The Greatest Aporia in the Parmenides (133b-134e) and the Reciprocity of Pros Relations

Abstract

The extant attempts in the literature to refute the greatest difficulty argument in the Parmenides have focused on denying the parallelism between the pros relations among Forms and those among particulars. However, these attempts are unsatisfactory, for the argument can reach its conclusion that we cannot know any Forms without relying on this parallelism. I argue that a more effective strategy is to deny the more essential premise that the knowledge-object relation is a pros relation. This premise is false because pros relations require definitional and ontological codependence between the relata, and the knowledge-object relation does not satisfy this reciprocity condition.

Résumé

Les tentatives existantes dans la littérature de réfuter l'argument de la plus grande difficulté dans le Parménide ont surtout entrepris de nier le parallélisme entre les relations de type pros entre les Formes et celles entre les particuliers. Par contre, ces tentatives sont insatisfaisantes, parce que l'argument peut mener à sa conclusion selon laquelle on ne peut connaître les Formes sans s'appuyer sur ce parallélisme. Je soutiens qu'une stratégie plus efficace consiste à nier la prémisse plus essentielle selon laquelle la relation objet-connaissance est une relation de type pros. Cette prémisse est fausse parce que les relations de type pros requièrent une codépendance définitionnelle et ontologique entre les relata, et la relation objet-connaissance ne satisfait pas à cette condition de réciprocité.

1. Introduction

In Plato's Parmenides, after shooting down various attempts by a young Socrates to offer a tenable account of the relationship between Forms and sensible particulars, Parmenides announces that the most destructive objection to the theory of Forms, “the greatest aporia,” is yet to come (133a8, b1).¹ Parmenides argues that the failure to provide such an account might lead to a deeper chasm between the world of Forms and the world of sensibles, such that the two worlds would be not only causally, but also epistemically isolated from each other, leading to the disastrous consequence that Forms are unknowable by us. Unlike the other, preceding objections, this last salvo presents a meta-critique of the theory of Forms. For the unknowability of Forms by us undermines the theory's epistemic motivation in positing Forms as the ultimate objects of knowledge with explanatory powers over their sensible homonyms. Our cognitive inaccessibility to Forms also has fatal consequences for Plato's other major philosophical commitments, such as the erotic conception of philosophy, the idea of a moral life, and the doctrines of the immortality of the soul and recollection, all of which rely on the very possibility of us having access to Forms. Moreover, if the epistemic isolation between Forms and sensibles is conceived as working both ways, it poses another devastating problem, perhaps not so much for the theory of Forms itself, but certainly for Greek piety: gods, who reside in the realm of Forms, cannot know us.

Therefore, at least considering the threat it purports to pose, the “greatest difficulty” argument seems to merit its name. However, whether the argument's actual logical construction lives up to this name is a matter of dispute. Can the argument be refuted? This is in fact part of the broader discussion about whether the logical challenges levelled against the theory of Forms in Part I of the dialogue should be regarded as damning, or to what extent Plato intends the reader to take them seriously, and, whether the exercises in Part II provide us with clues to refute them. At least four different views concerning the refutability of the greatest difficulty argument are found in the literature:

1. I. Given the argument's unacceptable consequences for the theory of Forms, a convenient interpretive option is to dismiss its logical strength entirely. Francis M. Cornford, for instance, radically underestimates the argument as “almost grossly fallacious,” and Walter G. Runciman suggests that “Plato cannot have thought this final argument valid, since it would inevitably destroy the whole theory of forms.”²

2. II. Some commentators, like Chung-Hwan Chen and Constance C. Meinwald, maintain that the argument can be turned into a refutable one that causes no serious difficulty, if the clues drawn from Part II are applied to its premises. However, neither of them offers a specific demonstration of how those clues apply to the argument based on a logical reconstruction.³

3. III. Some others hold that the argument fails, but the flaw in the argument is by no means a gross one. It takes some subtle considerations by “an able man” to refute the argument on its own terms. James W. Forrester, Frank A. Lewis, Mark L. McPherran, Reginald E. Allen, Byeong-uk Yi and Eunshil Bae offer various subtle considerations yielding different proposals for a refutation.⁴

4. IV. Yet another and larger group of commentators claim that if reconstructed with enough logical rigour and sympathy, the argument is in fact valid and presents a great difficulty for the theory of Forms. Part II does not offer a relevant answer to solve it and whether Plato has a genuine solution is a matter of further discussion. For instance, Harold F. Cherniss takes the greatest difficulty to be a “difficulty Plato has always recognized, and the complete solution of it has never been found.”⁵ Sandra Peterson concludes her well-known reconstruction of the argument by stating that she does not yet have an account of what Plato might have thought were its flaws.⁶ Samuel C. Rickless, following Peterson's reconstruction, says that the clue that he draws from the deductions in Part II helps solve the other objections but does not work in the case of the greatest difficulty.⁷ Matthew Duncombe, after providing a defence of the argument against the alternative refutations in the literature, claims that the argument is a legitimate reductio that is neither invalid nor question begging.⁸

In this paper, I will offer a novel refutation of the argument by working through a rigorous and valid reconstruction of it, which is what the proponents of (IV) would claim is missing in the current defences of (III). I will show that one of the transitional premises of the argument rests on a misconception of ‘knowledge’ or ‘knowing’ as instantiating the kind of reciprocal relationship necessary to run the argument. This premise has been almost universally accepted by commentators. Although a few have complained about its oddness, none has provided an articulate scrutiny of the problem and developed a refutation based upon it, as I hope to do here. So the position I will defend can be formulated as the following:

1. V. The argument can be reconstructed as a logically valid one, but because of at least one false premise it employs, it remains unsound. Thus, it can be refuted by denying the false premise, without having to employ any clues that can be drawn from Part II. However, the difficulty that the argument aims to point out is not trivial at all, but a serious one regarding the presentation of the theory, if not its essence.

2. The Argument is Refutable

A number of hints in the presentation of the argument in the dialogue constitute a reasonably strong ground to hold that the argument is assumed to be refutable by Plato. The first one lies in the non-committal nature of Parmenides’ announcement of the argument. He puts it through the mouth of an anonymous objector and distances himself from it:

[S]uppose someone were to say that if the forms are such as we claim they must be, they cannot even be known … you wouldn't be able to show him that he is wrong, unless the objector happened to be widely experienced and not ungifted, and consented to pay attention while in your effort to show him you dealt with many distant considerations. (133b 5–9)

While earlier in Part I Parmenides repeatedly points out Socrates’ lack of sufficient experience and training for a successful defence of the theory of Forms against objections, the party who is required to be widely experienced, gifted, and attentive here is not the defender of the theory but the objector. Only if the objector has such qualities, can he or she see that the objection against the knowability of Forms is unjustified. One may even wonder whether Parmenides implies that the greatest difficulty is not to refute the objection, but to persuade an inexperienced and ungifted objector that his or her objection fails. At any rate, the passage suggests that the argument can be refuted, though the kind of “distant considerations” to which the objector must attend in order to be persuaded remains unclear.

The second hint comes in Parmenides’ consolation of Socrates following the completion of the presentation of the argument:

… the forms inevitably involve these objections …. whoever hears about them is doubtful and objects that they do not exist, and that, even if they do, they must by strict necessity be unknowable to human nature; and in saying that he seems to have a point: and, as we said, he is extraordinarily hard to win over. Only a very gifted man can come to know that for each thing there is some kind, a being itself by itself; but only a prodigy more remarkable still will discover that and be able to teach someone else who has sifted all these difficulties thoroughly and critically for himself. (135a-b)⁹

Once again noting the extraordinary difficulty of convincing the objector of the knowability of Forms unless he or she is gifted enough to grasp the truth of the theory, Parmenides this time claims that the defender of the theory is required to be even more gifted to convey the theory to others. This lends further credence to the idea that the difficulty is not with the substance of the theory itself but with its presentation and reception.

Finally and most importantly, Parmenides abandons his critical mood entirely and makes a case for the indispensability of Forms:

… if someone … won't allow that there are forms … he won't have anywhere to turn his thought, since he doesn't allow that for each thing there is a character that is always the same. In this way he will destroy the power of dialectic entirely …. What then will you do about philosophy? Where will you turn, while these difficulties remain unsolved? (135b9-c5)

The necessity of positing Forms that Parmenides underlines here points to the epistemic motivation behind the theory: for there to be any knowledge or understanding (nous), there must be invariant and stable entities out there as intentional objects of knowledge. This epistemological assumption is consistently asserted in Plato's other dialogues as well. For instance, in Republic V, the proper object of knowledge is characterized as “what is completely” (477a3), and knowledge is said to be “set over what is” (477a9) as opposed to ignorance, which is set over what is not. In the Cratylus, Socrates insists that what is changing or changeable cannot be known by anyone, and that “it isn't even reasonable to say that there is such a thing as knowledge … if all things are passing on and none remain” (440a). Again, in the Sophist, it is emphasized that without things that are at rest and thus remain the same, “knowledge, understanding and intelligence” are impossible (249b-d). This has an important implication for the greatest difficulty argument. If Forms are thus epistemically necessary, then the conclusion of the argument that they are unknowable must be false, which entails that either the argument is invalid or that the argument employs at least one false premise even if the inference to the conclusion may be formally valid.

3. A Reconstruction of the Argument

That the argument ought to be refutable does not mean that it is easily refuted. On the contrary, all the aforementioned hints at its refutability also suggest that its refutation is “extraordinarily hard” and requires a “gift” and “many distant considerations.” Doing justice to this narrative thus requires one to reconstruct the argument as charitably and rigorously as possible. One such reconstruction in the literature has been offered by Peterson.¹⁰ Here I will adopt a version of her reconstruction.

The initial premise of Parmenides’ argument is a restatement of the separation that the theory introduces between Forms and things in or among us:

1. (1) Forms are not in us (alternatively, if x is a Form, x is not in us).

Parmenides leads Socrates to confirm this basic point about Forms: “I think you, Socrates, and anyone else who posits that there is for each thing some being, itself by itself (autēn kath’ hautēn), would agree, to begin with, that none of those beings is in us (en hēmin)” (133c2-5). Socrates’ answer includes a confirmation of the idea underlying the initial premise: “Yes — how could it still be itself by itself (autē kath’ hautēn)?” So the Forms are “not in us” because they are “themselves by themselves” (133c6). While, as I will point out later on, Plato's phrase ‘auta kath’ hauta’ has in fact a stronger ontological connotation, Socrates’ answer suggests that here it refers to Forms’ existence separately and independently from the things “in us.” “In (or among) us” (en hēmin) obviously refers to the sensible world in which we, humans reside. However, it is ambiguous whether “things in us” designate the sensible particulars themselves or the ‘immanent characters’ (or ‘likenesses’), which the sensible particulars possess by participating in the Forms and in virtue of which they are homonymous with them. Since the logical structure of the argument is not directly impacted by it, I will retain the ambiguity of “things in us” for now but will revisit this issue in the last section. Parmenides goes on to build the rest of the argument on the implications of the separation between Forms and things in us regarding a certain kind of relation:

And so all the characters that are what they are in relation to (pros) each other have their being in relation to themselves but not in relation to things that belong to us …. These things that belong to us, although they have the same names as the forms, are in their turn what they are in relation to themselves but not in relation to forms; and all the things named in this way are of themselves but not of the forms.

If one of us is somebody's master or somebody's slave, he is surely not a slave of master itself — of what a master is — nor is the master a master of slave itself — of what a slave is. On the contrary, being a human being, he is a master or slave of a human being. Mastery itself, on the other hand, is what it is of slavery itself; and, in the same way, slavery itself is slavery of mastery itself. Things in us do not have their power in relation to forms, nor do forms have theirs in relation to us; but, I repeat, forms are what they are of themselves and in relation to themselves, and things that belong to us are, in the same way, what they are in relation to themselves. (133c7-134a1):

Thus, Parmenides offers a principle of relationality:

If x is what it is in relation to (pros) y, and

1. (2) x is a Form, then y is a Form,

2. (3) x is a thing in us, then y is a thing in us.

However, the scope of the isolation that this principle introduces between Forms and “things in us” is a subject of debate. One view, defended by Forrester, Lewis, and Duncombe, is that this principle applies only to relational Forms and relational things in us, and that the isolation that it entails is quite a restricted one: a Form does not bear that relation to a non-Form but only to another Form, and the same holds for relational things in us.¹¹ However, Peterson, Ian Mueller, and Mary L. Gill maintain that the principle applies to all kinds of Forms and things in us, relational or not, and, consequently, that the isolation in question is a broader one.¹² Accordingly, a Form can be what it is with respect only to another Form, and likewise a thing in us can be what it is with respect only to another thing in us; in other words, Forms and things in us are mutually excluded from the explanations of what they are. Some, like Runciman, Chen, and Allen go as far as reading the principle as suggesting a radical divorce between Forms and things in us as to exclude all kinds of relations across the two realms.¹³ While there may be some textual ground for a broader interpretation of the principle (e.g., general statements at 133c7-9, 133e6–134a1, and the mention of Beauty, a non-relational Form, at 134c1), the restricted interpretation is more dialectically plausible. For, although the exclusion of all kinds of relations between Forms and things in us warrants the intended conclusion, given that ‘knowledge’ obviously is some kind of relation, such a premise would make the argument question begging and would be unacceptable for a Platonist, experienced or not, in the first place.¹⁴ As I noted in the Introduction, the argument sets out to show the radical and undesirable consequences of separation between Forms and things in us, triggered by a failure to offer an account of the nature of the relation between them, and would not have even a prima facie persuasive force if it were to start from the presupposition of a radical separation. I will thus adopt a restricted interpretation of the principle introducing only an intuitively plausible kind of isolation between Forms and things in us with respect to a specific kind of relation. Of course, the other crucial question here is precisely what kind of relation this is, and I will say more on this. It is, however, important to recognize that, whichever interpretation one adopts, the principle is tailored to replace the vertical relationship, which obtains between Forms and their participants in us with a horizontal relationship obtaining between the entities of the same realm.

Having laid down the principle as the ground premise of the argument, Parmenides presents a further, transitional premise:

So too … knowledge (epistēmē) itself, what knowledge is, would be knowledge of that truth itself, which is what truth is …. Furthermore, each particular knowledge, what it is, would be knowledge of some particular thing, of what that thing is …. But wouldn't knowledge that belongs to us be of the truth that belongs to our world? And wouldn't it follow that each particular knowledge that belongs to us is in turn knowledge of some particular thing in our world? …. And surely the kinds themselves, what each of them is, are known by the form of knowledge itself? (134a3-b5)

‘Knowledge’ or ‘knowing’ is treated here as falling under the extension of the kind of relation stated by the principle as obtaining only between entities of the same realm. That is, the relation between knowledge and its object, like the master-slave relation, is the kind of relation that is captured by the formula, ‘__ is what it is in relation to (pros) __.’ Accordingly:

1. (4a) If x is a knowledge of y, then x is what it is in relation to y.

However, even though the text explicitly specifies ‘knowledge’ as the first term or relatum here and ‘the object of knowledge’ as the second, Peterson suggests, without an argument, that the converse is also implied (though not stated) by the text, such that:

1. (4b) If x is a knowledge of y, then y is what it is in relation to x.¹⁵

I will offer the missing argument for (4b) in the last section, by demonstrating that any instance of pros relation must be construed as reciprocal and if ‘knowledge’ is indeed a pros relation, then (4b) is entailed by (4a). If (4a) and (4b) are the case, then knowledge and its object are what they are in relation to one another, and thus we have the transitional premise:

1. (4) If x is a knowledge of y, then x is what it is in relation to y and y is what it is in relation to x.

Parmenides claims that from these premises follows the conclusion that “none of the forms is known by us” (134b8) and the even “more shocking” one that “gods (who are supposed to reside in the realm of forms) could not know us or anything that belongs to us” (134e1). Since I take the main aim of the greatest difficulty argument to be to undermine the theory of Forms by demonstrating the unknowability of the Forms by us, I will focus here only on the alternative ways in which the argument can yield that conclusion.

Let our notations be:

F: (__ is a Form);

I: (__ is in or among us);

R: (__ is what it is pros __);

K: (__ is a knowledge of __). The premises are then:

1. (1) Fx → ~ Ix

2. (2) (Rxy ∧ Fx) → Fy

3. (3) (Rxy ∧ Ix) → Iy

4. (4) Kxy → Rxy ∧ Ryx

First Path:

1. (5) Ix ∧ Kxy → Ix ∧ Rxy (4)

2. (6) Ix ∧ Rxy → Iy (3)

3. (7) Iy → ~ Fy (1)

4. (8) ~∃x (Ix ∧ Kxy ∧ Fy) (5, 6, 7).

Thus, there is nothing in us such that it is the knowledge of a Form, or nothing we have knowledge of is a Form.

Second Path:

1. (9) Fx ∧ Kyx → Fx ∧ Rxy (4)

2. (10) Fx ∧ Rxy → Fy (2)

3. (11) Fy → ~ Iy (1)

4. (12) ~∃x (Fx ∧ Kyx ∧ Iy) (9, 10, 11).

Thus, there is no Form such that something in us is a knowledge of it, or simply, no Form is known by us.¹⁶

On this reconstruction, the argument is valid and the conclusion that Forms are not known by us does indeed follow from the premises. The only way to block the argument is then to demonstrate the falsity of one of its premises.

4. Some Alternative Suggestions for a Refutation

A number of different approaches have been offered regarding where the argument fails. Some commentators like Cornford, Runciman, and Gregory Vlastos claim that the fallacy of the argument lies in its employment of self-predication.¹⁷ Parmenides’ formulation of the master-slave relation at the level of Forms does not in fact involve self-predication, i.e., “Mastery itself … is what it is of slavery itself; and, in the same way, slavery itself is slavery of mastery itself” (133e 5–6). However, as I will show, rendering the relation between knowledge and its object as such a pros relation inescapably leads to self-predication, i.e., the Form of Knowledge is the knowledge of the Form of Truth (134a 3–4). Yet, as I will also argue, this is only a consequence of the more fundamental problem with premise (4). Thus, although self-predication is not excluded by the argument, it is not something the argument relies on as its driving force. Assigning a major role to self-predication in the greatest difficulty argument would also make the argument somewhat repetitive, and it would lead to a less natural reading of the dialogue, since self-predication and the difficulties that it leads to were the focus of another objection in Part I (i.e., the “third man” 132a1-b2).

Those proposals that engage with the steps of the argument more specifically target the parallelism between (2) and (3), and they suggest blocking the argument by rejecting (3). Two different reasons have been offered to do so. First, Mueller, Meinwald, and Charles H. Kahn argue that, while (2) seems to be in conformity with the theory's idea of the definitional priority of Forms to their sensible participants in us, (3) breaks the vertical link between Forms and their participants and replaces it with a horizontal one by wrongly treating the factual relations between correlate sensibles, e.g., human master and human slave, as definitional in the way correlate Forms, e.g., Form of Mastery and Form of Slavery, are related to each other.¹⁸ As Kahn claims, “What in fact links together masters and slaves ‘among us’ is not a conceptual relation or a matter of definition but a power relation of legal possession and physical intimidation.”¹⁹ In this respect, (3) violates the spirit of the theory of Forms, which aims to account for real, factual relations between sensible things in terms of the definitional, conceptual relations between Forms.²⁰ Thus, the parallel between (2) and (3) is fallacious and (3) should be rejected.

Second, Forrester and Yi and Bae defend the view that, while (2), according to which if two Forms, F and G, are correlates, then an instance of F can only bear the same relation to an instance of G, and not to G itself, is intuitively plausible, (3) is based on mistaking all instances of Forms to be sensible instances and should be rejected as it is. For, due to the doctrine of the communion of Forms, some Forms do indeed participate in and thus instantiate other Forms. Forrester argues that all Parmenides can legitimately reach is the rather “harmless conclusion that we cannot know any Form defined by the knowing relation,” which is the correlate of the Form of Knowledge, the Form of Truth, or, as he calls it, “Object-of-Knowledge.”²¹ Since the argument does not rule out the individual Forms, which are not defined by the knowledge relation, entering into knowledge relations and thereby instantiating “Object-of-Knowledge” and being objects of knowledge, our knowledge, which itself is an instance of the Form of Knowledge, can extend to those Forms as its objects. Yi and Bae argue that (3) falsely assumes that all instances of Forms are sensible particulars (i.e., instances that are “among us”), whereas in fact higher-order Forms like One, Being, Good and Truth are instantiated by other Forms as well.²² Accordingly, when (3) is appropriately revised as a principle of ‘correlation of instances,’ i.e., where Forms F and G are what they are in relation to one another, an instance of F is what it is in relation to an instance of G, it remains possible that human knowers, as instances of the Form of Knowledge, can know any Form that is an instance of the Form of Truth. Contrary to Forrester's restricted concession to the unknowability of the Form of Truth by humans, Yi and Bae²³ suggest that even this is not ruled out since the Form of Truth is also an instance of itself. This, I think, is an important improvement over Forrester's reading, for the unknowability of the Form of Truth would not be as “harmless” for Plato's theory of Forms as Forrester contends it would be. Since, as is a central thesis of the theory, knowing an instance of a property F as such involves knowing what F is an instance of, without knowing the Form of Truth or “Object-of-Knowledge” itself, it would not be possible to know any of its instances, whether a Form or a sensible, and thus anything at all! Forrester's restricted concession ends up giving Parmenides much more than what he sets out to prove. However, even Yi and Bae's more careful objection to (3) does not decisively block the argument, for, once (4) is allowed, (2) alone is sufficient to establish the damning conclusion that the Form of Truth can only be known by (or be an object of knowledge to) the Form of Knowledge. This is the common downside of both views that target (3): even if reasons offered to deny (3) are legitimate, the argument can still take the second path without having to rely on (3).

My suggestion for the refutation of the argument consists in denying not the parallel between (2) and (3), but the other parallel between these two and (4), which has generally been taken for granted by commentators, except for a few statements of reservation.²⁴ (4) is the transitional premise that is commonly used by both of the alternative paths to the conclusion, and is introduced to the argument through the assumption that the relation between knowledge and its object is an instance of the pros relation that holds between correlates like the Form of Mastery and the Form of Slavery or the human master and the human slave. My claim is that the knowledge-object relation is not such a relation.

5. Knowledge and the Nature of Pros Relations

I claimed earlier that the isolation entailed by (2) and (3) between Forms and things in us is a restricted one, and pertains only to the bearers of a specific kind of relation: a Form can bear this relation only to another Form, and a thing in us only to another thing in us. But what kind of relation is this and what does it imply for its bearers? The most natural reading of the formula ‘__ is what it is of or in relation to (pros) __’ is that the statement of this relation is a correct answer to the question of ‘what is __?’²⁵ In other words, the second term is part of the definition of the first term. The text also suggests that this definitional relation has an ontological significance such that the item in the first term would have its “being” (ousia) (133c7) or “power” (dynamis) (133e5) in relation to that in the second term. One concern here is that this relation would have to have the same significance for Forms and things in us. For, while positing a definitional or conceptual relation between Forms is appropriate, sensible particulars are not definable as such and relations between them are factual rather than definitional, as is stated by those who reject the parallel between (2) and (3). However, the definiendum in (3) could also be taken as the relevant property or ‘immanent character’ of the sensible particular. For instance, the mastery of the particular human master is definable in relation to the slavery of the particular human slave (and not in relation to the Form of Slavery). It would then also make sense to say that the mastery of the human master has its “being” or “power” in relation to the slavery of the human slave. The pros relation is then a relation of dependence. If x bears this kind of relation to y, x's being the kind of thing x is depends on y's being the kind of y is. But is this unidirectional dependence a sufficient condition of the pros relation? The master-slave example suggests that it is not. For the master-slave relationship is reciprocal in that neither mastery can be what it is without reference to slavery, nor slavery can be what it is without reference to mastery. If one direction of this relation is instantiated (of mastery to slavery) by two individuals, then the converse is also instantiated (of slavery to mastery) by the same two individuals. For one is always of the other, i.e., they are codependent. Thus, if master-slave is the paradigm of the pros relation, one might hold that the argument employs the additional implicit premise (3*) Rxy ↔ Ryx, which would support (4) Kxy → Rxy ∧ Ryx, the transitional premise that renders ‘knowledge’ to be a pros relation and invokes a reciprocity between knowledge and its object. For [(Rxy ↔ Ryx) ∧ (Kxy → Rxy)] → (Kxy → Ryx).

The problem, however, is that the knowledge-object relation does not reflect such reciprocity, as the two items are neither definitionally nor ontologically codependent. One direction of the relation, i.e., of knowledge to its object, satisfies this kind of dependence: it is reasonable to say that knowledge is what it is with reference to its object, or more generally, to truth. But the same cannot be said for the opposite direction of the relation, i.e., of the object to its knowledge. While it is grammatically sound to say that ‘x is an object of y,’ when y is a certain body of knowledge, this proposition would not state anything instructive as to what x really is. For being known or being an actual object of knowledge does not express an essential property of any object as such. For example, ‘Arithmetic is the knowledge of Numbers’ expresses the kind of knowledge arithmetic is with respect to its object, but ‘Numbers are the objects of knowledge to Arithmetic,’ although true, is circular and useless as an explanation of what numbers really are. Again, while ‘weather is the object of knowledge to meteorology’ is true, it does not tell us what weather is but only that it is the subject matter of a science, which, in turn, is what it is with reference to weather itself. The problem of circularity gets even starker in yet more specific cases: ‘This paper before me is the object of knowledge to the knowledge of this paper before me.’

It is easier to identify the lack of reciprocity in the knowledge-object relation by considering the ontological implications of the pros relation. There cannot be mastery without slavery or masters without slaves, and vice versa. However, while, again, knowledge is always of an object and thus can only exist as dependent on the existence of its object, this dependence is not reciprocated by the object. Objects of knowledge can exist without actual knowers having actual knowledge of them. It is reasonable to assume that meteorology cannot exist without weather events, but absurd to think that there would be no weather event without the science studying them. One might argue that there would be no weather events qua objects of knowledge without meteorology. However, this would just be stating a trivial fact about relations in general: once a thing is conceived as a relatum in a relation, it can exist qua that relatum only if the other relatum exists. While it is trivially true that weather events cannot exist qua objects of knowledge without meteorology or meteorologists, weather events can exist qua what they are, i.e., weather events, without the science or scientists studying them, and meteorology cannot exist qua what it is, i.e., the science of weather, without weather events.

The unidirectionality of knowledge relations is somewhat obscured at the most general level, when the relata are not specified as a certain kind or body of knowledge and a certain kind of object. One might thus suggest that there is a sense in which objects of knowledge can be rendered as dependent on knowledge when the relation in question is reformulated as holding between Knowledge and “Truth,” as Parmenides actually has it at 134a, or “Object-of-Knowledge,” as Forrester has it, or even better, “The Known,” as McPherran has it.²⁶ For The Known is of Knowledge as much as Knowledge is of The Known. However, this rendition would only operate upon a broadly idealist presupposition, construing objects to be dependent on or identical to their mental representations, or, as, for instance, in the case of Kant's transcendental idealism, defining objects and objective reality as having to conform to and thus in relation to the conditions of knowledge or the knowing subject. And Plato adopts a strong realist position, especially when it comes to Forms, taking them to be ontologically prior to and independent of the knower and the knowing relation and thus defining the knowledge of Forms as unidirectionally dependent on the Forms. In fact, it is what exists in such an independent manner that determines the scope of the knowable, according to Plato.

Now, if my proposition here is correct, obviously the lack of reciprocity in the knowledge-object relation undermines (4) Kxy → Rxy ∧ Ryx, which is used by both of the two paths to the conclusion of the argument. Yet, aside from the example of master-slave relation, Parmenides’ presentation of the argument does not seem to make reciprocity a necessary condition of pros relations and seems at least compatible with taking unidirectional (definitional and ontological) dependence as a sufficient condition of a pros relation between two items. Accordingly, one might want to resist (3*) Rxy ↔ Ryx, and revise (4) to (4a) Kxy → Rxy, by eliminating (4b) Kxy → Ryx, which was suggested by Peterson as an unsaid premise. This revision would rule out the second (passive) path, but leave the first (active) path intact as the latter does not rely on the knowledge-object relation being reciprocal. Of course, since, as I explained above, (3) is already suspect for reasons pointed out by other commentators, and the first path relies on (3), one could argue that the rejection of (3) and the revision of (4) to (4a) jointly form a complete refutation of the argument, blocking both paths to the conclusion.

However, I propose that reciprocity should in fact be a necessary condition of the pros relation, if it were to obtain between Forms. The pros relation, no doubt, implies a definitional and ontological dependence, whether it is only a unidirectional dependence of one item on another or it is a reciprocal dependence of two items on each other. So, the pros relation between two items entails that at least one of them is what it is in relation to something else, which, in turn, entails that it is not what it is in virtue of itself. Now, one central thesis in Plato's very introduction of Forms is that each Form is “auto kath'hauto,” “itself by itself” (Phaedo 66a3). As I noted earlier, this expression is rather ambiguous and can be construed modestly as referring only to the ‘separate’ (chōris) existence of Forms in the sense that they can and do exist independently of their particular instances.²⁷ Yet even this modest reading of “itself by itself” ends up entailing a stronger kind of independence or basicness to Forms, when considered together with the core premise of the theory: each particular that has a predicate or immanent character F has that predicate in virtue of (its relation to, whether it is that of participation or likeness or another) the Form F, that is, each particular instance of F is what it is in (some) relation to the Form F. For, provided that the Form F is predicated of itself and thus is also F, nothing but the Form F itself, which exists independently of all other particular instances of F, must be the source of its own F-ness. Thus, the Form F, with respect to its F-ness, which is what it essentially is, is in virtue only of itself and nothing else. Each Form is then what it essentially is independently of everything else, not only of its (other) sensible instances but also of other Forms.

How can a Form (F), which enters into the pros relation with another Form (G) and thus is what it is in relation to G, have the property of ‘being what it is in virtue only of itself’? The two modes of being, in relation to (pros) and by itself (kath'hauto), seem incompatible.²⁸ This strongly suggests that if two Forms (F, G) are indeed in a pros relation to each other, then they should be conceived not as two distinct Forms in their own rights but as two aspects of a single pair or relational Form structure (F-G). And F and G are conceivable as such only if F and G are inseparable from each other and thus reciprocally dependent on each other. Accordingly, it is the F-G pair that is what it is in virtue of itself, and F and G are what they are in relation to one another.

What I argue here is that such bidirectional and inseparable F-G structures are the logical consequence of pros relations between Forms. In other words, if Plato commits to (2), which posits the possibility of pros relations between Forms, then he should also commit to the idea that those Forms that enter into pros relations constitute such structures with their correlates and that pros relations, at least between Forms, are reciprocal, unless he is willing to give up one of the pillars of his theory of Forms, i.e., the ontological basicness of the Forms. There is also significant textual ground to think that Plato does indeed commit to the inseparable multi-Form structures. Héctor-Neri Castañeda argues that Plato's theory of relations in the Phaedo turns on the idea of “Form-chains.”²⁹Accordingly, certain relational facts between particulars, such as Simmias’ being taller than Socrates, are two-pronged and involve two Forms, Tallness and Shortness, each instantiated by one particular, Simmias and Socrates, respectively. The fact that Simmias is taller than Socrates is constituted neither by Simmias’ participation in Tallness alone, nor by Socrates’ participation in Shortness alone, but by both jointly. The Forms involved in these multi-pronged facts cannot enter into single-pronged facts. Such Forms constitute unbreakable “Form-chains” or relations and must always be jointly or simultaneously instantiated. That is, if F-G is a Form-chain, then there is no fact that is constituted by the instantiation only of F or only of G. In other words, F and G share the same domain of facts. Castañeda calls this “the law of factual enchainment.”³⁰

McPherran goes one step further and argues that the greatest difficulty argument in the Parmenides builds on this theory of relations introduced in the Phaedo. He claims that while the Pheado leaves the question of which Forms are bound by the law of factual enchainment unanswered, the idea of a pros relation, formulated by the expression “[being what it is] in relation to” (133c7-8) or “[having its] power (dynamis) in relation to,” provides the missing criterion for the enchainment among Forms:

A Form φ is governed by the law of factual enchainment for two-pronged facts if and only if there is a correct answer to the question ‘What is φ?’ which has the form ‘φ is what it is (is φ) (in respect) of ψ (a Form), where the converse of this (‘ψ is what it is of φ’) is also true.³¹

Forms are parts of Form-chains only if they are what they are in relation to one another and thus are reciprocally dependent on each other. For two Forms cannot be instantiated on their own but must always be coinstantiated, only if they are so dependent on each other. Mastery-Slavery is such a Form-chain.

The reciprocity criterion of enchainment would block one important possible objection to my view that pros relations, at least between Forms, must be reciprocal. Appealing to the doctrine of the communion or blending of Forms (Sophist 259e4-6), one might argue that a pros relation between Forms could be unidirectional if one of the Forms is a participant of the other, where the latter is a higher-order Form, such as Good, Being, and One in which all Forms participate unidirectionally. For instance, the Form of Beauty participates in the Form of Good and any fact that involves the instantiation of Beauty also involves the instantiation of Good, but not vice versa. However, Beauty-Good and other relations of conceptual subordination between Forms would not be subject to the law of factual enchainment and thus would not constitute Form-chains, for the latter necessarily require codependence or coinstantiation of the Forms involved in them. Unidirectional (definitional) dependence is then not sufficient for pros relations between Forms. Even though the definition of the Form of Beauty will involve reference to the Form of Good, as well as other higher-order Forms such as Being and One, it is still the case that the Form of Beauty is kath'hauto or ontologically basic: it has its being in virtue of itself. Or more simply, the Form of Beauty can enter into single-pronged facts. But the Forms that are parts of the Form-chains are inseparable and cannot be involved in distinct single-pronged facts. This should thus show that the pros relations between Forms are not to be construed in terms of the participation of one Form in another. The Form of Mastery is what it is in relation to the Form of Slavery not because it participates in the other. In fact, it would be quite absurd to say that Mastery instantiates Slavery or possesses the property of slavery or is a (non-sensible) slave itself, while it would be simply true to say that Beauty instantiates Good or is good (or is one or is). Instead, Mastery is what it is of Slavery because they are the opposite aspects of the same unique and exclusive reciprocal relation, and thus must always be coinstantiated.³²

One further objection here might be that even if Forms that are in pros relations are subject to the law of factual enchainment and thus are codependent, this does not mean that particulars should be so. That is, even if (2) should be revised as (Fx ∧ Rxy) ↔ (Fy ∧ Ryx), this still does not warrant a parallel revision of (3) as (Ix ∧ Rxy) ↔ (Iy ∧ Ryx). However, this objection would not hold water either, since the law of factual enchainment should apply also at the level of things in us. For enchainment is supposed to account for multi-pronged relational facts between particulars in the first place. Particulars enter into such multi-pronged relations in virtue of participating in enchained Forms and thereby possessing properties or immanent characters that are similarly enchained. In other words, the relational fact between a particular human master and a particular human slave is explained by two parallel chains, the Form-chain between Mastery and Slavery and the character chain between the mastery in the particular master and the slavery in the particular slave.³³ Thus, just like the Forms that are involved in two-pronged relational facts, the particulars that are involved in these relational facts by participating in them must also be what they are of each other and codependent.

More importantly, the reciprocal enchainment in pros relations is in fact what grounds the separation between Forms and things in us that the argument aims to maintain. For the same necessity that enchains two Forms or characters together also excludes all else, whether a Form, a character, or a particular, from the chain. This is why the Form of Mastery is what it is only of the Form of Slavery, and not of the slavery in the human slave or of the human slave or of anything else, and vice versa. Similarly, the particular human master is what it is only of the human slave (and not of the Form of Slavery or anything else) because the mastery in the human master is exclusively enchained with the slavery in the human slave. In other words, factual separation follows from factual enchainment.³⁴

Therefore, (3*) Rxy ↔ Ryx obtains and Mastery-Slavery is not just any example but one that illustrates this necessary condition of reciprocity in pros relations. As I argued previously, the knowledge-object relation fails to satisfy this condition of reciprocity and thus is not an instance of the pros relation, even though knowledge is indeed both definitionally and ontologically dependent on its object. Thus, because ~ (Kxy → Ryx), (4) Kxy → (Rxy ∧ Ryx) is false. Moreover, the more modest (4a) Kxy → Rxy is also untenable, since provided that (3*) Rxy ↔ Ryx, it entails the false consequence that (Kxy → Ryx).

Not only is (4) false, but it also leads to profound problems with regard to the Form of Knowledge. At 134a, where (4) is introduced, the Form of Knowledge is formulated as self-predicated, as the knowledge of the Form of Truth: “knowledge itself, what knowledge is, would be knowledge of that truth itself, which is what truth is.” And (4), when taken together with (2), introduces a particular Form of Knowledge for each particular Form: “each particular knowledge, what it is, would be knowledge of some particular thing, of what that thing is.” Now, this leads not only to a needless duplication of all Forms, but also another “third man” kind of regress that Parmenides himself does not raise. For the Form of Knowledge, qua specifically the Form of Knowledge, is a knowledge (of something other than itself, i.e., the Form of Truth or Object-of-Knowledge or The Known), but qua a particular Form, would be an object of knowledge to another Form of Knowledge*, which, in turn, would have to be known by yet another Form of Knowledge**.

Although the Parmenides is not the only place where the Form of Knowledge gets mentioned, one wonders whether Plato's theory of Forms really needs a Form of Knowledge.³⁵ In his discussions of what knowledge is both in the Republic and in the Theaetetus, Plato takes knowledge to be a kind of “thought” (Republic 476d), a certain “mental power” or state (477d6-8), whether it is a “perception” (Theaetetus 151e-187a) or “true judgment” (187b-201c) or “true judgment with an account” (201d-210a). Thoughts or mental states, however, are always of a certain content, i.e., intentional objects (other than themselves), and cannot exist or be defined “themselves by themselves.”³⁶ Unless mental states can enter into dyadic Form structures or chains with their objects, which they cannot because of the lack of reciprocity between them, it is not clear how mental states can be Forms. Interestingly enough, the fact that thoughts are always of some intentional object is precisely the ground for Parmenides’ criticism of Socrates’ hypothesis that Forms are thoughts (noēmata) earlier in the Parmenides (132b-c). Parmenides argues that if every Form itself were to be a thought, then it would have another Form* as its object, for if a thought is one or unitary, its object must also be one or unitary. But, then, again, this Form* would have to be the thought of another Form**, and so goes the infinite regress. Thus, the notion of Forms as thoughts in the soul is in fact incompatible with the theory's fundamental idea of positing the Forms as true, stable, and unitary objects of thought or knowledge.³⁷

6. Conclusion

The greatest difficulty argument can be decisively refuted by rejecting premise (4), which identifies the knowledge-object relation as one of reciprocity and codependence. Such reciprocity is neither supported, nor required by Plato's theory of Forms. While the theory presents Forms as proper objects of knowledge, it does so precisely because of the metaphysical status it attributes to Forms as explanatorily and ontologically fundamental and complete entities and thus as prior to and independent of any kind of mental or noetic representation of them by any cognitive subject. In other words, the epistemic role of Forms derives from their metaphysical status. Parmenides’ statement (135c1-2) on the necessity of the existence of Forms for there to be any thought at all is a confirmation of this unidirectionality.

The greatest difficulty argument thus fails to reach the ambitious conclusion that Forms are unknowable by us. However, the argument should still be taken seriously as it aptly points to the urgency of a positive account of how Forms and sensible particulars are related. For, despite the argument's failure to prove the unknowability of Forms, given the theory's introduction of a separation (chōrismos) between Forms and sensible particulars, the burden of proof that they are indeed related, and that this relation can live up to the epistemic and metaphysical roles that the theory attributes to it is still on the proponents of the theory. For, if the two kinds of entities are separate, the claim that they are related in a particular way requires a positive account. The lack of such an account would be a natural target for the opponents of the theory, as will be exemplified by Aristotle, who takes the separation of Forms and their sensible participants to be responsible for the main difficulties of the theory (Metaphysics M 4, 1078b 30–1, and M 9, 1096b32-5).³⁸ It is then quite plausible to read the greatest difficulty argument in particular, and the Parmenides in general, to be the expression of Plato's critical awareness of this problem in his own early and middle period presentation of the theory, which leaves the nature of separation and relation between Forms and sensible particulars underdefined.