Parmenides, a forefather of the tradition that Plato is engaged in, is quoted in the Sophist as advising people to avoid trying to show the existence of non-being, because such a task is doomed from the start. What Parmenides means by this is that, because being has to do with things that exist, non-being has to do with things that don’t exist. You can’t prove that things that don’t exist do exist. In a classic case of disobeying one’s elders, Plato goes against Parmenides’ advice and, by using the concept of motion, proceeds to prove the existence of non-being. The proof goes roughly as follows:
- Things are either in motion or at rest.
- Rest and motion don’t mix.
- Since we are talking about non-being, we have to talk about being.
- Being applies to both motion and rest.
- Being can’t be motion or rest, because if it were (for example, rest), then motion would participate in rest, which is impossible.
- We now have three categories: being, motion, and rest.
- Being is the same as itself and different from motion and rest.
- Rest is the same as itself and different from motion and being.
- Motion is the same as itself and different from rest and being.
- We have used “the same” and “different” to categorize being, rest, and motion. Which categories do “the same” and “different” belong to?
- “The same” and “different” must be their own categories. For example, if “the same” were motion, then point number 9 would say “Rest is motion, which is different from motion and being,” which is impossible.
- We now have five categories: being, motion, rest, the same, and different.
- Building on point number 8: If being is different from motion, then motion is different from being.
- This is the same as saying “Motion is non-being with respect to being.”
- Motion exists, and it is not being.
- Non-being exists, and it is difference—not the opposite of being—thus refuting Parmenides’ claim.