The following philosophical work is called Popularity Theory. Its primary assumption is an inverse relationship between social popularity and goodness. It’s sketchy, but content heavy.
What’s your opinion of it? What’s your opinion of its two postulates? What’s your opinion of its two definitions? Do you think it’s useful?
Popularity Theory.
Definition 1. A friend of a person P is a person that P likes.
Definition 2. An enemy of a person P is a person that P dislikes.
Postulate 1. A person P1 dislikes a person P2 who is better than P1.
Postulate 2. A person P1 likes a person P2 who is not better than P1.
Theorem 1. A person P1 who is better than a person P2 is an enemy of P2.
Proof. Consider a person P1 who is better than a person P2. By Postulate 1, P2 dislikes P1. Therefore by Definition 2, P1 is an enemy of P2.
Theorem 2. A person P1 who is not better than a person P2 is a friend of P2.
Proof. Consider a person P1 who is not better than a person P2. By Postulate 2, P2 likes P1. Therefore by Definition 1, P1 is a friend of P2.
Theorem 3. If a person is the best, then he is nobody else’s friend.
Proof. Consider a person P who is the best. P is better than everybody else, by Definition of Best. It follows by Theorem 1 that P is an enemy of everybody else.
Theorem 4. If a person is nobody else’s friend, then he is the best.
Proof. Consider a person P who is nobody else’s friend. P is thus, by Definition 1 and Definition 2, an enemy of everybody else. It follows that P is disliked by everybody else, by Definition 2. Assume P is not better than somebody else. Then it follows by Postulate 2 that somebody else likes P. But this contradicts the fact that P is disliked by everybody else. Therefore the assumption that P is not better than somebody else must be false, so P is better than everybody else. P is thus the best by Definition of Best.
Theorem 5. If a person is the best, then everybody is his friend.
Proof. Consider the best person. By Definition of Best, everybody is not better than him. It follows by Theorem 2 that everybody is a friend of him.
Theorem 6. If everybody is a person’s friend, then he is the best.
Proof. Let it be given that everybody is a person P’s friend. Assume that somebody, A, is better than P. Then by Theorem 1, A is an enemy of P. But this contradicts the fact that everybody is P’s friend. Therefore the assumption that somebody is better than P must be false, so nobody is better than P. P is thus the best by Definition of Best.
Theorem 7. If a person is the worst, then he is everybody’s friend.
Proof. Consider the worst person, P. By Definition of Worst, P is not better than any person X. By Theorem 2, P is a friend of X.
Theorem 8. If a person is everybody’s friend, then he is the worst.
Proof. Consider a person P who is everybody’s friend. It follows that everybody likes P, by Definition 1. Assume that P is better than somebody. Name this person X. By Postulate 1, X dislikes P. But this contradicts the fact that everybody likes P. Therefore the assumption that P is better than somebody must be false; hence P is better than nobody. By Definition of Worst, P is the worst.
Theorem 9. If a person is the worst, then nobody else is his friend.
Proof. Consider the worst person, P. By Definition of Worst, nobody else is not better than P. By Theorem 2, nobody else is P’s friend.
Theorem 10. If nobody else is a person’s friend, then he is the worst.
Proof. Consider a person P who has no other friends. By Definition 1, P does not like anybody else. Assume P is better than somebody else. Name this person X. It follows, by a property of Goodness, that X is not better than P. By Postulate 2, P likes X. But this contradicts the fact that P does not like anybody else. This contradiction means that the assumption that P is better than somebody else must be false, hence P is not better than anybody else. By Definition of Worst, P is the worst.
Theorem 11. A person cannot both be the best and be another’s friend. Being the best and being another’s friend are mutually exclusive.
Proof. Evident from Theorem 3 and Theorem 4.
Theorem 12. A person cannot both be the worst and have another friend besides himself. Being the worst and having another friend besides himself are mutually exclusive.
Proof. Evident from Theorem 9 and Theorem 10.
All comments and questions welcome!