Our desire to establish one way of doing things is a weakness of being human, for we find it hard to comprehend opposing theories as being simultaneously valid. Because the human condition dictates that something be either true or false, rules were made for establishing halachah . Yet, in the world of real truth, both theories might be valid or the first might apply to one situation and the second to another. The debates of the sages show different sides of the same coin. The same is true of the Torah.
Nothing is totally impure, without a pure aspect.
If someone declares an item pure and presents his arguments to prove it, he has only demonstrated one aspect of reality. Similarly, when one declares an item impure, he has only considered another aspect. One who declares something pure and one who declares it impure have both learned the Torah, but each one saw the item differently. However, God created that item as a whole, and He has the ability to create it in a manner that can be viewed in different ways. The school of Shammai was generally strict whereas the school of Hillel preferred leniency.