Aporia vol. 20 no. 1—2010 Plato’s Parmenides:Interpretations and Solutions to the Third ManMi c h a e l J. ha n s e nPlato’s Parmenides fulfills a special role in understanding Plato’s the-ory of Forms. Throughout his dialogues, Plato is pleased to invoke the theory intermittently, often as ancillary support for related pieces of his philosophy. But the theory itself remains concealed, even mysteri-ous, never giving up a broad exposure for the reader’s immediate analysis. Hence, what we know of Plato’s Forms comes from piecemeal assembly.1 Given this situation, the attacks levied against the Forms in the Parmenides may reveal vital aspects of the theory and perhaps even betray Plato’s own misgivings about the Forms. In this paper I will not consider whether Plato maintained his theory in the face of these critiques. Rather, I will pres-ent the famous Third-Man Argument from the Parmenides, review some competing perspectives on the argument among Plato’s interpreters, and defend Gregory Vlastos’ interpretation due to its historical plausibility.I. The Third-Man ArgumentThe Third-Man Argument (hereafter TMA) appears early in the Parmenides (132A1–B2). It is given in two varieties in the dialogue, but 1 For a comprehensive listing of Plato’s comments regarding the Forms, see “Chapter 28: Form” of The Great Ideas: A Synopticon of Great Books of the Western World (Vol. II). Encyclopaedia Britan-nica (1952), 536–41.Michael Hansen is a senior majoring in philosophy at Brigham Young University with minors in logic, linguistics, and ancient near eastern studies. His primary academic interests include ancient philosophy and philosophy of religon. He will be applying to graduate programs in the fall. This essay placed third in the 2010 David H. Yarn Philosophical Essay Contest.Mi c h a e l J. ha n s e n66there is rarely any instructive difference between the first and the sec-ond formulation. In this section I will formalize Parmenides’ argument so it can be analyzed apart from the redundancy of the text. My for-malization is not groundbreaking, and it inherits much of the work from previous formulations.2To understand the initial premises of the TMA we need only con-sult Plato’s arguments for the Forms. First, Parmenides begins with the principle of Uniqueness. The principle appears in Plato’s One-Over-Many argument. The basic idea is that when a number of things share a com-mon property, they do so by virtue of a single Form of that commonality. However, my wording here may betray certain modern nuances of the universal-particular relationship, so I will quote from the Parmenides to establish exactly what Plato has agreed to, which is commonly understood as Uniqueness:I suppose you think each form is one on the following ground: whenever some number of things seem to you to be large, perhaps there seems to be some one character, the same as you look at them all, and from that you con-clude that the large is one. (132A)3This passage is the final summary of the Forms from the dialogue, and it appears immediately before Parmenides presents the TMA. We will soon examine other references to the Forms found earlier in the Parmenides, but this passage illustrates the Uniqueness premise nicely. I will formalize this premise in the following way:(U) If multiple things (a, b, and c) are all F, then there must be a single Form, F-ness, by which they exhibit the same quality.Thus, where multiple things are large, there must be a unified Form of “Largeness” to account for their commonalty. This is intended to hold for any quality shared among things: where things are F, there is F-ness.Second, Parmenides establishes the principle of Participation. The principle has roots in Plato’s Phaedo where he proposes that “each of 2 My rendering of these premises is modeled after Vlastos’ original formulation of the TMA, with the added benefit of being informed by subsequent articles responding to his, with only a few of my own stylistic differences.3 It is interesting that Plato has allowed such a contextual relation as “large” to represent his Forms in the above passage, but this is no new development since we read about tall and tallness in the Phaedo. In fact, it is readily agreed upon that “the theory that Socrates presented at the beginning of the [Parmenides] is plainly the one developed in Symposium, Phaedo, and Republic” (Cooper 359). For this reason I feel no need to inspect these accounts of the Forms to look for develop-ments in Plato’s thought in the Parmenides: let us proceed by treating the theory as a mostly stable target to this point.Pl a t o ’s Pa r m e n i d e s 67the Forms existed, and that other things acquired their name by having a share in them” (102B). We find similar summaries of Participation from the Parmenides in 128E6–129A6, 130B, and 130E5–131A2.4 This estab-lishes the link between Forms and particulars. Thus, let us formulate the Participation premise in the following way:(P) If multiple things (a, b, and c) exhibit F, then they exhibit that quality through participation in the Form of F-ness. There is no need to be suspicious of conflating the appearance of F and the name of F in (U) and (P) respectively: both regard our account of things in the world. But it appears that (P) is an extension of (U), then. Notice that (U) proposes the singleness of F-ness, while (P) supplies the means by which things exhibit the characteristic F. Plato has distinguished these premises in his dialogue, but for simplicity’s sake let us collapse (U) and (P) into a single premise:(UP) If multiple things (a, b, and c) are all F, then there must be a single Form, F-ness, by which they exhibit the same quality through partici-pation. However, the above premises are alone insufficient to effect Par-menides’ conclusion that any given Form is not one, but infinitely many. The TMA itself requires additional, tacit premises in order to address the theory of Forms. This is where the TMA becomes especially informative about Plato’s theory. Because it appears that Plato thought the argument was a valid one, we are justified in searching out whatever assumptions are necessary to justify the TMA and how they impact the Forms. The greatest potential for understanding