Plato’s Contest: Answering the Challenge of the Parmenides

Ryan Haecker

Plato’s contest for the early Academy was to answer Parmenides’ criticisms of the Theory of the Universal Forms through an interpretation of the dialectical exercises presented in the Parmenides (§I). Plato’s Theory of the Universal Forms and the objections of the Parmenides may now be more precisely formulated in predicate logic using the notational convention developed by Edward Zalta (§II). The two principal objections to Plato’s Theory of the Universal Forms are the Third Man Argument and the Greatest Difficulty Argument: the Third Man Argument can be answered by Constance Meinwald’s distinction of Self-Predication and Gail Fine’s distinction of Non-Identity (§III); and the Greatest Difficulty Argument implies the inconsistent set of Russell's Paradox, yet may be answered through the construction of a hierarchical set theoretical model that subsumes and restricts the semantic scope of each subordinate hypothesis (§IV). The rejection of the external predication of the Third Man Argument and the two-world ontology of the Greatest Difficulty Argument suggests a monistic ontology of internal relations in the Concrete Universal form of all forms (§V).

Plato’s Contest: Answering the Challenge of the Parmenides By Ryan Haecker December 2014 Abstract: Plato s o test for the early Academy was to answer Parmenides riti is s of the Theory of the Universal Forms through an interpretation of the dialectical exercises presented in the Parmenides (§I). Plato s Theory of the Universal Forms and the objections of the Parmenides may now be more precisely formulated in predicate logic using the notational convention developed by Edward Zalta (§II). The two principal objections to Plato s Theor of the U i ersal For s are the Third Ma Argu e t a d the Greatest Diffi ult Argu e t: the Third Man Argument can be answered by Constance Mei ald s disti tio of Self-Predication and Gail Fi e s disti tio of Non-Identity (§III); and the Greatest Difficulty Argument implies the inconsistent set of Russell's Paradox, yet may be answered through the construction of a hierarchical set theoretical model that subsumes and restricts the semantic scope of each subordinate hypothesis (§IV). The rejection of the external predication of the Third Man Argument and the two-world ontology of the Greatest Difficulty Argument suggests a monistic ontology of internal relations in the Concrete Universal form of all forms (§V). I. The Challenge of the Parmenides The Parmenides recounts a dispute between the elderly Parmenides (age 65), the mature Zeno (age 40), and the young Socrates concerning the Theory of the Universal Forms, or Ideas.1 After reading a treatise on the absurdity plurality of beings, Zeno is questioned by Socrates as to whether the same argument might also repudiate the universal forms, (127e) and confirms that Socrates has correctly understood that the purpose of his argument is to defend Parmenides by responding to “those who assert plurality” by showing that the assumption “that there is a plurality leads to even more absurd consequences than the hypothesis of the one.” (1β8d) Parmenides then joins the controversy with a battery of explosive criticisms to challenge the Theory of the Forms. (130a-134e) He concludes that “these difficulties and many more besides are inevitably involved in the forms” (1γ5a), and recommends a “severe training” of dialectical exercises by which the theory might yet be saved. (135d) No consensus has yet been reached on how to interpret these bewildering exercises.2 Thomas K. Seung describes them as “the most obscure and enigmatic pieces Plato ever wrote.”3 William F. Lynch, S.J. calls the Parmenides the “supreme puzzle of ancient philosophy.”4 Many ancient scholars interpreted the Parmenides as a discourse on theology which described "all things that get their reality from the One."5 Some modern scholars have - more modestly – interpreted the dialogue as either a “record of honest 1 Although Plato seems to prefer the term Idea (ἰ έα or ἶ ος), this essay will assume the Aristotelian o e lature of u i ersal for s to ore learl disti guish u i ersal for s fro parti ular i sta es. 2 Kenneth M. Sayre reports that despite al ost t o ille ia of do u e ted o e tar , ho e er, s holars today are still struggling to make sense of the dialogue. Cf. Parmenides' Lesson, 1996: XI 3 Seung, Thomas K. Plato Re-Discovered: Human Value and Social Order, 1994: 185 4 Lynch, William F. An Approach to the Metaphysics of Plato through the Parmenides, 1959: 3 5 Proclus' Parmenides Commentary 638.18-19; For a summary of Neo-Platonist interpretations of the Parmenides see Joh Dillo s i trodu tio to Proclus' Commentary on Plato's Parmenides. perplexity” or as merely a “gymnastic exercise, not a disclosure of supreme divinity.”6 Since the criticisms of the Parmenides present Plato’s most explicit examination of the Theory of the Forms, interpretations of this dialogue may establish the place of the universal forms in Plato's mature philosophy: if Parmenides’ criticisms may be answered then Plato could have affirmed, but if not then Plato should have rejected, the Theory of the Universal Forms.7 Scholarly disagreement on the interpretation of this dialogue thus pivots on the gigantomachy of Plato’s Academy between the idealist 'gods' who defended the universal forms and the materialist 'giants' who wished to “drag everything down to earth out of heaven.”8 (246a) If the Platonic dialogues may be read as a dramatic conflict, in which each unresolved aporia ends in tragedy, then the Parmenides concludes at the height of tragic agony: for not only do the criticisms of Parmenides deal a devastating blow to the central pillar of Plato’s ontology, but neither do the dialectical exercises clearly provide any satisfactory answer. Plato's contest for the philosophers of the future was to discover a satisfactory interpretation of the dialectical exercises, to save the Theory of the Universal Forms, and to answer the challenge of the Parmenides. In the celebrated dialogues the Phaedo, the Republic, and the Symposium, Plato expounds his famous Theory of the Universal Forms. Heraclitean Flux had implied that at every moment any sensible object must possess some contrary opposite properties.9 (402a) Plato argues to the contrary (96a) that, if sensible objects must possess contradictory properties F and not-F then nothing can be explained; yet since explanations should be possible, there should be some supersensible universal forms with which to explain all properties in sensible objects.10 (72c) The Theory of the Universal Forms is thus the hypothesis that, if we are to ever explain a plurality of sensible objects that each share some common property we should, to avoid contradictions, postulate there to be one unchanging supersensible universal form, which is itself the prior ground of being and necessary condition for knowledge of each property in each particular sensible object. To explain the transcendental conditions for the possibility of knowledge, Plato's Theory of the Universal Forms thus postulates an indefinite multitude of supersensible universal forms in a transcendent realm beyond the sensible realm of all appearances.11 6 Vlastos, The Third Ma Argu e t i Plato s Par e ides, The Philosophical Review Vo. 63, No. 3, 1954: 343; Cornford, Francis M. Plato and Parmenides, 1939: 131 7 Plato scholars can be roughly divided according to their interpretation of the the status of the Theory of the Universal Forms after the Parmenides: Unitarians believe that Plato did not revise his theory, while Revisionists elie e he did. Aristotle s o te porar testi o Metaphysics Α a29 & M1078b9) suggests that Plato neither answered these criticisms nor revised the Theory of the Universal Forms. The result is an apparent interpretive paradox: if Plato recognized the criticisms to be valid then he should have revised his theory, yet there is no explicit evidence for such a revision; while if Plato did not revise his theory then he should not have thought the criticisms to be valid, yet Plato gives no answer to the criticisms. Cf. John Pepple, Plato s A s er to “peusippus: 18 8 John N. Findlay plausibly associates this allusion to the gods and giants with idealist friends and materialist enemies of the Theory of Universal Forms. Cf. Plato and Platonism, 1978:12 9 Aristotle reports Plato respo ded to Hera litea Flu : Plato a epted [“o rates ] approa h ut as lead it to think that it must be concerned with things other than the sensible. For it is impossible to formulate a general defi itio of a se si le thi g, si e all is i flu . Cf. The Metaphysics, 987a 29 10 Gail Fine describes how since se si les suffer o prese e [of o trar opposite properties], there ust e o se si le for s that es ape o prese e. Cf. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 54-57 11 Gail Fine observes that both Aristotle and his commentator Alexander of Aphrodisias regarded this argument for the possibility of knowledge as the primary motivation for the Theory of the Universal Forms. Cf. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 49 Ryan Haecker © 2014 2 The resulting picture is a two-world ontological dualism in which the being of the supersensible universal forms are located in a transcendent world that is separated beyond the being of all sensible particular instances. (508e) However, this ontological dualism conflicts with Plato's epistemological monism. Since every property of a plurality of objects must have some single explanation, there must also be one explanation for these two-worlds of beings; yet since any explanation must postulate the being of one universal form over many particular beings, the consequence is a set of beings that is inconsistently both one and many.12 Moreover, since each universal form is itself a particular being when conceived of in relation to other universal forms, and every plurality of particular beings must be explained by some further universal form, this plurality of all universal forms must also be explained by one further universal form over all universal forms. The many criticisms against the Theory of the Universal Forms in the second part of the Parmenides (130a-134e) each result from Plato’s two-world ontological dualism.13 Thus Aristotle thus reports that "it is not possible to acquire knowledge without the universal, but separating is the cause of the difficulty arising." (Metaphysics 1086a32) Plato never explicitly answers the challenge of the Parmenides.14 Paul Elmer More writes: “We have the whole doctrine of Ideas subjected to a process of destructive logic to which Plato makes no direct answer either here or elsewhere.”15 If the dialectical exercises of the third part of the Parmenides fail to answer the challenge of second part of the Parmenides, then the Parmenides must conclude in a tragic aporia that leaves the greatest objections to Plato’s Theory of the Universal Forms unanswered. Socrates then asks: “What are you going to do about philosophy, then? Where will you turn while the answer to these questions remains unknown?” (1γ5c) The clearest indication of how Plato intends to answer this challenge is briefly hinted at in the transitional passage when Parmenides speculates: “Only a very gifted man can come to know that for each thing there is some kind, a being by itself; but only a prodigy more remarkable still will discover that and be able to teach someone else who has sifted through all these difficulties thoroughly and critically for himself.”16 (135a-135b) For the purpose of this instruction, Parmenides prescribes a "severe training" of dialectical exercises, to explore the semantic implications of supposing that "such and such a thing is" and "is not", so that the truth may not escape us.17 (135d-136a) 12 Thomas K. Seung observes that set theory continues the ancient Pythagorean priority of being while formal logic continues the ancient Eleatic tradition of pure form. Cf. Plato Re-Discovered, 1994: 215 13 Giovanni Reale co urs that all of the o je tio s of the Par e ides tur i their arious a d o ple a s o the o eptio of the i telligi le Ideas as separate fro se si le thi gs. Toward a New Interpretation of Plato, 1996: 226. For a similar opinion see also More, The Parmenides of Plato, 1916: 135 14 Harold F. Cherniss claims that Plato suggested an answer to the Third Man Argument at Republic (597c) and the Timaeus (31a). Unfortunately, these texts do not explicitly answer the argument without additional assumptions. 15 More, Paul Elmer, The Parmenides of Plato, The Philosophical Review, Vol. 25, No. 2, Mar., 1916: 128 16 This allusion to a great dialectician of the future echoes Socrates' prognostication of a future "great man" who may resolve the aporiae of the Charmides (169a). 17 Constance Meinwald observes that the third part of the Parmenides contains the longest single stretch of uninterrupted argument in the Platonic corpus (30 Stephanus pages) and concurs that it was meant to resolve the aporiae of the second part. Cf. Goodbye to the Third Man. In The Cambridge Companion to Plato: 366-7 Ryan Haecker © 2014 3 II. Definitions Edward N. Zalta has developed a notational convention for the expression of Plato's Theory of the Universal Forms in Fregean quantified predicate logic: ΦF signifies the universal form of property F; Fn signifies the set of all particular instances of F; and nF signifies the set of all particular instances of F that participate in ΦF.18 n = A discrete set of objects, e.g. {n| n1, n2, n3 ...} F = The property instantiation of a form, e.g. Fn Fn = Ordinary predication, i.e. F(n) or Fn nF = Participation predication, i.e. (n)F or nF Φ = The universal form in which all properties participate, e.g. ΦF Plato’s Theory of the Universal Forms, or Ideas, may accordingly be defined as the conjunction of seven hypothetical assumptions: One-Over-Many (OM); Uniqueness (U); Non-identity (NI); Separation (S); Self-Predication (SP); Participation (P); and Explanation (E): Theory of the Universal Forms ≝ OM & U & NI & S & SP & P & E OM = One-Over-Many: For any set of many particular instances of F, i.e. {Fn}, there exists one universal form of F, i.e. ΦF: ∃Φ∀F∀n((F = {Fn | Fn ≠ Fn−1}) → (n=ΦF)) U = Uniqueness: For any set of particular instances of F, i.e. {Fn}, there exists one and only one universal form of F, i.e. ΦF: ∃Φ∃x∀F∀y(x=ΦF & (y= ΦF → y=x)) NI = Non-Identity: The universal form of F, i.e. ΦF, is not equivalent to the instances of F, i.e. Fn: ∃Φ∀F∀n(ΦF ≠ Fn) S = Separation: The universal form of F, i.e. ΦF, can exist without the particular instances of F, i.e. Fn: ∃x∀n∀F(¬Fn → ΦF) SP = Self-Predication: The universal form of F, i.e. ΦF, is F: 18 ∃Φ∀F(FΦ) Zalta, Edward N. Abstract Objects: An Introduction to Axiomatic Metaphysics, 1983; and Ho to “a Good e to the Third Ma , co-authored with Francis Jeffry Pelletier, Nous, 34/2, 2000: 165–202 Ryan Haecker © 2014 4 P = Participation: For each particular instance of F, i.e. Fn, the predication of property F of instance n is equivalent to the universal form of F, i.e. ΦF: ∃Φ∀F∀n(Fn=Φ & ΦF) E = Explanation: A universal form, i.e. ΦF, explains every particular instance of that form, i.e. Fn: ∃Φ∃n∀F(ΦF ⊨ Fn)) One-Over-Many (OM) hypothetically postulates one universal form ΦF over every plural set of particular instances Fn for the purpose of giving an Explanation (E) of the particular instances. Uniqueness (U) stipulates that this universal form is one and only one. Non-Identity (NI) distinguishes and Separation (S) separates this universal form ΦF from the particular instances Fn. Self-Predication (SP) and Participation (P) predicatively conjoin the property instances Fn with the universal forms ΦF in participative identity. These seven assumptions thus signify three consecutive moments in the process of conceiving of the Theory of the Universal Forms: (α) One-Over-Many (OM) and Uniqueness (U) signify the first moment of the unity of the plural particular property instances Fn in one unique universal form ΦF; ( ) NonIdentity (NI) and Separation (S) signify the second moment of the logical and ontic division of this universal form ΦF from its particular instances Fn; and ( ) Self-Predication (SP) and Participation (P) signify the third moment of the re-unification of the particular property instances Fn with the universal form ΦF by the predication of the property F in participation with the universal form ΦF. The challenge of the Parmenides (130a-134e) results from the conflict between the (α) first and ( ) third moments of unity, uniqueness, and self-predicative participative identity with the ( ) second moment of division, separation, and non-identity.19 The six arguments against Plato’s Theory of the Forms are arrayed in ascending order of difficulty: (1) the Extent of the Forms Argument (130a-e); (2) the WholePart Dilemma (130e-131e); (3) the Third Man Argument (132a-b); (4) the Conceptualism Argument (132b-c); (5) the Resemblance Regress (132c-133a); and (6) the Greatest Difficulty (133a-134a).20 The first two arguments (1 & 2) suggest some problems defining the extent (1) and composition (2) of the forms. The next three arguments (3, 4, & 5) argue that the assumptions of One-Over-Many (OM), SelfPredication (SP), and Non-identity (NI) initiate an infinite regress of universal forms, regardless of whether the universal forms are defined as predicative (3), conceptual (4), or resembling (5) entities. The final and greatest difficulty (6) argues that, even if this infinite regress were blocked, then the assumptions of One-Over-Many (OM) and Non-Identity (NI) would nonetheless produce a two-world ontological dualism to render knowledge of the universal forms impossible. Answering the challenge of the Parmenides thus requires some demonstration of how the semantic distinctions of the Third Man Argument (§III) and - more importantly - the ontological divisions of the Greatest Difficulty Argument (§IV) may be altogether subsumed in-and-through an integrally united monistic ontology. 19 It is interesting to note that the structure of the Parmenides in many ways mirrors the structure of the criticisms of Parmenides: both begin by discussing the Theory of the Universal Forms, then proceed to discuss the semantic modalities of the universal forms, and conclude with some i pli atio s for Plato s ontology. 20 For a su ar of Par e ides s riti is s see ‘i kless, “a uel C., Plato s Par e ides, : http://plato.stanford.edu/entries/plato-parmenides/ Ryan Haecker © 2014 5 III. Answering the Third Man Argument The Third Man Argument argues that for any plurality of particular instances Fn1 (e.g. men) of a universal form ΦF2 (e.g. manhood), in which ΦF2 is self-predicated (i.e. ΦF2 is F, or F(ΦF2)), then One-Over-Many (OM) implies that there must be some third universal ΦF3 that is the universal form of both the universal form Fn1 and ΦF2 (i.e. Fn1(ΦF2(ΦF3))). However, since this third universal form ΦF3 is not identical (NI) to either ΦF2 or Fn1, One-Over-Many further implies that there must be some fourth universal form ΦF4, and so on ad infinitum (i.e. Fn1(ΦF2(ΦF3... (ΦF∞)))). This infinite regress of universal forms contradicts the assumption of the Uniqueness (U), such that there is one and only one universal form (e.g. manhood) for each plurality of property instances (e.g. men). More importantly, if there is an infinite regress of universal forms for any property, then no purported explanation, which assumes universal forms to explain particular instances, may ever explain anything at all for the simple reason that for any universal form we should require some further explanation and some further universal form, and so on forever.21 The devastating consequence implied by the Third Man Argument is that the universal forms, which purport to explain particular instances, could neither explain the particular instances nor even the universal forms themselves. The Third Man Argument is, however, an ignorant refutation that could only ever succeed in refuting a vulgarized version of Platonism in which the universal forms are re-conceived as particular substances rather than as universal forms. This is how, for example, Aristotle re-cast Plato's universal forms as separate and particular substances when he publicized the Third Man Argument in On Ideas, the Metaphysics, and the Sophistical Refutations.22 This re-conception of universal forms as particulate substances mistakenly assumes that each of Plato's universal forms adventitiously predicate itself as an externally related particular substance rather than as an internally and essentially self-related universal form. Plato's universal forms cannot, however, be understood to be externally predicated in this way because, according to Participation (P), each particular instance must be qualitatively determined by its own internal participation in a universal form.23 The Third Man Argument thus conflates two distinct kinds of predication: the ordinary predication of externally related particular instances and the extraordinary predication of internally-related universal forms. Bertrand Russell recognized this error and proposed to distinguish the meaning of the ‘is’ of identity that implies the equivalence of all the properties of the terms from the meaning of the ‘is’ of predication that merely implies that the predicate term signifies some properties of the subject term.24 21 Fine, Gail. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 204 Cf. Aristotle s On Ideas 93.1; The Metaphysics, 990b17, 1079a13, 1039a2; Sophistic Refutations 178b36 23 John Niemeyer Findlay, Plato: The Written and Unwritten Doctrines, 1974: 33 "The [Third man argument is] among the most total ignorationes elenchi in the whole of philosophical history. For they assume that Plato believed, in full seriousness, in a world of firmly identical particular existents, sorted into classes by their intrinsic character and behavior, and that he then gratuitously invented a second world of detached Eide [Ideas]... Whereas the whole thrust of Platonism... was to deny that there was anything genuinely seizable and knowable, or anything truly causative and explanatory, in the flowing realm of particular things and matters of fact as such: what was seizable, what was knowable, what could truly imprint itself on and maintain itself in flux, and give purchase to our recognition, was always an Eidos [Idea]." 24 Hansen, Michael J., Plato s Par e ides: I terpretatio s a d “olutio s to the Third Man, Aporia vol. 20 no. 1, 2010: 69 22 Ryan Haecker © 2014 6 Constance Meinwald has further refined this semantic distinction into a distinction between garden variety pros ta alla predication in relation to others and tree type pro heauto predication in relation to itself: pros ta alla predications are “our common garden predications” in which the meaning of the predicate is extrinsic and not essential to the meaning of the subject, while pro heauto predications are special genus-species tree predications in which the meaning of the predicate is intrinsic and essential to the meaning of the subject “in virtue of a relation internal to the subject’s own nature.” Meinwald explains that pro heauto predication is exemplified in the genus-species tree in which “what it is to be S [species] is to be P [genus] with a differentia” so that the genus P may be predicated of the species S just as S is self-predicated of S because pro heauto predication is “grounded in the structure of the nature” of S.25 Meinwald claims that her pro heauto predication “straightforwardly answers” the objection of the Third Man Argument because the meaning of pro heauto predications are intrinsic rather than extrinsic to the essential structure of the subject, so that the self-predication of the universal forms “will always be true” in virtue of their essential and self-related nature. Constance Meinwald concludes that Plato composed the third part of the Parmenides to guide the reader to recognize how Socrates’ immature “mistake in semantics” could be answered by Plato's mature semantic distinction.26 However, Bryan Frances objects that any pros ta alla self-predicated universal form would initiate an infinite regress of universal forms, and so Meinwald’s answer to the Third Man Argument would fail if there were even one universal form that could be predicated pros ta alla. Frances observes at least four examples of universal forms (e.g. Being, Rest, Eternality, and the One) that can be predicated pros ta alla on Meinwald’s interpretation of the Parmenides, and thus concludes that “Meinwald has [only] proven that Plato had a plausible partial response to the third man argument.” Frances recommends instead, as there is “both reason to reject and no reason to accept” this assumption, Plato could have “solved the third man argument in the Parmenides” by rejecting Non-identity.27 Gregory Vlastos argues, because the Third Man Argument (TM) crucially depends on the premises of One-Over-Many (OM), Self-Predication (SP), and Non-Identity (NI) (i.e. TM ≡ OM & SP & NI), and Non-Identity is inconsistent with Self-predication (SP ⊥ NI), that no universal form can be consistently predicated of itsef (viz. Self-Predication) and the Third Man Argument is invalid.28 Vlastos concludes that he “can show that Plato had a perfectly good way of refuting the Third Man Argument” by rejecting non-identity, but that this very rejection of non-identity “would have been fatal to the separation theory and the degrees of reality theory which are central to [Plato’s] explicit metaphysics.”29 However, Gail Fine argues, to the contrary, that Plato’s ontology can be saved by the distinction between the logical nonidentity and the ontic separation of the universal forms and the particular instances: logical Non-Identity (NI) is minimally the assumption that the particular instances of a form (nF) are not identical in every respect to the universal form itself (i.e. NI ≡ ΦF ≠ nF); while ontic Separation (S) is the additional 25 Meinwald, Constance. Goodbye to the Third Man, in The Cambridge Companion to Plato: 378-80 Meinwald, Constance. Goodbye to the Third Man, in The Cambridge Companion to Plato: 374 27 Fra es, Br a . Plato s ‘espo se to the Third Ma Argu e t i Par e ides, Published in Ancient Philosophy v. 16,1996: 47-64 28 Vlastos, Gregor , The Third Ma Argu e t i Plato s Par e ides, The Philosophi al ‘e ie Vo. , No. , : 326 29 Vlastos, Gregor , The Third Ma Argu e t i Plato s Par e ides, The Philosophi al ‘e ie Vo. , No. , : 343,348 26 Ryan Haecker © 2014 7 existential assumption that the universal form is separate from its particular instances if the universal form can exist without their particular instances (i.e. S ≡ ¬⋄∃x∀F(nF) → □∃Φ∀F(ΦF)). Fine concurs that the Third Man Argument requires the assumptions of Self-Predication and Non-identity, but suggests two important distinctions. Fine distinguishes between narrow and broad Self-Predication: Narrow Self-Predication (NSP) is the assumption that universal forms are self-predicated as sensible particular instances are predicated (i.e. the garden variety pros ta alla predication); while Broad SelfPredication (BSP) is the assumption that universal forms are self-predicated, not as sensible particular instances, but to explain that the universal form is a member of the class of itself (i.e. the tree-type pro heauto predication of essential genus-species membership with itself).30 Fine also distinguishes strong from weak Non-Identity: Strong Non-Identity (SNI) is the assumption that nothing is identical to itself in virtue of itself, not even the universal forms; while Weak Non-Identity (WNI) is the assumption that only sensible objects are not identical to themselves in virtue of themselves.31 She argues that while either ontic Separation (S) or Strong Non-Identity (SNI) may imply Weak Non-Identity (WNI), neither Weak nor Strong Non-Identity implies ontic Separation (i.e. ¬ (SNI ⊨ S) & ¬ (WNI ⊨ S)). Hence, because the logical Non-Identity of universal forms and particular instances does not imply ontic Separation, logical Non-Identity does not imply the rejection of ontic Separation or the “degrees of reality theory” of Platonic ontology. Fine concludes that the Third Man Argument may thereby be satisfactorily answered by rejecting Strong Non-Identity in favor of Weak Non-Identity (WNI), and Narrow Self-Predication in favor of Broad Self-Predication (BSP).32 The Third Man Argument alleges that the self-predication of universal forms must initiate an infinite regress of universal forms that contradicts both their Uniqueness and Explanation. However, this argument ignorantly conflates the externally-related predication of ordinary discourse with the internallyrelated predication of Platonic Participation. Meinwald distinguished between garden variety pros ta alla and tree-type pro heauto predication, for which only the former may initiate the infinite regress of the Third Man Argument. Frances found this distinction of Self-Predication to be inadequate to block every instance of pros ta alla Self-Predication, and recommended the further rejection of Non-Identity. Vlastos objected that rejecting Non-Identity would imply the rejection of Separation and the whole edifice of Platonic ontology, but Fine answered that the rejection of Strong Non-Identity does not imply the rejection of either ontic Separation or Platonic ontology. Therefore, the Third Man Argument may be answered with Meinwald’s semantic distinction for Self-Predication and Fine’s ontological distinction for Non-Identity. The consequence of these distinctions is the concrete identity of abstractly differentiated particular instances in universal forms: Meinwald's pro heauto Self-Predication allows universal forms to predicate themselves as particular differentiated species within a universal and identical genus, while Fine's Weak Non-Identity allows universal forms to be asymmetrically identical to sensibly differentiated particular instances. However, even if the Third Man Argument is satisfactorily answered, there remains the challenge that Plato himself names the ‘Greatest Difficulty’. 30 Fine, Gail. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 62 Fine, Gail. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 207 32 Fine, Gail. On Ideas: On Ideas: Aristotle's Criticism of Plato's Theory of Forms, 1995: 225-6 31 Ryan Haecker © 2014 8 IV. Answering the Greatest Difficulty Argument The Third Man Argument has come to be regarded as the most decisive objection to Platonism, yet Plato passes over it without reply to address an even greater difficulty. The Greatest Difficulty pierces the explanatory heart of Platonism by alleging the impossibility of knowing the universal forms according to the hypotheses of the Two-World Theory of the Forms. It purports to show that the assumptions of ontic Separation and logical Non-Identity imply an unbridgeable ontological duality of the beings of the universal forms and particular instances which renders impossible knowledge of the particular instances of human affairs for gods (134c), and knowledge of universal forms for men (134a). The devastating consequence of the Greatest Difficulty is thus an unbridgeable dualism between two mutually exclusive sets of beings: the set of all universal forms and the set of all particular instances. Since the universal forms are both epistemic and ontic, this unbridgeable dualism can be interpreted in two aspects: as an epistemic dualism between knowledge of particular instances and universal forms that precludes knowledge of the universal forms; or as an ontological dualism between the beings of the particular instances and universal forms. Meinwald and Fine’s distinctions (§III) might be thought to bridge this divide because weak non-identity (WNI) and broad pro heauto self-predication (BSP) may allow many pros ta alla self-predicated instances to be vertically predicated under pro heauto selfpredicated universal forms, and predicative judgments may allow knowledge of universal forms through the predication of themselves and their particular instances. Since, however, the two-world theory of the forms implies there are at least two mutually exclusive set of beings, this ontological dualism implies an epistemic dualism between two exclusive domains of reference and knowledge.33 Meinwald and Fine's semantic answer to the Third Man Argument, therefore, cannot answer the Greatest Difficulty Argument because it assumes the possibility of referring to the very integrally united ontology that the Greatest Difficulty purports to divide. Answering both the epistemic and the ontic aspects of the Greatest Difficulty requires some further demonstration of how the differentiated sets of beings may be horizontally re-united in a monistic ontology. Pelletier and Zalta have shown how, with the additional assumptions of One-Over-Many (OM), NonIdentity (NI), and Self-Predication (SP), Platonic universal forms (ΦF) can be translated into the mathematical sets that include all of the members of any instance of a form (e.g. ΦF = { Fn | Fn1, Fn2,… Fnx}). While the universal forms are each one universal form over many particular instances, the consequence of the Greatest Difficulty is that the set of all universal forms and the set of all particular instances are divided as particular sets of beings. Since One-Over-Many (OM) implies that for any set of many particulars Fn there must be one corresponding universal form ΦF, and Plato’s ontological dualism implies there are two sets of particular beings (which minimally share the property of being), there must be some further universal form (e.g. Being) that is both over and within the set of all universal forms.34 The consequence of the Greatest Difficulty for Plato’s Theory of the universal forms is thus a universal form ΦF that designates a set G that is both included and excluded from the set of all universal forms. 33 Plato affirms a correspondence theory of referential knowledge in the Cratylus (385b), such that epistemic reference is only possible for that which is minimally some intellective being of hypothetical thought, and the possibility of referential knowledge is restricted by the necessary structure of being itself. 34 The importance of being in the Parmenides is disclosed by the first dichotomy that the One either is or is not. Ryan Haecker © 2014 9 This consequence of the Greatest Difficulty argument, in which two sets of beings both share and do not share a common universal form, can be translated into Bertrand Russell’s famous paradox, in which there is a set that is both included in and excluded from itself.35 If the universal form ΦF designates a set G such that G contains all member Fx with the property F that cannot be predicated of the universal form ΦF, then G is the set of Fx such that Fx is excluded from the set of Fx (i.e. G = {Fx | Fx Fx}). Since, however, Self-Predication (SP) requires all universal forms to predicate the property that they exemplify of themselves, F must be predicated ΦF (i.e. F(ΦF)). Predication implies that the predicated property (Fx) is included ( ) in the set of the properties that constitute the subject of predication (i.e. F(ΦF) ⊨ F ΦF). Thus, Fx must be included in set G that is designated by the universal form ΦF.36 This also implies that Fx is included in the set G that is included in itself (i.e. Fx (ΦF = G = {Fx}) ⊨ Fx Fx ⊨ G G). However, since G is the set of Fx such that Fx is not included in itself, the set of G must be both included and excluded from itself (i.e. G G ≡ G G). This paradoxical set of beings is therefore just as much a consequence of a naive Platonic Theory of Universal Forms, which allows for the self-predication of nonself-predicable properties, as it is a consequence of a naive set theory with an unrestricted comprehension axiom, which allows for the inclusion of sets that do not include themselves.37 Pelletier and Zalta warn that the “first and foremost worry for a theory of Forms is to avoid the Russell paradox."38 There are two standard answers to Russell's Paradox: the axiomatic answer that introduces an axiom to artificially restrict the range of possible sets to consistent sets; and the hierarchical answer that constructs a hierarchy of sets in which each superset restricts the predicative scope of its subsets. Pelletier and Zalta propose the axiomatic answer to restrict the scope of the predication of universal forms by a comprehension principle such that “any predicate which is formulable without pros heauto predication designates a property" (i.e.∃F∀x(Fx ≡ φ)).39 This restriction is meant to eliminate the inconsistent set of Russell’s Paradox by the introduction of a further axiom which excludes universal forms of the property corresponding to pro heauto and not pros ta alla predication. In Principia Mathematica, Bertrand Russell proposed the hierarchical answer by replicating this restriction to consistent sets at every level of the set theoretic model. Russell's Type Theory consists in a hierarchy of sets in which the predication of each set is defined by a higher level set, such that supersets always define the predicative scope of their subsets. Unfortunately, both of these answers to Russell’s Paradox fail to resolve the paradox for Platonism because each attempts to eliminate the inconsistent set by introducing a higher universal form to restrict the scope of predication to consistent sets: the axiomatic answer introduces the restricting 35 Joh N. Fi dla re og izes ‘ussell s Parado for Plato i o tolog a d re o e ds the o ti su su ptio of the inconsistent sets. Cf. Plato: The Written and Unwritten Doctrines, , : It is this supra-absurd situation [of Being Non-self-i sta tiati g] hi h led ‘ussell to hold su h Ideal o te ts to e ea i gless . 36 According to the rule of the substitution of equivalents, i.e. ΦF = G = {Fx}. 37 Naive set theory is characterized by the Comprehension Principle Axiom hi h states that for e er for ula φ o tai i g the free aria le , there is a set { | φ } hose e ers are φ . “et theoreti odels that assu e this axiom are called naïve set theories because, in the absence of some restrictions upon possible sets, there may e erge ‘ussell s Parado , i hi h there is a set that o tradi toril oth i ludes a d e ludes itself. “i e, moreover, in classical logic any consequence may be trivially derived from a contradictory antecedent, this contradictory set renders unrestricted set theoretic models inconsistent and trivial. 38 Pelletier, Francis Jeffry & Zalta, Edward N. How to Say Goodbye to the Third Man Nous, 34/2, 2000: 165-202: 24 39 Pelletier, Francis Jeffry & Zalta, Edward N. How to Say Goodbye to the Third Man Nous, 34/2, 2000: 165-202: 27 Ryan Haecker © 2014 10 axiom,40 while the hierarchical answer introduces the ramified hierarchy.41 Since, however, each of these higher principles must be one further universal form that stands over and is non-identical to many particular forms, and Russell’s Paradox for Platonism may be generated from any instance of One-OverMany and Non-Identity, each of these answers reiterates the paradox that it claims to resolve. I propose that Plato intended the enigmatic dialectical exercises of the Parmenides to guide students of philosophy to answer the Greatest Difficulty Argument by subsuming the semantic scope of all possible hypotheses including the inconsistent sets described by Russell’s Paradox.42 The semantic scope of each hypothesis is the extent of possible meanings signified the hypothesis. A set with a lesser range of possible meanings (A) can be subsumed by a set with a greater range of possible meanings (B), when all of the members of the lesser subset are contained within the greater superset (i.e. A B).43 Rather than dogmatically restricting the scope of the sets designated by universal forms, Plato proposes a hierarchical answer to Russell's Paradox in which inconsistencies may be located at every level within a hierarchy of sets. Plato was, of course, ignorant of both modern set theory and Russell’s Paradox. However, since universal forms designate the set of their particular instances, and the ontic dualism resulting from the Greatest Difficulty implies an inconsistent set of beings that can be translated into Russell’s Paradox, Plato may have recognized this possibility of answering the challenge of the Parmenides through the reunification of the differentia of all sets of beings designated by universal forms.44 In the bewildering third part of the Parmenides, Parmenides proposes a severe training of eight dialectical exercises to explore the hypothetical possibilities of predicating one of many and many of one. Plato indicates that the one and the many are not meant to specifically refer to one universal form and many sensible objects, but are rather meant to more generally refer to every possible object of predication; first, when Parmenides praises Socrates for not restricting the scope of his previous discourse with Zeno to visible objects (135e); and, second, when Zeno describes Parmenides’ exercises, in similar terms, as a 40 Since all universal forms are one-over-many, and the restricting Axiom is one universal form over many selfpredicating universal forms, this axiom stands over all universal forms as one universal of universals over many particular universals. However, since the Greatest Difficulty implies that the set of all universals is separated and excluded from the set of all particulars, this restricting axiom (R) must itself be inconsistently both included in and e luded fro the set of all u i ersals ΦF i.e. ‘ ΦF ≡ ‘ ΦF just as the i o siste t set of ‘ussell s Parado is both included and excluded from itself (i.e. G G ≡ G G). 41 Since the predicative scope of each subset can only be defined by its superset, and each superset is one-overmany and non-identical to its subsets, for any finite set theoretic model there must be one supreme superset at the ape of all sets. This supre e superset o er all sets ge erates ‘ussell s Parado e ause its predicative scope is not defined by any higher superset. Since this supreme set (S) must be a universal form that is both included and excluded from the set of all universals (i.e. S ΦF ≡ “ ΦF , the i o siste of ‘ussell s Parado i.e. G G ≡ G G) is thereby elevated to apex of Platonic ontology. 42 This requires an indirect proof of the consistency of Platonism (P) with the Greatest Difficulty (GD) by de o strati g that ‘ussell s Parado ‘P is ot i o siste t ith Plato is P : GD ⊨ RP; ¬(RP ⊥ P); ∴ ¬(GD ⊥ P) 43 Plato suggests supersets and subsets by his choice of the example of the forms of the slave and the master. (133e) 44 Willia F. L h, “.J. o urs that the ai task of the Par e ides is the re-unification of the differentia of beings: "[T]he arguments against the Ideas in the first half of the dialogues had indicated that the unity of the Idea has ee destro ed its dispersio i to a i sta es of parti ulars… No the ai task for Plato is reall to stress that this 'dilemma' of a one-become- a … [that] o urs he e er ou ha e a ki d of o e, a d o urs within the very inner ambit of such a one. Such a one is always a curious mixture of unity and multiplicity." Cf. An Approach to the Metaphysics of Plato through the Parmenides, 1959: 10 Ryan Haecker © 2014 11 “devious passage through all things.” (1γ6e) Parmenides then proposes to explore the hypotheses that: (a) the One either is or is not; (b) the One is itself or other than itself (i.e. Many); and (c) the One or the Many is either related (R) to or not related (NR) to itself. The eight hypotheses are the product of these three dichotomies (i.e. a x b x c = 2 x 2 x 2 = 23 = 8), which may be read in any one of at least one hundred and seventy non-linear sequences.45 The One is itself The One is not itself (Many) The One is (First Tetrad) H1 (NR) H2 (R) H3 (R) H4 (NR) The One is Not (Second Tetrad) H5 (R) H6 (NR) H7 (R) H8 (NR) Francis M. Cornford describes two major ancient interpretations of the dialectical exercises: the logical or semantic interpretation of the Middle Platonists (c. 130 – 68 BC) in which the hypotheses are logical exercises for the construction of possible predications of the One and the Many; and the metaphysical or ontological interpretation of the Neo-Platonists (c. AD 200 – 526) in which the hypotheses are the building blocks of the structure of being that constituted “Plato’s own metaphysical doctrine.”46 For example, Plotinus famously identified the first three hypotheses with the three divine Neo-Platonic hypostases or supreme beings of the One, the Logos, and the World-Soul.47 However, Cornford objects that this ontological interpretation fails to even consistently conjoin the first two hypotheses: the first hypothesis concludes that the One is simple and not predicable, while the second hypothesis concludes that the One is complex and predicable.48 Moreover, not only is each individual hypothesis internally contradictory, but also each pair, and pair of pairs of hypotheses (i.e. tetrads), is riddled with an insoluble multitude of contradictions. This multitude of contradictions threatens to trivialize the dialectical exercises and render them totally incapable of consistently of answering Parmenides' challenge.49 Contradictions can, however, be resolved in either a hard or a soft way: the hard way is to reject one of the contradictory propositions (e.g. if q = not-p, and p & q, then not-q); while the soft way is to distinguish some further quality of at least one of the apparently contradictory propositions so that it does not genuinely contradict the other (e.g. if q = not-Ap, and p & q, then p = Bp where Ap ≠ Bp).50 Accordingly, the hard solution to the contradictions of the Parmenides is to reject some of the propositional assumptions of the Theory of the Universal Forms.51 The soft solution is to show how, 45 Let (a) be the first combinatory pair of hypotheses; (b) be the combination of (a) and another pair of hypotheses; and (c) be the total combinatory sequence of (a) and (b); then a = (8x8)-8; b=(ax4)-1a; c=(bx2)-1b; c=170. 46 Cornford, Francis M. Plato and the Parmenides, 1939: v-xi 47 Cf. Stamatellos, Giannis. Plotinus and the Presocratics: Presocratic Influences in Plotinus' Enneads. SUNY Press, 2012: 12 48 Graham Priest counts over sixteen contradictions in the first pair of hypotheses alone! Cf. The Parmenides: A Dialetheic Interpretation. The electronic Journal of the International Plato Society, 12, 2012. 49 Contradictions render the dialectical exercises trivial according to the principle of explosion in classical logic, by which anything may be trivially proved from a contradiction. However, Graham Priest correctly observes that e ause Plato s Par e ides likel a tedated oth the a o i al for ulatio s of the La of No -Contradiction by Aristotle and the Principle of Explosion by Psuedo-Scotus, Plato cannot be unquestionably assumed to have intended the contradictory conclusions of the dialectical exercises to be rejected according to the classical laws of logic. Cf. The Parmenides: a Dialetheic Interpretation, 2012: 3-4 50 Seung, Thomas K. Plato Re-Discovered: Human Value and Social Order, 1994: 203-4 51 Samuel C. Rickless argues (1998, 2007, 2011) that the contradictions of the Parmenides can be resolved by rejecting the assumptions of the uniqueness and purity of the universal forms, where purity is defined as the non- Ryan Haecker © 2014 12 because the contradictions are merely an apparent degree of contrary opposition, the more contrary hypotheses may be subsumed by the less contrary hypotheses. While the hard solution may revise the Theory of the Universal Forms by rejecting some of its underlying assumptions, it cannot provide any instruction to answer Russell’s Paradox for Platonism. Only the soft solution of subsuming the less into the more consistent hypotheses allows for the construction of a satisfactory hierarchical model of sets. Thomas K. Seung proposes a semantic interpretation in which the semantic scopes of some of the hypotheses may be “incorporated” as subsets of the semantic scopes of other hypotheses. He argues, for example, that H3 is semantically equivalent to H2 because the many of H3 partake of the one of H2. Seung concludes that the positive result of the dialectical exercises of the Parmenides is to force an “ontological decision” between the semantic subsumption of the pair Hβ & Hγ into the pair H5 & H7 or vice versa; and recommends H5 & H7 to account for both the “changing material objects and the unchanging eidetic objects”.52 Each of the individual hypotheses and pairs of hypotheses may thus be interpreted to be subsumed into any of the other hypotheses or pairs of hypotheses so long as the semantic scope of the former subset is included in the semantic scope of the latter superset.53 Through this method of subsuming the subset hypotheses into superset hypotheses, and pairs of hypotheses into tetrads of hypotheses, we can construct a hierarchical set theoretic model that formally represents a semantic interpretation of the dialectical exercises. This method of subsuming subset hypotheses into superset hypotheses further implies that the subordinate hypotheses may be predicated of the superordinate hypotheses simply because any subset is analytically contained in its superset and whatever is analytically contained may be predicated of the containing subject. Consequently, each of the individual hypotheses, i.e. Hn, may be predicated of their containing individual, pair, or tetrad of hypotheses (e.g. Hnx… Hnb(Hna)). Since Meinwald’s tree-type pro heauto mode of predication is “strictly tied to objects with definitions,” the semantic subsumption of each subordinate hypothesis into a superordinate hypothesis can be translated into Constance Meinwald’s treetype pro heauto predication according to Pelletier and Zalta participation predication of (i.e. xF).54 Since Platonic participation is both ontic and epistemic, the construction of such a hierarchical set theoretic model of the sequence of pro heauto participative predication also signifies an ontology in which the being of each of the subordinate hypothesis participates in the being of the superordinate hypotheses (e.g. (xFa)Fb… Fx). A semantic interpretation of the participative predication of the hypotheses of the Parmenides may therefore anticipate a “skeletonized and ordered summary” of Plato’s ontology.55 Answering the challenge of the Parmenides minimally requires a demonstration of the possibility of a single non-contradictory interpretation. The Greatest Difficulty purported to show that the assumptions of ontic separation and logical non-identity implied an ontological dualism which rendered knowledge of coincidence of contrary opposite properties and uniqueness is defined as there being only one universal form for any multitude of particular instances. Cf. http://plato.stanford.edu/entries/plato-parmenides/ 52 Seung, Thomas K. Plato Re-Discovered: Human Value and Social Order, 1994: 202-14 53 Seung rejects hypotheses H1, H4, H6 and H8 because each make predication impossible (200). However, since none of these hypotheses may be predicated, and the scope of their predication is a nothing, the scope of nothing may be subsumed as a null set {Ø} into the semantic scope of any of the other predicable hypotheses: i.e. Øx ⊂ Hn. 54 Fra es, Br a . Plato s ‘espo se to the Third Ma Argu e t i Par e ides, Published in Ancient Philosophy Vol. 16, 1996: 47-64 55 Lynch. William F. An Approach to the Metaphysics of Plato through the Parmenides, 1959: viii Ryan Haecker © 2014 13 universal forms and Plato’s Theory of the Universal Forms impossible.56 The devastating consequence of this ontological dualism was a universal form that designated the inconsistent set of Russell’s Paradox, which, because it is both one and many, was both included and excluded from the set of all universal forms. Plato could have responded to this challenge by constructing a hierarchy of sets, in which each superset hypothesis restricts the predicative scope of its subset hypotheses and the inconsistent set is universally replicated at every level as a basic constituent of the set theoretic model. Since, therefore, the ontological dualism of the Greatest Difficulty implies the inconsistent set of Russell’s Paradox, and this paradox may be answered by the construction of set theoretic hierarchy, the Greatest Difficulty may be answered by hypothetically re-uniting all of the differentiated sets of beings into a monistic ontology. 56 Since the objection of Greatest Difficulty alleges that the Theory of the Universal Forms is inconsistent, invalid, and impossible, and even the possibility of a single instance contradicts the necessary impossibility of such an instance, answering this objection minimally requires a single consistent interpretation: ⋄ ∃ F∀ x(Fx) ⊥ □∃ F∀ x(Fx). Ryan Haecker © 2014 14 V. Plato’s Concrete Universal The challenge of the Parmenides arose from an inconsistency within the Theory of the Universal Forms between two-world ontological dualism and one-world epistemological monism: a divided logic of being was implied by the ontic Separation and logical Non-Identity of the universal forms from their sensible particular instances; while a unified logic of knowledge was implied by the requirement for one universal form to explain every multiple set of beings. The principal objections in the second part of the Parmenides had each emerged from the interstices of this conflict between unity and division: the Third Man Argument alleged that an infinite regress of universal forms would result from the reunion of the dual and non-identical elements in self-predication; while the Greatest Difficulty Argument alleged that ontological dualism implied an inconsistent set that is both included and excluded from itself. While the Third Man Argument could be answered by distinguishing between other-related pros ta alla and selfrelated pro heauto modes of Self-Predication, answering the Greatest Difficulty requires the hypothetical construction of a grand unified set theoretic hierarchy. Since the universal forms are both ontic and epistemic, this set theoretic construction suggests the further possibility of constructing a grand unified ontological hierarchy. This ontology would be monistic and united rather than dualistic and divided because every set of beings - even the non-being signified by negative statements - would be thoroughly subsumed by participation within some universal form. Yet unlike the three-tiered ontology of the Republic, in which the beings at lower ontic levels remained external with properties distinct from and inexplicable by the beings at higher ontic levels, every being within this ontology is internally related.57 Francis H. Bradley’s Theory of Internal Relations is the hypothesis that the relationships between subject and predicate terms are internal to the terms when the terms are essentially defined by their relations.58 While the grammar of ordinary language externally applies predicates to subjects (e.g. birds fly), the meaning of some predicate terms seems to be analytically contained in their subject terms (e.g. triangles are three-sided). The Theory of Internal Relations holds the relation of the terms to be analytically contained in the essential nature of the terms themselves.59 The rejection of the external predication of the Third Man Argument and the extrinsically divided ontology of the Greatest Difficulty implies the internal predication and intrinsically united ontology of this Theory of Internal Relations: Meinwald’s predicative subsumption of all external pros ta alla in internal pro heauto predication implies that all garden-variety predicates are internally rooted in the tree-type essence of the universal forms; and the semantic subsumption of all subordinate into superordinate hypotheses implies that each successive hypothesis is analytically contained in another within a grand hierarchy of sets. Since any subset may be predicated of its containing superset, each subordinate hypothesis may be predicated to participate in its 57 In the Republic (509b), Plato proposes a three-tiered ontological hierarchy consisting of (1) the Good, (2) the universal forms, and (3) the particular instances. Since, however, the form of the Good is simply Goodness itself, and essentially distinct from any of the subordinate universal forms, the universal form of the Good cannot explain its essential relations of all its subordinate universal forms. 58 F.H. Bradley, Appearance and Reality, 1893: 26-7: "So far as I can see, relations must depend upon terms, just as much as terms upon relations... Hence the qualities must be, and must also be related. But there is hence a diversity which falls inside each quality. Each has a double character, as both supporting and as being made by the relation... Every quality in relation has, in consequence, a diversity within its own nature, and this diversity cannot immediately be asserted of the quality. Hence the quality must exchange its unity for an internal relation." See also P.B. Blanshard The Nature of Thought, 1939, and A.C. Ewing, Idealism, 1934. 59 For a recent discussion and defense of F. H. Bradley's Theory of Internal Relations see Stewart Candlish, The Russell/Bradley Dispute and its Significance for Twentieth-Century Philosophy, Palgrave Macmillan, 2007: 141-73 Ryan Haecker © 2014 15 superordinate hypotheses (e.g. if Hx Hy then (Hy)Hx). This participative predication (e.g. xF) describes an internal relation between the predicate and the subject because Participation is a forteriori defined as the predication of an instance that is formally equivalent to the essential nature of a universal form (i.e. ∃Φ∀F∀n(Fn=Φ & ΦF)). Every subordinate hypothesis thus participates in a superordinate hypothesis such that the construction of a grand hierarchy of sets may signify an absolutely internally integrated ontology. Such a totalizing integration of all beings is tantamount to a universal form of all forms for which the abstract divisions of all particular beings are united in-and-through the One concrete ground of being itself.60 Georg Wilhelm Friedrich Hegel defined the Concrete Universal (konkrete Allgemein) as a universal form that, by containing particular beings within itself, actively particularizes itself into an individual substance.61 Where Plato is often understood to have prioritized the abstract objects of universal forms (e.g. manhood) over the particular instances (e.g. men), and Aristotle is often understood, to the contrary, to have prioritized the concrete objects of individual substances (e.g. Socrates), Hegel describes the universal forms as, not merely the passive abstract aggregate of their particular instances, but - more vivaciously - as the active concrete agents that particularize themselves into individual substances.62 The Concrete Universal is thus, not merely abstractly divided as one universal form over many particular instances, but rather the very concrete ground of being in which the particular instances are each rooted and united.63 Its abstract divisions are, moreover, not merely nullified, but rather subsumed (aufheben) and preserved in the self-particularizing motion of the Concrete Universal form of all forms.64 Hegel thus 60 Francis H. Bradley similarly concludes that nothing at all is external to the whole of being: "Nothing in the whole and in the end can be external, and everything less than the Universe is an abstraction from the whole, an abstraction more or less empty, and the more empty the less self-dependent. Relations and qualities are abstractions, and depend for their being always on a whole, a whole which they inadequately express, and which remains always less or more in the background." Cf. Appearance and Reality, Oxford, 1930: 520–1. 61 G.W.F. Hegel, The Encyclopedia of Philosophical Sciences, ,§ : But the u i ersal of the otio is ot a mere sum of features common to several things, confronted by a particular which enjoys an existence of its own. It is, on the contrary, self-particularizing or self-specifying, and with undimmed clearness finds itself at home in its a tithesis. 62 Robin G. Collingwood summarizes the Concrete U i ersal as the ie that o rete u i ersalit is i di idualit , the individual being simply the unity of the universal and the particular. The absolute individual is universal in that it is what it is throughout, and every part of it is as individual as itself. On the other hand it is no mere abstraction, the abstract quality of individualness, but an individual which includes all others. It is the system of systems, the world of worlds." Cf. Speculum Mentis, Oxford, 1924: 220–1 63 G.W.F. Hegel, The Encyclopedia of Philosophical Sciences, 1817, § : the i di idual as i di idual i the Singular judg e t , is a u i ersal… parti ularit is thus e larged to u i ersalit : or u i ersalit is odified through the i di idualit of the su je t… B ea s of particularity the immediate individual comes to lose its independence, and enters into an interconnection with something else… [T]he universal has the aspect of an external fastening, that holds together a number of independent individuals, which have not the least affinity towards it. This semblance of indifference is however unreal: for the universal is the ground and foundation, the root a d su sta e of the i di idual. 64 G.W.F. Hegel, The Encyclopedia of Philosophical Sciences, 1817, §164 No complaint is oftener made against the notion than that it is abstract. Of course it is abstract, if abstract means that the medium in which the notion exists is thought i ge eral a d ot the se si le thi g i its e piri al o rete ess… Although it e a stract therefore, it is the concrete, concrete altogether, the subject as such. The absolutely concrete is the mind — the notion when it exists as notion distinguishing itself from its objectivity, which notwithstanding the distinction still continues to be its o . Ryan Haecker © 2014 16 summarized the conclusion of the Parmenides as the “union of two determinations” which shows the “pure Platonic doctrine of Ideas” as the dialectical “identity with their ‘other'.”65 The Internal Relations of particular beings to universal being in the Concrete Universal are defined as instances of the unity of identity and difference, or identity-in-difference (Identität in Differenz), in which the non-identical differences present at lower ontological levels are elevated and united in-and-through the self-identity of higher ontological levels, just as all of the particular instances are united in-and-with the Concrete Universal form of all forms.66 All instances of contrary opposition are then thought to be so many different aspects of the particular internal divisions that are rooted in the essence of the Concrete Universal.67 The many contradictions of the Parmenides may thus seem to be merely apparent contradictions because, in each case, some further distinctions may perhaps show that neither of the contradictories genuinely contradicts.68 In order to answer Russell’s Paradox for Platonism, construct a set theoretic hierarchy, and preserve the robust ontological significance of the dialectical exercises, I recommended such a soft resolution of the apparent contrariness of the hypotheses through the subsumption of the semantic scope of the more contrary subordinate hypotheses by the less contrary superordinate hypotheses.69 The objection may yet be raised that this method of subsuming contrary hypotheses does not genuinely resolve, but rather wantonly incorporates, the rampant contradictions of the dialectical exercises.70 Since even a single contradiction would trivialize the construction of such a set theoretic hierarchy, answering this objection seems to require the resolution of every contradiction. This objection can now be answered by examining the status of contrariety in Plato’s ontology.71 65 G.W.F. Hegel, Lectures on the History of Philosophy, 1837, Vo. 1, Plato, Dialectic. Graham Priest translates identity-in-difference as a=a & a≠a , or (^A=^¬A) & (^A≠^¬A) where ^ signifies contrary opposition. This can be adapted to Zalta s otatio of Plato i parti ipatio as F = F =F & F ≠F . Cf. Dialectic and Dialethic, Science & Society, Vo. 53, No. 4, 1989/90: 410-12 67 Hegel writes of Plato: We see that Plato, i respe t of o te t, e presses othi g e epti g hat is alled indifference in difference, the difference of absolute opposites and their unity... differentiating unity in which the two moments are separate, as standing in different aspects." Cf. Lectures on the History of Philosophy, Plato 68 John N. Findlay interprets the pri ar se se o trareit i Plato s riti gs to e o trar oppositio . Cf. Plato: The Written and Unwritten Doctrines, : , , : It is lear that for Plato, as for a other Greek philosophers, contrariety and tension enter into his conception of what ultimately is, and that without disrupting its unity, or involving anything like the head-on self-destructive contradiction which Plato and Aristotle alike o de a d es he . 69 Paul E. More describes a similar interpretation by Alfred J.E. Fouillee in La philosophic de Platon: Parmenides shows that the union of contraries in the sensible world implies a similar union of contraries in the Ideas, and that the difficulties which concern the participation of sensible things in Ideas have their solution in the participation of Ideas among themselves. Hence the second part takes up this point, and demonstrates that whatever hypothesis you start with, it always involves the primitive union of contraries, the radical union of the one and the many. Thus, whatever pair of Ideas you may consider, positive and negative, you will always find a mediating term in some third Idea, so that all Ideas, even those mutually contradictory, enter into one another and are reconciled in the supre e U it . Cf. The Parmenides of Plato: 121 70 Graha Priest o ludes that Plato is i deed suggesti g that the o e has o tradi tor properties. The Parmenides: A Dialetheic Interpretation. The Electronic Journal of the International Plato Society, 12, 2012: 2 71 Plato employs contrariety in various ways throughout his dialogues. Aristotle defined four modes of contrariety (e.g. contradiction, privation, contrary, and relation); called those terms contraries "which, in the same genus, are separated by the greatest possible difference" (Categories 6-18); and those terms contradictory which both elo g a d do ot elo g at the sa e ti e to the sa e thi g a d i the sa e respe t Metaphysics 1005b19). 66 Ryan Haecker © 2014 17 Parmenides presented a proto-type of the Law of Non-Contradiction in On Nature: “Never will this prevail, that what is not is.”72 In response to Heraclitean Flux, Plato extended this ontic prohibition on the coincidence of being and non-being to the epistemic coincidence of contrary opposite properties in one and the same object of thought: “It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time. So that if ever we find these contradictions in the functions of the mind we shall know that it was not the same thing functioning but a plurality.” (436b) The Theory of the Forms was meant to establish the non-contradictory ground of being and knowledge, yet, in the Parmenides, the first part promises (130a) and the third part delivers (137c) a demonstration of how even the universal forms may possess contrary properties. Finally, the Phaedrus (265e), the Sophist (219a), the Statesman (258d), and the Philebus (15b) each describe a method of division (diaeresis) of dividing reality at the “natural joints.”73 This method of division is itself an instance of identity-in-difference, in which particular species are differentiated within an identical and universal genus.74 Identity-in-difference is, however, not in conflict with the Law of Non-Contradiction, but rather with the Law of the Excluded Middle, which holds that “of any one subject, one thing must be either asserted or denied” (Metaphysics 1011b24).75 Where the Law of Excluded Middle prohibits any middle option between exclusive and oppository terms (i.e. Fx v ¬Fx), identity-in-difference implies this middle option as a mediating identity (i.e. xF = (Fx=Fx) & (Fx≠Fx)). Furthermore, since the assumption of Participation (i.e. Fn=Φ & ΦF) implies there to be some mediating participative identity between distinct particular instances in universal forms (i.e. Fn=Φ), Plato’s Theory of the Universal Forms must be inconsistent with the Law of the Excluded Middle.76 To answer the Third Man and Greatest Difficult arguments by the participative predication of universal forms, Plato must therefore restrict the Law of the Excluded Middle to the relation between particulars instances, and exclude it from the relation of universal forms to particular instances. This restriction of the Law of the Excluded Middle to particular instances is, moreover, intuitively appropriate because only particular instances are separately excluded from one another by their mutually exclusive particularity. If the Law of Excluded Middle may thus only apply to particular instances, then it may not apply to the participative relation between universal forms and particular instances (i.e. xF) and is not at all inconsistent with the Platonic Participation of particular instances in universal forms. Plato may avoid every contradiction in the construction of a grand set theoretic hierarchy by restricting the Law of the Excluded Middle to the relations amongst particular instances, and paraphrasing every instance of apparently contradictory predication into the participative predication of many particular 72 Priest, Graham. The Parmenides: A Dialetheic Interpretation. The Electronic Journal of the International Plato Society, 12, 2012: 61 73 Hans Joachim Krämer distinguishes between the genuine contradiction between unity and multiplicity and the contrariety of the unlimited multiplicity (great-small). Cf. Plato and the Foundations of Metaphysics, 1990: 78 74 John Dillon reports that the Speusippus and the heirs of Plato in the Old Academy enthusiastically continued defining genus and species according to specific difference. Cf. The Heirs of Plato: A Study of the Old Academy (347-274 BC), 2003: 80-82 75 Ha s Joa hi Krä er o urs that the pri iples of o tradi tio a d the e luded iddle are esta lished solely within the constitution of determined [particular] being. Therefore in a strict sense, they can be valid for the highest pri iples the sel es O e, D ad o l i the protot pi al se se a d through a alog . Cf. Plato a d the Foundations of Metaphysics, 1990:78 76 F =ΦF & F ≠ΦF = F =F & F ≠F ⊨ (Fn & ¬Fn) ⊥ (Fx v ¬Fx) Ryan Haecker © 2014 18 species within one universal genus: the restriction of the Law of Excluded Middle to the relation between particular instances allows Plato to consistently predicate particular instances of universal forms; while the paraphrasing of contradictory statements into statements of participation allows Plato to consistently predicate the specific differences present at lower ontological levels in-and-through the essential selfidentity of universal forms at higher ontological levels.77 Plato’s monistic ontology thus avoids both contradiction and triviality by restricting the application of the Law of the Excluded Middle to the relation of particular instances under superordinate universal forms, just as a Platonic set theoretic hierarchy restricts the inconsistent set of Russell’s Paradox under supersets. Plato’s the Parmenides may now be reread as an outline of a proto-logic - prior to Aristotle’s formal syllogistic logic78 - in which being and knowing are organically conjoined in the monistic ontology of the Concrete Universal; the Law of the Excluded Middle is restricted to particular instances; and every particular instance is determined by Participation in a universal form that is essentially connected to all universal forms in a totalizing system of Internal Relations.79 Plato’s contest for the gods and giants of the Academy has been to interpret the dialectical exercises, to save the Theory of the Universal Forms, and to answer the challenge of the Parmenides. Plato evidently declined to answer Parmenides' criticisms but hinted (135a) that they might be answered by students who had first endured initiation by a severe dialectical training. The Theory of the Universal Forms had postulated an indefinite multitude of individual and transcendent universal forms to explain the possibility of knowledge, but had inadvertently produced a two-world ontological dualism, which both initiated an infinite regress of universal forms and allowed for the inconsistent set of Russell’s Paradox. This placed Plato in a fatal dilemma: affirming the transcendence of the universal forms meant the impossibility of any explanation of universal forms, while rejecting it meant the impossibility of any explanation of the self-contradictory particular instances. To avoid these perilous extremes of gods and giants Plato needed to critically revise the Theory of the Universe Forms to preserve the transcendence of the universal forms and re-unite all of the differentiated beings in a monistic ontology. This essay has shown how the Third Man Argument (§III) can be answered with Meinwald’s semantic distinction between garden variety pros ta alla and tree-type pro heauto predication; and the Greatest Difficulty Argument (§IV) can be answered by constructing a grand hierarchy of sets to re-unite all of the differentiated sets of beings. The consequence of rejecting the external predication of the Third Man Argument and the extrinsically divided ontology of the Greatest Difficulty is the replacement of two-world ontological dualism with the one-world ontological monism of the Concrete Universal form of all forms, in which all particular universal forms are internally related in a unified system of forms.80 77 Cf. Lynch, William F. An Approach to the Metaphysics of Plato through the Parmenides, 1959: 240; and Mora sik, J.M. For s, Nature, a d the Good i the Phile us , Phronesis, Vol. 24, No. 1, 1979: 90 78 Susanne Bobzien further describes some proto-t pi al logi al de elop e ts i pli it i Plato s theories of predication, contradiction, and participation, as well as how Middle Platonists, such as Alcinous, claimed to have de eloped a spe ifi all Plato i logi . Cf. A ient Logic: http://plato.stanford.edu/entries/logic-ancient/ 79 David W. Hamlyn similarly concludes that Plato believed the possibility of predication to require an identity-indifference communion of universal forms. Cf. The Communion of Forms and the Development of Plato's Logic, The Philosophical Quarterly, Vol. 5, No. 21, 1955: 289-302 80 Gustav Emil Müller similarly o ludes that the Par e ides assu es the diale ti al u it of ei g a d nonei g a d applies it to a h pothesis… the result is a diale ti al s ste of the hole of hi h there is o e essar starti g poi t e ause ea h part i plies a d postulates all of the others i the hole s ste . Cf. Plato, The Founder of Philosophy as Dialectic, 1965: 194 Ryan Haecker © 2014 19 Scholars have long disputed the identity of the divine craftsman, or Demiurge (dêmiourgos), who is described in the Timaeus (28a6) to have impressed an eternal model upon the cosmic World-Soul: Cornford (1937), Grube (1935), and Cherniss (1944) interpret the Demiurge as a symbol of universal reason immanent in the World Soul; while Hackforth (1936), Guthrie (1978), Mohr (1981), and Menn (1995) interpret the Demiurge as the transcendent author of the World Soul.81 Stephen Menn argues, against Cornford and Cherniss, that the Demiurge of the Timaeus is identical to the Nous of the Philebus, the Phaedo, and the Laws, but logically distinct and ontologically superior to the World Soul.82 He observes that there “seems to have been a broad consensus [among Platonists] that ‘nous’, in the proper sense, names only one being, and that this being is the highest, or very near the highest God.”83 The primary textual objection to this transcendent interpretation is derived from a passage in the Timaeus which describes how “it is impossible for Nous to come-to-be in anything apart from soul.” (γ0b) This and other parallel passages has led Conford to conclude that “the Demiurge is to be identified with the Reason in the World-Soul” and Cherniss to conclude that “Nous cannot exist apart from the soul.”84 Reginald Hackforth has argued, to the contrary, that here Plato is “speaking of the universe, not of its creator”, and thus does not mean that Nous cannot exist independently of the World-Soul, but merely that the World-Soul cannot be ordered according to reason without Nous.85 The Demiurge may thus be identified as the author of the World-Soul who eternally orders the cosmos by Nous. Texas Health and Science University 81 Maso , A dre . Plato s God: E ide e of the Phile us, : Menn des ri es ho , after riefl i trodu i g A a agoras Nous i the Phaedo (96a), Plato critically revised the concept of Nous in the Statesman (273b), the Philebus (26e), the Timaeus (28a6), and the Laws (966e) to become the functional equivalent of the Demiurge because it had assigned material causes to the formal cosmic order indifferent to the best, the Good, and any form of reason. The Demiurge and Nous are each the formal cause of order in the World-Soul (26e, 27b, 273b, 530a, 265c, 28a, 28c) and the king (basileus) of the world (274e, 28c, 28d, 272e), who gives order to heaven and earth (28d, 30c, 97c, 98a, 30c, 97c, 28e, 966e, 967b) for the highest good (30a, 37d). Cf. Plato on God As Nous, 1995: 4-7 83 Menn, Stephen. Plato on God As Nous, 1995: xii 84 Cornford, F.M. Plato’s Cosmology, The Classical Review, 1937: 197; and Cherniss, H.F. Aristotle’s Criticism of Plato and the Academy, 1944: 425 85 Hackforth, R. Plato’s Theism, 1936: 445 82 Ryan Haecker © 2014 20 Works Cited Aristotle. The Complete Works of Aristotle: The Revised Oxford Translation. Ed. Barnes, Jonathan. Princeton, N.J.: Princeton University Press, 1984. Benn, Alfred W. "Plato's Later Ontology." Mind, New Series, 11, No. 41, 1902: 31-53. Blanshard, Percy Bland. The Nature of Thought. London: G. Allen & Unwin, 1939. Bobzien, Susanne. "Ancient Logic." Stanford University. December 13, 2006. Accessed December 7, 2014. http://plato.stanford.edu/entries/logic-ancient/. Bradley, Francis H. Appearance and Reality: A Metaphysical Essay. 2nd ed. Oxford: Clarendon Press, 1930. Candlish, Stewart. The Russell/Bradley Dispute and Its Significance for Twentieth-century Philosophy. Houndmills, Basingstoke, Hampshire: Palgrave Macmillan, 2007. Cherniss, Harold F. Aristotle's Criticism of Plato and the Academy. Baltimore: Johns Hopkins Press, 1944. 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