Sandra Peterson's 'The Parmenides' in Oxford Handbook to Plato.
This is an eclectic sampling from Peterson for this class, and is not meant to accurately convey her full argument, which is much more detailed and nuanced than this document reflects. Please go read her article.

• Socrates is "very young" and yet has a theory of Forms
  
• The fictional date is before the Republic, the Phaedo, etc.
• It is strange to put a "very young" Socrates in charge of a theory of Forms:
• This theory of Forms is refuted
  
• If youth indicates immaturity, perhaps the theory of Forms in the later dialogues is more mature and escapes the refutation
• And yet, the timing of Socrates' meeting with Parmenides can only have plausibly occurred when Soc. was very young, so maybe it's just a 'young Socrates' because of that.
• Parmenides was born somewhere between 540 and 515 BCE, so he was indeed a fair bit older than Socrates. A good 45-75 years older.
• The upshot is that the fictional time of the dialogue is suggestive that maybe this is not a farewell to forms.
• And yet, the only way to have Socrates talk to Parmenides was to make Socrates very young and Parmenides very old.

• Socrates' reaction to Zeno at 127e8ff.:
• Then this is what your arguments are aiming at: nothing else but to contend, against all the things commonly said, that there are not many things? And do you think that each of the arguments is a proof for you, so that you also offer as many proofs as you have written arguments, that there are not many things? Hugely many and very great proofs. 127e8 ff.
• The plurals sort of drip with irony, given that Parmenides and Zeno want to claim that all is one.
• Zeno says that he says the same things as Parmenides.
• He does so in a different way: first he assumes Parmenides is wrong, then derives all sorts of ridiculous conclusions from that, and so concludes that Parmenides must be right.
• 127e1 has Socrates' summary of a Zenonian argument:
1. Suppose that beings are many. (i.e. assume for a moment that Parmenides is wrong in claiming that there is only one thing)
   
2. If they are many, they are both like and unlike.
3. So beings are both like and unlike.
   
4. It is impossible that unlike things be like or like things unlike.
5. Therefore, beings are not many.
• Peterson's reaction: Premise 4 is implausible: the same things can easily be both like (one thing) and unlike (another thing).
• Later, at 148a3, Parmenides defines being like as bearing the same attribute.
• So being unlike is bearing a different attribute or at least not bearing the same attribute?
• And it's perfectly possible for two things to be like (both be red) and unlike (one is big, the other small)
• Socrates' full response to the argument above:
• "Don't you acknowledge that there is itself by itself, a form of likeness, and again something contrary to such a thing, what unlike is, while the things we call many--me and you and the other things--together get these two things that are? Don't you and I and the other things we call many together have those two things that are? And don't things that together get likeness become like in that way and to the extent that they together get it, and the things [that together get] unlikeness [in that way and to the extent to which they together get it become] unlike, and [things that jointly get] both, both? Even if all things together get both contrary beings and are, by together having both, like and unlike, themselves to themselves, what is amazing [about that]?" 128e6-129b1
• First, some points about Socrates' language:
• If Socrates is meant to be here for the first time introducing technical things from a Theory of Forms, that will not be convincing to Zeno and Parmenides, who reject that theory of forms. Technical things need explanation and preparation: this must be "ordinary" things, things Zeno and Parmenides can readily understand.
• I.e. there ought to be a "dialectical requirement" operative here: Socrates should only introduce easily plausible initial claims.
  
• If he is to argue Zeno and Parmenides into accepting his ideas,  he needs to start by using ordinary concepts that they accept, not technical ones that they don't accept.
  
• Socrates speaks of a "form" of likeness: this means one of two things in English:
• "a specific variety," a "kind" of likeness
  
• "no specific one, but just the generic thing 'likeness' which applies to all things that carry the same name, that are like"
• Socrates' wording 'itself by itself' makes it likely that he means the second one.
• So Socrates is focusing on likeness or unlikeness considered apart from any particular likeness or unlikeness.
• what unlike is is parallel to itself by itself a form of likeness and so must also be itself by itself a form of unlikeness.
  
• Clearly this is language that reminds us of the FORMS.
• (Peterson tries to make do with only things that Plato is committed to in the Parmenides.)
• As such, she is elucidating a theory of Forms, not the theory of Forms.
• By speaking of "in that way and to the extent", Socrates is showing that he is aware that like and unlike are incomplete (they need a further term: like what? unlike what? and by what measure or parameter?). 
• they are "relational" and must relate things to each other: X cannot be like without being like something,  call it Y.
  
• Socrates is also aware that like and unlike are opposites.
• By together getting  and together having, Socrates means something like "acquire jointly" or "both have" or "jointly become like" or "together are unlike": this is ordinary Greek.
  
• This is not technical language in Greek, but rather ordinary Greek, so we cannot use the terminology here to justify importing THE theory of Forms as it is reconstructed in secondary literature and occurs in other dialogues.
• Next, what moves does Socrates make:
1. First, he shows that Zeno is conspicuously inconsistent (see the bolded enlarged plurals in the first text above)
   
2. More specifically, he says that Zeno is committed to there being two opposites, like and unlike, each of which is a being.
   
• So Zeno's premises imply the denial of his conclusion.
  
3. He denies Zeno's premise that a thing cannot be both like and unlike:
   
• Repeated from above: And don't things that together get likeness become like in that way and to the extent that they together get it, and the things [that together get] unlikeness [in the respect and to the extent to which they together get it become] unlike, and [things that jointly get] both, both? 
• Some scholars think this introduces technical entities, Forms, but Socrates is not doing so, says Peterson. He is just talking about ordinary likeness and unlikeness in a non-technical ordinary-language way.
  
• If Socrates were introducing technical entities in need of defense, Parmenides and Zeno would not be impressed (and would ask for that defense), says Peterson.
  
• note that her argument for her interpretation hinges at least partially on the interpretation of Parmenides' reaction (he is impressed): she makes no argument that Parmenides' reaction must be taken the way she takes it or that P's reaction must mean what she takes it to mean: the proof is in the pudding, as the saying goes. If the pudding works, produces good pudding, that's a point in its favor.

• Socrates' challenge to Parmenides and Zeno:
• If someone showed the likes themselves coming to be unlikes or the unlikes like--that, I think, would be a marvel; but if he shows things that jointly acquire both as having undergone both, it seems to me, Zeno, nothing out of the way, nor if someone shows all things are one by jointly having acquired unity and these same things many by again jointly having acquired multiplicity. But if--what one is--he will show this itself many and again the many one, I would at that point wonder at it. And so for all the others if he should show the kinds and forms themselves in themselves bearing these conditions that are contraries, it would be worthy of wondering at (129b1-c3)
• Peterson's formulation of the principle Socrates has invoked:
• 1. It is not the case that an opposite itself is in itself its opposite.
• here, "in itself" means 'as necessitated by its definition'
• The idea seems to be that the definition of a thing such as "like" does not mention that it is a being or is like other beings or is a unity or anything else apart from what "like" is, and certainly that definition is not unlikeness and does not employ unlikeness.
  
• Socrates wants to point out things such as "like" and "unlike" and distinguish them as separate. He uses the word for "separate" at 129d7 for the first time.
  
• Peterson says all that Socrates wants here is that these things such as "like" and "unlike" be separate from each other.
• At 129d8-e1, we get several examples of things that are separate from each other: likeness, unlikeness, multitude, one, rest, motion, "and all the rest."
  
• Note that they are separate from each other but not necessarily opposite to each other.
• How are they 'separate'?
• At 142b7-c5, much later, Parmenides suggests that each one supplies different information: if you say "one," it conveys different information from "like" or "motion" or "whole"
• that may be what is meant here, at this early stage
• It seems that Socrates wants to say that:
• the thing that "like" is is probably not its own opposite: that would be 'surprising'
  
• the thing that "like" is probably does not contain or convey both of a pair of opposites: that would be surprising.
• the thing that "like" is probably is not something else such as motion or one : that too would be surprising.
• Why would those things be surprising?
• Because they involve incompatibility.
• Because they would mean that the informational content of things such as "like" or "motion" would convey more than just "like" or "motion"
• An interesting aside:
  
• A new claim about "separation": the things that have "likeness" or "motion" or "oneness" are separate from "likeness" or "motion" or "oneness"
• This suggests what some thinkers have called "immanent" "characters":
• Namely, the "oneness" that I have is different from the "oneness" that you have, and both are different from "oneness" itself, which we both have.
• We could label them: my oneness is Oneness_me whereas your oneness is Oneness_you: so they are only 'different' from each other in that there are two of them and one is attached to me and the other to you: otherwise, they are identical to each other.
• In the language of the theory of Forms from other dialogues, there is the form and there are instances of it in particulars. Those instances are "immanent characters"
• Parmenides, around 130b, presses Socrates to say what things have forms:
• Does "human"? Socrates hesitates.
• Mud and hair, however, have no forms! But Socrates is worried that he might be wrong.
• Parmenides suggests that this is a mistake: Socrates should admit forms of hair and mud! Why?
• Parmenides' Spatial Conundrums
  
• 131a-b: do the things that share in a form share in it as a whole or only in a part of it?
• Parmenides' question would apply very well to a pie: we can't all have the whole pie if we are sharing it. Each person gets only a part, and that requires that the pie be divisible.
• But there are things we can "share" that are not like that: Socrates suggests we all share "a day." Peterson gives more examples: sharing a name, sharing a grandmother, etc.
• Is the whole thing in each of the things that share it? asks Parmenides.
• This too is an objection that applies only to certain sorts of things: think of the pie above and the grandmother above. We cannot share a pie without it being in us, but we can share a grandmother without her being in us either as a whole or in part (assuming certain things about what a grandmother is and that people don't extend in whole or in part into other people...).
• Parmenides is assuming that forms take up and exist in space and are instantiated in space.
• There is no reason to assume that, however.
  
• The "Third Man"
  
• 132a: I suppose you think that each form is one on the following ground: whenever some number of things seems to you to be large, perhaps there seems to be some one aspect [idea], the same as you look at them all, and from that you conclude that the large is one.
• Socrates accepts this: it is the "one-over-many premise":
• Whenever several things are F, there is one form, F-ness, that is over them all.
• Socrates also accepts that this one F over many F's is itself F: self-predication.
• F-ness is itself F.
• Socrates also accepts the need for additional F-nesses:
• The F-ness that is over Fs is not over itself.
• Thus to explain how F-ness and the other F's are F, we need a further F-ness.
• Let's call it F-ness*
• and then we'll need F-ness**
• and F-ness***
• etc.
• Peterson suggests that Socrates should not have accepted the need for additional F-nesses, for the following reasons:
• F-ness itself may be F, but there is no reason why it isn't F by virtue of itself.
• For instance, it seems obvious that a triangle is (or is not) a triangle by virtue of whether it fits the definition of triangle.  And the definition of triangle is, says Socrates, triangular (it contains nothing other than what a triangle is) by virtue of itself.
  
• In other words: If the form of triangle is triangular and also the thing to which all particular triangles must conform as long as they are triangles, there is no problem.
  
• 132b5: Socrates suggests that each form is a thought and occurs only in minds!
• He does so in an attempt to avoid the third man argument but still be able to claim that each form is one (not many)
• Socrates immediately drops this interesting suggestion.
• But he clearly wants to retain the claim that each form is one.
• 133a8: Do you see, Socrates, how great the impasse is if one marks off forms that are themselves by themselves... You almost do not yet touch on it-- how great the impasse is if you will posit one form for each of the things that are whenever you make a distinction.
• Interestingly, right after this, Parmenides hints that it still may be possible to meet his objections.
  
• Parmenides gets Socrates to admit that
  
• Some things are relational: knowledge is knowledge of something; being a master is being a master of something; etc.
• Generally, relational things take the form: X of Y, where X is the form in question and Y is the thing it is of.
  
• But the form of knowledge itself relates it to subject matter itself, the form 'subject matter,' not any particular subject matter.
• The idea is that the form of "father" itself  relates it to "offspring" itself but not to any particular offspring.
• For any given relational X,  the X itself is defined in relation to the Y itself.
  
• Then Parmenides ties Socrates into knots about whether a particular can relate to a thing itself
• He asks whether a particular instance of a relational, say Master Jones, is master of Slave itself.
• Socrates says "no.".
• Then Parmenides asks whether a particular human's knowledge can be of X itself.
• Socrates feels compelled to say "no" here too.
• That means that there can be no knowledge of the forms in us, because we always have particular knowledge, and a particular cannot be of an X itself (a form).
• Parmenides asks if god has the most precise knowledge (i.e. the form knowledge?) and gets Socrates to admit that that knowledge cannot be of us (we are particulars, not forms).
  
• Peterson thinks that Parmenides has taken advantage of Socrates:
• Maybe Socrates should not have said "no" to the question whether Master Jones is master of Slave itself.
• He should have pointed out that considered in and of themselves, the forms of relational things can only employ the forms of the things they relate to, but particular instances of relational things are different...
  
• It's not clear what Socrates' solution should have been, but Parmenides still encourages Socrates (135a7ff), which Peterson takes to be a cryptic sort of message that forms may still be salvageable. Remember that cryptic message in Euthyphro?
  

• The rest of Peterson's article is devoted to that long and difficult part of the Parmenides that comes in the late 130's and goes to the end of the dialogue, the longest argumentative passage in Plato.
• Parmenides suggests that Socrates needs practice.
• The drills will consist of the following.
• 135e: first hypothesize that a thing is and examine the consequences, then hypothesize that it is not and examine the consequences.
• And do this about any thing that you think is or has any other attribute!
• 136b8ff: You must examine the consequences for the thing you hypothesize in relation to itself and in relation to each one of the others ... and in turn you must examine the others, both in relation to themselves and in relation to whatever other thing you select on each occasion, whether what you hypothesize you hypothesize as being or as not being.
• So, one must go through all the permutations of consequences and positions.
• From the late 130's on, Parmenides examines the one itself, in 195 arguments.
• First from the hypothesis that the one is: call it H
  
• Then from the hypothesis that the one is not: call it ~H
  
• There are 8 clearly announced sections: 137c, 142b, 157b, 159b, 160b, 163b, 164b, 165e.
• The first four are H, the last four are ~H.
• Within these two halves, there are two about the one itself, and two about "the others."
• At 160b and 166c, at the end of each half, we are told that apparently contradictory results were achieved.
• But Constance Meinwald has come up with a way that the results are only apparently contradictory:
• The first section tells in detail what the one is not in itself.
• The second section tells in detail what the one is in relation to others.
• basically, these drills respond to Socrates' challenge to Parmenides and Zeno.

• What of the theory of Forms?
• Peterson thinks that the Parmenides depicts a "youthful" Socrates who adopts various positions that are untenable, such as the position that because a form of a relational can only employ the form of what it is related to, there can be no knowledge of the forms in a particular human. Purified of those positions, he can continue to speak of forms.
  
• In the various dialogues, Socrates either deals explicitly with interlocutors' admitted beliefs OR uses claims that they find acceptable, and so we cannot take every individual statement about forms in every passage at face value: they have contexts.
  
• To figure out PLATO'S theory of forms, Peterson suggests, we must look at what persistently survives examination in all the dialogues. This article does not claim to identify those things, but she thinks that Socrates' challenge to Parmenides and Zeno near the beginning of the Parmenides does contain some of them.
  
• There must be distinguishable aspects (forms themselves) that make discourse possible. Self-predication, carefully formulated, seems to work.
  
• Evidence? Why create a youthful Socrates to be disabused of problematic aspects of a theory of forms after having created a middle-aged one who still held those problematic aspects? That's pretty bad narrative!
• The bad aspects that Socrates is cured of in the Parmenides may recur in later dialogues, but they do so only because they are persuasive to or avowed by interlocutors.