is what it is in relation to mastery
itself, but it is not the case that slavery itself is what it is in
relation to a human master. Moreover, a human master is what he is in
relation to a human slave, but it is not the case that a human master
is what he is in relation to slavery itself. Similarly, a human slave
is what he is in relation to a human master, but it is not the case
that a human slave is what he is in relation to mastery
itself. Parmenides then instantiates P3 using the example of
knowledge:



> 
> 
> (P3K) If *X* is knowledge in humans and *X* is what it is in
> relation to *Y*, then *Y* is in humans.




And finally, Parmenides assumes that knowledge has merely relative
being:



> 
> 
> (P4) Knowledge is what it is in relation
> to what it is knowledge of.




The reasoning to the first conclusion is straightforward. By P4,
knowledge is what it is in relation to what it is knowledge of.
Consequently, by P3K, if *X* is knowledge in humans, then the
object of *X* (i.e., what *X* is knowledge of) is in
humans. Now, according to Separation, every form is itself by
itself. But, by P1, nothing that is itself by itself is in
humans. Consequently, whatever is in humans is not a form. So
if *X* is knowledge in humans, then the object of *X* is
not a form. That is to say, no knowledge in humans (i.e., no knowledge
that humans have) has any form as its object. Thus, from Separation,
P1, P3K, and P4, it follows that humans do not know any forms.




The second argument begins with two assumptions: (i) that any knowledge
that is a form is more precise than any knowledge that is in humans,
and (ii) that the gods have any knowledge that is more precise than any
knowledge that is in humans. From these two assumptions, what
follows is:



> 
> 
> (P5) If *X* is a knowledge and *X* is a form,
> then the gods have *X*.




Parmenides then reasons as follows. By P2, if *X* is a form and
*X* is what it is in relation to *Y*, then *Y* is
a form. By P4, knowledge is what it is in relation to what it is
knowledge of. So P2 and P4 together entail P6:



> 
> 
> (P6) If *X* is a form and *X* is a knowledge
> of *Y*, then *Y* is a form.




Now, by P1, anything that is itself by itself is not in humans, and,
by Separation, every form is itself by itself. Hence P1 and
Separation entail that no form is in humans, i.e., that if *Y* is a form,
then *Y* is not in humans. This result, taken together with P6,
entails P7:



> 
> 
> (P7) If *X* is a form and *X* is a knowledge
> of *Y*, then *Y* is not in humans.




Parmenides then infers the following conclusion from the conjunction
of P5 and P7:



> 
> 
> (C) If *X* is a knowledge of *Y*
> and the gods have *X*, then *Y* is not in humans.




If this inference were valid, then Parmenides would have shown that
the object of any knowledge the gods have is not in humans, i.e., that
the gods do not know human affairs. However, C does not follow
validly from the conjunction of P7 and P5. Rather C follows
validly from the conjunction of P7 and P5\* (for details, see Rickless
(2007, 92)):



> 
> 
> (P5\*) If *X* is a knowledge and the gods have *X*, then *X*
> is a form.




There are three main possibilities here: (i) Plato simply missed the
fact that the second argument is invalid; (ii) Plato intended his
readers to recognize the argument as invalid; and (ii) Plato
unintentionally misstated P5\* as P5. (In any event, it is
interesting to note that, whereas P3 but not P2 functions as a premise
in the first argument, P2 but not P3 functions as a premise in the
second argument.)


## 5. How to Save the Forms: The Plan of the Deductions 134e-137c




After having articulated potentially devastating criticisms of the
theory of forms, one might expect Parmenides to conclude that the
theory is a lost cause and should be abandoned. But,
surprisingly, Parmenides does exactly the opposite. He claims,
rather, that one who does not "allow that for each thing there is
a character that is always the same" (a clear reference to
One-over-Many) will "destroy the power of dialectic
entirely" (135b8-c2). Here Parmenides means one of two
things, depending on whether "dialectic" is taken in a
technical sense (as meaning the process by which a philosopher is
supposed to acquire knowledge of the forms--see *Republic*
534b3-c5) or in a non-technical sense (as meaning the ability to
converse or communicate).




In any event, Parmenides makes it clear that the power of dialectic
(however this is understood) cannot be saved unless the forms
themselves are saved. As a means of saving the forms, Parmenides
recommends a process of training that focuses on forms and takes note
of the fact that forms wander (in the sense of having contrary
properties, such as being like and unlike: 135e1-7). In
particular, Parmenides suggests that the training process take the
following shape. First, concerning some form, it must involve
extracting consequences from the hypothesis that that form is; second,
concerning the very same form, it must involve extracting consequences
from the hypothesis that that form is not (135e8-136a2).
Parmenides goes on to say that it is also important to consider
different sorts of consequences: first, consequences for the form that
is hypothesized to be (or to not be), and second, consequences for
things other than the form that is hypothesized to be (or not to
be). Parmenides also says that the training process should
involve extracting consequences for the relevant form in relation to
itself and in relation to the others, and consequences for things other
than the relevant form in relation to themselves and in relation to the
relevant form.


 As most commentators agree, the arguments that occupy the second
part of the dialogue may be grouped into eight distinct stretches of
reasoning or Deductions, with an additional Appendix to the first two
Deductions: 155e4-157b5. (Most Neoplatonists take there to be
nine Deductions. Rangos (2014), building on remarks in Grote (1865)
and a seminar of Heidegger from 1930-31, argues that the Appendix is
not merely a ninth Deduction, but also plays a special role in the
overall scheme of Deductions.) There is some controversy over the
principles of division that Plato uses to generate the groupings.
Some, notably Meinwald (1991; 2014), Peterson (1996; 2000; 2003), and
Sayre (1996), argue that the division into eight Deductions should be
explained by the three principles of division announced by Parmenides
in his description of the method of training. On this (non-standard)
picture, the Deductions should be understood as aiming at the
following conclusions (where 'con-*F*' refers to the
property contrary to the property of being *F*):



> 
> 
> 
> |  |  |
> | --- | --- |
> | (D1) | If the *G* is, then the *G* is not *F* and
> not con-*F* in relation to itself.
>  |
> | (D2) | If the *G* is, then the *G* is *F* and
> con-*F* in relation to the others. |
> | (D3) | If the *G* is, then the others are *F* and
> con-*F* in relation to the *G*. |
> | (D4) | If the *G* is, then the others are not *F* and not
> con-*F* in relation to themselves. |
> | (D5) | If the *G* is not, then
> the *G* is *F* and con-*F* in relation to the
> others. |
> | (D6) | If the *G* is not, then
> the *G* is not *F* and not con-*F* in relation to
> itself. |
> | (D7) | If the *G* is not, then the
> others are *F* and con-*F* in relation to
> the *G*. |
> | (D8) | If the *G* is not, then the others are not *F* and
> not con-*F* in relation to themselves. |
> 
> 
> 




Others, including Miller (1986), Gill (1996; 2014), Allen (1997), and
Rickless (2007), do not agree with this way of representing the proper
way of generating eight Deductions. In particular, these scholars take
issue with the claim that the third principle of division concerns
whether the consequences for the relevant form (or for things other
than the form) are *relative to itself* or *relative to
things other than it*. As they see it, the third principle of
division concerns whether the consequences for the relevant form (or
for things other than the form) are *positive* or
*negative*. According to this standard picture, the Deductions
should be understood as aiming at the following conclusions:



> 
> 
> 
> |  |  |
> | --- | --- |
> | (D1) | If the *G* is, then the *G* is not *F* and
> not con-*F* (in relation to itself and in relation to the
> others). |
> | (D2) | If the *G* is, then the *G* is *F* and
> con-*F* (in relation to itself and in relation to the
> others). |
> | (D3) | If the *G* is, then the others are *F* and
> con-*F* (in relation to themselves and in relation to the
> *G*). |
> | (D4) | If the *G* is, then the others are not *F* and not
> con-*F* (in relation to themselves and in relation to the
> *G*). |
> | (D5) | If the *G* is not, then the *G* is *F* and
> con-*F* (in relation to itself and in relation to the
> others). |
> | (D6) | If the *G* is not, then the *G* is not *F*
> and not con-*F* (in relation to itself and in relation to the
> others). |
> | (D7) | If the *G* is not, then the others are *F* and
> con-*F* (in relation to themselves and in relation to the
> *G*). |
> | (D8) | If the
> *G* is not, then the others are not *F* and not
> con-*F* (in relation to themselves and in relation to the
> *G*). |
> 
> 
> 




(D7 represents something of an anomaly here, because many of the
conclusions actually derived in that Deduction are of the form: If the
*G* is not, then the others *appear to be* *F*
and con-*F*. See section 6.8 below.) Parmenides then offers to
engage in the training exercise himself, taking "one" as
the relevant instance of "*G*", and considering a
range of properties as instances of "*F*" (being
many, being a whole, being limited, having shape, being in oneself,
being in another, being in motion, being the same as oneself, being
the same as another, being like oneself, being like another, touching
oneself, touching another, being equal to oneself, being equal to
another, being (or coming to be) older than oneself, being (or coming
to be) older than another, being in time, being, being named or spoken
of, and being the object of an account, opinion, knowledge, or
perception).




One of the primary motivations for adopting the non-standard picture
is that the standard picture makes it difficult to understand the
second part of the dialogue as anything other than incoherent. The
problem is this. On the standard picture, D1 and D2 together appear to
entail that if the one is, then the one is *F* and is
not *F* (and the one is con-*F* and is not
con-*F*), and hence that it is not the case that the one
is. Similarly, D3 and D4 together appear to entail that if the one is,
then the others are *F* and not *F* (and the others are
con-*F* and not con-*F*), and hence again that it is not
the case that the one is. On the other hand, according to the same
picture, D5 and D6 together appear to entail that if the one is not,
then the one is *F* and is not *F* (and the one is
con-*F* and is not con-*F*), and hence that it is not
the case that the one is not. And similarly, D7 and D8 appear to
entail that if the one is not, then the others are (or at least appear
to be) *F* and not *F* (and the others are, or at least
appear to be, con-*F* and not con-*F*), and hence again
that it is not the case that the one is not. Putting all eight
Deductions together, the overall result on the standard picture is a
straightforward contradiction. One advantage of the non-standard
interpretation, then, is that it avoids reading the Deductions as an
extended argument for a necessary falsehood.




However, there are also good textual reasons to think that the
standard picture is superior to the non-standard proposal. For
example, in D2, Parmenides argues that if the one is, then the one is
both different and the same in relation to itself (147b7-8),
both like and unlike in relation to itself (148d3-4), and both
older and younger in relation to itself (152e2-3). But according
to the non-standard picture Parmenides should not be using D2 to argue
for consequences about the one *in relation to itself*; rather,
Parmenides should be using D2 to argue for consequences about the one
*in relation to the others*. (In reply to this sort of
criticism, Meinwald (1991, 46-75; 2014) and Sayre (1996, 114)
argue, though in different ways, that Plato uses the
*in-relation-to* qualifications in a technical, rather than
ordinary, sense. For criticisms of Meinwald's influential proposal,
see Gill (1996, 56, fn. 90; 2014), Sayre (1996, 110-113), and
Rickless (2007, 102-106).)




Another way out of the problem posed by the seeming incoherence of the
Deductions is to suppose that the subject of one Deduction is
numerically distinct from the subject of some of the other
Deductions. Multisubjectist interpretations of this kind have
been defended by the Neoplatonists (including Plotinus and Proclus),
Cornford (1939), Miller (1986), and Sayre (1996). One of the
major problems facing multisubjectism is the fact that Parmenides is
quite explicit about the fact that the subject form of each Deduction
is identical to the subject form of each of the other Deductions.
(For further criticisms of multisubjectism, see Meinwald (1991,
24-26). For particular criticisms of the Neoplatonist version of
multisubjectism, see Allen (1997, 211-215 and 218-224).)




The standard view (if there is one) is that the Deductions have an
aporetic purpose: their aim is to perplex, to set problems that must
be solved, either by rejecting some of the premises that lead to the
master contradiction or by finding fault with the relevant
reasoning. Aporetic interpretations of this sort have been defended by
Schofield (1977), Gill (1996; 2012) and Allen (1997), with additional
support provided by Patterson (1999). On this kind of view, Plato does
not commit to any particular way of solving the problems: the second
part of the dialogue is simply meant to serve as a challenge to the
reader.




Yet another alternative is that detailed logical analysis of the
Deductions reveals arguments sufficient to establish both that the one
is and, importantly, that principles such as Purity-F and Uniqueness
are false. As Rickless (2007, 136-137 and 211) argues, Purity-F
is a background premise of both D1 and D4. Taken together, then, D1
and D2 are sufficient to establish that a contradiction follows from
the hypothesis that the one is, *on the assumption that Purity-F is
true*. That is, D1 and D2 together entail that if the one is, then
Purity-F is false. But D5 and D6 together entail that the one is. It
then follows directly that Purity-F must be false. (Gill (2014) argues
that the result of D1 and D8, taken together, is that the one must be
acknowledged to be both one and many. She also thinks that Plato
means for his readers to recognize that the same acknowledgment is
needed to overcome the problematic results of D4.) Moreover, D2
contains an argument for the claim that if the one is, then there are
infinitely many forms of oneness. Given that the one is (thanks to D5
and D6), it follows directly that there must be more than one form
corresponding to the property of being one, and hence that Uniqueness
is false. (Rickless (2007, 238-239) also argues that the second
part of the dialogue provides sufficient reason to reject No Causation
by Contraries.)


## 6. The Deductions 137c-166c




Each Deduction consists of separate stretches of reasoning (call
them "Arguments") leading to a number of logically
interconnected results (call them "Conclusions"). The
summary to follow is governed by the following notational
conventions. Each Deduction receives a number ("D1"
for the first Deduction, "D2" for the second Deduction, and
so on) and each Argument within each Deduction receives a number
("A1" for the first Argument, "A2" for the
second Argument, and so on). (The Appendix to the first two
Deductions will be called "App".) If an Argument has
exactly one Conclusion, the single Conclusion will be referred to as
"C". If an Argument has more than one Conclusion, the
Conclusions will be numbered ("C1" for the first
Conclusion, "C2" for the second Conclusion, and so
on).


### 6.1 The First Deduction 137c-142a




The aim of D1 is to establish, for a variety of different properties
*F*, that if the one is, then the one is neither *F* nor con-*F*.
Within D1, there are eighteen separate Arguments with the following
Conclusions: If the one is, then the one is not many (D1A1C), the one
has no parts (D1A2C1), the one is not a whole (D1A2C2), the one has no
beginning, middle, or end (D1A3C), the one is unlimited (D1A4C), the
one has no shape (D1A5C), the one is not in another (D1A6C1), the one
is not in itself (D1A6C2), the one is nowhere (D1A6C3), the one is not
in motion (D1A7C), the one is not at rest (D1A8C), the one is not
different from itself (D1A9C1), the one is not the same as another
(D1A9C2), the one is not different from another (D1A10C), the one is
not the same as itself (D1A11C), the one is not like another or itself
(D1A12C), the one is not unlike itself or another (D1A13C), the one is
not equal to itself or another (D1A14C1), the one is not unequal to
itself or another (D1A14C2), the one is not the same age as itself or
another (D1A15C1), the one is neither younger nor older than itself or
another (D1A15C2), the one is not in time (D1A16C), the one neither
comes to be nor ceases to be (D1A17C1), the one does not partake of
being (D1A17C2), the one is not (D1A17C3), the one is not one
(D1A17C4), the one is not named or spoken of, nor is it the object of
an account, knowledge, perception, or opinion (D1A18C).




Most of the individual Arguments in D1 are logically
interconnected. The exceptions are D1A1C, D1A9C1, D1A9C2, D1A10C,
and D1A11C, which do not depend on any previously established
Conclusions. (For details, see Rickless (2007, 112-137).)




The Arguments of D1 also rest on a large number of independent
premises, among which we find the following: (i) the property of being
one and the property of being many are contraries; (ii) anything that
has parts is many; (iii) a whole is a thing with parts from which no
part is missing; (iv) the beginning, middle, and end of *X* are
parts of
*X*; (v) the beginning and end of *X* are the limits
of *X*; (vi) to be round is to have extremities that are
equidistant in every direction from the middle; (vii) to be straight
is to have a middle that stands in the way of the two extremities;
(viii) anything that has shape must be either round or straight; (ix)
if *X* is in *Y*, then *X* is contained all
around by
*Y* and *X* touches *Y* in many places with many
parts; (x) if *X* is contained all around by *Y*
and *X* touches *Y* in many places with many parts,
then *X* is round; (xi) if *X* touches *Y* in
many places with many parts, then *X* has parts; (xii)
if *X* both contains itself and is contained by itself,
then *X* is two; (xiii) if *X* is two, then *X*
is many; (xiv) whatever is neither in itself nor in another is
nowhere; (xv) if *X* is in motion, then *X* alters from
itself or moves spatially; (xvi) if *X* moves spatially,
then *X* either spins in a circle in the same location or
changes from one place to another.




It appears that every Argument of D1 other than D1A9 is logically
valid. (However, see Gill (1996, 81, n. 134), who contends that
D1A12 and D1A13 are fallacious.)) D1A9 seems to commit the
fallacy of equivocation (see Rickless (2007, 121-123). It also
appears that almost all the independent premises of D1 are, at least
from Plato's point of view, either definitionally true or
self-evident. There are three exceptions to this claim. D1A10
depends on the questionable premise that if *X* is not different by
itself, then *X* itself is not different in any way (see Rickless (2007,
123-124)), and D1A11 depends on the questionable premise that
*if* the nature of the *F* is not identical to the
nature of the
*G*, *then* if *X* is *G* relative to
itself, *X* is not *F* relative to itself (see Rickless
(2007, 124-126)).




But the most interesting exception is D1A1. In D1A1,
Parmenides argues for D1A1C: that if the one is, then the one is not
many. The reasoning is simple. From Oneness or
Self-Predication, it follows that the one is one. Consequently,
if the one is, then the one is one. But the property of being one
and the property of being many are contraries, and, by Purity-F, no
form can have contrary properties. Given that the one is a form,
it follows that if the one is, then the one is not many. The
non-obvious premises here are Oneness (or Self-Predication) and
Purity-F. Whereas some might argue that D1A1 is unsound because
Oneness (or Self-Predication) is false, others might argue that the
very point of D1A1 is to show that if the one is and is many, then
Purity-F is false.


### 6.2 The Second Deduction 142b-155e




The aim of D2 is to establish, for a variety of different properties
*F*, that if the one is, then the one is both *F* and
con-*F*. Within D2 there are thirty-three separate Arguments
with the following Conclusions: If the one is, then the one partakes
of being (D2A1C1), the one is not the same as being (D2A1C2), the one
is a whole (D2A2C1), being and the one are parts of the one (D2A2C2),
the one is infinitely many (D2A3C and D2A5C), the different is not the
same as the one (D2A4C1), the different is not the same as being
(D2A4C2), the one has parts (D2A6C1), the one is a whole (D2A6C2), the
one is limited (D2A6C3), the one is unlimited (D2A6C4), the one has a
beginning, a middle, and an end (D2A7C1), the one has shape (D2A7C2),
the one is in itself (D2A8C1), the one is not nowhere (D2A8C2), the
one is in another (D2A8C3), the one is at rest (D2A9C1), the one is in
motion (D2A9C2), the one is the same as itself (D2A10C1), the one is
different from itself (D2A10C2), the one is different from the others
(D2A11C), the one is the same as the others (D2A12C), the one is like
the others (D2A13C and D2A15C1), the one is unlike the others (D2A14C
and D2A15C2), the one is like itself (D2A16C1), the one is unlike
itself (D2A16C2), the one touches itself (D2A17C1), the one touches
the others (D2A17C2), the one does not touch itself (D2A18C), the one
does not touch the others (D2A19C), the one is equal to itself
(D2A20C1), the one is equal to the others (D2A20C2), the one is both
greater than and less than itself (D2A21C1), the one is unequal to
itself (D2A21C2), the one is both greater than and less than the
others (D2A22C1), the one is unequal to the others (D2A22C2), the one
is more than, less than, and equal to itself in number (D2A23C1), the
one is more than, less than, and equal to the others in number
(D2A23C2), the one partakes of time (D2A24C), the one comes to be
older than itself (D2A25C1), the one comes to be younger than itself
(D2A25C2), the one always is older than itself (D2A26C1), the one
always is younger than itself (D2A26C2), the one is the same age as
itself (D2A27C1), the one is neither older nor younger than itself
(D2A27C2), the one neither comes to be older nor comes to be younger
than itself (D2A27C3), the one is older than the others (D2A28C), the
one is younger than the others (D2A29C), the one is the same age as
the others (D2A30C1), the one is neither older nor younger than the
others (D2A30C2), the one neither comes to be older nor comes to be
younger than the others (D2A31C), the one comes to be younger than the
others (D2A32C1), the one comes to be older than the others (D2A32C2),
the one partakes of time past, future, and present (D2A33C1), the one
is and comes to be, was and was coming to be, and will be and will be
coming to be (D2A33C2), and the one could be named and spoken of, as
well as be the object of an account, knowledge, perception, and
opinion (D2A33C3).




Most of the individual Arguments in D2 are logically
interconnected. The exceptions are D2A1C1, D2A1C2, D2A8C2,
D2A10C1, D2A12C, D2A18C, D2A19C, D2A20C1, D2A20C2, and D2A24C, which do
not depend on any previously established conclusions. (For
details, see Rickless (2007, 138-187).)




It appears that every Argument of D2 is logically valid.
(However, see Gill (1996, 64, n. 107), who contends that D2A8 and D2A9
are fallacious, and Patterson (1999, 98-100), who argues that D2A8 is
fallacious.)) Like the Arguments of D1, the Arguments of D2 rest
on a large number of independent premises, including many of the
independent premises of D1. Again, most of these premises are, at
least from Plato's point of view, either definitionally true or
self-evident. (Gill (1996, 83-84) objects to one of the premises
of D2A14 and Patterson objects to one of the premises of D2A28, but
Rickless (2007, 160 and 175) claims that it is unclear whether Plato
himself would have found these premises problematic.) There are
three exceptions to this claim: D2A4, D2A12, and D2A14. D2A4
depends on Causality and No Causation by Contraries, D2A12 depends on
No Causation by Contraries, and D2A14C depends on D2A12C. But
there are reasons for thinking that No Causation by Contraries is
false. When combined with Causality, No Causation by Contraries
entails that, for any property *F*, the *F* cannot be con-*F*. But,
*assuming that the one is*, this contradicts D2A3C, namely that
if the one is, then the one is (infinitely) many. So, if
Causality is true and the one is, then No Causation by Contraries must
be false. This result is important, for the falsity of No
Causation by Contraries would enable Plato to dispatch at least some of
Parmenides' earlier criticisms of the theory of forms, notably
the first two criticisms of the result of combining the theory of forms
with the Piece-of-Pie Model conception of partaking--see the end
of section 4.2 above.




Notice that D1 and D2 together entail that if the one is, then
Purity-F is false. To see why, consider the following. D1 establishes
that if Purity-F is true, then if the one is, then the one is
neither *F* nor con-*F*; and D2 establishes that if the
one is, then the one is both *F* and con-*F*. So if both
D1 and D2 are valid and based on acceptable premises (other than
Purity-F), then D1 and D2 together establish that if Purity-F is true
and the one is, then the one has
*contradictory* properties. Given that it is impossible
for something to have *contradictory* properties, it is a direct
consequence of the conjunction of D1 and D2 that if the one is, then
Purity-F is false.


### 6.3 The Appendix to the First Two Deductions 155e-157b




The function of the Appendix is to show that the Conclusions of D1 and
D2 together entail that, for a range of properties *F*, if the
one is, then there is a moment outside of time (the so-called
"instant") at which the one changes from being *F*
to being con-*F*. Within the Appendix, there are five Arguments
with the following Conclusions: If the one is, then there are
times *T1* and *T2* such that *T1* is distinct
from *T2* and the one partakes of being at *T1* and the
one does not partake of being at *T2* (AppA1C), there is a
definite time at which the one comes to be (AppA2C1), there is a
definite time at which the one ceases to be (AppA2C2), there is a time
at which the one ceases to be many (AppA3C1), there is a time at which
the one ceases to be one (AppA3C2), there is a time at which the one
is combined (AppA3C3), there is a time at which the one is separated
(AppA3C4), there is a time at which the one is made like (AppA3C5),
there is a time at which the one is made unlike (AppA3C6), there is a
time at which the one is increased (AppA3C7), there is a time at which
the one is decreased (AppA3C8), there is a time at which the one is
made equal (AppA3C9), there is something (call it "the
instant") (i) that is in no time at all and (ii) at which the
one changes both from being in motion to being at rest and from being
at rest to being in motion and (iii) at which the one is neither at
rest nor in motion (AppA4C), there is something (call it "the
instant") (i) that is in no time at all and (ii) at which the
one changes both from not-being to being and from being to not-being
and (iii) at which the one neither is nor is not (AppA5C1), there is
something (call it "the instant") (i) that is in no time
at all and (ii) at which the one changes both from being one to being
many and from being many to being one and (iii) at which the one is
neither one nor many (AppA5C2), there is something (call it "the
instant") (i) that is in no time at all and (ii) at which the
one changes both from being like to being unlike and from being unlike
to being like and (iii) at which the one is neither like nor unlike
(AppA5C3), and there is something (call it "the instant")
(i) that is in no time at all and (ii) at which the one changes both
from being small to being large and from being large to being small
and (iii) at which the one is neither large nor small (AppA5C4).




All of the individual Arguments within the Appendix are logically
interconnected. (For details, see Rickless (2007, 189-198).)




It appears that every Argument of the Appendix is logically
valid. Like the Arguments of D1 and D2, the Arguments of the
Appendix rest on a number of independent premises, including premises
of D1 and D2. Again, most of the premises are, at least from
Plato's point of view, either definitionally true or
self-evident. (Gill (1996, 86) objects to one of the premises of
AppA5. For a rejoinder, see Rickless (2007, 195, n. 2).)
There is one major exception to this claim, however. All of the
Arguments of the Appendix other than AppA1 depend for their soundness
on the soundness of AppA1. But AppA1 depends for its soundness on
the soundness of D1A17, which itself depends for its soundness on the
truth of Purity-F. So if Purity-F were false, then all the
Arguments of the Appendix would be unsound.


### 6.4 The Third Deduction 157b-159b




The aim of D3 is to establish, for a variety of different
properties *F*, that if the one is, then the others are
both *F* and con-*F*. Within D3, there are seven
Arguments with the following Conclusions: If the one is, then the
others are not the one (D3A1C), the others have parts (D3A2C1), the
others are a whole (D3A2C2 and D3A3C2), the others are one (D3A2C3 and
D3A3C1), the whole and the part of the others are many (D3A4C), the
whole and the part of the others are unlimited in multitude (D3A5C1),
the whole and the part of the others are unlimited (D3A5C2), the whole
and the part of the others are limited (D3A6C), each of the others is
like itself (D3A7C1), each of the others is like each of the others
other than itself (D3A7C2), each of the others is unlike itself
(D3A7C3), and each of the others is unlike each of the others other
than itself (D3A7C4). There is also the promise of a number of
Arguments establishing results of the form: If the one is, then the
others are both *F* and con-*F*.




Most of the individual Arguments in D3 are logically interconnected,
and connected to Arguments within previous Deductions. The only
exception is D3A2C1, which does not depend on any previously
established Conclusions. (For details, see Rickless (2007,
198-206).)




It appears that every argument of D3 is logically valid. Like the
Arguments of D1 and D2, the Arguments of D3 rest on a number of
independent premises, including premises of D1 and D2. Again, it
appears that the premises are, at least from Plato's point of
view, either definitionally true or self-evident. Thus it appears
that, from Plato's point of view, D3 establishes, without
reliance on Purity-F or any other potentially problematic assumption,
that if the one is, then the others have a host of contrary
properties.


### 6.5 The Fourth Deduction 159b-160b




The aim of D4 is to establish, for a variety of different properties
*F*, that if the one is, then the others are neither *F*
nor con-*F*. Within D4, there are four Arguments with the
following Conclusions: If the one is, then the others are not one
(D4A1C), the others are not many (D4A2C1), the others are not a whole
(D4A2C2), the others do not have parts (D4A2C3), the others are not
like (D4A3C1), the others are not unlike (D4A3C2), the others are not
both like and unlike (D4A3C3), the others are not the same (D4A4C1),
the others are not different (D4A4C2), the others are not in motion
(D4A4C3), the others are not at rest (D4A4C4), the others are not
coming to be (D4A4C5), the others are not ceasing to be (D4A4C6), the
others are not greater (D4A4C7), the others are not equal (D4A4C8),
and the others are not less (D4A4C9).




All of the individual Arguments in D4 are logically interconnected, and
connected to Arguments within previous Deductions. (For details,
see Rickless (2007, 207-211).)




It appears that all of the Arguments in D4 are valid. Like the
Arguments of the first three Deductions, the Arguments of D4 rest on a
number of independent premises, including premises of D1 and D2 (but
not D3). Again, it appears that most of the premises are, at
least from Plato's point of view, either definitionally true or
self-evident. There is one major exception to this claim,
however. All of the Arguments of D4 depend for their soundness on
the soundness of D1A2. But D1A2 depends for its soundness on the
soundness of D1A1, which itself depends for its soundness on the truth
of Purity-F.




So we can redescribe the Conclusion of D4 as follows: *If Purity-F
is true*, then if the one is, then the others do not have a host
of contrary properties. This means that D3 and D4 together entail that
if the one is, then Purity-F is false. For, by D3, if the one is, then
the others are both *F* and con-*F*. But, by D4, if
Purity-F is true and the one is, then the others are
neither *F* nor con-*F*. So, if Purity-F is true and the
one is, then the others have a host of *contradictory*
properties. Given that it is impossible for anything to have
contradictory properties, it follows directly that if the one is, then
Purity-F is false.


### 6.6 The Fifth Deduction 160b-163b




The aim of D5 is to establish, for a variety of different properties
*F*, that if the one is not, then the one is both *F* and con-*F*.
Within D5, there are twelve Arguments with the following Conclusions:
If the one is not, then the one is different from the others (D5A1C1),
we have knowledge of the one (D5A1C2), the one is different in kind
from the others (D5A2C), the one partakes of *something*,
*that*, and *this* (D5A3C), the one is unlike the others
(D5A4C1), the others are unlike the one (D5A4C2), the one partakes of
the unlike (i.e., has unlikeness) in relation to the others (D5A4C3),
the one partakes of the like in relation to itself (D5A5C1), the one is
like itself (D5A5C2), the one is unequal to the others (D5A6C1), the
others are unequal to the one (D5A6C2), the one partakes of the unequal
in relation to the others (D5A6C3), the one partakes of the large
(D5A7C1), the one partakes of the small (D5A7C2), the one partakes of
the equal (D5A7C3), the one partakes of being (D5A8C1), the one
partakes of not-being (D5A8C2), the one is in motion (D5A9C), the one
is not in motion (D5A10C1), the one is at rest (D5A10C2), the one is
altered (D5A11C1), the one is not altered (D5A11C2), the one comes to
be (D5A12C1), the one ceases to be (D5A12C2), the one does not come to
be (D5A12C3), and the one does not cease to be (D5A12C4).





Most of the individual Arguments in D5 are logically interconnected,
and connected to Arguments within previous Deductions. The
exceptions are D5A1C1, D5A1C2, D5A3C, D5A5C1, D5A8C1, D5A8C2, and
D5A10C1, which do not depend on any previously established
Conclusions. (For details, see Rickless (2007, 212-223).)




It appears that all of the Arguments in D5 are valid. Like the
Arguments of the first four Deductions, the Arguments of D5 rest on a
number of independent premises, including premises of D1, D2, and the
Appendix (but not D3 or D4). Again, it appears that most of the
premises are, at least from Plato's point of view, either
definitionally true or self-evident. Thus it appears that, from
Plato's point of view, D5 establishes, without reliance on
Purity-F or any other potentially problematic assumption, that if the
one is not, then the one has a host of contrary properties. In
other words, as Plato sees it, D5 establishes that if the one is not,
then Purity-F is false.




When combined with the results of D1 and D2 (or, alternatively, with
the results of D3 and D4), this result entails that Purity-F is in fact
false. To see this, recall that D1 and D2 (as well as D3 and D4)
together entail that if the one is, then Purity-F is false. Now,
by D5, if the one is *not*, then Purity-F is false. So,
*whether the one is or is not*, Purity-F is false. So
Purity-F is false.


### 6.7 The Sixth Deduction 163b-164b




The aim of D6 is to establish, for a variety of different properties
*F*, that if the one is not, then the one is neither *F*
nor con-*F*. Within D6, there are four Arguments with the
following Conclusions: If the one is not, then the one in no way is
(D6A1C1), the one in no way partakes of being (D6A1C2), the one in no
way comes to be (D6A2C1), the one in no way ceases to be (D6A2C2), the
one is not altered in any way (D6A2C3), the one is not in motion
(D6A2C4), the one is not at rest (D6A2C5), the one does not partake of
the small (D6A3C1), the one does not partake of the large (D6A3C2),
the one does not partake of the equal (D6A3C3), the one does not
partake of the like (D6A3C4), the one does not partake of the
different (D6A3C5), the others are not like the one (D6A4C1), the
others are not unlike the one (D6A4C2), the others are not the same as
the one (D6A4C3), the others are not different from the one (D6A4C4),
none of the following (namely, *of that*,
*to that*, *something*, *this*, *of this*,
*of another*, *to another*, time past, time future, time
present, knowledge, perception, opinion, account, and name) is
applicable to the one (D6A4C5), and the one is in no state at all
(D6A4C6).




Most of the individual Arguments in D6 are logically interconnected,
and connected to Arguments within previous Deductions. The
exceptions are D6A1C1 and D6A4C5, which do not depend on any previously
established Conclusions. (For details, see Rickless (2007,
223-228).)




It appears that all of the Arguments in D6 are valid. Like the
Arguments of the first five Deductions, the Arguments of D6 rest on a
number of independent premises, including premises of D5 (but not D1,
D2, D3, D4, or the Appendix). Again, it appears that most of the
premises are, at least from Plato's point of view, either
definitionally true or just plain obvious. Thus it appears that,
from Plato's point of view, D6 establishes, without reliance on
Purity-F or any other potentially problematic assumption, that if the
one is not, then the one has neither of a host of contrary
properties.




When combined with the results of D5, this result entails that the one
is. To see this, consider the following. D5 establishes that if the
one is not, then the one is both *F* and con-*F*; and D6
establishes that if the one is not, then the one is neither *F* nor
con-*F*. So if both D5 and D6 are valid and based on acceptable
premises, then D5 and D6 together establish that if the one is not,
then the one has *contradictory* properties. Given that it is
impossible for something to have *contradictory* properties, it
is a direct consequence of the conjunction of D5 and D6 that it is not
the case that the one is not. That is to say, D5 and D6 together
entail that the one is.




This is a significant result, for two reasons. First, it
reinforces the earlier result obtained from the conjunction of D1, D2,
and D5 (and from the conjunction of D3, D4, and D5), namely that
Purity-F is false. For D1 and D2 (as well as D3 and D4) together
entail that if the one is, then Purity-F is false. But D6
establishes that the one is. It then follows directly by modus
ponens that Purity-F is false. Second, the result that the one is
can be generalized to establish the being of any form whatever.
The reason is that none of the reasoning in the Deductions up to this
point depends on the one's having been chosen as the main subject
of the Deductions. Every Argument of D1-D6 would go through if
some other form were substituted for the one as the main topic of
discussion. Earlier, Parmenides had said that "only a very
gifted man can come to know that for each thing there is some kind, a
being itself by itself" (135a-b). So he has now revealed
himself to be the "very gifted man" of whom he had
spoken.


### 6.8 The Seventh Deduction 164b-165e




The aim of D7 is to establish, for a variety of different properties
*F*, that if the one is not, then the others are
both *F* and con-*F*. Within D7, there are six
Arguments with the following Conclusions: If the one is not, then the
others are (D7A1C1), the others are other (D7A1C2), the others are
different (D7A1C3), the others are other than each other (D7A1C4), the
others are infinitely many (D7A2C), each of the others appears to be
one (D7A3C1), each of the others is not one (D7A3C2), the others
appear to be infinitely many (D7A3C3), some of the others appear to be
even, others odd (D7A3C4), none of the others is either even or odd
(D7A3C5), among the others there appears to be a smallest (D7A4C1),
each of the others (even the other that appears smallest) appears
large in relation to its parts (D7A4C2), each of the others appears to
come to the equal (D7A4C3), each of the others appears to have no
beginning, middle, or end in relation to itself (D7A5C1), each of the
others appears unlimited in relation to itself (D7A5C2), each of the
others appears limited in relation to another (D7A5C3), each of the
others appears to be like itself and each of the others (D7A6C1), and
each of the others appears to be unlike itself and each of the others
(D7A6C2).




Most of the individual Arguments in D7 are logically interconnected,
and connected to Arguments within previous Deductions. The
exceptions are D7A1C1 and D7A3C5, which do not depend on any previously
established Conclusions. (For details, see Rickless (2007,
228-236).)




It appears that all of the Arguments in D7 are valid. Like the
Arguments of the first six Deductions, the Arguments of D7 rest on a
number of independent premises, including premises of every Deduction
other than D4 and the Appendix. Again, it appears that most of
the premises are, at least from Plato's point of view, either
definitionally true or just plain obvious. Thus it appears that,
from Plato's point of view, D7 establishes, without reliance on
Purity-F or any other potentially problematic assumption, that if the
one is not, then the others appear to have a host of contrary
properties. (Thus D7 does not conform exactly to
Parmenides' earlier description of the method of the
Deductions. What we would expect from that description is not
that D7 would establish that if the one is not then the others
*appear to have* contrary properties, but rather that D7 would
establish that if the one is not then the others *actually have*
contrary properties. This discrepancy remains something of a
mystery.)




Still, it is possible to extract an interesting result from D7, when
combined with a result from D3. D7A1 establishes that if the one is
not, then the others are (D7A1C1). But D3 establishes (for
many *F*'s) that if the one is, then the others
are *F*. But to be *F* is to be in some way or other. So
if the others are *F*, then the others are. So D3 establishes
(for many *F*'s) that if the one is, then the others are. Thus,
D3 and D7A1C1 together entail that, whether the one is or is not, the
others are. Hence D3 and D7 together entail that the others
are. Assuming that the others are (or, at least, include) all the
forms other than the one, it follows, in conjunction with the
previously established result that the one is (see section 6.7 above),
that *every form is*. This reinforces the previous claim (see
section 6.7 again) that the result that the one is can be generalized
to all the forms.


### 6.9 The Eighth Deduction 165e-166c




The aim of D8 is to establish, for a variety of different properties
*F*, that if the one is not, then the others are
neither *F* nor con-*F*. Within D8, there are two
Arguments with the following Conclusions: If the one is not, then none
of the others is one (D8A1C1), the others are not many (D8A1C2), the
others cannot be conceived to be either one or many (D8A2C).




Most of the individual Arguments in D8 are logically interconnected,
and connected to Arguments within previous Deductions. The lone
exception is D8A2C, which depends on none of the Conclusions
established in previous Deductions. (For details, see Rickless
(2007, 236-238).)




It appears that all of the Arguments in D8 are valid. Like the
Arguments of the first seven Deductions, the Arguments of D8 rest on a
number of independent premises, including premises of D1. Again,
it appears that most of the premises are, at least from Plato's
point of view, either definitionally true or just plain obvious.
Thus it appears that, from Plato's point of view, D8 establishes,
without reliance on Purity-F or any other potentially problematic
assumption, that if the one is not, then the others are, at the very
least, neither one nor many.




Taken together, D7 and D8 establish that the one is. For D7A2
shows that if the one is not, then the others are many. But D8A1
shows that if the one is not, then the others are not many. Thus
D7 and D8 show that if the one is not, then the others have
contradictory properties. Given that nothing can have
contradictory properties, it follows directly that the one is.
This reinforces the result previously established in section 6.7.


## 7. Conclusion




Scholars are deeply divided about the central interpretive questions
relating to a proper understanding of Plato's
*Parmenides*. What is the point of Parmenides'
criticisms of Socrates' theory of forms, taken both individually
and collectively, in the first part of the dialogue? What is the
point of Parmenides' instantiation of his own recommended method
of training in the second part of the dialogue? And how, in
particular, is the second part of the dialogue supposed to bear on the
first?




It is possible that the dialogue as a whole is a kind of satire (Tabak
(2015)), but given the logical connections both within and across its
parts, the satirical hypothesis is unlikely.




For some, Parmenides' criticisms are no more than a "record
of honest perplexity" (see Vlastos (1954, 343); Gill (1996; 2012);
Allen (1997)). On this view, Plato's intention was simply
to put forward difficulties for the theory of forms that he himself did
not, at least at the time he wrote the dialogue, see a way to
resolve. The main problem for this interpretation is that, after
having laid out his criticisms of the theory, Parmenides says that it
should be possible for a "very gifted man" to defend the
existence of the forms, and thereby explain the possibility of
dialectic, through a method of training that Parmenides himself goes on
to instantiate in the second part of the dialogue.




For others, Parmenides' criticisms are fallacies that someone who
follows Parmenides' recommended method of training will thereby be in
a position to diagnose. On this view, the theory of forms emerges
relatively intact at the end of the dialogue. The most influential
version of this view belongs to Meinwald (1991; 1992; 2014) and Peterson
(1996; 2000; 2003). According to Meinwald, Plato meant us to recognize
the invalidity of Parmenides' criticisms of the theory of forms by
having us focus on the *in-relation-to* qualifications that are
supposed to serve as one of the principles of division that explain
the fact that the second part takes the shape of eight separate
Deductions. These qualifications, properly understood, reveal that
subject-predicate sentences (of the form "*X*
is *F*") are ambiguous: to say that *X*
is *F* is to say
*either* that *X* is *F* *in relation to
itself* (i.e.,
*pros heauto*) *or* that *X* is *F* *in
relation to the others* (i.e., *pros ta alla*), where to
say that *X* is *F*
*pros heauto* is to say that the *F* is definitionally
true of *X*, and to say that *X* is *F* *pros
ta alla* is to say that *X* displays the feature of
being *F*. As Meinwald argues, if Plato meant us to recognize
the existence of such an ambiguity, then he probably meant us to
recognize that self-predicational sentences (of the form "The
*F* is *F*") are also ambiguous, and that the
ambiguity of such sentences reveals that the Third Man argument and
the Greatest Difficulty commit the fallacy of equivocation. The main
problem with this particular interpretive strategy is that it is
provably false that *all* versions of the Third Man argument
(or Greatest Difficulty) come out fallacious if self-predicational
sentences are ambiguous as between *pros heauto* and *pros
ta alla* readings. (For details, see Frances (1996).)




There are other interpretations that are similar to the one defended by
Meinwald and Peterson. Miller (1986), for example, argues that a
discerning reader who is able to look beneath the surface of the text
is in a position to recognize that Parmenides' criticisms are
effective only on the wrong-headed supposition that forms are
fundamentally similar to the sensible, material things that partake of
them. The point of the dialogue, on this view, is to help the
discerning reader see the forms for what they really are, transcendent
beings that should be accessed by reason rather than with the help of
categories drawn from sense experience. One of the problems with
such an interpretation is a problem that is common to esoteric readings
in general: once one has left the surface of the text, there are no
interpretive constraints on what one might find beneath the
surface. Virtually *any* interpretation will turn out to
be justified by the text. Another problem with this approach is
that it pays insufficient attention to the logical interconnections
among individual criticisms of the theory of forms, and between the
criticisms as a whole and the Deductions.




There is another way of answering the three central interpretive
questions, one on which Parmenides' criticisms as well as the
Deductions come out as serious and valid. (This is the interpretation
defended in Rickless (2007), and one aspect of which is defended,
though on different grounds, in Gill (2014).) What Parmenides'
criticisms reveal is that, whether combined with the Pie Model
conception of partaking or with Paradigmatism, Plato's middle period
theory of forms is internally inconsistent. It turns out that there
are three principles the abandonment of which would eliminate all
inconsistencies apart from the Greatest Difficulty: Purity-F,
Uniqueness, and No Causation by Contraries. Careful logical analysis
of the second part of the dialogue then reveals that the Deductions
establish not only that the forms posited by the middle period theory
exist, but also that Purity-F, Uniqueness, and No Causation by
Contraries are all false. It is then reasonable to suppose that Plato
meant the reader to recognize that the proper way to save the forms is
by abandoning these three basic assumptions. And, importantly, this
can be done without abandoning the most important principles at the
heart of the middle period theory, namely One-over-Many and
Separation. The aptly-named Greatest Difficulty is then left as a
challenge for future work.