How Do the Eight Hypotheses in Plato’s Parmenides Come to Light? Chiasmus

In this paper


Introduction
This paper aims to articulate the structure of the dialectic exercise presented in the second part of the Parmenides.In this regard, two questions are discussed.First, how many hypotheses are included in the exercise?Most scholars have maintained the existence of eight 1 The present article is published within the research project 'Research on Division and Methods of Division in Ancient Greek Philosophy', supported by the National Social Science Foundation of China,  General Program [Grant Number: 21BZX088].I presented the draft at Christoph Horn's colloquium when I was a visiting scholar at the University of Bonn in 2019.Many thanks to Christoph Horn and Walter Denis for their valuable advice on the early draft.I presented the revised paper at the Symposium Platonicum XII on Plato's Parmenides, held in Paris in 2019.Many thanks to the organizers and participants of the symposium.I would like to thank Luc Brisson, Noburu Notomi, Verity Harte, and William Altman for their helpful questions and comments.Special thanks are given to George Rudebusch for reading early and final drafts and discussing many details with me.I would also like to thank the anonymous reviewer for helpful comments and constructive suggestions that improved the paper.
hypotheses differ from one another?Some scholars have suggested that they differ in subject 4 , whereas others have claimed that they differ in the manner of predication. 5 engage with these questions and illuminate Plato's thinking, I delve into the transitional section of the Parmenides (136a4-c5).I agree with Meinwald that this contains crucial methodological remarks that play a key role in structuring the exercise.In analyzing this section, I begin by noting that in addition to diairesis (διαίρεσις), another method of division is used, namely, cross-division, which Porphyry terms chiasmus (χιαστή).In answer to the first question, I pursue an alternate proposal that Plato uses chiasmus to divide the exercise into eight hypotheses (Section 1).In answer to the second question, I disagree with Meinwald, who proposed that the eight hypotheses differ in terms of predication (Section 2).
Chiasmus (χιαστή) is the technical term that Porphyry invents for designating crossdivision. 12In commenting on Aristotle's Cat. 2, 1a20-b6, Porphyry declares that Aristotle makes a fourfold division by means of chiasmus: "According to which method [order] did Aristotle make the division?I say according to the chiasmus" (ὁ Ἀριστοτέλης κατὰ ποίαν τάξιν τὴν διαίρεσιν ἐξέθετο; κατὰ τὴν χιαστὴν λέγω, 78.35-6). 13Porphyry reconstructs the chiasmus in two steps.First, Porphyry sets up two pairs of opposites according to Aristotle's dictum.He identifies Aristotle's formulation, 'that which is in a substrate' with accident (ἐν ὑποκειμένῳ εἶναι = συμβεβηκόϛ); correspondingly, 'that which is not in a substrate' refers to substance (οὐκ ἐν ὑποκειμένῳ εἶναι = οὐσία). 14Substance and accident are opposed in such a way that the former is not in a substrate and the latter is in a substrate.Porphyry further identifies 'that which is said of a subject' with the universal (καθ' ὑποκειμένου λέγεσθαι = καθόλου); correspondingly, 'that which is not said of a subject' refers to the individual (οὐ καθ' ὑποκειμένου λέγεσθαι = μερικόν). 15Universal and individual are opposed in such a way that the former is said of a subject and the latter is not said of a subject.Thus, Porphyry establishes the two pairs of opposites by replacing Aristotle's own formulations 'not being in a substratebeing in a substrate' and 'said of a subject-not said of a subject' with his own terms, i.e., 'substance-accident' and 'universal-individual' (λέγω ὅτι ἡ οὐσία συμβεβηκὸς ἀντιδιαιροῦσα καὶ τὸ καθόλου ἐπὶ μέρους, 79.1-2).Then, these two pairs are cross-combined, thus constituting a 2 x 2 chiasmus.Although Porphyry does not include a diagram, his two successors, Ammonius and Philoponus, draw a diagram in their commentaries. 16Boethius preserves a Latin version, which is exactly the same as the original diagram drawn by Ammonius and Philoponus. 17  There are no essential differences between the two diagrams; nevertheless, the two exhibits certain differences, which warrants explanation.First, instead of using the term  in each pair cannot be combined. 18Given that an accident is in a substrate and a substance is not in a substrate, they cannot be combined.Given that the universal is said of a subject and the individual is not said of a subject, the two cannot be connected.Ammonius and Philoponus fully realize that two of the six combinations are invalid and thus append ἀσύστατον to the two horizontal lines that ostensibly combine substance with accident and the universal with the individual, which are marked in red in Diagram 1*.In Diagram 1, καθόλου and συμβεβηκὸς reverse their positions, so the ἀσύστατον-lines would have been two vertical lines, which are absent.To indicate that the opposites in a pair cannot be combined, I do not draw ἀσύστατονlines in the chiasmus shown in Diagram 1, nor do I include ἀσύστατον-lines in other instances of the chiasmus.
To summarize this section, Plato divides the exercise into eight hypotheses using chiasmus, which establishes the four subjects of the hypotheses by cross-combining the two subjects with the two relations (2 x 2).The four subjects of the hypotheses are distributed under opposite antecedents to generate eight hypotheses (4 x 2), as observed in the methodological remarks (136a4-c5) and intermediate (160b2-3) and final summaries (166c3-5).Meinwald appropriately notes that these three passages echo one another and indicate the structure of the exercise24 ; she also properly maintains that the two relations play a crucial role in structuring the exercise.25However, she does not realize that the two relations contribute to the establishment of the subjects of the hypotheses and instead improperly identifies them with two kinds of predication.

Pros heauto-Pros ta alla: Qualification of the Predicate
Regarding my proposal that Plato uses chiasmus to generate eight hypotheses, it is reasonable to consider how these hypotheses differ from one another.Meinwald proposes that they differ in terms of predication and that the qualification pair pros heauto-pros ta alla qualifies the predicate differently, thereby referring to two kinds of predication.Meinwald regards the former as 'tree predication' and the latter as 'ordinary predication'. 26Ordinary predication highlights the feature of the subject so that an individual as the subject is predicated by the feature that the individual has, e.g., 'Aristides is just'.Tree predication reveals the internal nature of the subject in such a way that a species as the subject is predicated by the genus to which the species belongs, e.g., 'Justice is a virtue'.Although scholars have criticized Meinwald, few have gone deep into her main arguments.I delve into the main arguments to explore how she improperly constructs pros heauto-pros ta alla as entailing two kinds of predication.

Pros heauto Predication
To identify pros heauto as tree predication, Meinwald gives attention to the argument concerning Difference and Identity in Hypothesis I (139b4-e6).From this argument, Plato draws four negative consequences step by step: The One is not different from itself (139b5-7), the One is not identical to another (139b7-c3), the One is not different from another (139c3-d1), and the One is not identical to itself (139d1-e6).Meinwald focuses on the third consequence.
According to Plato, the One cannot be different from another because to be different from another does not belong to one thing or anything else but rather belongs only to different-interpretation is problematic.In her view, the pros heauto predication, as a tree predication, explicates the nature of the subject in such a way that a species as the subject is predicated by the appropriate genus.In Hypothesis I, the tree predication appears in the negative form, such that no genera are predicated of the One.According to Meinwald, the negative formulation of the tree predication reflects the metaphysical fact that the One cannot be subordinate to any of the genera because the One is the principle of the genera and thus transcends all of them. 28ewing the One as the principle, Meinwald first puts the One that transcends all of the genera into the species-genus tree and then denies that the One belongs to the tree.Plato would acknowledge the One to be the principle; nevertheless, he would not locate the One in the species-genus tree because it, as the principle, does not belong to but rather remains beyond the tree and cannot be allocated to the species-genus tree initially or be explained by tree predication.
In applying the pros heauto predication in Hypothesis I, Meinwald has particular difficulty explaining the phrase 'the One is not one pros heauto'.The nature of the subject is supposed to be revealed by the pros heauto predication, according to which the nature of the One is not one.This conclusion is absurd.Peterson defends Meinwald's interpretation by explaining that 'the One is not one pros heauto' in the sense that "it is not the case that the One is by definition one" (1996: 190); put briefly, the One is not one by definition. 29 Fundamentally, Meinwald's predicative interpretation of pros heauto is especially inappropriate for interpreting Hypothesis I, according to which the One is negated by multiple pairs of predicates, such as whole-part, static-moved, and identical-different.Plato not only negates multiple pairs of predicates but also completely rejects the possibility of predication, as seen in the radical consequence that 'the One is not one'.This radical consequence can be derived directly from the antecedent of Hypothesis I; that is, there is nothing other than the One, so it cannot participate in any other Form.In the corresponding logical sense, the One cannot be attached to any predicate even though the predicate is one itself, for the single, indivisible One cannot be split into subject and predicate.Given that Plato completely rejects the possibility of predication by deconstructing the structure of predication, how could one interpret Hypothesis I in terms of predication?

Pros ta alla Predication
In the passage selected from Hypothesis V, Meinwald distinguishes between two uses of pros ti (1991: 57-63).Based on the second use of pros ti (2), she constructs the pros ta alla predication.
As Meinwald notes, Plato uses the dative to designate participation in the Form (2014: 476).In the Phaedo, Plato uses the dative τῷ καλῷ to reference participation in the Beautiful; accordingly, all beautiful things are beautiful by participating in the Beautiful (τῷ καλῷ πάντα τὰ καλὰ καλά, Phaed.100d7-8).Similarly, unlike things are unlike by participating in Unlike[ness] (ἀνομοίῳ τά γε ἀνόμοια ἀνόμοια ἂν εἴη, Parm.161b1-2), with ἀνομοίῳ referring to participation in Unlikeness.To designate participation in the Form F-ness, Plato uses not only the dative of a term (which signifies F-ness) but also pros with the accusative of a term (which signifies F-ness).Pros with the accusative πρὸς ἣν, that is, πρὸς ἀνομοιότητα, is used in the same sense as the dative ἀνομοίῳ, so πρὸς ἀνομοιότητα refers to participation in Unlikeness.On this account, (2) can be formulated as follows: Unlike things are unlike by participating in Unlikeness (ἀνομοίῳ), so the Others are unlike the One by participating in Unlikeness (πρὸς ἀνομοιότητα).The same reasoning holds for the argument concerning Inequality.In Plato's dictum, "Aren't the unequal things unequal by the Unequal?How [are they] not?Thus, the One participates in Inequality, in relation to which the Others are unequal to it?"(τὰ δὲ ἄνισα οὐ τῷ ἀνίσῳ ἄνισα; Πῶς δ' οὔ; Καὶ ἀνισότητος δὴ μετέχει τὸ ἕν, πρὸς ἣν τἆλλα αὐτῷ ἐστιν ἄνισα; 161c8-d1), pros with the accusative, πρὸς ἀνισότητα, is equivalent to the dative τῷ ἀνίσῳ and refers to participation in Inequality.Thus, the argument can be reconstructed: Unequal things are unequal by participating in the Unequal (τῷ ἀνίσῳ), so the Others are unequal to the One by participating in Inequality (πρὸς ἀνισότητα).
In my view, Meinwald is correct in noting that πρὸς ἀνομοιότητα and πρὸς ἀνισότητα are used to designate participation in Unlikeness and Inequality, respectively (1991: 56-63).
She also properly notes that pros ti in this case is associated with predication because it refers to participation.According to the theory of Forms, participation points to the metaphysical fact that an individual participates in the Form; e.g., Socrates participates in Justice.The metaphysical fact-the participation of Socrates in Justice-can be articulated through the statement 'Socrates is just'.Thus, the theory of Forms highlights the correspondence of the statement 'Socrates is just' to the metaphysical fact that Socrates participates in Justice.Based on the correspondence of predication to participation, pros ti is associated with predication insomuch that it refers to participation.Thus, pros ti, that is, A pros F-ness, refers to the participation of A in F-ness, which is represented by the statement 'A is F'.Applying this scheme to the arguments concerning Unlikeness and Inequality, one can arrive at the same conclusion as Meinwald, namely, that pros ti is associated with participation and predication.
The Others pros Unlikeness and pros Inequality refer to the participation of the Others in Unlikeness and Inequality, respectively, which are articulated by the statements 'the Others are unlike [the One]' and 'the Others are unequal [to the One]'.Meinwald properly notes that in the arguments concerning Unlikeness and Inequality, Plato uses πρὸς τι, that is, πρὸς ἀνομοιότητα at 161b3 and πρὸς ἀνισότητα at 161d1, to reference the metaphysical fact of participation and the logical correspondence of predication.Meinwald may be correct to regard the statements 'the Others are unlike [the One]' and 'the Others are unequal [to the One]' as ordinary predications in the sense that the Others exhibit the features of Unlikeness and Inequality.However, Meinwald errs when she mixes πρὸς ἀνομοιότητα and πρὸς ἀνισότητα, which appear in the arguments concerning Unlikeness and Inequality in Hypothesis V, with πρὸς τὰ ἄλλα, which is presented in the deduction and consequence of Hypotheses II, IV, V, and VIII, and thereby improperly identifies πρὸς τὰ ἄλλα in these hypotheses as referring to ordinary predications.If πρὸς τὰ ἄλλα were used in the sense of πρὸς ἀνομοιότητα or πρὸς ἀνισότητα, τὸ ἓν πρὸς τὰ ἄλλα in Hypotheses II and V would have referred to the participation of the One in the Others, which could be formulated as 'the One is the Others', and τὰ ἄλλα πρὸς τὰ ἄλλα in Hypotheses IV and VIII would have referred to the participation of the Others context is not predicate B but rather subject A. Pros ti qualifies subject A by connecting A with C, so subject A and relation pros C constitute the subject of the sentence.Clearly, the use of pros ta alla at 161a6 corresponds to the common usage of pros ta alla presented in the deduction and consequence of Hypotheses II, IV, V, and VIII.Since it refers to connection, the relation (pros) can be signified using the word 'and'.Thus, (1*) 'the One pros the Others is unlike' can be equivalently formulated as 'the One and the Others are unlike'; in general, 'A and B are C' (1**).
(1) The One is unlike pros the Others (A is B pros C).
(1*) The One pros the Others is unlike (A pros C is B).
(1**) The One and the Others are unlike (A and C are B).
As shown, pros ti is used in two ways.In the construction of 'A is B pros C', pros ti qualifies subject A by connecting the two elements of the subject, A and C, with each other, so the relation (pros) refers to connection and can be expressed as 'and'.In the construction of 'A is F pros F-ness', pros ti is associated with participation and predication.The metaphysical relation of individual to Form (A pros F-ness) is reflected by the logical relation of subject to predicate ('A is F'), so the relation (pros) refers to predication and can be expressed using the copula 'is'.Thus, pros ti is used either in the sense of connection or in the sense of predication.
Rickless referred to pros ti in the sense of predication as the 'Meinwald Reading' and to pros ti in the sense of connection as the 'Straightforward Reading'.Rickless endorsed the 'Straightforward Reading' while rejecting the 'Meinwald Reading' because in his view, Plato does not use pros ti in the sense of predication but rather merely in the sense of connection (2007: 102).According to Rickless, pros ti in the sense of connection is in accordance with its ordinary usage in the Greek language and the statements quoted from the Parmenides.As he properly noted, statements such as 'Simmias is taller pros Phaedo', 'Simmias is different pros Phaedo', 'Simmias is the same pros Phaedo', and 'Simmias is equal pros Phaedo' are equivalent to 'Simmias is taller than Phaedo', 'Simmias is different from Phaedo', 'Simmias is the same as Phaedo', and 'Simmias is equal to Phaedo'.Pros appears in different forms, namely, than, from, as, and to, because in English, different adjectives require different prepositions.In all cases, pros is used in the sense of connection and is used to connect the two elements of the subject with each other.The two subjects, Simmias and Phaedo, are combined using pros and compared in terms of height, substance (identity-difference), and quantity. 31In the Parmenides, similarly, two Forms, the One and the Other, are combined using pros and compared in substance (identity-difference), in quality (similar-dissimilar), in quantity (equalunequal), and in time (older-younger-same age).In Hypotheses I and II, Plato examines whether the One is identical to (pros) or different from (pros) itself and the Other; whether the One is similar to (pros) or dissimilar to (pros) itself and the Other; whether the One is equal to (pros) or unequal to (pros) itself and the Other; and whether the One is older than (pros), younger than (pros), or the same age as (pros) itself and the Other. 32Notably, in the statements that Rickless quotes from the Parmenides, Plato does not use pros with the accusative but instead uses the dative to designate the sense of connection. 33Given that pros with the accusative is used interchangeably with the dative, Rickless's interpretation is proper and valid.
I agree with Rickless that in all the cases mentioned above, pros ti (which appears as the equivalent dative) is used in the sense of connection.Nevertheless, I disagree with him regarding the claim that pros ti is used only in this sense.As analyzed, Meinwald is correct in claiming that pros ti in the two sentences (εἴη δὴ ἂν καὶ τῷ ἑνὶ ἀνομοιότης, πρὸς ἣν τὰ ἄλλα ἀνόμοια αὐτῷ ἐστιν, 161b1-4; καὶ ἀνισότητος δὴ μετέχει τὸ ἕν, πρὸς ἣν τἆλλα αὐτῷ ἐστιν ἄνισα, 161c8-d1) is used in the sense of predication.Rickless and Rudebusch suggested that to understand these two sentences properly, one should omit the comma before pros, which does not exist in the original text. 34This approach may be a solution, but the omission of the comma cannot change the fact that pros ti is not used in these two sentences to connect the two elements of the subject (as other cases show) but is rather used to indicate predication.
To summarize, a relation (pros ti) can refer either to the relation between the two elements of the subject, e.g., Simmias and Phaedo or the One and the Other[s], or to the relation between the subject and predicate, e.g., a predicative relationship such as 'Socrates is just' or 'the Others are unlike'.It is used not only in the sense of connection to connect the two elements of the subject but also in the sense of predication to connect the subject with the predicate.Thus, I believe that Meinwald is correct in noting that pros ti has a wider use than Rickless supposes.Meinwald appropriately distinguishes between the two uses of pros ti and properly notes that pros ti in its second use is associated with predication.However, she errs in identifying pros ti, used in the sense of predication, with pros ta alla as presented in the deduction and consequence of Hypotheses II, IV, V, and VIII.In my view, pros ti, which appears either as pros ta alla in the deduction and consequence of Hypotheses II, IV, V, and 32 Rickless 2007: 105-6. 33 Parm. 139b4-5, 139e7-8, 140b6-7, 140e2-5, 146a9-b2, 147c1-2, 149d8-9. 34 Rickless 2007: 104, n. 7. VIII or as pros to hen in the deduction and consequence of Hypotheses I, III, VI, and VII, is used in the sense of connection.That is, pros ti qualifies the subject by connecting the subject with the subject's relation.
3. Pros to hen-Pros ta alla: Qualification of the Subject Having argued with Meinwald in her philosophical context, in this section, I return to my own interpretation.With the help of chiasmus, I finally prove that pros ti qualifies the subject, and fundamentally argue that the qualification pair cannot be expressed one-sidedly as pros heauto-pros ta alla but must be articulated comprehensively and precisely as pros to henpros ta alla.Accordingly, I note that pros to hen-pros ta alla qualify the subject by combining two subjects with those two subjects' relations in a chiastic way.
Let us return to the first four hypotheses: if the One is, (I) what is to hen pros heauto; (II)   what is to hen pros ta alla; (III) what are ta alla pros to hen; and (IV) what are ta alla pros heauta?From I-II to III-IV, the subject shifts from to hen to ta alla; correspondingly, the qualification pair changes its form from pros heauto-pros ta alla to pros heauta-pros to allo (i.e., pros to hen).Meinwald appropriately notes that regarding the qualification pair, there is "a switch in singular and plural forms" (1991: 182n1) because "we have moved [from getting results for the One] on to get results for the [O]thers" (1991: 182n1).Meinwald's explanation cannot be correct, as she means that the qualification pair qualifies the predicate; if the qualification pair could have qualified the predicate, it would not change its form with the change of the subject. 35The qualification pair changes from pros heauto-pros ta alla to pros heauta-pros to allo because it qualifies the subject, shifting from qualifying to hen to qualifying ta alla.
Furthermore, the qualification pair, pros to hen-pros ta alla, qualifies the subject by connecting the two subjects with the two subjects' relations in a chiastic way.Pros to hen-pros ta alla qualify to hen by connecting to hen with to hen and with ta alla, leading to the two combinations of to hen pros to hen and to hen pros ta alla.To avoid duplication, that is, the appearance of to hen twice in the combination to hen pros to hen, the reflexive pronoun heauto is used to replace to hen in pros to hen; thus, to hen pros to hen-to hen pros ta alla becomes to 35 Even if the qualification pair could have been used as a sentential operator to qualify the entire sentence (as someone might argue), it would still not change its form with the change of the subject.The fact that the qualification pair changes its form with the change of the subject obviously proves that it qualifies the subject.
hen pros heauto-to hen pros ta alla, abbreviated as pros heauto-pros ta alla.Again, pros to hen-pros ta alla qualify ta alla by connecting ta alla with to hen and with ta alla, resulting in the two combinations of ta alla pros to hen and ta alla pros ta alla.To avoid duplication, that is, the appearance of ta alla twice in the combination ta alla pros ta alla, the reflexive pronoun heauta is used to replace ta alla in pros ta alla; thus, ta alla pros to hen-ta alla pros ta alla becomes ta alla pros to hen-ta alla pros heauta, abbreviated as pros heauta-pros to hen (i.e., pros to allo).Thus, pros ti has the connective characteristic and connects the two subjects with the two subjects' relations in a chiastic way, as seen in Diagram 3: Crucially, pros to hen-pros ta alla has the 'in relation to' structure, which implies X in relation to Y, X pros Y.When subject X changes, the subject's relation, i.e., the relation of X to Y, must change correspondingly.Thus, one cannot consider the subject's relation without considering the subject.Depending on whether to hen or ta alla are taken as a subject, the subject's relation appears either as pros heauto-pros ta alla or as pros heauta-pros to hen.The form in which the subject's relation appears depends on which subject-to hen or ta alla-is qualified, but both forms are generated from the original pair pros to hen-pros ta alla.
Therefore, the qualification pair, as the subject's relation, cannot be expressed one-sidedly as pros heauto-pros ta alla but must be articulated comprehensively and precisely as pros to henpros ta alla.Despite interpreting pros heauto-pros ta alla properly as the subjects' relations, Gill one-sidedly proposes that only pros heauto-pros ta alla designate the subject's relations and ignores the fact that the subjects' relations also appear in the form of pros heauta-pros to hen. 36Meinwald appropriately notes that the qualification pair changes its form from pros heauto-pros ta alla to pros heauta-pros to hen; however, she does not realize that these two forms are derived from the original pair pros to hen-pros ta alla.Moreover, Meinwald incorrectly treats pros heauto-pros ta alla as two types of predication by identifying pros 36 Gill 2012: 52-3, 53, n. 20;2014: 504-5.to hen ta alla pros to hen pros ta alla heauto-pros ta alla in the Parmenides with kath' hauto-pros allo in the Sophist. 37As analyzed, pros heauto cannot be identified with kath' hauto because pros heauto, which originates from to hen pros to hen, qualifies the subject to hen by connecting to hen with itself, whereas kath' hauto highlights the essential relationship of the predicate to the subject such that the subject (whatever it is) is characterized by a predicate that belongs to the subject's nature.

Conclusion
In the second part of the Parmenides, Plato constructs an exercise featuring two subjects and two relations.The two subjects, the One-the Others, and the two relations, in relation to the One-in relation to the Others, are cross-combined, thus constituting a 2 x 2 chiasmus.The I draw the same conclusion as Sayre.We both focus on three pairs but treat them differently.These three pairs are (1) the pair of antecedents 'if the One is-if the One is not', (2)   the pair of subjects 'the One-the Others', and (3) the pair of relations 'in relation to the Onein relation to the Others'.By combining (2) the pair of subjects with (3) the pair of relations, I address the fact that these two pairs are cross-combined, thus constructing a 2 x 2 chiasmus; in addition, the two 2 x 2 chiasmata are incorporated into the framework established by (1) the pair of antecedents.Sayre takes another approach by analyzing (1) the pair of antecedents and (2) the pair of subjects jointly and considering the consequences simultaneously.In this way, Sayre offers a tentative interpretation by positing that there are two subjects of eight hypotheses, that is, the One (I, II, V, VI) and the Others (III, IV, VII, VIII). 38Sayre considers (3) the pair of relations 'in relation the One-in relation to the Others' by analyzing the eight hypotheses meticulously and fully considering their consequences.He thus discovers that the One is related to the One in Hypotheses I and VI, the One is related to the Others in Hypotheses II and V, the Others are related to the One in Hypotheses III and VII, and the Others are related 37 Meinwald 1991: 75;1992: 381;2014: 480-1. 38 Sayre 1978: 134-6: 134-6, especially the scheme at 135; 1983: 37-42, especially the scheme at 41-2.
to the Others in Hypotheses IV and VIII. 39On this basis, Sayre corrects his tentative interpretation by maintaining that there are four subjects of eight hypotheses: the One in relation to the One (I-VI), the One in relation to the Others (II-V), the Others in relation to the One (III-VII), and the Others in relation to the Others (IV-VIII).In conclusion, the subjects in each of the paired hypotheses (I-VI, II-V, III-VII, and IV-VIII) are the same. 40Sayre analyses the text, whereas I use the chiastic method.Although we take different approaches, we arrive at the same conclusion; this fact fully demonstrates the propriety and validity of this interpretation.Following in the footsteps of Porphyry and the Alexanderian commentators, I pursue an alternate proposal by discussing how Plato employs a special method of chiasmus to design the exercise as a complex of eight hypotheses and structure the eight hypotheses in a chiastic way.With the help of chiasmus, I finally show that Meinwald's predicative interpretation is incorrect, not only because pros heauto-pros ta alla does not qualify the predicate, but also fundamentally because the qualification pair does not appear in the form of pros heauto-pros ta alla, but rather in the form of pros to hen-pros ta alla.
To summarize the eight hypotheses, whether the One is or is not, the One in relation to itself or the Others in relation to themselves are neither H nor con-H for many values of H, whereas the One in relation to the Others or the Others in relation to the One are both H and con-H for many values of H. 41 Hypotheses II, III, V, and VII positively note that when the One and the Others are combined with each other, their combination is equipped with and predicated by contrary attributes, while Hypotheses I, IV, VI, and VIII show the negative results of separating the One and the Others from each other.If I set aside all of the hypotheses that have a negative consequence and focus on all of the hypotheses that have a positive consequence, I can further conclude that a Form and its Opposite cannot be related solely to themselves but must be combined with each other.'The One-the Others' represent opposite Forms in the sense that they are equivalent to 'the One-the Many', with 'the Others' (ta alla) being equivalent to 'the Many' (ta polla). 42'The Others' differ from the One but cannot be anything other than the One.Rather, 'the Others' are identified with 'the Many', which can be clearly seen in the statement that "the Others that differ from the One is [the] Many" (τὰ δ' ἕτερα τοῦ ἑνὸς πολλά που ἂν εἴη, 158b2-3).In constructing the exercise, Plato is concerned with the opposite Forms 'the One-the Many'; this fact is most evident in the paradigm that Plato's character Parmenides 39 Sayre 1978: 136-141, especially the scheme at 140;1983: 42-9, especially the scheme at 44. 40 Sayre 1978: 139-141, 147-8, especially the scheme at 140;1983: 42-5, especially the scheme at 44;1996: 116-9, 124-6, especially the scheme at 119.See also Kutschera 1995: 51;Scolnicov 2003: 25-9. 41 See also Sayre 1978: 143-4;1983: 46-7;1996: 119, 126-133. 42 See also Walker 1938: 493-7; Halfwassen 1992: 299-300; Scolnicov 2003: 26, Fig. 2 at 28.  invokes in the transitional section.Since Parmenides explicitly states that in the exercise of the Many, we investigate the consequences of the Many and those of the One (136a5-b1), in the exercise of the One, correspondingly, we should inquire into the consequences of the One and those of the Many.
In the first part of the Parmenides, Plato's character Socrates encounters the difficulty (aporia) posed by the question of whether the intelligible entities, that is, the opposite Forms, such as One-Multitude/Many (τὸ ἓν-πλῆθος), Similarity-Dissimilarity, and Motion-Rest, should be combined with or separated from each other (129d6-e4).In the second part, Plato's character Parmenides examines the combination and separation of the opposite Forms of 'the One-the Many'.Based on the conclusion drawn from the second part of the Parmenides, that is, that a Form and its Opposite cannot be related solely to themselves but must rather be combined with each other, the aporia presented in the first part can be solved as follows.The components of the opposite Forms 'the One-the Many' cannot be separated from each other (as the negative consequences of Hypotheses I, IV, VI, and VIII show) but must rather be connected with each other (as the positive consequences of Hypotheses II, III, V, and VII show).
Moreover, the confusion regarding how it is possible for the Kinds and Forms in themselves to have contrary properties (εἰ μὲν αὐτὰ τὰ γένη τε καὶ εἴδη ἐν αὑτοῖς ἀποφαίνοι τἀναντία ταῦτα πάθη πάσχοντα, ἄξιον θαυμάζειν, 129c2-3) can ultimately be dispelled by the positive consequences of Hypotheses II, III, V, and VII such that the combination of the opposite Forms, the One and the Many, has contrary properties and is both H and con-H for many values of H.

Appendix
In the transitional section of the Parmenides, Plato designs a universal scheme for an exercise using chiasmus.Based on Plato's use of chiasmus, I reconstruct the universal scheme in the following way.Having assumed a Form, I identify 'the Form-its Opposite' as one pair of opposites and 'in relation to the Form-in relation to its Opposite' as another pair of opposites.
These two pairs cross each other, thus establishing a 2 x 2 chiasmus that connects the Form or its Opposite with itself or with its Opposite in a chiastic way.As previously noted, pros in the sense of connection can be expressed by the word 'and'; visually, the relationship (pros) of the Form or its Opposite to itself or to its Opposite can be represented by a one-way arrow 43 , as shown below: 43 Many thanks to Christoph Horn for correcting me by noting that the arrows (which signify the relation of the subject to itself or to its opposite) are not two-way but should rather be one-way.

1.
Plato's Division of the Exercise: An Application of Chiasmus After encountering many difficulties concerning the theory of Forms, in the transitional section of the Parmenides, Plato's character Parmenides notes that to understand Forms, gymnastic training (dialectic exercise) is required.The exercise consists of multiple hypotheses, deriving apparently contradictory consequences from two contradictory antecedents.Plato's character Parmenides explains the design of the exercise as follows: hypothesizes as an example, if Many are, what must follow for the Many themselves in relation to themselves and in relation to the One and for the One in relation to itself and in relation to the Many.Conversely, if 6 I use the Greek text of the Parmenides edited by Burnet, I. (1901, Oxford: Oxford University Press) and refer to the English translation by Gill, M. L. and Ryan, P. (1997, In Cooper, J. M. [ed.] Plato Complete Works, Indianapolis, Cambridge: Hackett Publishing Company: 359-397) and the German translation by Schleiermacher, If the One is, what follows for the One in relation to itself and to the Others, and what follows for the Others in relation to the One and to themselves?If the One is not, what follows for the One in relation to itself and to the Others, and what follows for the Others in relation to the One and to themselves?
I present the original below in Diagram 1*.Diagram 1* To clarify this issue, I also modify the original Diagram 1* and draw an alternative Diagram 1, as shown below.Diagram 1
Ammonius and Philoponus, I use Aristotle's term, the synonym καθ' ἕκαστόν, to signify the individual because καθ' ἕκαστόν and καθόλου (i.e., καθ' ὃλον) are similar in construction to κατὰ τινος and can be regarded as a prepositional pair.Second, I switch the positions of καθόλου and συμβεβηκὸς in Diagram 1* so that in Diagram 1, καθόλου is located in the position of συμβεβηκὸς and συμβεβηκὸς in the position of καθόλου.My reason for this switch is that I use Diagram 1 (which is concerned with the classification of things) to illuminate Diagram 2 (which is concerned with the exercise of the One).As Diagram 2 below shows, in the exercise of the One, it is more natural to arrange the pair of prepositional structures πρὸς τὸ ἓν-πρὸς τὰ ἄλλα vertically up and down than to place them left and right on the same horizontal line.The same reasoning holds for καθόλου-καθ' ἕκαστόν, which can be regarded as prepositional structures analogous to πρὸς τὸ ἓν-πρὸς τὰ ἄλλα.As with the position of πρὸς τὸ ἓν-πρὸς τὰ ἄλλα (see Diagram 2 below), I arrange καθόλου-καθ' ἕκαστόν vertically up and down (see Diagram 1 above), thereby switching the positions of καθόλου and συμβεβηκὸς.Third, as Diagram 1* shows, Ammonius and Philoponus (as well as Boethius) draw six lines.A 2 x 2 chiasmus requires two pairs of opposites to cross each other.Two pairs of opposites are cross-combined, thus producing six possible combinations, which are represented by the six lines drawn in Diagram 1*.The cross-combination of two pairs of opposites generates six possible combinations, two of which are invalid because the opposites 2, Aristotle classifies things into four kinds using a 2 x 2 chiasmus; Ammonius and Philoponus take this step further by drawing a diagram for chiasmus.Following in the footsteps of Porphyry and the Alexanderian commentators, I propose that in the Parmenides, Plato uses the same kind of 2 x 2 chiasmus to divide the exercise of the One into eight hypotheses.The exercise of the One includes opposite antecedents.Regarding each antecedent, Plato performs a 2 x 2 chiasmus.The two subjects of the inquiry 'the One-the Others' (which appear as 'for the One-for the Others') and the two subjects' relations 'in relation to the One-in relation to the Others' cross each other, thus establishing a 2 x 2 chiasmus, as illustrated in Diagram 2: Diagram 2 Under opposite antecedents, two chiasmata emerge.They are constructed in the same way and exhibit the same 2 x 2 structure.They differ in terms of their antecedents: one has a positive antecedent (see Table One and the Others.By 'the subject of the hypothesis', I refer to the grammatical subject identified in the deduction and consequence of each hypothesis, which takes the form of a question and answer in each hypothesis.If the One is, e.g., the deduction of Hypothesis I is formulated as the question 'what follows for the One in relation to itself?'The consequence of Hypothesis I is expressed as the answer 'the One in relation to itself is neither H nor con-H (abbreviation of the contrary of H) for many values of H'.Thus, the subject of Hypothesis I is the One in relation to itself.In general, the subject of the hypothesis as a whole is composed of a subject of the inquiry (namely, the One or the Others) and its relation to itself or to its opposite (namely, its relation to the One or to the Others).On this account, a 2 x 2 chiasmus constitutes the subjects of the hypotheses by cross-combining the two subjects of the inquiry 'the One-the Others' with those two subjects' relations, i.e., 'in relation to the One-in relation to the Others'.Using the 2 x 2 chiasmus, the four subjects of the hypotheses are established: the One in relation to itself, the One in relation to the Others, the Others in relation to the One, and the Others in relation to themselves.These four subjects are distributed under opposite antecedents; thus, eight hypotheses are generated.In this manner, Hypotheses I and VI share the subject 'the One in relation to itself', Hypotheses II and V share the subject 'the One in relation to the Others', Hypotheses III and VII share the subject 'the Others in relation to the One', and Hypotheses IV and VIII share the subject 'the Others in relation to themselves'.According to the 2 x 2 chiasmus, therefore, the subjects in each of the paired hypotheses (I-VI, II-V, III-VII, and IV-VIII) are the same.Notably, some scholars, in their reconstructions of the structure of the exercise, have applied 'in relation to the One-in relation to the Others' to each of the hypotheses (as Rickless, Polansky, and Cimakasky have done) or applied them to the third hypothesis (as Gill has done).As said, 'in relation to the One-in relation to the Others' are cross-combined with 'the One-the Others' to establish four subjects for the eight hypotheses; thus, it is impossible for the two relations to be applied to each of the eight hypotheses.constitutions; see Liu 2021: 28-31.In History of Animals (487b34-488a2), similarly, Aristotle establishes a 3 x 2 chiasmus by cross-combining the triple 'walking-flying-swimming' with the pair 'gregarious-solitary', thus classifying animals into six classes; see Liu 2021: 41-2.Many thanks to George Rudebusch for suggesting the use of the mathematical terms m-tuple and n-tuple in this context, which express the meaning of the m x n chiasmus more precisely.Chiasmus provides strong evidence to prove that in Plato's design of the exercise, each of the eight hypotheses examines the consequences of the One or those of the Others-either in relation to the One or in relation to the Others and that none of the hypotheses examines the consequences of the One or those of the Others-both in relation to the One and in relation to the Others. 22hat the 2 x 2 chiasmus establishes four subjects of eight hypotheses is evident not only in the methodological remarks but also in the intermediate and final summaries.In these two summaries, the four subjects of the hypotheses are the grammatical subject of the consequence, formulated as 'the subject of the hypothesis is negated or affirmed for many values of H', while in the methodological remarks, they appear to be the grammatical subject of the deduction, expressed as 'what follows for the subject of the hypothesis'.Echoing the methodological remarks that elucidate the eight deductions, the final summary summarizes the eight consequences as follows:[…], ἓν εἴτ' ἔστιν εἴτε μὴ ἔστιν, αὐτό τε καὶ τἆλλα καὶ πρὸς αὑτὰ καὶ πρὸς ἄλληλα πάντα πάντως ἐστί τε καὶ οὐκ ἔστι καὶ φαίνεταί τε καὶ οὐ φαίνεται.(Parm.166c3-5)   […], whether [the] One is or is not, it and the Others in relation to [itself/]themselves and to each other are all things in all ways and are not as well as appear and appear not.In the final summary, the four subjects of the hypotheses are distributed under opposite antecedents, appearing as the grammatical subjects of the eight consequences.Since the subjects in each of the paired hypotheses are the same (I-VI, II-V, III-VII, and IV-VIII), the final summary can be reconstructed as follows.(a) Whether the One is or is not, the One in relation to itself (Hypotheses I-VI) and the Others in relation to themselves (Hypothesis IV) are nothing in the sense that they are neither H nor con-H (οὐκ ἔστι).(b) Whether the One is or is not, the One in relation to the Others (Hypotheses II-V) and the Others in relation to the One (Hypothesis III) are all things in all ways in the sense that they are both H and con-H (πάντα πάντως ἐστί).(c) If the One is not, the Others in relation to the One (Hypothesis VII) appear to be both H and con-H (φαίνεται).(d) If the One is not, the Others in relation to themselves (Hypothesis VIII) appear to be neither H nor con-H (οὐ φαίνεται).The intermediate summary explicates the first four consequences, which are derived from the positive antecedent, as follows:Οὕτω δὴ ἓν εἰ ἔστιν, πάντα τέ ἐστι τὸ ἓν καὶ οὐδὲ ἕν ἐστι καὶ πρὸς ἑαυτὸ καὶ πρὸς τἆλλα, καὶ τἆλλα ὡσαύτως.(Parm.160b2-3) Thus, if [the] One is, the One is all things and not one in relation to itself and to the Others, and the Others likewise.Consistent with the final summary, the intermediate summary should appear as follows: if the One is, the One in relation to itself (Hypothesis I) and the Others in relation to themselves (Hypothesis IV) are nothing in the sense that they are neither H nor con-H, while the One in relation to the Others (Hypothesis II) and the Others in relation to the One (Hypothesis III) are all things in all ways in the sense that they are both H and con-H.On this basis, what Plato literally demonstrates in the intermediate summary can be reconstructed as follows: Since it is nothing, the One in relation to itself is not one, as Hypothesis I shows (τὸ ἓν […] οὐδὲ ἕν ἐστι […] πρὸς ἑαυτὸ); since it is all, the One in relation to the Others is all things, as Hypothesis II shows (πάντα τέ ἐστι τὸ ἓν […] πρὸς τἆλλα).The same reasoning holds for the Others (τἆλλα ὡσαύτως): the Others in relation to the One are all things, as Hypothesis III shows; the Others in relation to themselves are nothing, as Hypothesis IV shows.As Meinwald properly notes,

2 x 2
chiasmus constitutes four subjects of eight hypotheses: 'the One in relation to itself' is the subject of Hypotheses I and VI, 'the One in relation to the Others' is the subject of Hypotheses II and V, 'the Others in relation to the One' is the subject of Hypotheses III and VII, and 'the Others in relation to themselves' is the subject of Hypotheses IV and VIII.In conclusion, the eight hypotheses differ in subject such that the subjects in each of the paired hypotheses (I-VI, II-V, III-VII, and IV-VIII) are the same.
Many are not, you must inquire what will follow for the One and for the Many both in relation to [itself/]themselves and in relation to each other.(2) And again, if you hypothesize, if Similarity is or if it is not, what will follow from each hypothesis for the things hypothesized themselves and for the Others both in relation to [itself/]themselves and in relation to each other.(3) And the same scheme is applied to Dissimilarity, to Motion, to Rest, to Generation and Destruction, and to Being itself and Nonbeing.(4) And in one word, in the case of whatever you hypothesize as Being or as Nonbeing or something that has another property, you must inquire about the consequences in relation to itself and in relation to each one of the others, whatever you select, and it is likewise with several [of the Forms] and all [of the Forms].And again, [you must inquire about] the Others in relation to themselves and in relation to another whatever you always select, whether you hypothesize that what you hypothesize is or is not.When you are completely trained, you will authoritatively see the truth."
This claim is true, but it is not what Plato meant.Indeed, the Sophist is not Sophist by definition, and the Statesman is not Statesman by definition since Plato does not use self-predication to produce a definition.Rather, Plato defines the Sophist or Statesman by dividing a certain genus, expertise or knowledge, into multiple differentiae, so the Sophist or Statesman is ultimately defined as a combination of a genus with multiple differentiae.It is true that self-predication does not serve as a mode of definition; in the Parmenides, however, Plato is not concerned with definition.Peterson defends Meinwald's interpretation by eliciting a definition, which Plato does not consider here; Meinwald interprets 'the One is not one' by adding pros heauto, which does not exist in Plato's text (τὸ ἓν οὔτε ἕν ἐστιν [οὔτε ἔστιν], 141e12).Meinwald's interpretation and