Cognition, Objects, and Proportions in the Divided Line

In a recent publication, Nicholas Smith discussed some elements of the Republic’s divided line (Rep. 509d6-511e4) to demonstrate that they create an unresolved problem. I tackle Smith’s argumentation to show that elements of the divided line that are mentioned by him do not create problems in interpreting this passage. On the contrary, these features convey one of the most important doctrines behind this passage. This is the idea that the world of sensible things holds a dependence upon the world of Forms in the same way that shadows and reflections depend on the things that are shadowed and reflected. Following this line of reasoning, I propose an interpretation of the divided line in which both knowledge and opinion are set over the same kind of objects F. One has an opinion about F whenever apprehending F by means of its effects, and one has knowledge about F whenever apprehending F itself.

In a recent publication in Plato Journal, Nicholas Smith (2018) proposes a problem of interpreting the Republic's divided line. According to Smith, the relationship between line segments and the degrees of clarity and truth that these segments intend to indicate are stated in such a way that the platonic doctrine behind this passage becomes troublesome. Smith begins his argument by indicating that for both versions of the divided line (Rep. VI. 511d6-e4 and Rep. VI. 509d6-510b1), the proportions between line segments are intended to indicate different degrees of clarity and truth. However, as Smith duly notes, it is unclear how Plato relates truth and clarity with the objects and cognitive states that are mentioned in the passage. For example, consider how Plato first explains the line: "It is like a line divided into two unequal sections. Then divide each section-namely, that of the visible and that of the intelligible-in the same ratio. In terms now of relative clarity and opacity (σαφηνείᾳ καὶ ἀσαφείᾳ), one subsection of the visible consists of images (εἰκόνες). And by images (εἰκόνας) I mean, first, shadows (σκιάς), then reflections in water (τὰ ἐν τοῖς ὕδασι φαντάσματα) and in all closepacked, smooth, and shiny materials, and everything of that sort, if you understand. I do.
In the other subsection of the visible, put the originals of these images, namely, the animals around us, all the plants, and the whole class of manufactured things. Consider them put. Would you be willing to say that, as regards truth and untruth (ἀληθείᾳ τε καὶ μή), the division is in this proportion: As the opinable (τὸ δοξαστὸν) is to the knowable (τὸ γνωστόν), so the likeness is to the thing that it is like? Certainly."  In this first passage, the degrees of clarity apply to the objects with which the line segments are associated (shadows, ref lections, animals, plants, and manufactured things). Truth, in contrast, is brought in at the end to be applied to "the opinable" (τὸ γνωστόν) and "the knowable" (τὸ γνωστόν). In the subsequent second version of the divided line, however, clarity applies to cognitive states (παθήματα ἐν τῇ ψυχῇ; i.e., to νόησις, διάνοια, πίστις, and εἰκασία, respectively), whereas truth applies to the objects that these παθήματα ἐν τῇ ψυχῇ are "set over" (ἐφ᾽ οἷς ): "There are four such conditions in the soul (παθήματα ἐν τῇ ψυχῇ), corresponding to the four subsections of our line: Understanding (νόησις) for the highest, thought for the second (διάνοια), belief (πίστις) for the third, and imaging (εἰκασία) for the last. Arrange them in a ratio, and consider that each shares in clarity (σαφήνεια) to the degree that the subsection it is set over shares in truth (ἀλήθεια)." (Rep. VI.511d6e41) These two passages, when considered together, indicate that Plato appears to be somewhat unsure about what precisely truth and clarity are supposed to measure. To resolve this lack of precision, Smith goes back to Socrates' discussion about the merits of knowledge, opinion, and ignorance in Book V, simply to find the same kind of inexactness. He then suggests that we take the quality of kinds of objects as fundamental and the quality of different cognitive states as explicable in terms of the quality of these objects. In this case, Plato would remain consistent in the middle books of the Republic in applying degrees of truth to kinds of objects, whereas the measure of clarity of cognitive states would "co-vary with the truth of the objects" (Smith, 2018, p. 100).
So far, this line of reasoning is good, but Smith's "nightmare" begins when he considers proportions between different line segments. Whatever the exact construction of the line that one adopts, there is a feature of it that seems to be inescapable. For both versions of the line, there must be a proportion between the two upper segments (taken together) and the two lower segments (taken together) that also applies to the two lower segments relative to each another. In the two figures below, that means (I1 + I2) / (V1 + V2) = V2/V1. 2 However, this mathematical feature of the line is supposed to create a philosophical problem of considerable importance: "As far as I know, there has been no notice in the literature about the problem that this seems to create, namely, that V1 + V2 (that is, the entire lower section of the original division) must be clearer (and, as we soon learn, given the association of clarity and truth, also truer) than either V1 or V2 by themselves. But this seems to me to create nonsense: How can V1 + V2 be clearer or truer than either V1 or V2? Why would adding the relative lack of clarity (and truth) in V1 to whatever we find in V2 make V1 + V2 clearer (and truer) than V2 just by itself? Plato tells us that V1 consists in shadows and reflections in water and other reflective surfaces. Why would adding these to the visible originals give us a collection of things that is clearer or truer than the collection of visible originals without shadows and ref lections added to that collection?" (Smith, 2018, p. 104) Again, some excerpts below: "The problem is that it seems absurd to think that visible originals taken together with their visible images (V1 + V2) will be clearer or truer than the visible originals alone […] So, too, the epistemic deficiencies we are supposed to associate with the lower subsections of the line, relative to the subsections just above them in each of the original divisions, make it absurd to suppose that Plato intends whatever epistemic condition we should apply to the entire lower line (V1 +V2)-δόξα in Book's VII recapitulation-to be clearer and truer than either εἰκασία or πίστις by themselves. Why would adding the (less clear/less true) εἰκασία to πίστις yield a clearer (or truer) cognitive condition (taken as a whole) than that enjoyed by πίστις alone?" (Smith, 2018, p. 104) So, Smith concludes: "We are left with the unhappy result that Plato makes proportions of clarity and truth the focus of the comparisons he makes in the divided line passage, but in doing so, he creates an image that has both mathematical and also philosophical entailments that do not seem to represent views he would accept." (Smith, 2018, p.105) In the following sections, I challenge Smith's conclusion by providing the explanation he demands. However, given the cleverness of Smith's argumentation, to explain why grasping V1 + V2 represents a clear and truer apprehension of reality than grasping V2 alone, I will briefly discuss two of the most famous-or should I say infamous-dogmas of 20th century scholarly platonism: the "two worlds theory" and the doctrine of "degrees of reality."

DEGREES OF REALITY
In 1965, Gregory Vlastos established what appears to be one of the most important tenets of analytically inspired interpretations of Plato. The so-called doctrine of degrees of reality was first put forward in the essay Degrees of Reality in Plato (1965) and then developed somewhat further in Vlastos' presidential address before the American Philosophical Association, later published as A Metaphysical Paradox (1966). The central hypothesis is that degrees of being in Plato could never mean degrees of existence be-cause the very notion of grades of existence is complete nonsense. According to this idea, whenever Plato says that a given Form F is "really real," he is not asserting something about its existence-he is just categorizing its way of being F. The platonic thesis that sensible things "are and are not" means that sensible things "are and are not" p for a given predicate p. However, it would be extremely difficult for Plato to make sense of these expressions for the existential sense of being since the very notion of existence "rules out as monstrosity a tertium quid between existence and non-existence" (Vlastos, 1966, p. 10). Even if Plato had wished to follow this difficult path, then he "would have had to fight his native language all the way, and some sign of the combat would have shown up in the text" (Vlastos, 1966, p. 10).
Vlastos advises contemporary platonists to stop talking about degrees of existence and instead give attention to different ways by which Forms and sensible particulars are related to their predicates. The importance of this lesson for platonism can hardly be understated. Inspired by these remarks, a whole generation of scholars further developed an interpretation of Plato that places predication in the center of his doctrine, making his philosophy more relevant to contemporary philosophical discussions and turning passages that really seemed at odds for older interpreters into clearly understandable texts.
Nevertheless, I dare to say that it is time for us to reconsider Vlastos' absolute interdiction of the idea of degrees of existence in Plato. I do not mean that we should take the notion of predication from the center of platonic metaphysics. Rather, my point is that to answer Smith's aforementioned questions and correctly understand the divided line, we must consider degrees of existence.
One feature of the first segment of the line (V1) that is almost never noticed by interpreters is that Plato takes a considerable amount of time explaining exactly what kinds of images he has in mind here. Although the word εἰκόνες could be used to characterize statues, pictures, or any other kind of representations, Plato makes it clear that he is populating this segment of the line with shadows and reflections (σκιάς; φαντάσματα). Later, he takes care to mention that these are shadows and ref lections that are caused by animals, plants, and objects that populate the segment V2 (Rep. 510a 5-6). We then must ask why he is so careful in describing the nature of these images and also what features shadows and ref lections have in common when considered relative to the original of which they are images.
What these kinds of images have in common is that they are all direct effects of their models in a way that a painting or a statue is not. Therefore, shadows and ref lections depend on their models for their existence in a manner that statues and paintings do not depend. If someone draws a caricature of me and then takes it away, then it will continue to exist, the same way that a statue of Fidel Castro exists now in Cuba although the man is now gone. Conversely, a shadow or reflection does not hold this kind of independent existence and can only exist while its model is effectively causing it.
I submit that this kind of dependent being of shadows and reflections represents a lesser degree of existence. Its essential feature is that these kinds of images can only exist as a dependent effect of their models, in opposition to other kinds of representations that we usually find in Plato, such as paintings and statues, that can exist independently of their models. A picture of me depends on me to be recognized as an image of myself, but it does not depend on me to exist. Yet, my image that is reflected in a mirror ceases to exist as soon as I am gone. If I cease to exist, then there can no longer be a shadow or reflection of me.
Another feature of this lower degree of existence is that these entities (shadows and reflections) are usually not considered enumerable objects. If someone wants to enumerate how many things are involved in the situation of a man who sees himself in a mirror, then he would probably say that there are just two things: the man and the mirror. The man's image in this case is usually considered simply an effect of the relationship between these two things. Likewise, my shadow and I do not form a pair of objects in the same way that a statue of myself and I would form. Of course my shadow exists, but it is not usually enumerated, and it only exists as a dependent effect of myself. Conversely, a statue of me is as much enumerable as I am and can exist even if I cease to exist. In fact, the majority of paintings and statues exist for longer than their models.
As soon as we realize the specificities of the kinds of images that Plato uses to populate V1, it becomes clear why apprehending V1 + V2 represents a truer and clearer grasp of reality than apprehending V2 alone. The objects of V2 are direct causes of the objects of V1. Therefore, grasping both of them is equivalent to grasping something more than the mere collection of objects in V1 + objects in V2. It reflects grasping the causal relationship that these two classes of objects have between them. When the prisoner of the cave starts to climb his way out of the cave and sees the objects that cast their shadows on the wall, he perceives this higher class of things and also understands that the shadows that he had previously seen were caused by these objects (καθορᾶν ἐκεῖνα ὧν τότε τὰς σκιὰς ἑώρα: Rep. 515d1).
In fact, causal bonds that govern the relationship between different line segments is arguably one of the most important lessons of the line. Because of the mathematical features of the line that are pointed out by Smith, the relationship between V1 and V2 is necessarily proportional to the relationship between the whole realm of intelligible things (I1 + I2) and the whole realm of visible things (V1 + V2). Therefore, sensible things are images that are caused by intelligible things in the same manner that my shadow is caused by me. Just as my shadow depends on me to exist, sensible objects depend on Forms.
That the proportions of the divided line make the causal relations between line segments one of the most important lessons to be taken from this passage is such clear fact that only years of prejudice against the idea of degrees of existence could generate the problem proposed by Nicholas Smith. Only attributing the same degree of existence to every entity in the line it could be considered puzzling the fact that two consecutive segments taken together represent a clearer and truer apprehension of reality than just the upper segment.

TWO WORLDS THEORY
According to the doctrine of degrees of reality, images exist to the same degree as their models. Therefore, grasping the model is tantamount to apprehending the original, whereas grasping an image is tantamount to apprehending a different, independent object that just happens to be an image or imitation. If so, then why would I need the imitative version after being in contact with the original?
Smith's problem is such a good piece of scholarly reflection that it points to a subtle relationship between the two dogmas mentioned above (i.e., the doctrine of degrees of reality and the two worlds theory). According to the doctrine of degrees of reality, originals and images are two different independent entities. Therefore, apprehending one of them is never a way of apprehending the other. Applying this understanding to objects of the line, we arrive at the result that apprehension of the images (shadows and reflections) that populate V1 is in no way related to apprehension of the objects (plants and animals) that populate V2. Moreover, if objects of knowledge and objects of opinion populate different segments of the line, as indeed is the case, then there could be no opinion about objects of knowledge or knowledge about objects of opinion.
Gail Fine (1977) introduced the terminology "two worlds theory" as an indication that Plato distinguishes knowledge and beliefs by reference to their objects, such that one can have knowledge but not beliefs about Forms and beliefs but not knowledge about sensible things. Fine tries to save Plato from this theory by presenting a reading that makes some of the Republic's arguments about knowledge content-oriented rather than object-oriented. In her interpretation, "knowledge and belief are distinguished not by their different sets of objects, but by their truth implications" (Fine, 1977, p. 139). This movement has been severely criticized, mostly because many think that there are abundant, uncontroversial assertions of an object-oriented theory of knowledge in both the Republic and many other dialogues (c.f. Gonzales, 1996). I will not discuss Fine's arguments in this paper, but it is important to point out that my solution to Smith's problems provides an interpretation of the divided line in which we avoid the two worlds theory while maintaining an object-oriented theory of knowledge.
Smith's problem indicates that the two worlds theory follows naturally from the doctrine of degrees of reality. Once the idea of grades of existence is denied from the picture, it becomes necessary to assign a different, independent object for each segment of the line and for each cognitive state of the mind that these segments represent. If existence is never a matter of degrees, then the objects of different line segments exist independently, and grasping one of them is never a way of grasping the other. If I am acquainted with Achilles only by one of his pictures, then what I have seen is not Achilles himself but rather another object or an imitation of him. Therefore, I can only have an opinion. If I see Achilles himself, then what I see is not merely an image but rather another object: the original. Now I can only have knowledge of him.
However, if we break the interdiction of degrees of existence and start to recognize that objects in V1 do not exist by themselves but only as effects of objects in V2, then it becomes clear that πίστις and εἰκασία represent two different ways of apprehending the same set of objects. Again, it is important to think about the kinds of images that Plato has in mind here. Grasping my shadow or reflection is not the same as grasping a different object as it would be if we were talking about statues or paintings. My shadow and my ref lection are caused by me in a similar way that a flower causes its smell. To notice a flower by means of its smell is not to notice a different independent object but rather to grasp the flower through one of its direct effects. If I see myself in the mirror, then what I see is not a different object as it would be if I was seeing a statue of me. To see my ref lection in the mirror is just an indirect way of seeing myself.
Similarly, when the prisoner starts to climb his way out of the cave and sees the objects that cast shadows on the wall, he realizes that his previous experiences were nothing but a defective apprehension of these same objects that he now clearly sees. At this moment, he will "know each image for what it is and also of what it is the image" (γνώσεσθε ἕκαστα τὰ εἴδωλα ἅττα ἐστὶ καὶ ὧν: Rep. 520c5) Due to the proportions of the line, the relationship between images and objects of which they are images is analogous to the relationship between sensible things and Forms. Consequently, whenever I see a display of beauty in the sensible world, what I am apprehending is the very Form of beauty through one of its effects. Of course, this would be a defective apprehension. Restricted to sensible things, one cannot achieve knowledge. To have knowledge, one must grasp the Form of beauty itself. Nevertheless, opinion and knowledge are different cognitive states about the same set of objects. Furthermore, according to this interpretation, Forms are the primary objects of knowledge but not necessarily the only objects of knowledge. Knowing the causal bounds that govern relationships between Forms and sensible things, one can also know the sensible things as they are, namely direct effects of Forms.
By providing a reading of the divided line in which the distinction between knowledge and opinion does not depend on different kinds of objects, my interpretation saves Plato from the unwelcome consequences of the two worlds theory. The fact that, according to the two worlds theory, one cannot pass from the cognitive state of opinion to the cognitive state of knowledge about the same object is just one of them. Moreover, my interpretation avoids such kinds of unwelcome results without throwing the baby out with the bathwater (i.e., without giving up the idea of an object-oriented theory of knowledge in Plato). According to my interpretation of the divided line, one has an opinion about F whenever apprehending F by means of its effects, and one has knowledge about F whenever apprehending F itself. As explained in the previous section, direct effects of the kinds of shadows and reflections are not different, ontologically independent objects. These are non-enumerable, ontologically dependent manifestations of F itself.

CONCLUSION
In summary, I disagree with Nicholas Smith's conclusion that the divided line is misconstructed. Rather, I take the mathematical property of the line that he considers troublesome as entailing one of the most important pieces of doctrine behind this passage. This is the idea that the world of sensible things holds a dependence upon the world of Forms in the same way that shadows and reflections depend on the things that are shadowed and reflected. To understand how this doctrine is conveyed by the divided line, we must surpass Vlastos' interdiction of the notion of degrees of existence. As a benefit of this transgression, we save Plato from some negative outcomes of the two worlds theory, including the embarrassing idea that different cognitive states must have different objects.