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The Likeness Regress: Plato's Parmenides 132Cl2-133A7 THE LIKENESS REGRESS THE LIKENESS REGRESS PLATO'S PARMENIDES 132C12-133A 7 By KARL DARCY OTTO, B.A., M.A. A Thesis Submitted to the School of Graduate Studies In Partial Fulfilment of the Requirements for the Degree Doctor of Philosophy McMaster University © Copyright by Karl Darcy Otto, July 2003 DOCTOR OF PHILOSOPHY (2003) Mc Master University (Philosophy) Hamilton, Ontario TITLE: The Likeness Regress: Plato's Parmenides 132cl2-133a7 AUTHOR: Karl Darcy Otto, B.A. (Toronto), M.A. (McMaster) SUPERVISOR: Professor David L. Hitchcock NUMBER OF PAGES: x, 147 11 Abstract Since Forms and particulars are separate, Plato is left with the task of describing the way in which they are related. One possible way of con­ struing this relation is to suppose that particulars resemble Forms. Socrates proposes this and is refuted by Parmenides in the so-called Likeness Regress (Parmenzdes 132c12-133a7). This work comprises both an exposition and an analysis of the Likeness Regress. In the exposition, I work out the argument-form of the Like­ ness Regress in second-order logic (and later, show that first-order logic is sufficient). This symbolisation provides a baseline for the balance of the exposition, which has two focusses: first, I define what it means for par­ ticulars to resemble Forms, with the help of D. M. Armstrong's account of resemblance in A Theory of Unwersals; second, I demonstrate that the infinite regress argument of the Likeness Regress is indeed vicious, with the help of T. Roy's theory of regress arguments. In the analysis, I proceed with the premiss that an asymmetrical account of the resemblance relation would allow Socrates to escape Parmenides' refu­ tation. I examine various accounts of asymmetrical resemblance (including those accounts put forward by R. E. Allen, P. T. Geach and G. Vlastos), but reject these in favour of my own account. My account of asymmetrical resemblance is based on understanding the resemblance relation as a func­ tion that is not self-inverse. Finally, I argue that the Likeness Regress need not be considered definitive, since we find in the ontology of the Timaeus a conception of resemblance that fits my account of asymmetrical resem­ blance. lll Acknowledgements I gratefully acknowledge the guidance and encouragement of my supervi­ sor, Dr. David Hitchcock, without whose assistance this thesis would never have been completed. His timely and detailed comments were invaluable, and greatly improved the focus and quality of my work. I should also like to thank Dr. Ken Darter for his insightful comments, and for being willing to take a final draft of my work with him on vacation. And I am indebted to Dr. Spiro Panagiotou, both for his perceptive criticisms of this thesis, and his kind advice and friendship over the past several years. Thanks is also due to the many people who have discussed various as­ pects of this thesis with me, including Brian Burge Hendrix, Justin Busch, Rodney Cooper, Ravi Danesh, David Godden, Nick Griffin, Roland Packer, Conrad Rogers and Tony Roy. Finally, no less thanks is due to those who have supported and encouraged me during the writing process, including Kiersten Burge-Hendrix, Chris Richinson, James Steeves, Brian Switzer, Charissa Varma, my family: Aristotle, Mum, Dad, Kira, Dan and Bamma. And I should especially like to thank Helena Viveiros for her dedication and extraordinary patience over the past year (cptAW crs). lV To my parents. Without their support, this would not have been possible; without their love, this would not be meaningful. v Contents Abstract llI Acknowledgements IV Contents vi List of Figures IX List of Tables x Introduction 1 Participation . 2 Participation in the Parmenides . 3 Participation as Resemblance 5 Objectives and Structure . 7 I Exposition 9 1 The Likeness Regress 10 1.1 Socrates' Thesis (132c12-d5) . 12 1.1.1 First Sentence (132dl-3) .. 12 1.1.2 Second Sentence (132d3-5) . 18 1.2 Three Additional Premisses .... 20 1.2.1 First Premiss (132d6-9) .. 21 1.2.2 Second Premiss (132d10-e2) 27 1.2.3 Third Premiss (132e3-5) 27 1.3 The Regress (132e6-133a4) ... 32 1.4 The Upshot (132e6-7, 133a5-7) 36 VI 2 Resemblance and the Regress 38 2.1 Defining Resemblance . 39 2.1.1 A :dodern Account .. 39 2.1.2 Resemblance for Socrates . 42 2.1.3 Three Difficulties 44 2.2 The Infinite Regress 49 2.2.1 Two Difficulties . 50 2.2.2 Infinite Regress Arguments . 52 2.2.3 Benign and Vicious Regresses 57 2.2.4 Application to the Likeness Regress 61 2.2.5 Loose Ends ... 65 2.3 Conclusion of Exposition ......... 68 II Analysis 69 3 Analysis of Resemblance 70 3.1 Defending Socrates . 70 3.2 \i\lords for "Resemblance" 71 3.3 Forms as Jtcrpcrodyµcri:cr 74 3.3.1 Models ...... 76 3.3.2 Patterns . 95 3.4 Asymmetrical Resemblance 102 3.4.1 Expressing Resemblance as a Function 102 3.4.2 r() is not Self-inverse . 104 3.4.3 A New Defence of Socrates . 104 4 Resemblance in the Timaeus 106 4.1 The Three Kinds ... 108 4.1.l The Summary . 109 4.1.2 The Receptacle 111 4.1.3 Particulars . 122 4.1.4 Forms . 128 4.2 Resemblance ..... 129 4.2.1 That Which Resembles Form 129 4.2.2 Structural Patterns . 132 4.3 Evaluating the Regress . 135 4.3.1 Symmetry and Asymmetry. 137 Vll 4.3.2 Denial of Symmetry 139 Conclusion 140 Two Questions Answered . 141 Four Questions Raised 142 Literature Cited 144 Vlll List of Figures 2.1 Elements Related by R . 54 2.2 Infinite Series of Boxcars . 58 2.3 Force-body Diagram for Boxcar b 1 59 2.4 Infinite Series of Resemblance ... 63 3.1 Appassionata - Sonata No. 23 inf minor . 96 3.2 Curves on a Cartesian Plane . 101 4.1 Tetrahedron . 124 4.2 Equilateral Surface .... 124 4.3 Cube Defined by Vectors . 134 lX List of Tables 3.1 Socrates' Words for ·'Resemblance" .. 72 3.2 Parmenides' Words for "Resemblance" 73 4.1 Parts and Wholes . 125 4.2 Structural Patterns 133 x Introduction xcrl wGw o\hz µ~ rrtx.6CJY)Wl rro-rz o\.hz ~p~m:o vuv, &'A'A' fon "'CO "'COLOU"'COV, we; Eµol cptx.tvnm, "'CWV A6ywv C{U"'CWV &Scrvtx."'COV n xtx.l &y~pwv rr<X8oc; EV ~µtv. Plato, Phlb. 15d6 8 Helen and Penelope are both beautiful. We know from this statement. without being supplied with any additional information, that Helen and Penelope have something in common. But difficulties arise when we try to be more specific about what it is they hold in common, or how they do so. If either Helen or Penelope possesses beauty as one may possess a physical object, it would not be possible for them to have beauty in common: the beauty possessed by Helen would be different from the beauty possessed by Penelope. We might as well give the beauty of Helen and the beauty of Penelope different names, and consider them to be unrelated. So we may rule out the possibility of their possessing beauty as one may possess a physical object. But if beauty is not possessed in this way, we return to our initial question: what is this beauty that Helen and Penelope have in common, and how do they possess it? Socrates gives his "safe answer" to this question in the course of describ­ ing his oe:ui:e:pov n:AoOc; ("second sailing," Phd. 100d4-8):1 the particular "is made beautiful in virtue of nothing other than the presence of or the share in Beauty, or in whatever way [the particular] is attached [to the Form]." This assertion is not restricted to that which is beautiful, but is generalisable to an unspecified extent (cf. "I speak thus for all," Phd. 100c4-6). 1References to the Parmenides, the Timaeus and the Laws follow the line numbers in the Bude editions: all others follow the line numbers in the first OCT editions. 1 Ph.D. Thesis - K. Darcy Otto - l\JcJ\Jaster - Philosophy 2 Socrates' answer provides a solution to our question by distinguishing between the character of beauty, which both Helen and Penelope have, and that Form in virtue of which Helen and Penelope have this character. Helen and Penelope both have the same character of beauty not by possessing beauty as one may possess a physical object, but by being related in some way to Beauty. But this solution replaces our initial question with one that is almost as virulent: what is the Form, and how are particulars related to it? In what follows, we shall focus on the second part of this question, and attempt to determine how particulars are related to Forms. Participation The names Socrates uses to describe the relation between Forms and particulars are µs-csxe:tv (e.g., Phd. l00c5, Smp. 2llb3, R. 402d3) and µnaf.aµ~cX.ve:tv (e.g., Prm. 129a3) - verbs that we translate as "to partici­ pate" when employed in a technical sense. The non-technical sense of these verbs typically indicates a relation of sharing. For example, µe:-caf.aµ~cX.ve:t -co nsµn-cov µspoc; -cwv ~~c.pwv means that one shares in a fifth part ( ac­ cusative) of the votes (partitive genitive) (cf. Ap. 36a7-b2; Smyth 1956, §§1343-4). Thus, it seems reasonable to suppose that the technical sense of these words is parasitic upon the non-technical sense: Penelope has a share of Beauty. But it is doubtful that this does much to elucidate the nature of participation.
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