The "Divided Line" and the Four Stages of Truth from Plato's "Republic"

A famous passage from Plato's Republic that helps lay out Plato's theory of knowledge and the doctrine of the forms is the passage about the "divided line." It is at the end of Book VI of The Republic and takes place right before the Allegory of the Cave. Here Plato is going to use the simple image of a line being cut into four parts to represent a sort of ascent in knowledge from the lower to the higher forms of knowledge. The four parts also represent the spectrum of inquiry that human beings experience, with the highest being the most true and goal of knowledge. Reflected in the divided line one can begin to see the formation of formal logic, as well as the distinction between inductive and deductive knowledge. 

The Visible Realm

The line is first cut in two to represent the distinction between the tangible knowledge of the senses and the intelligible knowledge of the mind. 

                                                                        1                                2

                                               ------------------------ | ------------------------

                                                  Senses - Visible          Mind - Intelligible

Then he proceeds to say that each of the two sections is subdivided to represent two types of knowledge that take place within those categories of visible and intelligible. He says that the first section, A,  represents "images" or reflections of things which can be seen in shadows, water, and polished objects. Obviously seeing a reflection of something can make it very hard to even make out what the thing is. The second section, B, represents a more direct sight of material things such as animals, plants, living things, and artifacts (anything people make). Now, the identity of the thing seen becomes clearer as it is no longer simply a reflection but is seen directly and is a type of thing, whether that be a type of animal or a man made thing. 

                   A                                                     B                                               C                                    D

------------------------------ | ------------------------------|------------------------------ | ------------------------------

   Images/Reflections           Living things, Artifacts

                             Senses - Visible                                                      Mind - Intelligible

The Intelligible Realm

Plato continues toward true knowledge as one passes from the visible to the intelligible. Now, in the third part, C, the mind now has a grasp of the form of things, but it is a grasp that still depends on the image provided by the senses. It is not just sight but understanding of a thing in the intelligibility of the mind. Also Plato mentions that "hypothesis" are part of this section. Having comprehended the intelligible nature in things, one can begin to speculate about the higher nature of reality. As an example of this, Plato mentions the use of drawn images to help the mind grasp the absolute versions of those images, such as a drawn circle versus the intelligible idea of circle. One can then use these basic concepts to reason to higher mathematical laws which are further abstracted from physical images or shapes. 

"Do you not also know that, although they use visible figures and reason about them, they are not thinking about these, but rather about the things which they resemble? They do not reason about the figures which they draw, but about the absolute square and the absolute diameter, and so on. The visible figures which they draw or make [in section B], which have shadows and reflections in water of their own [in section A], are converted by them into other images [in section C]. They are really seeking to see the things themselves, which can only be seen with the eye of the mind." 

The four section, D, and highest form of knowledge is that which is most stripped from any physical representation. This happens when one is able to move from the hypothesis, still represented with the help of images, and to come to an understanding of first principles. First principles are those which are universally true and thus stripped of all material existence. Now one has reached the realm of the forms. From the forms one can then also extract the logical consequences and continue to reason without the help of the senses at all. 

      Imagination                              Belief                              Thought                     Understanding                   

        (eikasia)                                 (dianoia)                            (pistis)                             (Noesis) 

Expressed in Reflections  Expressed in Sense World     Expressed in Math         Expressed in Dialectic

                    A                                                  B                                       C                                       D

------------------------------ | ------------------------------|------------------------------ | ------------------------------

   Images/Reflections          Living things, Artifacts   Mental Ideas/Hypothesis           First Principles

                             Senses - Visible                                                      Mind - Intelligible

Now the divided line is complete and one can see the four stages of knowledge. The enlightened man recognizes that truth is to be led from the changing sensible world and into the unchanging realm of the intelligible and universally true, to have the soul return as much as possible to the realm of the forms. "There is, then, a scale of them, and let us suppose that the various faculties have clearness in the same degree that their objects have truth."