Rethinking Deduction Five of Plato’s Parmenides (160b5-163b6)1

The fifth “deduction” in Plato’s Parmenides (160b5-163b6) concerns the consequences that follow for a (or the) one from the hypothesis that it is not. I argue that the subject of this hypothesis is, effectively, any Form, considered just insofar as it is one Form. The hypothesis, I further argue, does not concern any essential aspect of a Form, but rather posits its contingent non-instantation (“a one is not” = “a Form is not instantiated”). The motion this deduction attributes to its one is a special type of motion: motion into and out of instantiation.

In the first part of the Parmenides, Socrates posits the existence of intelligible Forms in order to block Zeno's attempts to reduce to absurdity the view that there are many things. After raising problems for this proposal that leave Socrates at a loss, Parmenides remarks that Socrates' difficulties arise not so much from the proposal itself, but from Socrates' failure to submit it to the proper dialectical examination. In considering any novel ontological posit, one should investigate the consequences that follow, both for the thing potentially posited and for things "other" than it, first, from the hypothesis that it is, and then, from the hypothesis that it is not. Parmenides then agrees to illustrate this four-fold procedure by applying it to (as he says) his own hypothesis: the one (137b1-4). He proceeds to give two treatments of each of the four tasks he has distinguished, and these eight deductions (as they are usually denominated) generate many contradictory results, not only among themselves but also within individual deductions.
The arguments of these deductions have puzzled commentators since antiquity, and have in the past half-century received a great deal of critical scrutiny. This paper aims to advance the understanding of these arguments by sketching what I think is a novel interpretation of the Fifth Deduction, Parmenides' first go at deducing the consequences that follow for the one (or for "a" one) from the hypothesis that it is not. There are three important questions that an interpretation of this (or any other) deduction must address: (1) what is the referent of "one" in the hypothesis; (2) what sort of being is at issue in the hypothesis from which the deduction starts, and (3) whether "one" figures as the subject or the predicate in the hypothesis. As to this last question, I shall assume, like most scholars, that in our hypothesis "one" figures as subject. 2 The options for understanding what exactly "one" refers to are (a) the Form of Unity; 3 (b) any Form considered as a unit; 4 (c) any sensible object considered as a unit; 5 (d) any object at all, considered as a unit. 6 Lastly, as concerns the being at issue in the hypothesis, there are two major options. On one of these, the Fifth Deduction is concerned with the kind of non-being at play in negative predication, and which is explicated in the Sophist in terms of otherness (cf. 255c-260b). The Fifth Deduction would then be concerned, on this account, with the non-being involved when we say that the (or "a") one is not F. 7 The other major option takes the being denied to the one here as existence, more specifically, spatio-temporal existence. 8 In what follows I shall argue that the one should be understood as any Form, considered as a unit, and that the being denied it is spatio-temporal existence, construed in a special way: as (contingent) non-instantiation.
The Fifth Deduction can be divided into five sections: 1. 160b-5-d2: Introductory passage specif y ing the one in question in the deduction. 2. 160d3-161c2: Logico-ontological features of the one that is not. 3. 161c3-e2: Quantitative features of the one that is not. 4. 161e3-162b8: The being of the one that is not. 5. 162b9-163b5: Motion and rest of the one that is not.
The introductory section begins from the intelligibility of the hypothesis itself. Parmenides points out that the sentence "one is not" asserts something different from analogous sentences with a different subject term.
He offers as examples first the sentence "notone is not," which he calls the "complete opposite" of our hypothesis, and continues with the sentences "largeness is not" and "smallness is not." 9 I do not think much should be made of Parmenides' use of "not-one" in his initial example; his point is simply to show, as starkly as possible, that difference in subject-term produces a different meaning. The choices of "largeness" and "smallness," on the other hand, are, I think, very significant. They seem to be something like Forms; 10 since they are substituted for the one in our hypothesis, it is reasonable to suppose that it, too, is a Form. Our only question is whether Parmenides means the Form of Unity in particular, or any Form, considered as a unit. In favor of the former is that it has Parmenides invoke the contrast between Unity and (e.g.) Largeness, which seems a more straightforward contrast than that between any-Form-you-like and Largeness. In favor of the latter are translation considerations (εἰ ἓν μὴ ἔστιν at 160b7 is perhaps most naturally translated, "if a one is not") and the fact that no use is made of the specific nature of unity in the rest of the deduction. We may safely conclude that either Parmenides is discussing any Form as a unit, or (which amounts to much the same thing) he is using the Form of Unity as an example, the conclusions about which, since they do not depend on the specific nature of unity, can be generalized to any Form as such.
Parmenides argues that, since we know that the above sentences assert that in each case something different is "the thing that is not" (τὸ μὴ ὄν), we must have knowledge of the one of our hypothesis, a knowledge that is independent of whether we attribute being or not being to it (160c7-d2). In section two he expands the point about the one's difference from the others by saying that it possesses "difference in kind" (ἑτεροιότης) from them. This relatively rare term (used in our dialogue only here and in the Sixth Deduction 11 ) is most naturally taken as emphasizing that the one differs not only numerically from the others, but also qualitatively from them. 12 That is, it has some qualitative nature that differentiates it from the others and allows us to think and talk of it in distinction from them. Given the prior occurrence of Largeness and Smallness as the things the one is different from, we should take Parmenides to be talking about the different essential natures of different Forms.
In the remainder of section two Parmenides attributes other logico-ontological features to the one, all on the basis of the intelligibility of the hypothesis. The one is a definite object of reference (a "that"), and possesses likeness to itself and unlikeness to the others (i.e., other Forms). In the middle of this section he emphasizes that possessing these features is not at all incompatible with our hypothesis that the one is not: It is not possible for the one to be (εἶναι), if in fact it is not, but nothing prevents it from partaking of many things. Indeed, it's even necessary, if in fact it's that one and not another that is not. (160e7-161a2) The remarkable passage is important for the interpretation of our deduction, since it in effect gives us a gloss on what is meant by being in the hypothesis under consideration. 13 I suggest that the passage can best be understood if we take the being that is denied to the one (both here and in the hypothesis) to be existence, and in particular, spatio-temporal existence. From the beginning of the deduction Parmenides has insisted that to make any meaningful assertion about the one, including that it is not, requires that it have certain features: it must be knowable, be an object of reference, differ qualitatively from other things, bear unlikeness to them and likeness to itself. In the current passage Parmenides points out that the ascription of these features to the one we are talking about does not entail that it has spatio-temporal existence.
The theme of the relation between the one's (non-)being and its possession of attributes is taken up again, in perhaps even more paradoxical form, at the beginning of section three. There, at the beginning of his treatment of the quantitative attributes of the one, we read: Furthermore, it is not equal to the others either (οὐδ᾽αὖ ἴσον γ᾽ ἐστὶ τοῖς ἄλλοις); for if it were equal, it would both by that very fact (ἤδη) be, and be like them in respect of equality. But those are both impossible, if in fact one is not. (161c3-5) Parmenides emphasizes that to say that the one that is not is equal to the others involves two absurdities, the first of which is, quite simply, that it would be. In light of the earlier passage discussed above (160e7-161a2), Parmenides must here be making a distinction between the locutions "the one participates in F-ness" and "the one is F." The former locution lets us describe the features the one possesses without asserting that it exists; the latter does not. And indeed, looking back over the dialogue so far, we find that Parmenides has made a point of avoiding the latter locution. 14 Instead, he says either that the one participates in such-and-such a feature, or that such-andsuch a feature belongs to it (εἶναι + dative). 15 In fact, later in this section Parmenides argues that equality does indeed count as one of the features of the one: "To the one that does not exist, then, as it seems, there would belong a share of equality, too" (161e1-2). 16 The problem with the statement at the beginning of the section was the implication of existence that Parmenides attaches to the locution "x is F." This repeated insistence that the one's possessing various features does not entail its being (or its existence, as I have construed it) makes all the more surprising the assertion at the beginning of section four that the one of our deduction also "participates in being (οὐσίας), in a way." All but the last line of section four is dedicated to arguing for this claim; the argument ends with the assertion: "Being (οὐσία) too, then, seems [to belong] to the one, if it is not" (162b6-7). The argument is thus encased by both locutions Parmenides has used to ascribe attributes to the one. One might suppose that Parmenides is simply adding one more feature to those he has already argued the one possesses, a feature more or less on the same level with likeness-to-self and the rest. But the initial qualification that the one participates in being "in a way" alerts us to the possibility that this case is different.
What sort of being does this argument ascribe to the one? If this is not to constitute a bare contradiction of the hypothesis of the deduction, it must be a different sort of being from that at issue in the hypothesis. One possibility is that, while the hypothesis denies that the one has spatio-temporal being, the one is said here to participate in the kind of being appropriate to Forms: eidetic being, if you will. On this view, this eidetic being is a kind of being that is constituted by, or presupposed by, the one's participation in other Forms (such as likeness-to-itself). 17 While attractive, there are reasons to resist such an interpretation. One such reason consists in the fact that, in tying the being here assigned to the one to its participation in other Forms, this interpretation seems directly to contradict Parmenides' earlier claim (160e7-161a2) that participation in Forms does not entail that the one is. 18 One might insist that Parmenides there meant that spatio-temporal existence is not required for participation, and that now he is revealing that a different sort of being is. But if the point of our passage is that the participation we have been talking about actually requires that the one have a kind of being, it seems quite odd for Parmenides to have already drawn attention to the (non-obvious) issue of whether participation requires being, and to have answered that question in the negative. Though this consideration is not decisive, it does motivate looking for another interpretation; and as we shall see, there is one available that avoids this problem, and also accords better (as I shall argue) with the remainder of the deduction.
Parmenides begins his argument for the claim that the one of our deduction participates in being as follows: [The one] must have the condition we say it to have (ἔχειν αὐτὸ δεῖ οὕτως ὡς λέγομεν); for if did not have that condition, we would not be saying true things when we say that the one is not; but if we are saying true things, it is clear that we are saying things that are (ὄντα). (161e4- 6) In saying that the one is not, this passage asserts, we are saying that it is in a certain condition; and its being in that condition is precisely a kind of being. This seems to be a general point about predication; so it is natural to think that the being here referred to is the being that is sometimes thought to be expressed by the copula. 19 In fact, I think that Parmenides has a somewhat narrower notion in mind: that of the copula in a specifically accidental predication, that is, a predication that is only contingently true. The language of being in or having a certain condition (ἔχειν πως) is, I suggest, particularly well-suited to accidental predication. 20 Our deduction is not considering a one that by its nature is not; it is considering a contingency, the case where a one which we know and can talk about happens not to be.
Such a supposition fits well with a straightforward reading of how the argument proceeds: Therefore, as it seems, the one is a notbeing (οὐκ ὄν); for if it is not to be a notbeing, but is somehow to give up its being towards not-being, it will straightway be a being (ὄν). -Absolutely -So if it is not to be, it must have being a not-being (τὸ εἶναι μὴ ὄν) as a bond (δεσμόν) of its not being. (162a1-5) If the not-being to which the bond binds the one were (in effect) the eternal being that characterizes a Form, then the supposition that the one (as a Form) might lose this being without the necessary bond would be not only contrary to fact, but also metaphysically impossible. Though this is not a decisive consideration against such an interpretation, it is nonetheless true that if the bond here is, rather, the copula of an accidental predication, a much less extravagant scenario is envisioned. There is no impossibility in the one's ceasing to be in the condition in which it only contingently finds itself. Furthermore, it is at least prima facie plausible to maintain that there must be something keeping it in that condition, so long as it remains in it; and its ceasing to be in that condition may well be described as a breaking or destruction of that bond.
If the one of our deduction is a Form, how are we understand the hypothesis that it, contingently, is not? If we understand the not-being of our deduction as non-existence, and in particular spatio-temporal non-existence, then our hypothesis is that the Form is contingently not in space and time. Forms are, to be sure, essentially non-spatio-temporal; but it is, I suggest, only a slight stretch of language to say that when a Form is instantiated, it possesses spatio-temporal existence. Our deduction considers instantiation, as it were, from the side of the Form rather than the sensible. For the Form to be instantiated is for it, for some period of time and in some place, to be spatio-temporally. The hypothesis of our deduction posits a case where a Form is not instantiated; and our current passage points out that when we say such a thing, we are in fact ascribing a certain kind of being to the Form: its being not-instantiated-here.
The being in which the one is here said to participate is copulative: it connects the one to its condition of being un-instantiated. Whether or not this is ultimately a philosophically viable position is a question beyond the scope of this paper. Certainly Parmenides goes on to develop the notion in a paradoxical fashion -insisting that there is not only a positive copula, but also a negative one, so that what is instantiated is connected by copulative being to spatio-temporal existence and by negative copulative being to spatio-temporal non-existence (see 162a6-b3). But however we are to understand these developments, and whether Plato means them seriously or not, nonetheless understanding the being that the one shares in as that which contingently connects it to being (un)instantiated makes good sense of the deduction up to this point. And it will further prove itself in helping make sense of the assertions about the motion of the one that Parmenides makes in the fifth section of the deduction.
At the end of the fourth section, after proving that that the one that is not participates in being, Parmenides remarks that, since it is not, it also participates in non-being. At the beginning of section five he advances the claim that something can only be and not be in the same condition if it transitions from one to the other. Here, as at the start of the so-called appendix to the first two deductions (155e8-10), Parmenides makes a blatantly fallacious inference: since on any account the one has been shown to participate in being and non-being in different senses, there is no incompatibility in its participating in both at the same time. But although the inference is faulty, nonetheless, if the one's participation in being is, as I have proposed, its relation to (non)instantiation, then a consideration of motion from being to not-being is certainly in order: for a Form does (typically) 'move' back and forth between being instantiated and not being instantiated. And supposing that this is Parmenides' topic helps make sense of the many surface contradictions which mark this section.
If the one of our hypothesis is a Form, then the ascription to it of any sort of motion may seem particularly problematic. Before going on to show how this motion can be understood as the change from being un-instantiated to being instantiated, it will be useful to note an alternate interpretation that has been proposed in the literature. This view takes its cue from another Platonic passage where the possibility of ascribing motion to a Form is considered: Sophist 248b-249b, where a Form's coming to be known is construed as a kind of change or motion. On this view, the motion between being and not-being that the Form undergoes mirrors the change in the mind of the inquirer when, for example, she first subsumes a Form under a higher one (so contemplating an aspect of its being) and then distinguishes it from its congeners (so contemplating what it is not). 21 But the motions that would thus be ascribed to the Form depend crucially on the context of philosophical inquiry; and of that there is no explicit mention in our passage. This is, I think, a serious drawback to the interpretation under discussion.
After stating that the one must change from one condition to another, Parmenides investigates what sort of change this could be. He first considers locomotion, and rules that out, since the one that is not "is nowhere among beings" (162c7); he then rules out rotating in place, for much the same reason. These are obviously changes that instantiations of Forms undergo, and if the one of our deduction is a Form, it cannot undergo these. Parmenides then considers alteration, and rules it out on quite other grounds: And surely, the one isn't altered from itself either, whether as a being or as a notbeing. For the statement would no longer be about the one, but about something else, if in fact the one were altered from itself. (162d5-8) Here Parmenides reverts to the logicosemantic considerations he appealed to at the start of the deduction. The qualitative likeness-to-self and difference-from-others that the one must have in order for us coherently to talk about it must not change, if it is indeed to be a stable referent of our discourse.
Having ruled out all the possible kinds of change, 22 Parmenides concludes that the one stands fixed and at rest (162e1-2). But far from abandoning his earlier claim that the fact that the one is in two conditions requires it to change, Parmenides calmly concludes that the one is both moving and at rest (162e2-3).
Parmenides does not, however, just leave us with this bare contradiction. Rather, he reopens the question of the sort of motion the one undergoes, this time at a meta-level, as it were. Assuming that the one undergoes some motion, he argues that it must, thereby, necessarily undergo alteration: Furthermore, if in fact it moves, there is a great necessity for it to be altered; for in whatever way something is moved, to that extent it is no longer in the same condition as it was, but in a different one (οὐκέθ᾽ ὡσαύτως ἔχει ὡς εἴχεν, ἀλλ᾽ ἑτέρως). (162e4-163a2) The sort of alteration Parmenides has in mind here is clearly not the same as the change in qualitative character that he has just previously rejected. In particular, it is noteworthy that in this passage Parmenides reverts to the language of having (or being in) a condition, language with which he introduced the question of motion (162b9-10), and which made its first appearance in the deduction when he approached the question of the kind of being that the one that is not shares in "in a way" (161e4). If the "conditions" in those earlier passages were the conditions of being instantiated and being un-instantiated, the alteration of the one at issue here is its change from one such state to the other. Parmenides' subsequent inference, then, that the one both does and does not undergo alteration is thus only an apparent contradiction (163a6-7).
Parmenides goes on to explicate the sense in which the one does alter in terms of coming to be and perishing: Must not that which is altered come to be different from what it was before, and perish from its previous condition (ἐκ τῆς προτέρας ἕχεως)? … Therefore also the one, if it is not, comes to be and perishes, if it is altered, and does not come to be or perish, if it is not altered. (163a7-b4) The interpretation we have developed allows us to understand the coherent sense that lies behind this swirl of apparent contradictions. The one's progressing from the state of not-being to being is indeed a coming-to-be -not of the Form insofar as it is the qualitative nature it is and remains, but rather of its instances in the spatio-temporal world. 23

Bibliography
These sentences seem to me to settle beyond reasonable doubt that the one of our hypothesis is subject, not predicate. Parmenides is hardly asking us to consider the sentence "the world is not largeness." 10 Indeed, they are called forms (εἴδη) in the Second Deduction (149e9). 11 164a3. The adjective ἑτεροῖον is also found in deduction VII (165d2) in connection with the variegated appearances of the many when it is hypothesized that one is not. 12 See on this point the discussion in Cornford (1950) 222-223. 13 The importance of the passage is noted by  33-34, who suggests that the only way to make sense of it is to take "participation" here to refer to accidental (his "per aliud") predication, and to take the being denied of the one to be that involved in essential (per se) predication. On his view, this passage asserts that the one of our hypothesis has no essential attributes (a condition that, he argues, characterizes sensibles). 14 This has been noticed by several commentators, including ) 223-224, Miller (1986) 145, and Kahn (2013 Compare, for example, 161b3: "Then unlikeness would also belong to the one" (εἴη δὴ ἂν καὶ τῷ ἑνὶ ἀνομοιότης). 16 The idiom here combines both the language of participating-in and that of belonging-to. 17 For views of this sort, see  147149 and Sanday (2015) 159-161. 18 Miller does not seem to comment on 160e7-161a2. Sanday (2015) 157 quotes only the latter part of the passage (161a1-2), omitting the crucial point that participation does not entail being. 19 The question of whether treating the copula as expressing a kind of being leads to an infinite regress is one that cannot be addressed here. Gill and Ryan (1996) 94-99 argues that the purpose of this section is precisely to show that treating the copula as Parmenides does here leads to a vicious regress. (For a fuller statement of the position, see Gill (2002).) 20 Compare the Stoic category of πὼς ἔχοντα, of which a standard example was apparently a fist (a hand disposed a certain way); see Brunschwig (2003) 212. 21 See  149-153 with n. 41, Sanday (2015) 161-165. 22 Parmenides had divided exhaustively divided change into these kinds in the Second Deduction (138b8-c6). 23 I wish to thank the participants in the Virtual IPS Panel on Plato's Later Dialogues in June 2020 for valuable comments on the version of this paper presented there. I am particularly grateful to Mitchell Miller for his helpful comments on the written version of this paper; naturally, he should not be presumed to agree with any of its claims. I also wish to thank the students of my University of Kansas Plato seminar of fall 2018, who valiantly picked their way with me through the deductions of the Parmenides.