TheParmenides is, quite possibly, the most enigmatic ofPlato’s dialogues. The dialogue recounts an almost certainlyfictitious conversation between a venerable Parmenides (the EleaticMonist) and a youthful Socrates, followed by a dizzying array ofinterconnected arguments presented by Parmenides to a young andcompliant interlocutor named “Aristotle” (not thephilosopher, but rather a man who became one of the Thirty Tyrantsafter Athens’ surrender to Sparta at the conclusion of thePeloponnesian War). Most commentators agree that Socrates articulatesa version of the theory of forms defended by his much older namesakein the dialogues of Plato’s middle period, that Parmenidesmounts a number of potentially devastating challenges to this theory,and that these challenges are followed by a piece of intellectual“gymnastics” consisting of eight strings of arguments(Deductions) that are in some way designed to help us see how toprotect the theory of forms against the challenges. Beyond this, thereis precious little scholarly consensus. Commentators disagree aboutthe proper way to reconstruct Parmenides’ challenges, about theoverall logical structure of the Deductions, about the main subject ofthe Deductions, about the function of the Deductions in relation tothe challenges, and about the final philosophical moral of thedialogue as a whole.

TheParmenides inspired the metaphysical and mysticaltheories of the later Neoplatonists (notably Plotinus and later,Proclus), who saw in the Deductions the key to the hierarchicalontological structure of the universe.

• 1. Overview of the Dialogue
• 2. The Introductory Section: Zeno’s Argument 126a–128e
• 3. Socrates’ Speech: The Theory of Forms 128e–130a
• 4. Problems for the Theory of Forms 130a–134e
  • 4.1 The Extent of the Forms 130a–e
  • 4.2 The Whole-Part Dilemma 130e–131e
  • 4.3 The Third Man Argument 132a–b
  • 4.4 Forms as Thoughts 132b–c
  • 4.5 The Likeness Regress 132c–133a
  • 4.6 The Greatest Difficulty 133a–134e
• 5. How to Save the Forms: The Plan of the Deductions 134e–137c
• 6. The Deductions 137c–166c
  • 6.1 The First Deduction 137c–142a
  • 6.2 The Second Deduction 142b–155e
  • 6.3 The Appendix to the First Two Deductions 155e–157b
  • 6.4 The Third Deduction 157b–159b
  • 6.5 The Fourth Deduction 159b–160b
  • 6.6 The Fifth Deduction 160b–163b
  • 6.7 The Sixth Deduction 163b–164b
  • 6.8 The Seventh Deduction 164b–165e
  • 6.9 The Eighth Deduction 165e–166c
• 7. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Overview of the Dialogue

Plato’sParmenides consists in a critical examinationof the theory of forms, a set of metaphysical and epistemologicaldoctrines articulated and defended by the character Socrates in thedialogues of Plato’s middle period (principallyPhaedo,Republic II–X,Symposium). According to thistheory, there is a single, eternal, unchanging, indivisible, andnon-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., byparticipating in, or partaking of) the form of beauty that beautifulthings (other than beauty) are beautiful, it is by virtue of partakingof 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 thatpartake of them.

In the metaphysics of his middle period, Plato does not provide atheory of the nature of the partaking relation. But in theParmenides, Plato considers two accounts of the partakingrelation. According to the first “Pie Model” account,forX to partake ofY is for the whole or a partofY to be inX (as a part ofX). Accordingto the second “Paradigmatistic” account, forX topartake ofY is forX to resembleY. In thefirst part of the dialogue, Plato sets out reasons for thinking that,on either of these accounts of partaking, the theory of forms isinternally inconsistent.

Immediately following these criticisms, Plato describes a generalmethod of training designed to save the forms. The methodconsists of a series of eight Deductions (with an Appendix to the firsttwo) focusing on consequences that may be derived from positing thebeing of a particular form and consequences that may be derived frompositing 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, inparticular, that whether the one is or is not, the one (and also thingsother than the one: the others) havenone of a series of pairsof contrary properties (whole/divided, in motion/at rest,same/different, like/unlike, equal/unequal, older/younger). Platoalso shows that, whether the one is or is not, the one (and also theothers) have (or, at least, appear to have)all of thesecontrary properties.

2. The Introductory Section: Zeno’s Argument 126a–128e

The dialogue’s narrator is Cephalus, who has just arrived inAthens after a trip from his home in Clazomenae. Cephalus runsinto Plato’s brothers, Adeimantus and Glaucon, and asks themto confirm the existence of someone who has completely memorized aconversation that Parmenides and Zeno once had with Socrates.Adeimantus confirms that his half brother, Antiphon, can recite theconversationin toto, having heard it from a friend ofZeno’s, Pythodorus, in whose house the conversation tookplace. Cephalus, Adeimantus, and Glaucon then pay a visit toAntiphon, who, after a bit of prodding, agrees to replay theconversation. As most scholars agree, the conversation putativelyrecorded by Pythodorus, passed on to Antiphon, and then recounted toCephalus, is almost certainly fictitious. This hypothesis partlyexplains why Plato chose to frame the dialogue at thirdremove.

As Antiphon tells the story, the noted Eleatic philosophers, Parmenides(then around 65 years old) and Zeno (then around 40 years old), havecome to Athens for the Great Panathenaea. Having heard of theirarrival, a youthful Socrates (then around 20 years old) and somefriends of his have come to Pythodorus’s house to listen to Zenoread from his book. At the end of Zeno’s performance,Pythodorus, Parmenides, and Aristotle, who have been waiting outsidethe house, return and witness an exchange between Zeno andSocrates.

The exchange begins with Socrates’ summary of the generalstructure of one of Zeno’s arguments:

1. If the things that are are many, then they are both like andunlike.
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 ofdefense of Parmenides’ monism: just as others have argued thatmonism leads to absurd results (Zeno may be thinking here of the sortsof absurdities mentioned by Plato atSophist 244b–245e), sopluralism suffers consequences that are, if anything, even moreabsurd.

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 formsarticulated in Plato’s middle perioddialogues.

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

(Causality) Things that areF(other than theF) areF by virtue of partaking oftheF.

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

(Separation) TheF is itself by itself, atleast in the sense of being separate from, and hence not identicalwith, the things that partake of it.

According to Separation, like things are like by partaking of aseparate form of likeness and unlike things are unlike by partaking ofa separate form of unlikeness. Although the properties of beinglike and being unlike are contraries, they are notcontradictories. As Socrates emphasizes, it is possible forsensible things to partake of both likeness and unlikeness, and hencebe both like and unlike.

Socrates does not explain how this is supposed to be possible, butwe can extract an explanation from what he says later in his speechabout the properties of being one and being many, and from what thePhaedo tells us about the properties of being tall andshort. AtParmenides 129c, Socrates claims that hehimself is one (being one among the seven persons who are present) andmany (in having many parts: right/left, front/back, upper/lower).AtPhaedo 102b ff., Socrates points out that Simmias is tallerthan Socrates (and hence tall), but that Simmias is also shorter thanPhaedo (and hence short). Thus, Simmias is both tall andshort. Clearly Socrates envisages the possibility of sensiblethings being both like in one way and unlike in another. Forexample, Socrates is like Plato (in that each is a philosopher) andunlike Meletus (in that one, but not the other, is a poet). Beinglike Plato, Socrates is like. Being unlike Meletus, Socrates isunlike. Hence Socrates is both like and unlike. The generalprinciple here is Impurity-S:

(Impurity-S) Sensible things areimpureinasmuch as they can (and, in fact, often do) have contrary properties.

It follows from Impurity-S that premise (2) of Zeno’s argumentis false. But Socrates is willing to grant that Zeno is notentirely mistaken. He insists that although it is far fromsurprising to be told that sensible things have contrary properties, hewould beastonished to learn thatforms have contraryproperties. Socrates, then, holds that forms arepure,in the following sense:

(Purity-F) Formscannot have contrary properties.

According to Purity-F, not only is it the case that the one cannotbe both one and many and the like cannot be both like and unlike, butit is also the case that the one cannot be both like and unlike and thelike cannot be both one and many. (Notice that Purity-F andImpurity-S together entail Non-Identity, the claim that no form isidentical to any sensible thing. A similar form of argumentappears atPhaedo 74b–c, a passage in which Socrates arguesthat whereas sensible things that are equal are also unequal, the equalis not unequal, and hence the equal is not identical to any equalsensible thing.)

Socrates’ speech therefore articulates some of the basic elementsof the theory of forms, namely Causality and Separation, along withImpurity-S and Purity-F. The theory to which Socrates alludes isricher than his brief description of it suggests. As the dialogueproceeds, the interlocutors invoke a number of additional middle periodprinciples. One principle that plays an important role in thesequel is One-over-Many (seeRepublic 596a6–7):

(One-over-Many) For anyplurality ofF things, there is a form ofF-ness byvirtue of partaking of which each member of the plurality isF.

It is from this principle that Plato infers, by means of the ThirdBed Argument (Republic 597c1–d3), the principle ofUniqueness:

(Uniqueness)For any propertyF, there is exactly one form ofF-ness.

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

(Self-Predication) Forany propertyF, theF isF.

For example, the beautiful is beautiful (Euthydemus 301b5,Cratylus 439d5–6,Hippias Major292e6–7). It is clear that the point generalizes to allproperties, including the property of being large (seePhaedo100c4–6, 102e5–6). (Notice that Purity-F andSelf-Predication together entail the principle that theFcannot possess the property that is contrary toF, instancesof which appear throughout the middle dialogues (HippiasMajor 291d1–3,Phaedo 102e5–6) and inSocrates’ speech (Parmenides 129b1–3, 129b6–c1).)And finally, the sequel also repeatedly refers to the principle ofOneness:

(Oneness) Each form isone.

Oneness and Uniqueness are different principles: to say that a formisone is to say not that it isunique, but ratherthat it is something that can be counted. For Plato infers theoneness of the beautiful from the fact that the beautiful and the uglyare two, the oneness of the large from the fact that the large and thesmall are two, and so on (Republic 475e9–476a6,524b3–9). Thus the sense in which each form is one is similar tothe 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 articulatessix 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 acceptanceof the claim that there is a separate form corresponding toevery predicate or property. Socrates expressesconfidence in the existence of separate forms of justice, beauty,goodness, and every form of that sort, uncertainty about the existenceof separate forms of humanity, fire, and water, and outright skepticismabout the existence of separate forms for hair, mud, anddirt.

It is unclearwhy Socrates finds himself in doubt about theexistence of forms for natural kinds (such as humans and water) andstuffs or mixtures (such as hair and mud). For some commentators,Socrates simply makes “a wrong admission” as a result ofhis youth and inexperience (Allen 1997, 124; see also Sayre 1996,74). After all, Plato alludes to a form of bee atMeno 72b–c,a form of shuttle atCratylus 389d, and forms of bed and oftable atRepublic 596b. Although shuttles, tables, and bedsare artifacts, and hence perhaps relevantly different from naturalkinds, such as human beings and water, there seems no reason to thinkthat humans differ from bees in regard to whether they havecorresponding forms. However, it is difficult to understand whyPlato would pen a conversation in which a character whoembodies his own middle period theory would admit something he has nogood reason to admit.

One possibility (see Gill 1996, 22) is that Plato is alluding to themiddle period thesis that only certain types of properties summon theunderstanding to think about forms. For example, in theRepublic,Socrates claims that whereas “thesoul isn’t compelled to ask the understanding what a finger is,since sight doesn’t suggest to it that a finger is at the sametime the opposite of a finger” (523d2–6), the soul is compelledto ask what the large is and what the small is when sight suggests toit that the index finger is both big (relative to the pinky) and small(relative to the middle finger). If forms were merely posited toexplain the compresence of contrary properties in sensible things, thenthere would be no need to posit a form corresponding to properties(such as water and dirt) that have no contraries. However, thisis unlikely to be the source of Socrates’ worry here, for theRepublic passage does not discuss metaphysical reasons forpositing the existence of forms, but rather discusses the psychologicaland epistemic question of what prompts the soul to think of forms thathave already been posited.

Another option (Rickless 2007, 54–55; see also Miller 1986, 46) is thatPlato means us to recognize a tension between Self-Predication andSeparation (or Non-Identity) in the theory of forms. On the onehand, the fact that justice is just, beauty beautiful, and goodnessgood does not suggest that justice, beauty, and goodness are concrete,sensible things. That is, Self-Predication gives us no reason todeny that justice, beauty, and goodness are separate forms, numericallydistinct from sensible things. By contrast, if there are formsfor human and mud, then Self-Predication requires that the human be ahuman being and the mud be muddy. It is difficult to see howhuman things and muddy things could be non-sensible. SoSelf-Predication gives us at least some reason to deny that there is aform for human and mud that is distinct from every sensiblething. This interpretation meshes well with Socrates’remark that hair, mud, and dirt “are in fact just what wesee” and that it is for this reason that “it’s toooutlandish 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 formcorresponding to every property, Parmenides derives a number ofabsurdities from the result of combining the theory of forms with aparticular conception of the partaking relation, the Pie Model.According to the Pie Model, participants literally get a share of theforms of which they partake, in a way analogous to the way in whichthose who partake of a pie literally get a share of the pie. ThePie Model comes in two versions: according to the Whole Pie Model, forX to partake ofY is forX to get the wholeofY as its share ofY (i.e., for the wholeofY to be inX); according to the Piece-of-PieModel, forX to partake ofY is forX toget a (proper) part ofY as its share ofY (i.e.,for a (proper) part ofY to be inX). WhatParmenides goes on to argue is that the theory of forms is internallyinconsistent on either version of the Pie Model.

Suppose, first, that partaking conforms to the Whole Pie Model. Nowimagine that there are at one time three sensibleFthings,A,B, andC, each separate fromeach of the others. According to Causality, eachofA,B,C isF by virtue ofpartaking of theF, and hence eachofA,B,C partakes oftheF. Given the Whole Pie Model, it follows thattheF is, as a whole and at a single time, in eachofA,B,C. Thus, the whole oftheF is inA, the whole of theF isinB, and the whole of theF isinC. IfA,B, andC are inseparate places, then Causality and the Whole Pie Model togetherrequire that one and the same form be, as a whole, in separate placesat the same time. Parmenides concludes that, on this picture, therelevant 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 isseparate from itself as an absurdity in itself. On otherinterpretations (Teloh 1981, 155; Miller 1986, 48; Sayre 1996, 76),Plato does not treat this result as absurd in itself. Absurdityonly arises when this result is combined with the further thought thatnothing that is separate from itself could be a single thing. (Inthis case, the same form would have to be three things, rather than onething.) For the claim that the relevant form is not onecontradicts Oneness, the claim that every form is one.

Socrates tries to avoid the relevant absurdity, however it isunderstood, by supposing that a form is like a day, in the followingsense: just as a day can be in many separate places at the same timewithout being separate from itself, so a form can be in many separateplaces at the same time without being separate from itself.Parmenides does not think much of Socrates’ suggestion. Heimmediately counters that Socrates’ day is like a sail: a day canbe in many separate places at the same time only inasmuch as differentparts of it are over the separate places, just as a sail can cover manyseparate people only inasmuch as each person is covered by a differentpart of the sail. Parmenides’ point, then, is that the onlyway to make sense of Socrates’ day analogy is to reduce it to thePiece-of-Pie Model, the very model against which Parmenides goes on toargue.

A number of scholars have balked at Parmenides’ assimilation ofthe day analogy to the sail analogy (Cherniss 1932, 135; Peck 1953,132; Crombie 1963, 330–331; Sprague 1967, 96; Miller 1986, 49–50; Sayre1996, 76). They have assumed that, whether a day is thought of asa time-interval or as the sun’s rays (the light of day), it is infact possible for one and the same day to be in separate places at thesame time. However, it does not in fact make sense to supposethat a time-interval is in separate placesat the same time(Rickless 2007, 58). And it is not in fact true that the samepacket of rays shines on the separate places bathed by the light ofday; rather, different packets of rays shine on different places(Panagiotou 1987, 18). Moreover, it makes little sense to supposethat Plato would introduce a way out of the dilemma he himself hasconstructed without explicitly alerting his readers to that fact.To suppose otherwise would be to defend a particularly esoteric readingof 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 PieModel. Suppose, then, the same three sensibleFthings—A,B, andC—in separateplaces at the same time. According to Causality,A,B, andC areF by virtue of partaking oftheF, and henceA,B, andCpartake of theF. Given the Piece-of-Pie Model, there is apart of theF in each ofA,B, andC. If the same absurdity generated from the Whole Pie Modelis to be avoided, we must suppose that the part of theF thatis inA is numerically distinct from the part of theF that is inB and from the part of theFthat is inC, and also that the part of theF thatis inB must be numerically distinct from the part of theF that is inC. (Otherwise we would have the samepart of theF existing, as a whole, in separate places at thesame time; and hence we would have something that is separate fromitself.) Thus theF must have numerically distinct parts, andmust therefore be divided (or, at least, divisible). Parmenidesconcludes from this that theF cannot be one, a conclusionthat 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 littlesense in the wake of Socrates’ speech. There Socratesinsisted that he himself is one (in being one among many) even thoughhe has many parts (front and back, upper and lower, and so on).So Socrates does not suppose that it is truein general that athing with parts cannot be one. The hypothesis that makes mostsense of Socrates’ admission at the end of Parmenides’criticism of the Piece-of-Pie Model that theF cannot be one is thatSocrates is antecedently committed to Purity-F. For anything thathas many parts isipso facto many (just as Socrates’having many parts is sufficient for his being many), and yet, byPurity-F, no form can have contrary properties. Given that theproperty of being one and the property of being many are contraries, itfollows from Purity-F and the claim that theF is many that theFcannot be one (Rickless 2007, 59–60).

The upshot of the Whole-Part Dilemma is that absurdity orinconsistency follows from the theory of forms on either of the twopossible versions of the Pie Model conception ofpartaking. IfX’s partaking ofY amounts to thewhole ofY being inX (the Whole Pie model), thenCausality conjoined with the existence of sensible things in separateplaces at the same time entails the absurd conclusion that forms areseparate from themselves; but ifX’s partaking ofYamounts to a part ofY being inX (the Piece-of-PieModel), then Causality and Purity-F (along with the claim that havingmany parts is sufficient for being many and the claim that being oneand being many are contrary properties) are inconsistent withOneness.

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

1. By Causality, everyF thing (other than theF)isF by partaking of theF. But, by the Piece-of-PieModel, forX to partake ofY is forX toget a part ofY. Thus, everyF thing (other than theF) isF by getting a part of theF. NowletF be the property of being large. In that case, everylarge thing (other than the large) is large by getting a part of thelarge. But, Parmenides assumes, ifX is a part ofY,thenX is smaller thanY (andY is largerthanX), and henceX is (in some way) small. Soevery large thing (other than the large) is large by getting somethingsmall. But this is absurd: as Socrates himself emphasizesatPhaedo 101b, it would be monstrous to say that somethingis made larger by something small. This is an instance of a generalclaim we might call “No Causation by Contraries”:

   (No Causation by Contraries) For any propertyF, nothing thatisF could make something possess a property that is contraryto the property of beingF.

2. The result of combining Causality with the Piece-of-Pie Modelentails that equal things (other than the equal) are equal by getting apart of the equal. Given that any part ofX must be smaller thanX (see above), it follows that equal things (other than theequal) are equal by getting something that is smaller than theequal. But, Parmenides assumes, ifX is smallerthanY, thenX is unequal toY, andhenceX is (in some way) unequal. So every equal thing (otherthan the equal) is equal by getting something unequal. But, again byNo Causation by Contraries, this result is absurd: nothing that isunequal could make something be equal.

3. The result of combining Causality with the Piece-of-Pie Modelentails that small things (other than the small) are small by gettinga part of the small. This result entails that if there are any smallthings (as indeed there are), then the small must have parts. ButifX is a part ofY, thenY is largerthanX (see above), and henceY is (in some way)large. Consequently, the small must be large. But, bySelf-Predication, the small is small. So the small is both large andsmall. But this result contradicts Purity-F, according to which thesmall cannot have contrary properties, and hence cannot be both largeand small.

4. As before, the result of combining Causality with thePiece-of-Pie Model entails that small things (other than the small)are small by getting a part of the small. But, Parmenides assumes,forX to getY is just forY to be addedtoX. It follows that small things (other than the small) aresmall by having a part of the small added to them. But this is absurd:it is impossible to make something small byadding somethingto it.

These four quick arguments show that the result of combining Causalitywith the Piece-of-Pie Model does not sit well with other aspects ofthe theory of forms, in particular No Causation by Contraries (1 and2), 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 “ThirdMan”. The moniker derives from Aristotle, who in variousplaces (e.g.,Metaphysics 990b17 = 1079a13, 1039a2;Sophistical Refutations 178b36 ff.) discusses (something akinto) the argument atParmenides 132a–b in these terms.

Parmenides sets up the argument by pointing out that, according to thetheory of forms, Oneness is supposed to follow fromOne-over-Many. (Some, e.g., Fine (1993, 204), claim that Platomeans the sentence “each form is one” to expressUniqueness, not Oneness. But this is certainly not what arelevantly similar sentence expresses atRepublic 476a2–6 and524b7–11—see above.) From the existence of a plurality ofFthings and the fact that, for any such pluralityP, there is a form ofF-ness by virtue of partaking of which each memberofP isF, it follows that there is one form“over” the many members ofP (in the sense ofbeing that by virtue of partaking of which each member ofPisF). And given that anything that is one “over”many is (in some sense) one, it follows that any form that hasparticipants is one.

There is a vast literature on the Third Man argument, initiated by thegroundbreaking 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 agreethat 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 thelarge is large—an instance of Self-Predication, but denies thatthe argument, when generalized to forms other than the large, relieson Self-Predication.) They also agree that the reasoning generates aninfinite regress of forms of largeness, and that the argument could begeneralized to generate an infinite regress of forms corresponding toany predicate. But commentators differ over why Plato takes theregress to be vicious or problematic, and what Plato would haverecommended as a way of avoiding the absurdity generated by thereasoning.

Parmenides generates the infinite regress as follows. Consider aplurality of large things,A,B, andC. ByOne-over-Many, there is a form of largeness (call it“L1”) by virtue of partaking ofwhichA,B, andC are large. BySelf-Predication,L1 is large. So there is now a newplurality of large things,A,B,C, andL1. Thus, by One-over-Many, there is a formof largeness (call it “L2”) by virtue ofpartaking of whichA,B,C, andL1 are large. HenceL1 partakes ofL2. At this point,Parmenides assumes something like the following Non-Identityassumption:

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

(Notice that Non-Identity follows directly from Separation.) From thefact thatL1 partakes ofL2, Non-Identity entailsthatL2 is numerically distinct fromL1. Thus, theremust be at least two forms of largeness,L1andL2. But this is not all. By Self-Predication,L2is large. So there is now a new plurality of largethings,A,B,C,L1,andL2. Thus, by One-over-Many, there is a form of largeness(call it “L3”) by virtue of partaking ofwhichA,B,C,L1, andL2are large. HenceL1 andL2 both partakeofL3. But then, by Non-Identity,L3 is numericallydistinct from bothL1 andL2. Thus, there must be atleast three forms of largeness,L1,L2,andL3. Repetition of this reasoning, based on One-over-Many,Self-Predication, and Non-Identity, then generates an infinitehierarchy of forms of largeness, with each form partaking of everyform that lies above it in the hierarchy. (That is, for every m and nsuch that m<n,Lm partakes ofLn.)

In what way does the existence of an infinite regress of formsrepresent a problem for the theory of forms? One answer to thisquestion (see Vlastos (1954, 328, fn. 12; 1955), Goldstein and Mannick(1978), Penner (1987, 279–282), and Fine (1993, 204)) is thatthe nature of the problem is fundamentally epistemic. On this view,the theory of forms includes the thesis that, for anypropertyF, the primary function of theF is toexplain theF-ness ofF things, and hence to make itpossible for humans to apprehend and know things asF. But, so the story goes, Plato assumes that an infiniteregress of forms ofF-ness, each of which explainstheF-ness of the forms ofF-ness below it in the hierarchy, cannot explain theF-ness of theoriginal plurality ofF things: explanation must come to an endsomewhere. Although this interpretation makes sense of theepistemic language that Plato sprinkles throughout the Third Manpassage, it does not make sense of the fact that Parmenides sets up theargument by pointing out that the oneness of the large follows from itsbeing one “over” many large things (see above). So itis unlikely that the epistemic reading of the Third Man is what Platohad in mind.

Other scholars claim, quite correctly, that the existence of infinitelymany 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 ThirdMan argument is designed to uncover. But Plato seems to belooking to establish more than this. For in the last sentence ofthe relevant passage, Parmenides announces that the argument shows thateach form is no longer one, but infinitely many. Although mostcommentators gloss this comment as the claim that there is no longerone form per predicate, but rather infinitely many, this is not whatthe sentence actually says. What the sentence suggests is thatthe existence of infinitely many forms of largeness conflicts withOneness.

One way to make sense of this claim is by way of the following chain ofreasoning. As we’ve seen, One-over-Many, Self-Predication,and Non-Identity together generate an infinite hierarchy of forms oflargeness, each of which partakes of the forms above it in thehierarchy. Thus,L1 partakes of infinitely many forms,L2partakes of infinitely many forms,L3 partakes of infinitely manyforms, and so on. Now there are passages in which Plato appearsto assume that forms are as many as the predicates that can be trulyapplied to them (seePhilebus 14c8–d3, and Rickless (2007,71)). And if we assume that Parmenides is still working with thePiece-of-Pie model of partaking, then the fact that a form partakes ofinfinitely many forms entails that it has infinitely many parts, andhence is itself infinitely many. So from the existence of aninfinite regress of forms and from what appear to be dialecticallyappropriate assumptions, it is possible to argue that each form in thehierarchy is infinitely many. Given that the property of beingone and the property of being many are contraries, it then followsdirectly from Purity-F that each form in the hierarchy is notone. This interpretation explains why Parmenides announces at theend of the argument that each form is no longer one, but infinitelymany (see Rickless (2007, 64–75)).

Many commentators think that the fundamental inconsistency revealed bythe Third Man argument rests with the combination of One-over-Many,Self-Predication, and Non-Identity. For them, the Third Manrequires that Plato give up at least one of these principles. Buton the interpretation that best explains the set-up and final sentenceof the passage, Plato need not give up any of these principles in orderto avoid inconsistency: he can simply abandon Purity-F (and perhapsalso Uniqueness).

4.4 Forms as Thoughts 132b–c

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

Parmenides’ first argument appears to have the followingstructure. First, all thoughts have intentional objects: every thoughtisof something rather than nothing. Second, the object ofany thoughtT is something thatT thinks to be oneover all the instances. But anything that is thought to be one overall the instances is a form. Parmenides concludes that the intentionalobject of every thought is a form, and hence if every form is athought then every form is a thought of a form. Although Parmenidesdoes not make this explicit, it is plain that if every form isnumerically distinct from the form of which it is the intentionalobject, then (thanks to Self-Predication and Non-Identity) an infiniteregress of forms beckons (see Rickless (2007, 75–79), and alsoGill (1996, 40) and Sayre (1996, 84)). Again, there is nothing tosuggest that Plato finds the existence of an infinite regressproblematic in itself. Rather, the existence of a regress threatensUniqueness, and, when combined in the appropriate way with Purity-F,threatens Oneness. The reasoning that leads to conflict with Onenessis parallel to the relevantly similar portion of the Third Manargument (see section 4.3 above, and Rickless (2007, 79–80) fordetails).

As if this weren’t bad enough, Parmenides goes on to derive afurther absurdity from the result of combining the proposal that formsare thoughts with the Pie Model conception of partaking. Assumingthat thoughts do not have parts, the only way for an object to partakeof a thought in accordance with the Pie Model is for the object to getthe thought as a whole. So if forms are thoughts, then accordingto the Pie Model everything is composed of thoughts, and hence allthings think. But, Parmenides assumes, this panpsychist thesis isabsurd. Parmenides considers a way of avoiding this absurditythat depends on assuming that something’s having a thought as apart does not entail that it is a thinking thing. But, arguesParmenides, the only way to make sense of this proposal is to assumethat thoughts are unthinking, an assumption that is also absurd initself.

4.5 The Likeness Regress 132c–133a

At the conclusion of Parmenides’ criticism of Socrates’suggestion that forms might be thoughts, Socrates tries a completelydifferent tack: he suggests that forms are patterns set in nature(paradeigmata) and that partaking of a form amounts to beinglike it (call the combination of these claims“Paradigmatism”). Paradigmatism is incompatible bothwith the proposal that forms are thoughts and with the Pie Modelconception of partaking. The idea that forms are patterns thatserve as models for their participants is not new, for it appears invarious places in the dialogues of Plato’s middle period (seeRepublic 472b7–c7, 510a ff., 597a4–5,596b6–8—and alsoTimaeus 29c1–2, 48e5–49a1, 50c4–6).

Most commentators agree that Parmenides’ criticism ofSocrates’ Paradigmatism depends (at least in part) on theconstruction of an infinite regress. But scholars are dividedover the identity and structure of the regress: whereas some see thereasoning 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), andFine (1993, 211–215)), others see the reasoning as generating a regressof forms of likeness (e.g., McCabe (1994, 87–89), Schofield (1996),Allen (1997, 179–186), and Rickless (2007, 80–85)). (The maintextual and thematic reasons for preferring the latter reading to theformer 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 pluralityofF things,A,B, andC. ByOne-over-Many, each ofA,B,C isF by virtue of partaking of a formofF-ness (say,F1). BySelf-Predication,F1isF. HenceA,B,CandF1 are allF, and each is like the others inbeingF. Now consider the new plurality ofFthings,A,B,C, andF1. ByOne-over-Many, each ofA,B,C, andF1 isF byvirtue of partaking of a form ofF-ness (say,F2). By Non-Identity,F2 is numerically distinctfromF1. By Self-Predication,F2isF. HenceA,B,C,F1,andF2 are allF, and each is like the others in beingF. Nowconsider the new plurality ofFthings,A,B,C,F1,andF2. By One-over-Many, eachofA,B,C,F1, andF2isF by virtue of partaking of a form ofF-ness (say,F3). By Non-Identity,F3 isnumerically distinct from bothF1 andF2. Wetherefore have three distinct forms ofF-ness. Repetition of the same pattern of reasoning thenestablishes the existence of an infinite regress of forms ofF-ness. This reasoning is homologous to the Third Man argumentinasmuch 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, thereasoning relies explicitly on Paradigmatism and on an assumption thatParmenides emphasizes as he is setting up his criticism, namely thatthe relation of likeness is symmetrical: ifX islikeY, thenY is likeX(Parmenides 132d5–7). Consider two things,AandB, that both have the property of beingF.Given that there is a property thatA andB bothshare, it follows thatA is likeB and thatB islikeA. Thus,A is like something (and hence, insome way, like) andB is like something (and hence, in someway, like). But, by One-over-Many,A andB are bothlike by virtue of partaking of a form of likeness(say,L1). Now assume forreductio that something islikeL1 orL1 is like something. ClearlyifL1 is like something, thenL1 is (in some way)like. And if something is likeL1, then, by symmetry oflikeness,L1 is like it, and hence againL1 is (insome way) like. So, whether something is likeL1orL1 is like something,L1 is like. Now, by One-over-Many,L1 is like byvirtue of partaking of a form of likeness (say,L2), andhenceL1 partakes ofL2. By Non-Identity,L2 is numerically distinctfromL1. But also, by Paradigmatism,L1 islikeL2, and hence, by symmetry of likeness,L2 islikeL1. SoL2 is (in some way) like. ByOne-over-Many, then,L1 andL2 are like by virtue ofpartaking of a form of likeness (say,L3), andhenceL1 andL2 both partake ofL3. By Non-Identity,L3 is numerically distinct frombothL1 andL2. This gives us three distinct forms of likeness.Repetition of the same pattern of reasoning then establishes theexistence of an infinite regress of forms of likeness. Taking forgranted that the existence of such a regress is in some way absurd orproblematic, Parmenides infers that the assumption forreductio is false, i.e., that nothing is likeL1andL1 is like nothing. But this result is itselfunacceptable. For we already have it thatA andBare like by virtue of partaking ofL1, and hencethatA andB partake ofL1. ByParadigmatism,A andB are likeL1, andhence something is likeL1. Moreover, by symmetry oflikeness,L1 is likeA andB, andhenceL1 is like something. Thus the assumptionforreductio is true. Contradiction. Notice that, on thesecond view, the reasoning leading to the relevant regressisnot homologous to the Third Man argument: instead ofderiving the claim that each form of likeness is like fromSelf-Predication, Parmenides infers it from the conjunction ofParadigmatism and the symmetry of likeness.

On either interpretation of the identity and structure of the relevantregress, it is as yet unclear why Parmenides finds the regressproblematic. It is reasonable to assume that Parmenides’reason for finding the likeness regress problematic is the same as hisreason for finding the largeness regress problematic in the Third Manargument. The same three options canvassed in section 4.3 areavailable. Some contend that the regress is epistemic and viciousby its very nature, others that the regress conflicts with Uniqueness,and yet others that the regress leads to the claim that each form inthe relevant infinite hierarchy is many, and hence, by Oneness andPurity-F, both one and not one.

4.6 The Greatest Difficulty 133a–134e

At the conclusion of the Likeness Regress, Parmenides raises what hecharacterizes as the greatest difficulty for the theory of forms.This difficulty takes the form of two arguments, the first designed toshow that, if the forms are as Socrates has described them, they cannotbe known by human beings, the second designed to show that, if theforms are as Socrates has described them, then the gods cannot knowhuman affairs. Both of these conclusions, if true, would bedevastating to the theory of forms. For, first, in the middledialogues, Plato takes for granted that humans can know at least someforms (seeMeno 76a6–7 andPhaedo 74b2–3) andsketches a method (i.e., dialectic) that is designed to provide humanswith 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 cannotknow human affairs.

Commentators differ over the proper way to reconstruct and evaluate thetwo arguments. With respect to the first argument, some scholars(such as Lewis (1979)) claim that the argument is invalid, some (suchas Peterson (1981)) that there are two different valid ways ofreconstructing 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 usuallythought to be largely homologous to the first.

The first argument begins with the assumption (call it P1) that nothingthat is itself by itself is in (or among) humans (Parmenides133c3–6). This assumption reflects a particular understanding ofwhat Separation requires, a conception that is emphasized in thedialogues of the middle period (seeSymposium 211a8–b1 andTimaeus 52a1–3—for discussion, see Rickless (2007,19–20)). Parmenides then adds, and provides instances of, twofurther premises, P2 and P3:

(P2) IfX is a form andX is what it isin relation toY, thenY is a form.

(P3) IfX is in humans andX is what itis in relation toY, thenY is in humans.

For many commentators, P2 states that forms are related to otherforms, but not to sensible things, and P3 states that sensible thingsare 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 goodreasons for thinking that this interpretation is incorrect (seeForrester (1974), Lewis (1979), Peterson (1981), Yi and Bae (1998),and Rickless (2007, 88)). Plato’s formulation of P2 and P3presupposes a distinction between two ways of being: relative beingand absolute being. Something has relative being if it is impossibleto describe its nature without mentioning something else to which itis related. Something has absolute being if it does not have relativebeing. The point of P2 is that it is in relation to another form thatany form with merely relative being is defined. The point of P3 isthat it is in relation to another sensible thing that any sensiblething with merely relative being is defined. This reading is confirmedby Parmenides’ illustrations of P2 and P3. Mastery itself, hesays, is what it is in relation to slavery itself, but it is not thecase that mastery itself is what it is in relation to a humanslave. Similarly, slavery itself is what it is in relation to masteryitself, but it is not the case that slavery itself is what it is inrelation to a human master. Moreover, a human master is what he is inrelation to a human slave, but it is not the case that a human masteris what he is in relation to slavery itself. Similarly, a human slaveis what he is in relation to a human master, but it is not the casethat a human slave is what he is in relation to masteryitself. Parmenides then instantiates P3 using the example ofknowledge:

(P3K) IfX is knowledge in humans andX is what it is inrelation toY, thenY is in humans.

And finally, Parmenides assumes that knowledge has merely relativebeing:

(P4) Knowledge is what it is in relationto 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, ifX is knowledge in humans, then theobject ofX (i.e., whatX is knowledge of) is inhumans. Now, according to Separation, every form is itself byitself. But, by P1, nothing that is itself by itself is inhumans. Consequently, whatever is in humans is not a form. SoifX is knowledge in humans, then the object ofX isnot a form. That is to say, no knowledge in humans (i.e., no knowledgethat 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 knowledgethat 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 anyknowledge that is in humans. From these two assumptions, whatfollows is:

(P5) IfX is a knowledge andX is a form,then the gods haveX.

Parmenides then reasons as follows. By P2, ifX is a form andX is what it is in relation toY, thenY isa form. By P4, knowledge is what it is in relation to what it isknowledge of. So P2 and P4 together entail P6:

(P6) IfX is a form andX is a knowledgeofY, thenY 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 andSeparation entail that no form is in humans, i.e., that ifY is a form,thenY is not in humans. This result, taken together with P6,entails P7:

(P7) IfX is a form andX is a knowledgeofY, thenY is not in humans.

Parmenides then infers the following conclusion from the conjunctionof P5 and P7:

(C) IfX is a knowledge ofYand the gods haveX, thenY is not in humans.

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

(P5*) IfX is a knowledge and the gods haveX, thenXis a form.

There are three main possibilities here: (i) Plato simply missed thefact that the second argument is invalid; (ii) Plato intended hisreaders to recognize the argument as invalid; and (ii) Platounintentionally misstated P5* as P5. (In any event, it isinteresting to note that, whereas P3 but not P2 functions as a premisein the first argument, P2 but not P3 functions as a premise in thesecond argument.)

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

After having articulated potentially devastating criticisms of thetheory of forms, one might expect Parmenides to conclude that thetheory 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 isa character that is always the same” (a clear reference toOne-over-Many) will “destroy the power of dialecticentirely” (135b8–c2). Here Parmenides means one of twothings, depending on whether “dialectic” is taken in atechnical sense (as meaning the process by which a philosopher issupposed to acquire knowledge of the forms—seeRepublic534b3–c5) or in a non-technical sense (as meaning the ability toconverse or communicate).

In any event, Parmenides makes it clear that the power of dialectic(however this is understood) cannot be saved unless the formsthemselves are saved. As a means of saving the forms, Parmenidesrecommends a process of training that focuses on forms and takes noteof the fact that forms wander (in the sense of having contraryproperties, such as being like and unlike: 135e1–7). Inparticular, Parmenides suggests that the training process take thefollowing shape. First, concerning some form, it must involveextracting consequences from the hypothesis that that form is; second,concerning the very same form, it must involve extracting consequencesfrom the hypothesis that that form is not (135e8–136a2).Parmenides goes on to say that it is also important to considerdifferent sorts of consequences: first, consequences for the form thatis hypothesized to be (or to not be), and second, consequences forthings other than the form that is hypothesized to be (or not tobe). Parmenides also says that the training process shouldinvolve extracting consequences for the relevant form in relation toitself and in relation to the others, and consequences for things otherthan the relevant form in relation to themselves and in relation to therelevant form.

As most commentators agree, the arguments that occupy the secondpart of the dialogue may be grouped into eight distinct stretches ofreasoning or Deductions, with an additional Appendix to the first twoDeductions: 155e4–157b5. (Most Neoplatonists take there to benine Deductions. Rangos (2014), building on remarks in Grote (1865)and a seminar of Heidegger from 1930–31, argues that the Appendix isnot merely a ninth Deduction, but also plays a special role in theoverall scheme of Deductions.) There is some controversy over theprinciples of division that Plato uses to generate the groupings.Some, notably Meinwald (1991; 2014), Peterson (1996; 2000; 2003), andSayre (1996), argue that the division into eight Deductions should beexplained by the three principles of division announced by Parmenidesin his description of the method of training. On this (non-standard)picture, the Deductions should be understood as aiming at thefollowing conclusions (where ‘con-F’ refers to theproperty contrary to the property of beingF):

(D1)	If theG is, then theG is notF andnot con-F in relation to itself.
(D2)	If theG is, then theG isF andcon-F in relation to the others.
(D3)	If theG is, then the others areF andcon-F in relation to theG.
(D4)	If theG is, then the others are notF and notcon-F in relation to themselves.
(D5)	If theG is not, thentheG isF and con-F in relation to theothers.
(D6)	If theG is not, thentheG is notF and not con-F in relation toitself.
(D7)	If theG is not, then theothers areF and con-F in relation totheG.
(D8)	If theG is not, then the others are notF andnot con-F in relation to themselves.

Others, including Miller (1986), Gill (1996; 2014), Allen (1997), andRickless (2007), do not agree with this way of representing the properway of generating eight Deductions. In particular, these scholars takeissue with the claim that the third principle of division concernswhether the consequences for the relevant form (or for things otherthan the form) arerelative to itself orrelative tothings other than it. As they see it, the third principle ofdivision concerns whether the consequences for the relevant form (orfor things other than the form) arepositive ornegative. According to this standard picture, the Deductionsshould be understood as aiming at the following conclusions:

(D1)	If theG is, then theG is notF andnot con-F (in relation to itself and in relation to theothers).
(D2)	If theG is, then theG isF andcon-F (in relation to itself and in relation to theothers).
(D3)	If theG is, then the others areF andcon-F (in relation to themselves and in relation to theG).
(D4)	If theG is, then the others are notF and notcon-F (in relation to themselves and in relation to theG).
(D5)	If theG is not, then theG isF andcon-F (in relation to itself and in relation to theothers).
(D6)	If theG is not, then theG is notFand not con-F (in relation to itself and in relation to theothers).
(D7)	If theG is not, then the others areF andcon-F (in relation to themselves and in relation to theG).
(D8)	If theG is not, then the others are notF and notcon-F (in relation to themselves and in relation to theG).

(D7 represents something of an anomaly here, because many of theconclusions actually derived in that Deduction are of the form: If theG is not, then the othersappear to beFand con-F. See section 6.8 below.) Parmenides then offers toengage in the training exercise himself, taking “one” asthe relevant instance of “G”, and considering arange of properties as instances of “F” (beingmany, being a whole, being limited, having shape, being in oneself,being in another, being in motion, being the same as oneself, beingthe same as another, being like oneself, being like another, touchingoneself, touching another, being equal to oneself, being equal toanother, being (or coming to be) older than oneself, being (or comingto be) older than another, being in time, being, being named or spokenof, and being the object of an account, opinion, knowledge, orperception).

One of the primary motivations for adopting the non-standard pictureis that the standard picture makes it difficult to understand thesecond part of the dialogue as anything other than incoherent. Theproblem is this. On the standard picture, D1 and D2 together appear toentail that if the one is, then the one isF and isnotF (and the one is con-F and is notcon-F), and hence that it is not the case that the oneis. Similarly, D3 and D4 together appear to entail that if the one is,then the others areF and notF (and the others arecon-F and not con-F), and hence again that it is notthe case that the one is. On the other hand, according to the samepicture, D5 and D6 together appear to entail that if the one is not,then the one isF and is notF (and the one iscon-F and is not con-F), and hence that it is notthe case that the one is not. And similarly, D7 and D8 appear toentail that if the one is not, then the others are (or at least appearto be)F and notF (and the others are, or at leastappear to be, con-F and not con-F), and hence againthat it is not the case that the one is not. Putting all eightDeductions together, the overall result on the standard picture is astraightforward contradiction. One advantage of the non-standardinterpretation, then, is that it avoids reading the Deductions as anextended argument for a necessary falsehood.

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

Another way out of the problem posed by the seeming incoherence of theDeductions is to suppose that the subject of one Deduction isnumerically distinct from the subject of some of the otherDeductions. Multisubjectist interpretations of this kind havebeen defended by the Neoplatonists (including Plotinus and Proclus),Cornford (1939), Miller (1986), and Sayre (1996). One of themajor problems facing multisubjectism is the fact that Parmenides isquite explicit about the fact that the subject form of each Deductionis 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 ofmultisubjectism, see Allen (1997, 211–215 and 218–224).)

The standard view (if there is one) is that the Deductions have anaporetic purpose: their aim is to perplex, to set problems that mustbe solved, either by rejecting some of the premises that lead to themaster contradiction or by finding fault with the relevantreasoning. Aporetic interpretations of this sort have been defended bySchofield (1977), Gill (1996; 2012) and Allen (1997), with additionalsupport provided by Patterson (1999). On this kind of view, Plato doesnot commit to any particular way of solving the problems: the secondpart of the dialogue is simply meant to serve as a challenge to thereader.

Yet another alternative is that detailed logical analysis of theDeductions reveals arguments sufficient to establish both that the oneis and, importantly, that principles such as Purity-F and Uniquenessare false. As Rickless (2007, 136–137 and 211) argues, Purity-Fis a background premise of both D1 and D4. Taken together, then, D1and D2 are sufficient to establish that a contradiction follows fromthe hypothesis that the one is,on the assumption that Purity-F istrue. That is, D1 and D2 together entail that if the one is, thenPurity-F is false. But D5 and D6 together entail that the one is. Itthen follows directly that Purity-F must be false. (Gill (2014) arguesthat the result of D1 and D8, taken together, is that the one must beacknowledged to be both one and many. She also thinks that Platomeans for his readers to recognize that the same acknowledgment isneeded to overcome the problematic results of D4.) Moreover, D2contains an argument for the claim that if the one is, then there areinfinitely many forms of oneness. Given that the one is (thanks to D5and D6), it follows directly that there must be more than one formcorresponding to the property of being one, and hence that Uniquenessis false. (Rickless (2007, 238–239) also argues that the secondpart of the dialogue provides sufficient reason to reject No Causationby Contraries.)

6. The Deductions 137c–166c

Each Deduction consists of separate stretches of reasoning (callthem “Arguments”) leading to a number of logicallyinterconnected results (call them “Conclusions”). Thesummary to follow is governed by the following notationalconventions. Each Deduction receives a number (“D1”for the first Deduction, “D2” for the second Deduction, andso on) and each Argument within each Deduction receives a number(“A1” for the first Argument, “A2” for thesecond Argument, and so on). (The Appendix to the first twoDeductions will be called “App”.) If an Argument hasexactly one Conclusion, the single Conclusion will be referred to as“C”. If an Argument has more than one Conclusion, theConclusions will be numbered (“C1” for the firstConclusion, “C2” for the second Conclusion, and soon).

6.1 The First Deduction 137c–142a

The aim of D1 is to establish, for a variety of different propertiesF, that if the one is, then the one is neitherF nor con-F.Within D1, there are eighteen separate Arguments with the followingConclusions: If the one is, then the one is not many (D1A1C), the onehas no parts (D1A2C1), the one is not a whole (D1A2C2), the one has nobeginning, middle, or end (D1A3C), the one is unlimited (D1A4C), theone has no shape (D1A5C), the one is not in another (D1A6C1), the oneis not in itself (D1A6C2), the one is nowhere (D1A6C3), the one is notin motion (D1A7C), the one is not at rest (D1A8C), the one is notdifferent from itself (D1A9C1), the one is not the same as another(D1A9C2), the one is not different from another (D1A10C), the one isnot 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 isnot equal to itself or another (D1A14C1), the one is not unequal toitself or another (D1A14C2), the one is not the same age as itself oranother (D1A15C1), the one is neither younger nor older than itself oranother (D1A15C2), the one is not in time (D1A16C), the one neithercomes to be nor ceases to be (D1A17C1), the one does not partake ofbeing (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 ofan account, knowledge, perception, or opinion (D1A18C).

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

The Arguments of D1 also rest on a large number of independentpremises, among which we find the following: (i) the property of beingone and the property of being many are contraries; (ii) anything thathas parts is many; (iii) a whole is a thing with parts from which nopart is missing; (iv) the beginning, middle, and end ofX areparts ofX; (v) the beginning and end ofX are the limitsofX; (vi) to be round is to have extremities that areequidistant in every direction from the middle; (vii) to be straightis 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)ifX is inY, thenX is contained allaround byY andX touchesY in many places with manyparts; (x) ifX is contained all around byYandX touchesY in many places with many parts,thenX is round; (xi) ifX touchesY inmany places with many parts, thenX has parts; (xii)ifX both contains itself and is contained by itself,thenX is two; (xiii) ifX is two, thenXis many; (xiv) whatever is neither in itself nor in another isnowhere; (xv) ifX is in motion, thenX alters fromitself or moves spatially; (xvi) ifX moves spatially,thenX either spins in a circle in the same location orchanges from one place to another.

It appears that every Argument of D1 other than D1A9 is logicallyvalid. (However, see Gill (1996, 81, n. 134), who contends thatD1A12 and D1A13 are fallacious.)) D1A9 seems to commit thefallacy of equivocation (see Rickless (2007, 121–123). It alsoappears that almost all the independent premises of D1 are, at leastfrom Plato’s point of view, either definitionally true orself-evident. There are three exceptions to this claim. D1A10depends on the questionable premise that ifX is not different byitself, thenX itself is not different in any way (see Rickless (2007,123–124)), and D1A11 depends on the questionable premise thatif the nature of theF is not identical to thenature of theG,then ifX isG relative toitself,X is notF 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 notmany. The reasoning is simple. From Oneness orSelf-Predication, it follows that the one is one. Consequently,if the one is, then the one is one. But the property of being oneand the property of being many are contraries, and, by Purity-F, noform can have contrary properties. Given that the one is a form,it follows that if the one is, then the one is not many. Thenon-obvious premises here are Oneness (or Self-Predication) andPurity-F. Whereas some might argue that D1A1 is unsound becauseOneness (or Self-Predication) is false, others might argue that thevery point of D1A1 is to show that if the one is and is many, thenPurity-F is false.

6.2 The Second Deduction 142b–155e

The aim of D2 is to establish, for a variety of different propertiesF, that if the one is, then the one is bothF andcon-F. Within D2 there are thirty-three separate Argumentswith the following Conclusions: If the one is, then the one partakesof being (D2A1C1), the one is not the same as being (D2A1C2), the oneis 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 thesame as the one (D2A4C1), the different is not the same as being(D2A4C2), the one has parts (D2A6C1), the one is a whole (D2A6C2), theone is limited (D2A6C3), the one is unlimited (D2A6C4), the one has abeginning, a middle, and an end (D2A7C1), the one has shape (D2A7C2),the one is in itself (D2A8C1), the one is not nowhere (D2A8C2), theone is in another (D2A8C3), the one is at rest (D2A9C1), the one is inmotion (D2A9C2), the one is the same as itself (D2A10C1), the one isdifferent from itself (D2A10C2), the one is different from the others(D2A11C), the one is the same as the others (D2A12C), the one is likethe others (D2A13C and D2A15C1), the one is unlike the others (D2A14Cand D2A15C2), the one is like itself (D2A16C1), the one is unlikeitself (D2A16C2), the one touches itself (D2A17C1), the one touchesthe others (D2A17C2), the one does not touch itself (D2A18C), the onedoes not touch the others (D2A19C), the one is equal to itself(D2A20C1), the one is equal to the others (D2A20C2), the one is bothgreater than and less than itself (D2A21C1), the one is unequal toitself (D2A21C2), the one is both greater than and less than theothers (D2A22C1), the one is unequal to the others (D2A22C2), the oneis more than, less than, and equal to itself in number (D2A23C1), theone is more than, less than, and equal to the others in number(D2A23C2), the one partakes of time (D2A24C), the one comes to beolder than itself (D2A25C1), the one comes to be younger than itself(D2A25C2), the one always is older than itself (D2A26C1), the onealways is younger than itself (D2A26C2), the one is the same age asitself (D2A27C1), the one is neither older nor younger than itself(D2A27C2), the one neither comes to be older nor comes to be youngerthan itself (D2A27C3), the one is older than the others (D2A28C), theone is younger than the others (D2A29C), the one is the same age asthe others (D2A30C1), the one is neither older nor younger than theothers (D2A30C2), the one neither comes to be older nor comes to beyounger than the others (D2A31C), the one comes to be younger than theothers (D2A32C1), the one comes to be older than the others (D2A32C2),the one partakes of time past, future, and present (D2A33C1), the oneis and comes to be, was and was coming to be, and will be and will becoming to be (D2A33C2), and the one could be named and spoken of, aswell as be the object of an account, knowledge, perception, andopinion (D2A33C3).

Most of the individual Arguments in D2 are logicallyinterconnected. The exceptions are D2A1C1, D2A1C2, D2A8C2,D2A10C1, D2A12C, D2A18C, D2A19C, D2A20C1, D2A20C2, and D2A24C, which donot depend on any previously established conclusions. (Fordetails, 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 D2A9are fallacious, and Patterson (1999, 98–100), who argues that D2A8 isfallacious.)) Like the Arguments of D1, the Arguments of D2 reston a large number of independent premises, including many of theindependent premises of D1. Again, most of these premises are, atleast from Plato’s point of view, either definitionally true orself-evident. (Gill (1996, 83–84) objects to one of the premisesof D2A14 and Patterson objects to one of the premises of D2A28, butRickless (2007, 160 and 175) claims that it is unclear whether Platohimself would have found these premises problematic.) There arethree exceptions to this claim: D2A4, D2A12, and D2A14. D2A4depends on Causality and No Causation by Contraries, D2A12 depends onNo Causation by Contraries, and D2A14C depends on D2A12C. Butthere are reasons for thinking that No Causation by Contraries isfalse. When combined with Causality, No Causation by Contrariesentails that, for any propertyF, theF cannot be con-F. But,assuming that the one is, this contradicts D2A3C, namely thatif the one is, then the one is (infinitely) many. So, ifCausality is true and the one is, then No Causation by Contraries mustbe false. This result is important, for the falsity of NoCausation by Contraries would enable Plato to dispatch at least some ofParmenides’ earlier criticisms of the theory of forms, notablythe first two criticisms of the result of combining the theory of formswith the Piece-of-Pie Model conception of partaking—see the endof section 4.2 above.

Notice that D1 and D2 together entail that if the one is, thenPurity-F is false. To see why, consider the following. D1 establishesthat if Purity-F is true, then if the one is, then the one isneitherF nor con-F; and D2 establishes that if theone is, then the one is bothF and con-F. So if bothD1 and D2 are valid and based on acceptable premises (other thanPurity-F), then D1 and D2 together establish that if Purity-F is trueand the one is, then the one hascontradictory properties. Given that it is impossiblefor something to havecontradictory properties, it is a directconsequence of the conjunction of D1 and D2 that if the one is, thenPurity-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 andD2 together entail that, for a range of propertiesF, if theone is, then there is a moment outside of time (the so-called“instant”) at which the one changes from beingFto being con-F. Within the Appendix, there are five Argumentswith the following Conclusions: If the one is, then there aretimesT1 andT2 such thatT1 is distinctfromT2 and the one partakes of being atT1 and theone does not partake of being atT2 (AppA1C), there is adefinite time at which the one comes to be (AppA2C1), there is adefinite time at which the one ceases to be (AppA2C2), there is a timeat which the one ceases to be many (AppA3C1), there is a time at whichthe one ceases to be one (AppA3C2), there is a time at which the oneis 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 atime at which the one is increased (AppA3C7), there is a time at whichthe one is decreased (AppA3C8), there is a time at which the one ismade equal (AppA3C9), there is something (call it “theinstant”) (i) that is in no time at all and (ii) at which theone changes both from being in motion to being at rest and from beingat rest to being in motion and (iii) at which the one is neither atrest nor in motion (AppA4C), there is something (call it “theinstant”) (i) that is in no time at all and (ii) at which theone changes both from not-being to being and from being to not-beingand (iii) at which the one neither is nor is not (AppA5C1), there issomething (call it “the instant”) (i) that is in no timeat all and (ii) at which the one changes both from being one to beingmany and from being many to being one and (iii) at which the one isneither one nor many (AppA5C2), there is something (call it “theinstant”) (i) that is in no time at all and (ii) at which theone changes both from being like to being unlike and from being unliketo 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 bothfrom being small to being large and from being large to being smalland (iii) at which the one is neither large nor small (AppA5C4).

All of the individual Arguments within the Appendix are logicallyinterconnected. (For details, see Rickless (2007, 189–198).)

It appears that every Argument of the Appendix is logicallyvalid. Like the Arguments of D1 and D2, the Arguments of theAppendix rest on a number of independent premises, including premisesof D1 and D2. Again, most of the premises are, at least fromPlato’s point of view, either definitionally true orself-evident. (Gill (1996, 86) objects to one of the premises ofAppA5. For a rejoinder, see Rickless (2007, 195, n. 2).)There is one major exception to this claim, however. All of theArguments of the Appendix other than AppA1 depend for their soundnesson the soundness of AppA1. But AppA1 depends for its soundness onthe soundness of D1A17, which itself depends for its soundness on thetruth of Purity-F. So if Purity-F were false, then all theArguments of the Appendix would be unsound.

6.4 The Third Deduction 157b–159b

The aim of D3 is to establish, for a variety of differentpropertiesF, that if the one is, then the others arebothF and con-F. Within D3, there are sevenArguments with the following Conclusions: If the one is, then theothers are not the one (D3A1C), the others have parts (D3A2C1), theothers are a whole (D3A2C2 and D3A3C2), the others are one (D3A2C3 andD3A3C1), the whole and the part of the others are many (D3A4C), thewhole and the part of the others are unlimited in multitude (D3A5C1),the whole and the part of the others are unlimited (D3A5C2), the wholeand the part of the others are limited (D3A6C), each of the others islike itself (D3A7C1), each of the others is like each of the othersother than itself (D3A7C2), each of the others is unlike itself(D3A7C3), and each of the others is unlike each of the others otherthan itself (D3A7C4). There is also the promise of a number ofArguments establishing results of the form: If the one is, then theothers are bothF and con-F.

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

It appears that every argument of D3 is logically valid. Like theArguments of D1 and D2, the Arguments of D3 rest on a number ofindependent premises, including premises of D1 and D2. Again, itappears that the premises are, at least from Plato’s point ofview, either definitionally true or self-evident. Thus it appearsthat, from Plato’s point of view, D3 establishes, withoutreliance on Purity-F or any other potentially problematic assumption,that if the one is, then the others have a host of contraryproperties.

6.5 The Fourth Deduction 159b–160b

The aim of D4 is to establish, for a variety of different propertiesF, that if the one is, then the others are neitherFnor con-F. Within D4, there are four Arguments with thefollowing 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 notlike (D4A3C1), the others are not unlike (D4A3C2), the others are notboth 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 notcoming to be (D4A4C5), the others are not ceasing to be (D4A4C6), theothers 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, andconnected to Arguments within previous Deductions. (For details,see Rickless (2007, 207–211).)

It appears that all of the Arguments in D4 are valid. Like theArguments of the first three Deductions, the Arguments of D4 rest on anumber of independent premises, including premises of D1 and D2 (butnot D3). Again, it appears that most of the premises are, atleast from Plato’s point of view, either definitionally true orself-evident. There is one major exception to this claim,however. All of the Arguments of D4 depend for their soundness onthe soundness of D1A2. But D1A2 depends for its soundness on thesoundness of D1A1, which itself depends for its soundness on the truthof Purity-F.

So we can redescribe the Conclusion of D4 as follows:If Purity-Fis true, then if the one is, then the others do not have a hostof contrary properties. This means that D3 and D4 together entail thatif the one is, then Purity-F is false. For, by D3, if the one is, thenthe others are bothF and con-F. But, by D4, ifPurity-F is true and the one is, then the others areneitherF nor con-F. So, if Purity-F is true and theone is, then the others have a host ofcontradictoryproperties. Given that it is impossible for anything to havecontradictory properties, it follows directly that if the one is, thenPurity-F is false.

6.6 The Fifth Deduction 160b–163b

The aim of D5 is to establish, for a variety of different propertiesF, that if the one is not, then the one is bothF 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 kindfrom the others (D5A2C), the one partakes ofsomething,that, andthis (D5A3C), the one is unlike the others(D5A4C1), the others are unlike the one (D5A4C2), the one partakes ofthe unlike (i.e., has unlikeness) in relation to the others (D5A4C3),the one partakes of the like in relation to itself (D5A5C1), the one islike itself (D5A5C2), the one is unequal to the others (D5A6C1), theothers are unequal to the one (D5A6C2), the one partakes of the unequalin relation to the others (D5A6C3), the one partakes of the large(D5A7C1), the one partakes of the small (D5A7C2), the one partakes ofthe equal (D5A7C3), the one partakes of being (D5A8C1), the onepartakes of not-being (D5A8C2), the one is in motion (D5A9C), the oneis not in motion (D5A10C1), the one is at rest (D5A10C2), the one isaltered (D5A11C1), the one is not altered (D5A11C2), the one comes tobe (D5A12C1), the one ceases to be (D5A12C2), the one does not come tobe (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. Theexceptions are D5A1C1, D5A1C2, D5A3C, D5A5C1, D5A8C1, D5A8C2, andD5A10C1, which do not depend on any previously establishedConclusions. (For details, see Rickless (2007, 212–223).)

It appears that all of the Arguments in D5 are valid. Like theArguments of the first four Deductions, the Arguments of D5 rest on anumber of independent premises, including premises of D1, D2, and theAppendix (but not D3 or D4). Again, it appears that most of thepremises are, at least from Plato’s point of view, eitherdefinitionally true or self-evident. Thus it appears that, fromPlato’s point of view, D5 establishes, without reliance onPurity-F or any other potentially problematic assumption, that if theone is not, then the one has a host of contrary properties. Inother 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, withthe results of D3 and D4), this result entails that Purity-F is in factfalse. 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 isnot, then Purity-F is false. So,whether the one is or is not, Purity-F is false. SoPurity-F is false.

6.7 The Sixth Deduction 163b–164b

The aim of D6 is to establish, for a variety of different propertiesF, that if the one is not, then the one is neitherFnor con-F. Within D6, there are four Arguments with thefollowing 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 noway comes to be (D6A2C1), the one in no way ceases to be (D6A2C2), theone 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 ofthe small (D6A3C1), the one does not partake of the large (D6A3C2),the one does not partake of the equal (D6A3C3), the one does notpartake of the like (D6A3C4), the one does not partake of thedifferent (D6A3C5), the others are not like the one (D6A4C1), theothers are not unlike the one (D6A4C2), the others are not the same asthe 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, timepresent, knowledge, perception, opinion, account, and name) isapplicable 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. Theexceptions are D6A1C1 and D6A4C5, which do not depend on any previouslyestablished Conclusions. (For details, see Rickless (2007,223–228).)

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

When combined with the results of D5, this result entails that the oneis. To see this, consider the following. D5 establishes that if theone is not, then the one is bothF and con-F; and D6establishes that if the one is not, then the one is neitherF norcon-F. So if both D5 and D6 are valid and based on acceptablepremises, then D5 and D6 together establish that if the one is not,then the one hascontradictory properties. Given that it isimpossible for something to havecontradictory properties, itis a direct consequence of the conjunction of D5 and D6 that it is notthe case that the one is not. That is to say, D5 and D6 togetherentail that the one is.

This is a significant result, for two reasons. First, itreinforces the earlier result obtained from the conjunction of D1, D2,and D5 (and from the conjunction of D3, D4, and D5), namely thatPurity-F is false. For D1 and D2 (as well as D3 and D4) togetherentail that if the one is, then Purity-F is false. But D6establishes that the one is. It then follows directly by modusponens that Purity-F is false. Second, the result that the one iscan be generalized to establish the being of any form whatever.The reason is that none of the reasoning in the Deductions up to thispoint depends on the one’s having been chosen as the main subjectof the Deductions. Every Argument of D1–D6 would go through ifsome other form were substituted for the one as the main topic ofdiscussion. Earlier, Parmenides had said that “only a verygifted man can come to know that for each thing there is some kind, abeing itself by itself” (135a–b). So he has now revealedhimself to be the “very gifted man” of whom he hadspoken.

6.8 The Seventh Deduction 164b–165e

The aim of D7 is to establish, for a variety of different propertiesF, that if the one is not, then the others arebothF and con-F. Within D7, there are sixArguments with the following Conclusions: If the one is not, then theothers are (D7A1C1), the others are other (D7A1C2), the others aredifferent (D7A1C3), the others are other than each other (D7A1C4), theothers are infinitely many (D7A2C), each of the others appears to beone (D7A3C1), each of the others is not one (D7A3C2), the othersappear to be infinitely many (D7A3C3), some of the others appear to beeven, 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) appearslarge in relation to its parts (D7A4C2), each of the others appears tocome to the equal (D7A4C3), each of the others appears to have nobeginning, middle, or end in relation to itself (D7A5C1), each of theothers appears unlimited in relation to itself (D7A5C2), each of theothers appears limited in relation to another (D7A5C3), each of theothers appears to be like itself and each of the others (D7A6C1), andeach 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. Theexceptions are D7A1C1 and D7A3C5, which do not depend on any previouslyestablished Conclusions. (For details, see Rickless (2007,228–236).)

It appears that all of the Arguments in D7 are valid. Like theArguments of the first six Deductions, the Arguments of D7 rest on anumber of independent premises, including premises of every Deductionother than D4 and the Appendix. Again, it appears that most ofthe premises are, at least from Plato’s point of view, eitherdefinitionally true or just plain obvious. Thus it appears that,from Plato’s point of view, D7 establishes, without reliance onPurity-F or any other potentially problematic assumption, that if theone is not, then the others appear to have a host of contraryproperties. (Thus D7 does not conform exactly toParmenides’ earlier description of the method of theDeductions. What we would expect from that description is notthat D7 would establish that if the one is not then the othersappear to have contrary properties, but rather that D7 wouldestablish that if the one is not then the othersactually havecontrary properties. This discrepancy remains something of amystery.)

Still, it is possible to extract an interesting result from D7, whencombined with a result from D3. D7A1 establishes that if the one isnot, then the others are (D7A1C1). But D3 establishes (formanyF’s) that if the one is, then the othersareF. But to beF is to be in some way or other. Soif the others areF, then the others are. So D3 establishes(for manyF’s) that if the one is, then the others are. Thus,D3 and D7A1C1 together entail that, whether the one is or is not, theothers are. Hence D3 and D7 together entail that the othersare. Assuming that the others are (or, at least, include) all theforms other than the one, it follows, in conjunction with thepreviously established result that the one is (see section 6.7 above),thatevery form is. This reinforces the previous claim (seesection 6.7 again) that the result that the one is can be generalizedto all the forms.

6.9 The Eighth Deduction 165e–166c

The aim of D8 is to establish, for a variety of different propertiesF, that if the one is not, then the others areneitherF nor con-F. Within D8, there are twoArguments with the following Conclusions: If the one is not, then noneof the others is one (D8A1C1), the others are not many (D8A1C2), theothers 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 loneexception is D8A2C, which depends on none of the Conclusionsestablished in previous Deductions. (For details, see Rickless(2007, 236–238).)

It appears that all of the Arguments in D8 are valid. Like theArguments of the first seven Deductions, the Arguments of D8 rest on anumber of independent premises, including premises of D1. Again,it appears that most of the premises are, at least from Plato’spoint 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 problematicassumption, that if the one is not, then the others are, at the veryleast, neither one nor many.

Taken together, D7 and D8 establish that the one is. For D7A2shows that if the one is not, then the others are many. But D8A1shows that if the one is not, then the others are not many. ThusD7 and D8 show that if the one is not, then the others havecontradictory properties. Given that nothing can havecontradictory 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 questionsrelating to a proper understanding of Plato’sParmenides. What is the point of Parmenides’criticisms of Socrates’ theory of forms, taken both individuallyand collectively, in the first part of the dialogue? What is thepoint of Parmenides’ instantiation of his own recommended methodof training in the second part of the dialogue? And how, inparticular, is the second part of the dialogue supposed to bear on thefirst?

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 itsparts, the satirical hypothesis is unlikely.

For some, Parmenides’ criticisms are no more than a “recordof honest perplexity” (see Vlastos (1954, 343); Gill (1996; 2012);Allen (1997)). On this view, Plato’s intention was simplyto put forward difficulties for the theory of forms that he himself didnot, at least at the time he wrote the dialogue, see a way toresolve. The main problem for this interpretation is that, afterhaving laid out his criticisms of the theory, Parmenides says that itshould be possible for a “very gifted man” to defend theexistence of the forms, and thereby explain the possibility ofdialectic, through a method of training that Parmenides himself goes onto instantiate in the second part of the dialogue.

For others, Parmenides’ criticisms are fallacies that someone whofollows Parmenides’ recommended method of training will thereby be ina position to diagnose. On this view, the theory of forms emergesrelatively intact at the end of the dialogue. The most influentialversion of this view belongs to Meinwald (1991; 1992; 2014) and Peterson(1996; 2000; 2003). According to Meinwald, Plato meant us to recognizethe invalidity of Parmenides’ criticisms of the theory of forms byhaving us focus on thein-relation-to qualifications that aresupposed to serve as one of the principles of division that explainthe fact that the second part takes the shape of eight separateDeductions. These qualifications, properly understood, reveal thatsubject-predicate sentences (of the form “XisF”) are ambiguous: to say thatXisF is to sayeither thatX isFin relation toitself (i.e.,pros heauto)or thatX isFinrelation to the others (i.e.,pros ta alla), where tosay thatX isFpros heauto is to say that theF is definitionallytrue ofX, and to say thatX isFprosta alla is to say thatX displays the feature ofbeingF. As Meinwald argues, if Plato meant us to recognizethe existence of such an ambiguity, then he probably meant us torecognize that self-predicational sentences (of the form “TheF isF”) are also ambiguous, and that theambiguity of such sentences reveals that the Third Man argument andthe Greatest Difficulty commit the fallacy of equivocation. The mainproblem with this particular interpretive strategy is that it isprovably false thatall versions of the Third Man argument(or Greatest Difficulty) come out fallacious if self-predicationalsentences are ambiguous as betweenpros heauto andprosta alla readings. (For details, see Frances (1996).)

There are other interpretations that are similar to the one defended byMeinwald and Peterson. Miller (1986), for example, argues that adiscerning reader who is able to look beneath the surface of the textis in a position to recognize that Parmenides’ criticisms areeffective only on the wrong-headed supposition that forms arefundamentally similar to the sensible, material things that partake ofthem. The point of the dialogue, on this view, is to help thediscerning reader see the forms for what they really are, transcendentbeings that should be accessed by reason rather than with the help ofcategories drawn from sense experience. One of the problems withsuch an interpretation is a problem that is common to esoteric readingsin general: once one has left the surface of the text, there are nointerpretive constraints on what one might find beneath thesurface. Virtuallyany interpretation will turn out tobe justified by the text. Another problem with this approach isthat it pays insufficient attention to the logical interconnectionsamong individual criticisms of the theory of forms, and between thecriticisms as a whole and the Deductions.

There is another way of answering the three central interpretivequestions, one on which Parmenides’ criticisms as well as theDeductions come out as serious and valid. (This is the interpretationdefended 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 Modelconception of partaking or with Paradigmatism, Plato’s middle periodtheory of forms is internally inconsistent. It turns out that thereare three principles the abandonment of which would eliminate allinconsistencies apart from the Greatest Difficulty: Purity-F,Uniqueness, and No Causation by Contraries. Careful logical analysisof the second part of the dialogue then reveals that the Deductionsestablish not only that the forms posited by the middle period theoryexist, but also that Purity-F, Uniqueness, and No Causation byContraries are all false. It is then reasonable to suppose that Platomeant the reader to recognize that the proper way to save the forms isby abandoning these three basic assumptions. And, importantly, thiscan be done without abandoning the most important principles at theheart of the middle period theory, namely One-over-Many andSeparation. The aptly-named Greatest Difficulty is then left as achallenge for future work.

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