Part I. Introduction
Introductory Conversation (126a-127a)
Characters and Setting (127a-d)
Zeno's Paradox (127d-128e)
Socrates' Solution to Zeno's Paradox (128e-130a)
Part II: Problems with Forms (130a-135d)
a. The Extent of Separate Ideas (130a-e)
b. The Dilemma of Participation (131a-c)
c. The Paradox of Divisibility (131c-e)
d. The Largeness Regress (131e-132b)
e. Ideas as Thoughts (132b-c)
f. Ideas as Paradigms (the second regress) (132d-133a)
g. Separation and Unknowability (133a-134e)
h. Conclusion: Forms are necessary for thought and discourse. (134e-135d)
Part III: Parmenides gives Socrates training in dialectic
(Hypotheses about Unity) (135d-166c)
NOTE: This outline of the dialogue's structure follows that of R.E. Allen in his book, Plato's Parmenides: Translation and Analysis (Minnesota 1983).
Allen interprets the dialogue as "aporetic". An "aporia" (aporia) is a puzzle, knot, or blocked passageway. Aristotle often begins a work by stating a number of "aporiai" about a subject; these are initial puzzles that need to be resolved. Often they are apparent dilemmas that really offer a middle way out. Allen sees Plato as launching this method here in the Parmenides.
The "second half" (Part III) of the dialogue is much longer than the "first half" (Part II). Part III involves a complex succession of proofs about Unity (which is supposed to be Parmenides' own One Being, but seems more like a form of Unity). On the assumption that Unity is, a complex number of contradictory conclusions are proven to follow. This part of the dialogue has been seen as an exercise in dialectic for Plato's students, as a philosphical joke, or as a dense maze in which we must pick out certain pointers toward the truth about participation and Forms.
There is a tremendous literature about the so-called "Third Man Argument" in philosophical journals. For more references, consult my Parmenides bibliography.
Most commentators now begin from Vlastos' analysis of the argument's premises, and his discussion of how the argument works. They often diagnose the problems of both regress arguments to show that they require similar premises, often described in these ways:
There is exactly one form corresponding to every predicate that has a form.
Any form can be predicated of itself. (Largeness is large).
Non-Identity (NI) If a thing has a certain character, it cannot be identical with the form in virtue of which we apprehend that character.
One-Over-Many (OM) If there is a set of things with one predicate, then there is one form for that predicate, and the members of the set have that predicate in virtue of that form.
See also an outline of the argument.
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