A note of caution: Plato wrote dialogues not treatises. These dialogues show the life of the philosophical mind at work: questioning, arguing, speculating, imagining, wondering, struggling, and understanding. They do not show finished results that we can confidently attribute to Plato himself. Rather, they are invitations to join the dialogue and engage with the issues ourselves. So whenever we talk about an issue or idea in Plato’s work—in this case the Forms—we must be sensitive to the dramatic context out of which these ideas and issues come. That said, it is often helpful to isolate certain ideas suggested by Plato’s characters for consideration. This is what I want to do here: I want to point out a few important passages where some character argues for, rather than just describes or discusses, the existence of the Forms. We can then consider the soundness of these arguments. Just remember that there is no clear “theory of the Forms” (as it is often referred to by scholars) nor can we be sure what Plato thought of these arguments.
In a previous post we saw one of Plato’s arguments for those eternal, unchanging, immaterial, and totally objective entities called Forms. In his dialogue Phaedo Plato has his character Socrates give another argument for, on the one hand, the existence of Forms and, on the other hand, for our a priori (known without sense experience) possession of some Forms in our soul. The argument is often referred to as “the argument from equality”. The argument concerns the concept of equality but it can apply to other concepts we think we know a priori. S. Marc Cohen describes it well: “The argument tries to show that we cannot abstract the concept of equality from our sense-experience of objects that are equal. For we never experience (in sense-perception) objects that are really, precisely, equal, and we must already have the concept of equality in order to judge the things we encounter in sense-perception to be approximately, imperfectly, equal.” Cohen presents a formalization of the argument on a more general level where “F” stands for some property we think an object has:
Premise 1: We perceive sensible objects to be F.
Premise 2: But every sensible object is, at best, imperfectly F. That is, it is both F and not F. It falls short of being perfectly F.
Premise 3: We are aware of this imperfection in the objects of perception.
Premise 4: So we perceive objects to be imperfectly F.
Premise 5: To perceive something as imperfectly F, one must have in mind something that is perfectly F, something that the imperfectly F things fall short of. (E.g., we have an idea of equality that all sticks, stones, etc., only imperfectly exemplify.)
Premise 6: So we have in mind something that is perfectly F.
Premise 7: Thus, there is something that is perfectly F (e.g., Equality), that we have in mind.
Conclusion: Therefore, there is such a thing as the F itself (e.g., the Equal itself), and it is distinct from any sensible object.
We could go on and on with other examples. We could speak about how we compare one state to another in terms of how much justice they exemplify. How do we do this? Is justice in the physical world? Is it something we grasp through the five senses? Can we grab it, taste it, hear it, smell it, or see it? Plato would say we have access to images of justice: we think of just people, laws, and institutions. But not justice itself! To make sense of justice itself, we would have admit that it is not part of the physical world; we would have to believe in the Form of justice.
Now plenty of people will argue that abstract concepts like equality and justice are formed through abstraction. Our brain, presumably, can see similarities and eventually come up with a general concept from them. On this view, often referred to as nominalism, there would be no independent existence to the concepts of equality justice: they would be in our minds, created by us, and would cease to exist if all humans were to cease to exist. Plato would no doubt respond by saying: “But how did you see two things as similar or approaching equality in the first place if you didn’t have some real standard of real equality? Isn’t the process of abstraction something that can only take place if you are already in possession of an abstraction?” In Plato’s Meno, Socrates put forth the thesis of recollection: all learning is recollection of things our immortal soul already knows but has temporarily forgotten. Given the above, we see that these things our soul already knows are the non-physical Forms; and it is the Forms that enable us to make the comparisons we make.
 See The Republic of Plato. Translated by Francis MacDonald Cornford (New York: Oxford, 1945), p. 325.
 The following analysis of the argument is from http://faculty.washington.edu/smcohen/320/phaedo.htm